【作者】 贺强；

【导师】 史少云；

【作者基本信息】 吉林大学 ， 基础数学， 2014， 博士

【摘要】 多点边值问题是一类典型的非线性问题,它广泛地出现在物理、工程、生物等众多领域,可用于刻画多点支持桥梁、弹性稳定性理论以及有N部分不同密度组成均匀截面的悬链线等现象。本论文的第一部分主要研究一类多点边值问题,内容包括：一、多点边值问题解的存在性以及解的相关性质；二、由于此类问题一般无法求出解析解,我们给出一些有效的数值解法。第一部分主要研究多点支持桥梁所满足的二阶三点边值问题,我们分别在共振和非共振情形证明了此类问题无穷多解的存在性,并针对具体问题提出了有效的数值解法。多点支持桥梁满足以下二阶三点边值问题：其中λ≥0,β∈[0,1],α.β∈[0,1]我们分别考虑了以下三种情况：α·β=1,λ=0,(2) α·β=1,λ>0,(3) α·β＜1,λ>0,(4)通常称条件(2)为共振条件,而称条件(3)和(4)为非共振条件.定义1称函数u*∈C2[0,1]为(1)的下解,如果u*满足定义2称函数u*∈C2[0,1]为(1)的上解,如果u*满足引理1(定理1,[91])假设以下三个条件同时成立：(A1)λ≥0,β∈[0,1];(A2)f(·,·)是定义在(0,1)×R上的实函数,且满足(i)对每一个确定的u∈R,f(·,u)在(0,1)上是可测的,(ii)f(t,·)在t∈(0,1)上是几乎处处连续的,(iii)对任意给定的N>0都存在一个函数kN(t)∈E,使得|f,(t,u)|≤kN(t) t∈(0,1),u∈[-N,N],其中E:={h(t)∈Llocl(0,1)；‖h‖EE≤+∞}是完备的Banach空间(A3)(1)有下解u*(t)和上解u*(t).且当t∈[0,1]时u*(t)≤u*(t)则边值问题(1)存在一个解u0(t),且u*(t)≤u0(t)≤u*(t).我们首先利用上下解方法来证明二阶三点边值问题无穷多解的存在性,然后利用打靶法将边值问题转化为初值问题,再利用牛顿迭代法来进行数值求解,最后给出相应的数值模拟,从而验证了该方法的有效性及可行性。第一部分的主要结构如下：第一章是研究背景及现状。在1.1节中,简要介绍了多点边值问题的应用背景；1.2节中针对二阶三点边值问题解的存在性以及求解此类问题的数值解法,概述了相应的研究历史以及研究现状；在1.3节中,简述了本文的主要工作。第二章研究二阶三点边值问题解的存在性问题。首先简要介绍了研究二阶三点边值问题所需要的基本概念及其相关结果,然后利用上下解的方法,针对不同实际问题证明了二阶三点边值问题无穷多解的存在性。第三章提出了一种求解二阶三点边值问题的有效数值解法,并针对具体问题给出了相应的数值模拟,进而验证了该方法的有效性和正确性。第二部分主要研究拓扑作用函数。在文献[95]中,王俭将经典的作用函数推广到了哈密顿同胚映射F情形,其中F是亏格大于等于1的有向闭曲面M上的一个同痕的时间1-映射,他对测度也进行了推广,但是推广后的测度没有原子(即该测度在该集合上没有单点具有正测度)在F的可缩不动点上,且F的可缩不动点的缠绕数的集合满足某一种有界性条件.即测度有全支集情形,我们针对王俭结果的一种特殊情形,给出一个简单证明.王俭推广的经典作用函数的结果叙述如下：定理1([95])设M是一个亏格g≥1的有向闭曲面, F是M上的恒等同痕I的时间1-映射.假设μ∈M(F)在集合FixCont,I(F)上没有原子（即该测度在该集合上没有单点具有正测度）,并且满足ρ(μ)=0.则经典的作用函数可推广到如下情形：F是一个微分同胚映射(不必是C1);I满足WB-性质,且测度μ具有全支集;I满足WB-性质,且测度μ是遍历的.本文将给出上述定理中第二种情形的一个简单的证明，从而使得人们能更容易的理解推广的拓扑作用函数。第二部分的主要结构如下.第四章主要介绍拓扑作用函数的研究背景及现状，并阐述了我们的主要结果。在第五章中,我们介绍一些符号，基本概念和Brouwer同胚理论的一些重要结果。特别介绍弱有界性和正常返点上的缠绕数的定义。第六章阐述辛几何中的经典作用函数，并将其推广到一种简单的情形。然后，基于一个关键的命题，我们将作用函数推广到更一般的情况。具体来说，得到以下结果：定理2(主要结果)设M是一个亏格g≥1的有向闭曲面, F是M上的恒等同痕I的时间1-映射.假设μ∈M(F)在集合FixCont,I(F)上没有原子,具有全支集,并且满足ρ(μ)=0.如果I满足WB-性质,那么拓扑作用函数是良定义的.最后给出证明定理2需要的一个关键命题的证明。命题1(主要命题)假设F是M上的一个μ-辛映射,它相应的同痕I满足弱有界性条件.假设μ在FixCont,I(F)上没有原子,那么对F的任意两个不同的不动点a和b,以及对几乎每一个常返点z, i(a, b, z)存在且有界,并且这个界只依赖于a和b.更多还原

