The Study On A Class Of The Second-order Boundary Value Problem And The Topological Action Function

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.