【作者】 江卫华；

【导师】 何连法；

【作者基本信息】 河北师范大学 ， 基础数学， 2009， 博士

【摘要】 常微分方程多点边值问题起源于许多不同的应用数学和物理领域,例如:由N部分不同密度组成的均匀截面的悬链线的振动可以转化为多点边值问题;弹性稳定性理论的许多问题也可转化为多点边值问题处理.对多点边值问题的研究,始于二十世纪八十年代,由Il’in和Moiseev首次对二阶线性常微分方程进行研究.到二十世纪九十年代,Gupta开始讨论二阶非线性常微分方程三点边值问题,此后,许多作者研究了更一般的非线性多点边值问题,并取得了丰富的成果.对常微分方程来说,正解往往是人们注重研究的符合现实意义的一类解,人们常将研究微分方程正解的存在性问题转化为研究积分算子在锥上的不动点的存在性问题.研究积分算子不动点的存在性常用的理论是非线性泛函分析的度理论和不动点指数理论,其中最常用的定理是:Schauder不动点定理,Krasnosel’skii不动点定理,Leggett-Williams不动点定理和它的一般化—五个泛函不动点定理.尽管很多学者应用上述定理对多点边值问题正解的存在性进行研究,并取得了丰富的成果.但由于使用这些常见的不动点定理需要假设非线性项是连续的,且Green函数需满足特定的条件要求,使得这些常用的定理适用范围具有一定的局限性.因此,仍然存在许多未解决的具有挑战性的问题.抽象空间中的积微分方程理论是近三十年发展起来的一个重要的分支,现在已有专著多部.对于抽象空间中边值问题的研究始于二十世纪七十年代,但由于在抽象空间中研究积分算子不动点的存在性具有很大困难,使得这一领域的研究进程非常缓慢.目前未涉及到的需要研究的问题有很多.针对以上这些问题,本文分以下五个方面进行研究:1.对二阶和三阶微分方程边值问题,我们打破常规限制,采用新的方法和技巧,分别证明非线性项中含有一阶导数且非线性项可在[0,1]中任一点具有奇性以及在定义域内可以具有不连续性的二阶和三阶多点边值问题正解的存在性.我们使用的工具是Krein-Rutman定理和不动点指数理论.该结果扩展了现有的一些结论.2.我们利用H.Amnn不动点定理,研究非线性项中含有一阶导数,且可在[0,1]中任意一点具有奇性以及在定义域内具有不连续性的带有p-Laplacian算子的多点边值问题多个正解的存在性.这个新的方法和技巧,消除了目前多数结论必须以非线性项在定义域内连续作为条件的限制.3.对具体空间中(k,n-k)共轭边值问题的研究已经取得了很多的成果.然而,就我们所知,对抽象空间中的这一问题还未有研究.本文通过构造适当的锥,利用锥上严格集压缩算子不动点定理,研究抽象空间中积分算子A(u(t)):=integral from n=0 to 1 G(t,s)f(s,u(s))ds的一个、两个和多个不动点存在性.然后,利用这一结论研究抽象空间中(k,n-k)共轭边值问题的一个、两个和多个正解的存在性,并且这一结论可用于一类抽象空间中的边值问题研究.4.对于高阶多点边值问题,我们做了如下研究:(1)利用五个泛函不动点定理研究非线性项中含有各阶导数的2n阶多点边值问题至少三个正解的存在性;(2)利用五个泛函不动点定理研究非线性项中含有各阶导数的n阶多点边值问题多个正解的存在性.5.在抽象空间中,我们还研究了以下三种类型的二阶边值问题:(1)利用Darbo不动点定理,通过构造适当的锥,首先给出一个积分算子的不动点的存在和不存在条件,然后利用该结论研究抽象空间中具有非齐次边界条件的二阶两点和多点边值问题正解的存在性和不存在性条件;(2)利用锥上严格集压缩算子不动点定理,我们研究抽象空间中具有共振的二阶多点边值问题多个正解的存在性.该问题的特点是相应的齐次边值问题具有非零解,即共振性,从而不能用通常的方法来研究.难点在于找出与其等价的一个非共振边值问题的Green函数所满足的不等式;(3)对于抽象空间中二阶两点边值问题正解的存在性,在锥是正规的条件下已有研究.我们采用新的方法和技巧,去掉这一条件限制,证明了该边值问题正解的存在性,使得我们所得结论的使用范围更广.更多还原

【Abstract】 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.更多还原