几类非线性微分方程边值问题解的存在性及振动分析
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摘要
动力系统的概念,最早起源于十九世纪末,在经典力学和微分方程定性理论的研究中。动力系统是一种描述一个给定空间中的所有点随时间旅程的方法,关心的是微分方程解的长期行为。根据所研究的微分方程形式的不同,分为线性微分方程系统和非线性微分方程系统。对于线性系统,解的存在唯一性是显而易见的。但是对于非线性系统,情况比较复杂,并且也没有一种普遍适用的方法来求解非线性微分方程。所以研究非线性微分方程解的存在性就尤为重要。而微分方程边值问题,作为微分方程的一个重要分支,在力学、天文、物理中有着重要的应用,同时在化学、生物学、气象学、医学、经济学以及航空航天、水电能源、环境、动力和生物工程等领域中也有着广泛的应用。
本论文研究动力学系统中几类非线性微分方程共振与非共振条件下的多点边值问题、二阶差分方程以及无穷区间上的微分方程边值问题、具有时滞分布的偶数阶带阻尼项的微分方程的振动分析以及奇异摄动非线性微分系统边值问题。讨论解的存在性和唯一性、正解和多个正解的存在性,以及解的振动性,渐进性等。主要工具是Mawhin迭合度定理、微分不等式理论、拓扑度理论、不动点定理、奇异摄动方法等。全文分五章。
第一章介绍有关动力系统和微分方程边值问题以及非线性振动的背景知识和当前进展情况,对本文工作的创新之处作出说明。
第二章通过对Bananch空间进行直和分解,构造合适的投影算子,并进行有效的先验界估计,运用Mawhin迭合度理论研究核空间维数为一和核空间维数为二的三阶微分方程多点共振边值问题、高阶微分方程多点共振边值问题以及高阶非局部共振边值问题的可解性。与已有的工作相比,本章运用新的方法,获得了新的结果。
第三章运用上下解方法和Leray-Schauder度理论,研究具非线性边界条件的三阶微分方程多点边值问题解的存在性和唯一性。运用两对上下解方法研究非线性项依赖于高阶导数的n阶三点边值问题三解的存在性。我们研究问题的边界条件是非线性的,因而讨论的边值问题更具一般性。本章的结果推广和改进了已有的工作。
第四章运用Leggett-Williams不动点定理,研究二阶差分方程多点边值问题,通过给出离散边值问题相应的Green函数,并讨论Green函数的性质,得到二阶离散边值问题的多个正解的存在性。类似地运用Leggett-Williams不动点定理,研究无穷区间上二阶微分方程边值问题,得到了三个正解的存在性。我们的结果推广了以前的工作。
第五章运用广义的Riccati技巧、Hardy-Littlewood-Polya不等式研究具有时滞分布的偶数阶带阻尼项的微分方程的振动性,得到了一些新的振动性准则。运用微分不等式理论、不动点定理和奇异摄动理论研究二阶非线性奇异摄动微分系统边值问题。得到了解的存在性以及解的一致有效渐近估计。本章的主要结果推广和改进了已有文献的工作。
This dissertation is mainly concerned with multi-point boundary value problems of nonlinear differential equations with the resonance case and non-resonance case, discrete boundary value problems for difference equations, even-order distributed delay differential equations with damping and singularly perturbed boundary value problems for nonlinear differential systems. We study the existence of solutions and the uniqueness of solution, positive solutions and the multiplicity of positive solutions and the oscillation. The proofs of the results are based Mawhin's coincidence degree theorem, differential inequality technique, topology degree theory, fixed point theorems in cones and the methods of singular perturbation respectively.
The whole thesis contains five chapters.
The first chapter introduces concisely the historical situation, the present development and the research background of dynamical systems and boundary value problems for nonlinear differential equations and nonlinear oscillation, as well as the main work done in this thesis.
Chapter 2 devotes to the study of the solvability of third order multi point boundary value problems at resonance with the case dim Ker L=1 and dim Ker L=2, respectively. We also discuss the solvability of multi point and nonlocal boundary value problems at resonance for higher order differential equations. The method is based upon the theory of Mawhin's coincident degree. The approaches are different from those used in the past and the results improve and extend what are given in the relevant literatures.
In chapter 3, we first study a kind of third order boundary value problem with nonlinear multi-point boundary conditions. The existence and uniqueness of solutions are given by use of upon lower and supper solutions methods and the theory of Leray-Schauder degree. We also investigate the multiplicity of solutions for a n order three-point boundary value problem with nonlinear terms depending explicity on the higher order derivative. Compared with the work done by others, we mainly deal with more generalized nonlinear boundary conditions. Our results improve and extend some work in the past.
In chapter 4, by applying Leggett-Williams fixed point theorem, the Green's function and exploring its properties, the existence of the multiplicity of positive solutions for second order difference equations with three-point boundary value problems is given. Similarly, the multiplicity of second order boundary value problems on infinite intervals is discussed. Our theorems include some available results as special cases.
