Positive Solutions Of Multiple-Point Boundary Value Problems For Ordinary Differential Equations

The multi-point boundary value problems for ordinary differential equations arise in a variety of different areas of applied mathematics and physics.For example,the vibrations of a guy wire of uniform cross-section and composed of N parts of different densities can be set up as a multi-point boundary value problem;many problems in the theory of elastic stability can be handled by the method of multi-point problems. The study of multi-point boundary value problems for linear second-order ordinary differential equations was initiated by Il'in and Moiseev in 1980s.In 1990s,Gupta began discussing three-point boundary value problems of second-order nonlinear ordinary differential equations.Since then,many authors have studied more general nonlinear multi-point boundary value problems and obtained many achievements.For ordinary differential equations,positive solutions are always attached importance to and are a kind of practical solutions.The study of existence of positive solutions to differential equations is often transformed into investigating the existence of fixed points for integral operators on a cone.The theories most used for the research into the existence of fixed points for integral operators are that of the degree of nonlinear functional analysis and that of fixed point index.And the theorems widely used are Schauder fixed point theorem,Krasnosel'skii fixed point theorem,Leggett-Williams fixed point theorem and its generalization—the five functional fixed point theorem.Despite the fact that many authors have studied the existence of positive solutions by these theorems and obtained many achievements,in order to use these familiar fixed point theorems,we need to assume that the nonlinear terms are continuous and the Green's functions must satisfy given conditions,which makes the range of using these theorems restricted,so there are yet many challenging questions to be solved.The theory of differential-integral equations in abstract space has been an important branch in the past 30 years.There are several monographs on this theory.The boundary value problems in abstract space have been researched since 1970s.But the development of the study in this field is very slow because the study of existence of fixed point for integral operators in abstract space is very difficult.At present,there are many problems which have not been dealt with but need to be studied.Aiming at these problems,in the following,we will study them in five aspects.1.By using new methods and skills,we prove the existence of positive solutions to second-order and three-order multi-point boundary value problems depending on the first-order derivative,respectively,in which the nonlinear term may be singular at any point of[0,1]and non-continuous in its domain.The main tools used are Krein-Rutman theorem and fixed point index theory.This result extends some of the existing results.2.By using H.Amnn fixed point theorem,the existence of multiple positive solutions to multi-point boundary value problems with p-Laplacian is discussed.In the problem,the nonlinear term contains the first-order derivative,which may be singular at any point of[0,1]and may be non-continuous in its domain.At present,in most results,the nonlinear terms must be continuous in its domain.This new method and skill eliminate this restriction.3.In concrete spaces,the study on(k,n-k)conjugate boundary value problems has obtained many achievements.But,as far as we know,the study for these problems in abstract spaces hasn't been done.By constructing suitable cone,we research the existence of one,two and multiple fixed points for the integral operator A(u(t)):= integral from n=0 to 1 G(t,s)f(s,u(s))ds in abstract space by using fixed point theorem in a cone for strict set contraction operators.Then,using the obtained result,we investigate the existence of one,two and multiple positive solutions to the(k,n-k)conjugate boundary value problems in abstract space.And this result can be used to study a kind of boundary value problems in abstract space.4.For high-order multi-point boundary value problems,we study them as follows:(1)Using the five functional fixed point theorem,we discuss the existence of at least three positive solutions to 2n-order multi-point boundary value problem with all derivatives.(2)Using the five functional fixed point theorem,we discuss the existence of multiple positive solutions to n-order multi-point boundary value problem with all derivatives. 5.In abstract spaces,we also discuss the following three kinds of second-order boundary value problems.(1)Utilizing Darbo fixed point theorem,we firstly give the conditions of existence and non-existence of fixed point for an integral operator by constructing suitable cone. Then,applying the obtained results,we study the conditions of existence and non-existence for second-order two-point and multi-point boundary value problems with non-homogeneous boundary conditions in abstract space,respectively.(2)Using fixed point theorem in a cone for strict set contraction operators,we research the existence of multiple positive solutions for second-order multi-point boundary value problem at resonance in abstract space.The characteristic of this problem is that the corresponding homogeneous boundary value problem possesses nonzero solutions, i.e.,resonance.So,it can't be studied by using general methods.The difficulty lies in finding of the inequality satisfied by the Green's function of a non-resonance boundary value problem which is equal to the boundary value problem at resonance.(3)When the cone is normal,the existence of positive solutions to second-order two-point boundary value problems in abstract space has been researched.Eliminating this restriction,we prove the existence of positive solutions to this problem by using new methods and skills,which makes the range of using our result wider.