In this paper, the Glazman-Krein-Naimark theory for a class of discrete Hamiltonian systems is developed. A minimal and a maximal operators, GKN-sets, and a boundary space for the system are introduced. Algebraic characterizations of the domains of self-adjoint extensions of the minimal operator are given. A close relationship between the domains of self-adjoint extensions and the GKN-sets is established. It is shown that there exist one-to-one correspondences among the set of all the self-adjoint extensions, the set of all the d-dimensional Lagrangian subspaces of the boundary space, and the set of all the complete Lagrangian subspaces of the boundary space.

This paper is concerned with spectral problems for a class of discrete linear Hamiltonian systems with self-adjoint boundary conditions, where the existence and uniqueness of solutions of initial value problems may not hold. A suitable admissible function space and a difference operator are constructed so that the operator is self-adjoint in the space. Then a series of spectral results are obtained: the reality of eigenvalues, the completeness of the orthogonal normalized eigenfunction system, Rayleigh's principle, the minimax theorem and the dual orthogonality. Especially, the number of eigenvalues including multiplicities and the number of linearly independent eigenfunctions are calculated.

This paper is concerned with coupled boundary value problems for self-adjoint second-order difference equations. Existence of eigenvalues is proved, numbers of eigenvalues are calculated, and relationships between the eigenvalues of a self-adjoint second-order difference equation with three different coupled boundary conditions are established. These results extend the relevant existing results of periodic and antiperiodic boundary value problems.

研究Banachl空间上连续Frechet可微映射导出的离散动力系统之混沌。建立一个由正则非退化同宿轨道产生混沌的判定定理，并对n维实空间上的离散动力系统的混沌进行了讨论，建立了两个由非退化返回扩张不动点产生混沌的判定定理，其中一个为Marotto定理的修正定理。特别地，分别给出了一般Banach空间及n维实空间上的连续可微映射不动点为扩张的充分必要条件，彻底解决了多年以来人们对n维实空间上连续可微映射不动点的扩张性与其Jacobi矩阵特征值之间关系的困惑。

This paper is concerned with spectral problems of higher-order vector difference equations with self-adjoint boundary conditions, where the coefficient of the leading term may be singular. A suitable admissible function space is constructed so that the corresponding difference operator is self-adjoint in it, and the fundamental pectral results are obtained. Rayleigh's principles and minimax theorems in two special linear spaces are given. As an application, comparison theorems for eigenvalues of two Sturm-Liouville problems are presented. Especially, the dual orthogonality and multiplicity of eigenvalues are discussed.