【Abstract】 In recent years, the multi-point boundary value problem occurs widely in many fields, such as physics, engineering, biology, etc, which commonly used to describe the support multi-point bridge, the elastic stability theory, and the vibrations of a guy wire of uniform cross-section and composed of N parts of different densities, The problem is a typical non-linear differential equations. The first part of this thesis mainly concern on the theory analysis of this issue, moreover, study of this problem can be divided into two areas:On the one hand, studying on the existence of solutions and related properties of solutions; on the other hand, due to the analytical solution of the problem can not be solved, therefore, it is necessary to make an effective numerical method.The first part is concerned on the research of the support multi-point bridge, which satisefies the second-order three-point boundary value problem, we prove that the solutions of this kinds of problem in both resonance and nonresonance cases are countably infinite, and propose a valid numerical method for the practical problems.Consider the following second-order three points boundary problem which describes the multi-point bridge: where λ≥0,β∈[0,1],α·β∈[0,1]. We consider the following three cases: α·β=1,λ=0.(2) α·β=1,λ>0,(3) α·β＜1,λ>0,(4)Now let us introduce the upper and lower solution method for analyzing the existence of a solution of the three-point boundary value problem (1). Definition1A function u*∈C2[0,1] is a lower solution of equation (1) if u*satisfies Definition2A function u*∈C2[0,1] is an upper solution of equation (1) if u*satisfies Lemma1(Theorem1,[91]) Suppose that the following three assumptions are satisfied:(Al) λ≥0,β∈[0,1];(A2) f (.,.) is a real-valued function defined on (0,1)×R and satisfies (i) f (t., u) is a measurable function defined on (0,1) for each fixed u∈R,(ii) f (t,-) is a continuous function defined on R for almost all t∈(0,1),(ii) for every given N>0, there exists a function hN(t) E E, such that|f (t, u)|≤kN(t) for almost all t∈(0,1) and u∈[-N, N], where E:={h(t)∈Llocl(0,1);‖h‖E≤+∞is the Banach space equipped with the norm‖h‖E=f0s|上(要)|厅+δl(1-s)|h(s)|ds+fδl(1-s)|h(s)|ds;(A3) There exist two functions u.(t) and u*(t) that are lower and upper solutions to equation (1), respectively. Moreover, u*(t)≤u*(t), on [0,1].Then equation (1) has an exact solution uo (t) with u*(t)≤uo (t)≤u*(t).We introduce the upper and lower solution method for analyzing the existence of a solution of the three-point boundary value problem (1), based on which we will in-vestigate an example in next section. In this paper, we introduce the upper and lower solution method which plays an fundamental role in our analysis. Moreover, we prove the second-order three points boundary problem has infinitely many solutions. Next apply the shooting method and Newton iterative method to slove the prolem, and illustrate the Numerical simulations to verify the effectiveness and validity o this method.The main structure of the first part is as follows:The first chapter is mainly studied the background and actuality. In section1.1, we give a brief introduction on the application back for the multi-point boundary val-ue problems; whereafter, in section1.2, for the existence and the numerical method of the second-order boundary value problem, we give the corresponding study history andresearch status; in section1.3, we illustrate the structure of the work in this paper.The second chapter is studied on the solution number of second-order three pointboundary value problem. Firstly, we introduce briefy the basic concepts and the relatedtheorems of the second-order three-point boundary value problem. Then, we apply theupper and lower solution method to prove the solution of this problem are infnite in bothresonance and nonresonance cases.In the third chapter, we propose an efcient numerical method for solving the two-order three-point boundary value problem, moreover, give the corresponding numericalsimulation for the practical problems, and then verify the validity and efectiveness of thismethod.In the second part, we concern on the topological action function.The classical action is well known in the case of Hamiltonian difeomorphisms whichis the time-one map of the fow of the non-autonomous Hamiltonian equations and themeasure is defned by a volume form. In [95], Wang generalizes that action to the casethat the map F is Hamiltonian homeomorphism which is the time-one map of an identityisotopy on a closed oriented surface M with g≥1, the measure is general but withoutatoms on contractible fxed points of F, and the set of linking numbers of contractible fxedpoints of F satisfes certain boundedness condition. We will give an alternative proof ofthe case where the measure has a total support which is a special case of Wang’s result.Wang [95] defne a new action function which generalizes the classical function:Theorem1([95]) Let F be the time-one map of an identity isotopy I on a closed orientedsurface M with the genus g≥1. Suppose that μ∈M(F) has no atoms on FixCont,I(F)and that ρ(μ)=0. In all of the following casesF is a difeomorphism (not necessarily C1);I satisfes the WB-property, the measure μ has total support;I satisfes the WB-property, the measure μ is ergodic,an action function can be defned which generalizes the classical case.The goal of this article is to give a simple proof of the second case of the theoremabove and the second case is more natural and important.Theorem2(Main Theorem) We suppose F is a μ-Hamiltonian map on a closed orientedsurface M with g≥1and I is the corresponding identity isotopy. If I satisfes the WB-property and μ has no atoms on FixCont,I(F), then the function L is well defned. The second part is organized as follows.In chapter4, we mainly give the background and actuality. Moreover, we introducethe main work of the second part.In chapter5, we will frst introduce some notations and recall the precise defnitionsof some important mathematical objects. And then we will defne the Weak boundednessproperty and the linking number on positive recurrent points. And then we will recallthe classical action function in symplectic geometry and generalize the action to a simplecase. moreover, base on a key proposition (Proposition6.3), we will generalize the actionto a more general case.In chapter6, we will briefy recall some of the main results of the theory of Brouwerhomeomorphisms. Then we prove Proposition6.3.更多还原