Eigenvalue Problems Of Second-Order Difference Equations And Transformations Of Complex Discrete Linear Hamiltonian Systems

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].