The Existence Of Positive Solutions For Two-point Boundary Value Problems Of Fourth-order Nonlinear Ordinary Differential Equations

The boundary value problems(BVP)of differential equations always have broad application background in the fields of bioscience and engineering physics.The boundary value problems of fourth-order nonlinear ordinary differential equations as an important part of the boundary value problems of differential equations also are widely used in elastic beam theory.Therefore,it is very important for us to study the boundary value problems of fourth-order nonlinear ordinary differential equations.In this thesis,the main work is to research the existence and multiplicity of the positive solution for several different kinds of the two-point boundary value problems of fourth-order nonlinear ordinary differential equations by using Krasnosel'skii fixed point theorem,Leggett-Williams fixed point theorem and Avery-Peterson fixed point theorem.The thesis is divided into three chapters,and its main contents are as follows:Chapter one is an introduction,the background and status quo for the boundary value problems of differential equations researched in the thesis are briefly introduced,and its innovations are also illustrated.At the same time,the basic concepts and lemmas to be used in this thesis are presented.Finally,we focus on the methods for finding the Green functions of the boundary value problems of differential equations.In chapter two,we mainly research the existence of positive solutions for two kinds of the two-point boundary value problems of fourth-order nonlinear ordinary differential equations.Firstly,we aim at the following two-point boundary value problems of fourth-order nonlinear ordinary differential equations:(?)the existence of positive solutions for this boundary value problem is proved by using Krasnosel'skii fixed point theorem.Then via cone fixed point index theorem,we study the existence of positive solutions for the two-point boundary value problems of fourth-order nonlinear ordinary differential equations with integral boundary value conditions:(?).In chapter three,we mainly research the existence of multiple positive solutions for two kinds of the two-point boundary value problems of fourth-order nonlinear ordinary differential equations.First,we extend the Leggett-Williams fixed point theorem,and by which establish the existence and multiplicity of positive solutions for the two-point boundary value problems of fourth-order nonlinear ordinary differential equations:(?)Next,we prove the existence of multiple positive solutions for the following two-point boundary value problems of fourth-order nonlinear ordinary differential equations by extended Avery-Peterson fixed point theorem:(?)...