【作者】 张超；

【导师】 史玉明；

【作者基本信息】 山东大学 ， 基础数学， 2007， 硕士

【摘要】 本文主要讨论了三个方面的内容：分别是时间尺度上二阶对称线性方程周期与反周期边值问题的特征值，时间尺度上一类奇异二阶对称线性方程的极限型分类及时间尺度上的L’H(?)spital法则。1988年，德国数学家S．Hilger在他的博士论文中首次提出测度链(Measure chains)分析，即一个把连续与离散分析统一的数学方法。而在许多情况下，我们只需考虑测度链的一种特殊情况-时间尺度。近年来，时间尺度动力学的研究引起了广泛的兴趣，其研究内容涵盖了许多领域，如时间尺度上的微积分概念和理论、动力方程的振动性、特征值问题、边值问题、偏微分方程等。时间尺度上的动力学理论有极其重要的理论意义和广泛的应用前景。这一理论不仅能揭示连续与离散系统的共同点，为我们的研究提供新的强有力的理论工具，还能使我们能够更清楚地理解连续与离散系统以及其它复杂系统中的本质问题。现实问题中，有些过程有时依赖于连续时间变量，有时依赖于离散时间变量，而有些过程的时间变量是分段连续的。对这些问题，用时间尺度上的动力方程就可恰当的给出它们的数学模型。例如虫口模型，一类昆虫的数量从四月份到九月份以一定的增长率连续地增长，到了十月份突然全部死亡，但是它们的卵到来年四月份又开始孵化。这样，这种昆虫就又可以以一定的增长率增长。整个过程的时间变量是分段连续的，可以用一个时间尺度上的动力系统来描述，进而加以解决。再比如，一个由电阻，电容及自感线圈所组成简单串联电路[3，Example 1.40]，当电容以固定频率做周期闭合时，电路中电荷，电流的改变率恰好可以用时间尺度上的导数来刻画。另外，时间尺度上的动力系统在经济学领域也有着广泛的应用。例如，关于动态均衡分析经济学理论的蛛网模型。传统的蛛网模型，时间变量要么是离散的，要么是连续的，无法确切描述某一季节性产品的供求关系。当我们引入时间尺度上的蛛网模型后，就能较好的解决这一问题。E．A．Coddingtong和N．Levinson，J．K．Hale，W．Magnus和S．Winkler等数学家研究了二阶微分方程边值问题特征值的性质，并得到了周期与反周期边值问题特征值比较的结果。对于差分方程边值问题的特征值，F．V．Atkinson，M．Bohner，A．Jirari，史玉明，陈绍著等学者都做过深入的研究．2005年，王怡和史玉明对二阶差分方程周期与反周期边值问题特征值进行了比较。2006年，孙华清和史玉明又将其推广到耦合的边界条件中去。我们发现，虽然二阶微分和二阶差分方程周期与反周期边值问题特征值的个数有着本质的差别，但是它们的比较结果是非常类似的。既然，时间尺度理论是一种统一研究连续情况和离散情况的方法，因此就考虑能否将特征值的比较结果推广到一般的时间尺度中去。从而，不仅把已有的结果统一起来，而且包含了更复杂的时间尺度。本文的主要目的之一就是探讨此问题。对称线性微分算子和差分算子的谱问题都可分为两类：一类是定义在有限闭区间上，且算子系数具有较好性质的，这类称为正则谱问题。否则，称为奇异谱问题。1910年，H．Weyl开始了奇异微分算子谱理论的研究，发现了奇异二阶对称微分方程可分为极限点型与极限圆型两大类．随后，E．C．Titchmarsh．E．A．Coddington，N．Levinson等学者把他的结果进一步深化和完善，形成了Weyl-Titchmarsh理论。无限区间上的二阶形式自伴纯量差分方程的谱问题首先由F．V．Atkinson所研究。随后，D．B．Hinton．A．Jirari等人做了进一步研究。史玉明，陈绍著，S．L．Clark．B．Beckermann．M．Bohner等对二阶及高阶自伴的向量差分方程与离散Hamilton系统的谱问题进行了研究。2001年，陈景年和史玉明给出了实系数二阶奇异形式自伴差分方程极限点型与极限圆型的几个判定准则和一个充分必要条件。最近，史玉明建立了具有一个奇异端点的离散Hamilton系统的Weyl-Titchmarsh理论。另外，孙书荣将文献的工作推广到时间尺度上Hamilton系统的谱问题，建立了时间尺度上Hamilton系统的Weyl-Titchmarsh理论。文献是按最小算子的亏指数给出时间尺度上Hamilton系统按极限型分类。而本文将利用类似Weyl的方法，将时间尺度上奇异二阶对称线性微分方程分为极限点型与极限圆型。这是本文所讨论的另一个主要问题。关于时间尺度上微积分的基本概念和理论，M．Bohner和A．Peterson做了大量的工作。但由于连续与离散的本质不同，很多结论并不完善，不能直观体现出连续为时间尺度的特殊情况。众所周知，在经典微积分理论中，L’H(?)spital法则占有十分重要的地位。利用它可以帮助我们解决不定式等很多问题。文献已经给出了时间尺度上的L’H(?)spital法则，但其条件过于繁杂。我们将就此问题也进行了研究。本文分为四章。第一章，介绍时间尺度的有关预备知识及基本理论，为以下三章做准备工作。第二章，研究时间尺度上二阶对称线性方程周期与反周期边值问题的特征值．主要利用Dirichlet边值问题特征值的性质以及振动性结论，建立周期与反周期边值问题的特征值之间的关系。进而说明，这一结论不仅统一了E．A．Coddington与N．Levinson所得二阶对称线性微分方程和王怡与史玉明所得二阶线性差分方程之周期与反周期边值问题特征值比较的结论，而且拓广了所研究问题的范围。第三章，研究时间尺度上奇异二阶对称线性微分方程的极限型分类。首先，证明了L~2(I)是Hilbert空间。之后，利用分析的方法构造一个集族，并证明该集族构成圆环族。然后证明此圆环族具有嵌套性，从而得到一极限集。根据极限集的几何性质将方程分为极限点型与极限圆型。最后，建立几个极限点型与极限圆型判定准则。第四章，建立一个新的时间尺度上的L’H(?)spital法则。利用时间尺度上的一类链式法及中值定理，从而建立了在较弱条件下的时间尺度上L’H(?)spital法则。更多还原

【Abstract】 This paper mainly deals with three problems: eigenvalues of second-order symmetric linear equations with periodic and antiperiodic boundary conditions on time scales, classification for singular second-order symmetric linear equations on time scales, and L’Ho|^spital rules on time scales.As a tool for establishing a unified framework for continuous and discrete analysis, a theory of dynamic equations on measure chains was introduced by S. Hilger in his Ph.D. thesis [1] in 1988. In many cases, it is necessary to study a special case of measure chains-time scales. In the last decade, the investigation of dynamic systems on time scales has involved much interest in quite a few fields, such as calculus, oscillation of dynamic systems. eigenvalue problems, boundary value problems, partial differential equations on time scales, and etc [2. 