The Existence Of Positive Solutions Of Two Classes Singular Boundary Value Problems

With the development of the times, a great amount of physical phenomenon, such as Nuclear Physics, Electromagnetics Fluid Mechanics and Nonlinear Optics, evoked people’s great interest, and was widely researched. But one of the most important tools in studying these areas is differential equation. Yet boundary value problems of differential equation is a kind of research area with profound significance. At the recent decade, existence and uniqueness on positive solutions of differential equations get wide and deep research.In recent years, in many fields of Non-Newtonian Mechanics, Cosmic Physics and Theory of Elasticity, boundary value problems of differential equation with p-laplacian operator, super linear or sublinear have wide range of applications. The existence of the solution causes lots of researchers’ concern.This paper mainly discusses the existence of quasi-symmetric positive solutions of a class of singular fourth-order p-laplacian differential equations with three points boundary value problems:At the same time, it also studies the existence of positive solutions of a class of singular super linear boundary value problems of higher order differential equations:In the first chapter of this paper, we first introduce the historical background of differential equations coming into being. Next, we list the main work for the paper.In the second chapter, it gives some lemmas and theorems in the first place, then by using fixed point theorems on different cones, obtains the conclusion and proof of the existence of quasi-symmetric positive solutions of a class of singular fourth-order p-laplacian differential equations with three points boundary value problems. In the third chapter, for the higher order super linear singular boundary value problem, we use a fixed point theorems of cone expansion and cone compression in the form of norm to give the conclusion and proof of the existence of positive solutions of differential equation.In the fourth chapter, we summarize all the research work done for the paper, and bring some new questions for further research.