A Survey On The Multi-point Boundary Value Problems For Singular Differential Equations

Multi-point boundary value problem is an important branch ofdifferential equation theory, it has profound physical background andextensive theoretical application, so many authors had committed tomulti-point boundary value problem in resent years.In this thesis, we summarize some known results on the boundary valueproblem of differential equations, and excepte to get furtherunderstanding in this area.This paper is organized as follows:Firstly, the preamble describes the background and problems to beconsidered. Then we outlined some results in the references.In the first chapter, we consider the second-order singular boundaryvalue problem of differential equations. We summarize some existenceresults on the positive solutions for the two-order multi-point boundaryvalue problem by using the upper and lower solution method and coneexpansion and compression fixed point theorem.In the second chapter, we consider the third-order singular boundaryvalue problem of differential equations. We summarize some existenceresults on the positive solutions under different boundary conditions byusing various kinds of methods.In the third chapter, we consider the fourth-order singular boundaryvalue problem of differential equations. In recent years，the researchon the higher-order multi-point singular boundary value problem ofdifferential equations is increasing, and many results are obtained underthe monotonicity condition. In2010, Li Qin, Wang Haiping and Lu Huiqinused the reduction method and Leray-Schauder principle to get theexistence of solution without monotonicity assumption.In the fourth chapter, we consider the n-order singular boundary valueproblem of differential equations. We summarize some existence resultson the positive solutions by using cone expansion and compression fixedpoint theorem.Finally, we introduce an example, and also look into the future studyin this direction.