几类微分方程的同宿和异宿轨道的研究

发布时间：2018-07-23 10:15

【摘要】：本文利用变分方法研究了一阶Hamilton系统和几类二阶阻尼微分方程的同宿轨道和异宿轨道问题.在各种假设条件下,分别获得了同宿轨道和异宿轨道的存在性和多解性.主要内容如下：第一章介绍一些研究的背景、研究概况以及一些预备知识.第二章考虑如下的一阶Hamilton系统z=JHz(t,z),a.e.t∈R,这里的H(t,z)依赖于t,但关于t不是周期的.在一些比著名的(Ambrosetti-Rabinowitz)超二次条件(简称为(AR)条件)弱的超二次假设下,利用环绕定理得到了同宿轨道的存在性.另外,还讨论了次二次条件下同宿轨道的多解性.改进和推广了一些文献中的已有结果.第三章讨论如下带阻尼的微分方程ü+cu-L(t)u+Wu_(t,u)=0,其中c≥0是一个常数；对称矩阵L(t)关于t是非周期的；W(t,u)依赖于t,但关于t不是周期的.在W(t,u)为次二次或者超二次情形下,利用临界点理论得到了该方程的无穷多个同宿轨道或拟同宿轨道的存在性.改进了现有文献中的一些结果,同时对于Zhang,Yuan提出的公开问题给出了明确的回答.此外,利用Nehari流形还考虑了当W(t,u)不定号时该方程的同宿轨道或拟同宿轨道的存在性.第四章讨论如下带阻尼的微分方程ü+g(t)u-L(t)u+Wu_(t,u)=0,其中g∈C(R, R);对称矩阵L(t)关于t不是周期的；W(t,u)依赖于t,但关于t不是周期的.在关于g,L以及W的一些合理假设下,利用喷泉定理和对偶的喷泉定理讨论了当W(t,u)分别是超二次、次二次以及凹凸组合项情形时,该方程无穷多个同宿轨道的存在性问题,改进和推广了现有文献中的一些结果.第五章考虑如下方程的同宿轨道和异宿轨道问题ü+Au-L(t)u+Wu_(t,u)=f(t),其中A是一个反对称常数矩阵；L(t)∈C(R,RN2)是一个对称一致正定矩阵；函数f∈L2(R,RN)且w∈C1(R×RN,R)首先,考虑了强不定问题的同宿轨道的存在性和多解性,其中W(t,u)关于t是周期的且满足较弱的渐进二次条件.然后,在关于L(t),f(t)和W(t,u)的某些假设条件下,利用变分方法得到了异宿轨道的存在性.即对于某个子集m(?)RN, (?)x∈m都存在一个异宿轨道w,使得w(-∞)=x且w(+∞)∈m\{x}.
[Abstract]:In this paper, we use the variational method to study the homoclinic orbits of the first order Hamilton system and several classes of two order damped differential equations. Under various hypothetical conditions, the existence and multiple solutions of the homoclinic orbits are obtained respectively. The main contents are as follows: the first chapter introduces some research background, research situation and some of them. In the second chapter, the second chapter considers the following first order Hamilton system z=JHz (T, z) and a.e.t R, where H (T, z) depends on T, but t is not periodic. Under the hypothetical super two assumption that is weaker than the famous (Ambrosetti-Rabinowitz) two times condition, the existence of the homoclinic orbit is obtained by the surround theorem. In addition, the existence of the homoclinic orbit is obtained. We discuss the multiple solvability of homoclinic orbits under the two times. Improve and extend the existing results in some literature. The third chapter discusses the following differential equation u +cu-L (T) u+Wu_ (T, U) =0 with damping, where C > 0 is a constant; the symmetric matrix L (T) is non periodic on T; W (T,) is not periodic. In the case of two or two times, the existence of infinitely many homoclinic orbitals and quasi homoclinic orbitals of the equation is obtained by using the critical point theory. Some results in the existing literature are improved, and a clear answer to the open problem proposed by Zhang and Yuan is given. Furthermore, the use of the Nehari manifold is also considered when the W (T, U) indefinite number is considered. The existence of the homoclinic orbits of the equation. The fourth chapter discusses the following differential equations with damped +g (T) u-L (T) u+Wu_ (T, U) =0, wherein the G C (R, R); the symmetric matrix is not periodic; The fountain theorem discusses the existence of infinitely many homoclinic orbits when W (T, U) is super two, two times and concave convex combination terms, and improves and generalizes some results in the existing literature. The fifth chapter considers the homoclinic orbits of the following equation, the problem of u +Au-L (T) u+Wu_ (t, U) =f (T) of the following equation, and A is an inverse. A symmetric constant matrix; L (T) C (R, RN2) is a symmetric and consistent positive definite matrix; the function f L2 (R, RN) and w C1 C1 (R x), first, consider the existence and multi solvability of the homoclinic orbits of strongly indefinite problems. The existence of the heteroclinic orbit is obtained by the variational method. That is, there is a heteroclinic orbit w for a subset of M (?) RN and (?) x m, which makes w (- infinity) =x and w (+ infinity) m{x}. m{x}.
【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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