In chapter 5, by using the generalized Riccati technique and an inequality due to Hardy, Littlewood and Polya, several new oscillation criteria are established to even-order distributed delay differential equations with damping. We also apply differential inequality theory, fixed point theorem and singular perturbation theory to study the existence of solution and the asymptotic estimate of solution for a singularly perturbed three-point boundary value problem for nonlinear differential systems. The results obtained extend and improve earlier results in existing literature.
本论文研究动力学系统中几类非线性微分方程共振与非共振条件下的多点边值问题、二阶差分方程以及无穷区间上的微分方程边值问题、具有时滞分布的偶数阶带阻尼项的微分方程的振动分析以及奇异摄动非线性微分系统边值问题。讨论解的存在性和唯一性、正解和多个正解的存在性,以及解的振动性,渐进性等。主要工具是Mawhin迭合度定理、微分不等式理论、拓扑度理论、不动点定理、奇异摄动方法等。全文分五章。
第一章介绍有关动力系统和微分方程边值问题以及非线性振动的背景知识和当前进展情况,对本文工作的创新之处作出说明。
第二章通过对Bananch空间进行直和分解,构造合适的投影算子,并进行有效的先验界估计,运用Mawhin迭合度理论研究核空间维数为一和核空间维数为二的三阶微分方程多点共振边值问题、高阶微分方程多点共振边值问题以及高阶非局部共振边值问题的可解性。与已有的工作相比,本章运用新的方法,获得了新的结果。
第三章运用上下解方法和Leray-Schauder度理论,研究具非线性边界条件的三阶微分方程多点边值问题解的存在性和唯一性。运用两对上下解方法研究非线性项依赖于高阶导数的n阶三点边值问题三解的存在性。我们研究问题的边界条件是非线性的,因而讨论的边值问题更具一般性。本章的结果推广和改进了已有的工作。
第四章运用Leggett-Williams不动点定理,研究二阶差分方程多点边值问题,通过给出离散边值问题相应的Green函数,并讨论Green函数的性质,得到二阶离散边值问题的多个正解的存在性。类似地运用Leggett-Williams不动点定理,研究无穷区间上二阶微分方程边值问题,得到了三个正解的存在性。我们的结果推广了以前的工作。
第五章运用广义的Riccati技巧、Hardy-Littlewood-Polya不等式研究具有时滞分布的偶数阶带阻尼项的微分方程的振动性,得到了一些新的振动性准则。运用微分不等式理论、不动点定理和奇异摄动理论研究二阶非线性奇异摄动微分系统边值问题。得到了解的存在性以及解的一致有效渐近估计。本章的主要结果推广和改进了已有文献的工作。
This dissertation is mainly concerned with multi-point boundary value problems of nonlinear differential equations with the resonance case and non-resonance case, discrete boundary value problems for difference equations, even-order distributed delay differential equations with damping and singularly perturbed boundary value problems for nonlinear differential systems. We study the existence of solutions and the uniqueness of solution, positive solutions and the multiplicity of positive solutions and the oscillation. The proofs of the results are based Mawhin's coincidence degree theorem, differential inequality technique, topology degree theory, fixed point theorems in cones and the methods of singular perturbation respectively.
The whole thesis contains five chapters.
The first chapter introduces concisely the historical situation, the present development and the research background of dynamical systems and boundary value problems for nonlinear differential equations and nonlinear oscillation, as well as the main work done in this thesis.
Chapter 2 devotes to the study of the solvability of third order multi point boundary value problems at resonance with the case dim Ker L=1 and dim Ker L=2, respectively. We also discuss the solvability of multi point and nonlocal boundary value problems at resonance for higher order differential equations. The method is based upon the theory of Mawhin's coincident degree. The approaches are different from those used in the past and the results improve and extend what are given in the relevant literatures.
In chapter 3, we first study a kind of third order boundary value problem with nonlinear multi-point boundary conditions. The existence and uniqueness of solutions are given by use of upon lower and supper solutions methods and the theory of Leray-Schauder degree. We also investigate the multiplicity of solutions for a n order three-point boundary value problem with nonlinear terms depending explicity on the higher order derivative. Compared with the work done by others, we mainly deal with more generalized nonlinear boundary conditions. Our results improve and extend some work in the past.
In chapter 4, by applying Leggett-Williams fixed point theorem, the Green's function and exploring its properties, the existence of the multiplicity of positive solutions for second order difference equations with three-point boundary value problems is given. Similarly, the multiplicity of second order boundary value problems on infinite intervals is discussed. Our theorems include some available results as special cases.
In chapter 5, by using the generalized Riccati technique and an inequality due to Hardy, Littlewood and Polya, several new oscillation criteria are established to even-order distributed delay differential equations with damping. We also apply differential inequality theory, fixed point theorem and singular perturbation theory to study the existence of solution and the asymptotic estimate of solution for a singularly perturbed three-point boundary value problem for nonlinear differential systems. The results obtained extend and improve earlier results in existing literature.
引文
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