3, 4, 5]. The theory of dynamic systems on time scales is of very important theoretical significance and has a wide range of applications. It can not only reveal the similarity between the discrete case and the continuous case, but also explain the discrepancies that occur in parallel statements in continuous and discrete cases. In the real world, there are a lot of processes that depend on continuous time variable sometimes and discrete time variable sometimes, and there are many other processes that depend on piecewise continuous time variable. So we can work out more exactly mathematical models by using dynamic equations on time scales for these cases. For example, the time scales calculus can model insect populations that are continuous while in season, die out in winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population. An example of a simple electric circuit with resistance, inductance and capacitance is given in [3]. Recently, cobweb models on time scales are established and discussed.E. A. Coddingtong and N. Levinson, J. K. Hale, W. Magnus and S. Winkler [6, 7, 8] respectively studied properties of eigenvalues of second-order differential equations with periodic and antiperiodic boundary conditions and compared their eigenvalues.For eigenvalue problem of difference equations, F. V. Atkinson, M. Bohner, A. Jirari, Y. Shi, and S. Chen [9, 10, 11. 12, 13, 14] did a lot of profound and creative work. In 2005, Y. Wang and Y. Shi [15] made the comparison of eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions. In 2006, H. Sun and Y. Shi [16] extended these results to a coupled boundary condition. Although the numbers of eigenvalues of second-order differential and difference equations with periodic and antiperiodic boundary conditions are quite different, there comparison results are similar. So we wonder if the comparison results can be extended to time scales. This is one of the main aims of this paper.The spectral problems of symmetric linear differential operators and difference operators can both be divided into two cases. Those defined over finite closed intervals with well-behaved coefficients are called regular. Otherwise, they are called singular. In 1910, H. Weyl [17] gave a dichotomy of the limit-point and limit-circle cases for singular second-order symmetric linear differential equations. Later, many mathematicians, such as E. C. Titchmarsh, E. A. Coddington, N. Levinson [6, 18] developed his work and established the Weyl-Titchmarsh theory. Singular second-order formally self-adjoint scalar difference equations over infinite intervals were firstly studied by F. V. Atkinson [9]. His work was followed by D. B. Hinton, A. Jirari, etc.