Research On The Existence Of Positive Solutions For Several Classes Of Boundary Value Problems Of Nonlinear Ordinary Differential Equations

The boundary value problems?BVPs?of ordinary differential equations?ODEs?have wide applications and have been studied thoroughly.Generally,only positive solutions are significative in some practical problems,therefore it is of great theoretical significance and practical value to study the positive solutions of the BVPs of ODEs.In this thesis,the existence of positive solutions of several classes of BVPs of nonlinear ODEs is investigated.The thesis is divided into the following five chapters.In the first chapter,firstly,the background knowledge and research status of BVPs of ODEs are briefly introduced,then,the main contents of the thesis are expounded,finally,we present definitions and lemmas used in the thesis.In the second chapter,we discuss the Green's functions for two classes of BVPs of second order nonlinear ODEs.In the third chapter,the existence of positive solutions of Sturm-Liouville BVPs of second order nonlinear ODEs is studied under two different boundary value conditions.Firstly,by employing the Guo-Krasnosel'skill fixed point theorem,we study the existence of at least two positive solutions for a class of two point BVPs under the conditions of the nonlinear term f satisfying f₀=?and f_?=??f₀=0 and f_?=0?.Then,the existence of positive solutions for a class of m-point BVPs with sign changing nonlinearities is proved by applying the properties of fixed point index of compact operators.In the fourth chapter,the existence of positive solutions for two kinds of m-point BVPs of third order nonlinear ODEs is investigated.Firstly,by using the Guo-Krasnosel'skill fixed point theorem,we study the existence of positive solutions for a class of m-point BVPs with the nonlinear term f satisfying the super-linear or sub-linear conditions.Then,the existence of at least three positive solutions for a class of m-point BVPs with sign changing nonlinearities is discussed via using the Leggett-Williams fixed point theorem.The fifth chapter is the research summary and prospect of the thesis.