【作者】 王怡；

【导师】 史玉明；

【作者基本信息】 山东大学 ， 基础数学， 2005， 硕士

【摘要】 本文主要讨论两方面的内容。一是二阶差分方程周期和反周期边值问题的特征值,二是复离散哈密顿系统的Prüfer变换和三角变换。 对于微分方程周期和反周期边值问题的特征值,Coddington和Levinson,Hale,Magnus和Winkler[1,2,3]等人研究二阶微分方程边值问题特征值的性质,并比较周期和反周期两种边值问题的特征值,获得一些很漂亮的结果。这些结果被章梅荣[4]推广到一维p-拉普拉斯算子。 而对于差分方程边值问题的特征值,Atkinson,Bohner,Jirari,史玉明,陈绍著[5,6,7,8,9,10]等人做出很大贡献。但是,对于二阶差分方程的周期和反周期边值问题的特征值及其比较却鲜有讨论。因此,本文的一个主要目的就是探讨这一问题。 本文要讨论的第二个问题是复离散哈密顿系统的变换。在经典的Sturm-Liouville理论中,Prüfer变换和三角变换是研究方程振动性,谱理论等的重要工具。随着高维系统研究的深入,Barrett[11]建立连续的三角系统,从而为这两种变换在高维情形下的推广奠定基础。利用三角系统,Barrett,Reid,郑召文[11,12,13]得到连续哈密顿系统的Prüfer变换,并获得一系列哈密顿系统的振动性结论.而Do(?)l(?)[14]则把三角变换推广到连续哈密顿系统。 对于差分方程的研究,Anderson[15]首先建立一种特殊的离散三角系统,引入高维离散的正弦,余弦函数,从而使Prüfer变换和三角变换能够拓展到离散的情形。Bohner和Do(?)ly[16,17,18]对此做出主要贡献。他们定义一般实离散三角系统的概念,利用该系统给出实辛差分系统的Prüfer和三角变换,并由此获得实辛差分系统的一些振动性判定定理。 但是,Bohner和Do(?)l(?)的变换仅针对实差分系统,而对于更广泛的复差分系统,如复离散哈密顿系统则不适用。本文的另一个主要目的就是给出更一般的复离散三角系统的概念以及复离散哈密顿系统的Prüfer变换和三角变换。全文共分为三章。每章的第一节简要介绍所研究问题的背景。更多还原

【Abstract】 This paper are mainly divided into two parts. One is on eigenvalues of second-order difference equations with periodic and antiperiodic boundary conditions, the other is on transformations of complex discrete Hamiltonian systems.For eigenvalue problems of second-order differential equations with periodic and antiperiodic boundary conditions, Coddington and Levinson, Hale, and Magnus and Winkler [1, 2, 3] studied properties of eigenvalues of second-order differential equations and compared eigenvalues of the periodic and antiperiodic boundary value problems. They obtained some beautiful results. Zhang [4] extended these results to one-dimensional p-Laplacian.For spectral theory of difference equations boundary value problems, Atkinson, Bohner, Jirari, Shi, and Chen [5, 6, 7, 8, 9, 10] did a lot of good work. But, there is little literature which discusses eigenvalues of periodic and antiperiodic boundary value problems and compares them. This is one main aim of the paper.The other subject of this paper is to discuss transformations of complex discrete linear Hamiltonian systems. In the study of Sturm-Liouville theory and oscillation theory of second-order linear ordinary differential equations, both the Priifer transformation and the trigonometric transformation are very useful tools. With the development of the research on higher-dimensional systems, Barrett [11] established continuous trigonometric systems and provided a basis for extending the two transformations to higher-dimensional cases. Using trigonometric systems, Barrett, Reid, and Zheng [11, 12,13] obtained Priifer transformations of continuous linear Hamiltonian and a series ofoscillation results. Dosly [14] extended the trigonometric transformation to continuous linear Hamiltonian systems.For difference equations, Anderson [15] first established a kind of special discrete trigonometric systems and introduced discrete sine and cosine functions so that Prufer and trigonometric transformations can be extended to discrete higher-dimensional cases. Bohner and Dosly [16,17,18] defined general real discrete trigonometric systems, gave Priifer and trigonometric transformations of real symplectic difference systems, and obtained several oscillation criteria.However, in the research of complex linear Hamiltonian systems, those two transformations established by Bohner and Dosly are not available. The second aim of this paper is to establish a more general complex discrete trigonometric system, a Priifer transformation, and a trigonometric transformation for the complex discrete linear Hamiltonian system. This paper is divided into three chapters. The first section of each chapter introduces the relative background.This first chapter discusses eigenvalue problems of second-order difference equations with periodic and antiperiodic boundary conditions. Section two of Chapter One first introduces Wronskian Identity. By using results of Atkinson [5, Chapter 4], properties of eigenvalues of the Dirichlet boundary value problem and a special oscillation result are given. According to results of Shi and Chen [9, Lemma 2.1 and Theorem 4.1], existence and numbers of eigenvalues of the periodic and antiperiodic boundary value problems are discussed. A representation of solutions of a nonhomogeneous linear equation with initial conditions is given. Section three compares eigenvalues of the periodic and antiperiodic boundary value problems. This result can be regarded as a discrete analog of [1, Chapter 8, Theorem 3.1]. The last section proves an important lemma used in Section three.Chapter two and three discuss transformations of complex discrete linear Hamiltonian systems. Chapter two prepares for Chapter three. Since complex discrete trigonometric systems are special complex symplectic systems, Section two will first introduce the definition of complex symplectic systems and some useful lemmas, and give complex discrete trigonometric system. Several properties and a criterion which play an important role in the next section are given in Section three. In addition, trigonometric systems can be transformed to some special symplectic systems which have better properties.Section two of the third chapter remarks that any complex discrete Hamiltonian system can be written as a symplectic system, but a symplectic system may not be written as a Hamiltonian system. Lemmas and the Prufer transformation are given. Section three is devoted to trigonometric transformations which preserve oscillatory properties. The results of the Prufer and trigonometric transformations in this paper include those of [16, 18].更多还原