Existence And Multiplicity Of Positive Solutions For Several Classes Of Fourth-order Two-point Boundary Value Problems

The formation and development of ordinary differential equations is affected by complex variable function,combinatorial topology,lie group.Its wide application is not only reflected in some natural science disciplines,such as mechanics,physics,biology,astronomy,but also closely related to the development of engineering technology.At present,the development of computer technology has promoted the application and theoretical research of ordinary differential equations.In recent years,nonlinear functional analysis has become an important tool in the study of nonlinear problems in astronomy,physics,fluid mechanics,elastic mechanics,aerospace technology and biotechnology.Therefore,the study of nonlinear differential equations is concerned by the majority of scholars.In this paper,we study the existence of positive solutions of boundary value problems for three classes of nonlinear fourth-order ordinary differential equations.The paper consists of following four chapters.In chapter 1,the first part reviews the research background,status and significance of boundary value problems of nonlinear fourth-order ordinary differential equations.In the second part,the relevant symbols,definitions and theorems of this article are introduced.At the last part the main structure of the paper is briefly introduced.In chapter 2,we mainly investigate the existence and multiplicity of positive solutions of boundary value problems for two classes of fourth-order ordinary differential equations.Based on the fixed-point index theory,some sufficient conditions for the existence of positive solution and multiplicity of one simple-supported and the other sliding-supported fourth-order two-point boundary value problem with parameter is investigated.In chapter 3,we discuss the existence of positive solutions for another class of fourth-order boundary value problem in Banach space.By establishing comparison theorem and using pontryagin maximum principle and fixed point theorems for increasing operators,we obtain some sufficient conditions for the existence of solutions of the problem.In the last chapter,the full paper is summarized,and the prospects for future research are put forward.