Positive Solutions Of Boundary Value Problems For Higher Order Nonlinear Ordinary Differential Equations

In recent years,applied mathematics,physics,mechanics and other applied disciplines universally involve the boundary value problems.In recent decades,with the need of practical problems and the improvement of nonlinear functional analysis theory,new theoretical results on nonlinear boundary value problems have been emerging,which further points out the direction for the research of nonlinear ordinary differential equation boundary value problems in other fields.Among them,boundary value problems of high-order nonlinear ordinary differential equation are closely related to many mathematical models established by practical problems including missile flight stability,bridge engineering and other problems.Therefore,the research of the existence and multiplicity of solutions of boundary value problems for nonlinear ordinary differential equation has become one of the most important topics.In this paper,mainly according to the nonlinear functional analysis method,we discuss the existence and multiplicity of positive solutions of boundary value problems for several kinds of high-order nonlinear ordinary differential equations and our main results improve or generalize the results obtained in the literature.The paper includes four main contents:In first chapter,the historical background and significance of nonlinear boundary value problems are reviewed.The research status of boundary value problems for ordinary differential equations at home and abroad is analyzed.Finally,the work of this paper is brief introduction.In second chapter,we explore the existence,multiplicity and uniqueness for positive solutions of boundary value problems for 29)order nonlinear ordinary differential equation with all lower derivatives.In the chapter,the high-dimensional equation boundary value problem is transformed into a low-dimensional one by the order reduction method.Based on a priori estimate of the positive solution established by integral identity and integral inequality,the main results of the boundary value problem are obtained by means of the fixed index theory.The highlights of this chapter are two aspects: one is the introduction of order reduction method,and the other is the generalization of the literature on Lidstone problem.In third chapter,we study the existence and multiplicity of positive solutions of boundary value problems for systems of high order nonlinear ordinary differential equations concerning all lower order derivatives.The novelty of the boundary value problem is that,on the basis of order reduction method,two auxiliary linear functions are constructed to characterize the growth behavior of the nonlinear terms,and then we achieve a priori estimation via combining the properties of concave function and matrix theory.On that basis,the existence of positive solutions of the above boundary value problems is proved by utilizing the fixed point index theory.Besides,the results in the first chapter and the methods in related papers are extended and improved by the chapter.In fourth chapter,we are devoted to investigate the existence and multiplicity of positive solutions of boundary value problems for systems of high-order singular nonlinear ordinary differential equations.research literatures on this subject are not uncommon.The method used in this chapter is distinct from the relevant literatures.This chapter mainly takes the composition operator and skillfully links the integral equations of the two operators together.Then,we use fixed point index theory to establish our main results on the basis of a priori estimate generated by properties of concave function,Jensen’s inequality and non-negative matrix.In addition,the singular boundary value problem can be the same order or of different order.