满足非线性边值条件的哈密顿系统的解

发布时间：2025-08-12 15:01

　　设X是无限维可分的Hilbert空间,其中范数记为|| · ||,内积记为(·,·);A:D(A)(?)X → X是一个可逆的无界自伴算子且σ(A)=σd(A).Y是一个Banach空间,其范数记为||· ||Y,并且D(A)(?)Y(?)X,D(A)到Y的嵌入是紧的,Y到X的嵌入是连续的.N:Y → X是连续算子,M:Y → Y是一个紧的算子且存在ρ>0,使得||M(x)||Y ≤ ρ对于任意x ∈ Y成立.考虑算子方程:x = A-1Nx + Mx.(0.0.1)利用拓扑度方法和线性算子方程的指标理论,我们得到当Nx满足渐近线性条件时,(0.0.1)解和非平凡解的存在性结果.这些结果可以应用到满足不同的非线性边值条件的哈密顿系统,并得到在渐近线性条件下的解的存在性定理.我们研究了带有脉冲影响的,满足非线性Picard边值条件的二阶哈密顿系统,并且推广了 Lees的关于那些系统的存在性结果.我们建立了系数矩阵属于L1的,满足广义周期边值条件的线性哈密顿系统的指标.并且利用这些指标理论和临界点理论,我们研究了渐近线性哈密顿系统.最后,我们将带有周期能量的二阶哈密顿系统的一些存在性结...

【文章页数】：75 页

【学位级别】：博士

【文章目录】：
摘要
Abstract
Chapter 1 Preface
    1.1 Introduction
    1.2 Some known results of Index Theory
    1.3 Some Lemmas
Chapter 2 Nonlinear Boundary Value Conditions and Ordinary Differential Systems with Impulsive Effects
    2.1 Preliminaries and main results
    2.2 Proof of the Theorem 2.1.1
    2.3 Applications to Hamiltonian systems
        2.3.1 Applications to fibrst order Hamiltonian systems
        2.3.2 Applications to second order Hamiltonian systems
        2.3.3 Applications to first order Hamiltonian system with impulses
        2.3.4 Applications to second order Hamiltonian system with impulses
Chapter 3 Nonlinear Sturm-Liouyille Boundary Value Problems for Second Order Hamiltonian Systems with Impulsive Effects
    3.1 Main results
    3.2 Proof of Theorem 3.1.1
Chapter 4 Existence of Solutions for Second Order Hamiltoman Systems with Resonance
    4.1 Index theory for second order linear Hamiltonian systems
    4.2 A generalization of a theorem by Pipan-Schechter
    4.3 Existence of solutions for second order Hamiltonian systems
    4.4 Multiple solutions for symmetric Hamiltonian systems
    4.5 Three solutions for symmetric Hamiltonian systems
Chapter 5 Second Order Hamiltonian Systems with Periodic Potentials
    5.1 Main results
    5.2 Proof of Theorem 5.1.1
    5.3 Applications
Bibliography
Acknowledgements
Publications or Preprints



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