[10, 19]. The spectral problems of second-order and higher-order formally self-adjoint vector difference equations and discrete linear Hamiltonian systems were investigated systematically by Y. Shi, S. Chen. S. L. Clark, B. Beckermann, M. Bohner etc [11, 13, 14, 20, 21, 22]. In 2001. J. Chen and Y. Shi [23] obtained a sufficient and necessary condition and several criteria of limit-point and limit-circle cases for second-order formally self-adjoint linear difference equations with real coefficients. Recently, Y. Shi [24] established the Weyl-Titchmarsh theory of discrete linear Hamiltonian systems. More recently, S. Sun [25] extended Shi’s work to Hamiltonian systems on time scales and established Weyl-Titchmarsh theory of Hamiltonian systems on time scales. Sun give the classification of singular Hamiltonian systems on time scales in terms of the defect indices of the minimal operator. In the present thesis, we employ Weyl’s method to divide singular second-order symmetric linear differential equations on time scales into two cases: limit-point and limit-circle cases. This is another focus of this paper.M. Bohner and A. Peterson [3, 4] have made great progress on the basic calculus on time scales. But many results are not complete, such as L’Ho|^spital rules [3, 4, 26]. As we all know, the L’Ho|^spital rule plays an important role in classical calculus. It can help us deal with many problems. In this paper, we will give some revised L’Ho|^spital rules on time scales.This paper is divided into four chapters. In Chapter 1, the time scale calculus is introduced and some fundamental relative theories are given.In Chapter 2. we study eigenvalue problems of second-order symmetric linear equations with periodic and antiperiodic boundary conditions on time scales. We mainly employ the properties of eigenvalues of the Dirichlet boundary value problem and an os(?)illation result to compare eigenvalues of the periodic and antiperiodic boundary value problems on time scales. Finally, we will show our result not only covers those existing results in the differential and difference cases, which are studied by E. A. Cod-dington and X. Levinson [6] and Y. Wang and Y. Shi [15], but also covers other more complicated time scales.In Chapter 3. we focus on the classification of singular second-order symmetric linear differential equations on time scales. Firstly. L~2(I) is proved to be a Hilbert space. Secondly, we construct a family of nested circles. These circles converge to a limiting set. The dichotomy of the limit-point case and limit-circle case for singular second-order symmetric linear differential equations on time scales is given by geometric properties of the limiting set. Finally, several criteria of the limit-point case and limit-circle case are established, respectively.In Chapter 4. applying chain rule and the mean value theorems on time scales, we give two L’Ho|^spital rules on time scales under some weaker conditions.更多还原