This paper is concerned with 2-D discrete systems of the form xm-1, n=f(xm,n, xm,n+1), where f:R2→R is a function, m, n e N0={0, 1, 2, . . .}. Some sufficient conditions for this system to be stable and a verification of this system to be chaotic in the sense of Devaney, respectively, are derived.

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.

This paper is concerned with establishing the Weyl-Titchmarsh theory for a class of discrete linear Hamiltonian systems over a half-line. Fundamental properties of solutions, regular spectral problems, and the corresponding maximal and minimal operators are first studied. Matrix disks are constructed and proved to be nested and converge to a limiting set. Some precise relationships among the rank of the matrix radius of the limiting set, the number of linearly independent square summable solutions, and the defect indices of the minimal operator are established. Based on the above results, a classification of singular discrete linear Hamiltonian systems is given in terms of the defect indices of the minimal operator, and several equivalent conditions on the cases of limit point and limit circle are obtained, respectively. Especially, several problems in the limit point case are more carefully investigated, including fundamental properties of square summable solutions, properties of the Weyl function, which is the unique element in the limiting set in this case, and inhomogeneous boundary problems, self-adjointness of the corresponding Hamiltonian operator, relationship between the spectrum of the Hamiltonian operator and the analyticity of the Weyl function, as well as the dependence of the spectrum on the boundary data, in which some interesting separation results for the spectrum are obtained. Finally, another set of four equivalent conditions on the limit point case are established.

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.

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。