This paper is concerned with chaos in discrete dynamical systems governed by continuously Frech6t differentiable maps in Banach spaces.A criterion of chaos in-duced by a regular nondegenerate homoclinic orbit is established.Chaos of discrete dy-namica0 systems in the n-dimensiona0 rea0 space is also discussed.with two criteria de-rived for chaos Induced by nondegenerate snap-back repellers，one of which is a modified version of Marotto's theorem.In particular.a necessary and sufficient condition is obtained for an expanding fixed point Of a differentiable map in a general Banach space.and in an n-dimensiona0 real space.respectively.It completely solves a long-standing puzzle about the relationship between the expansion of a continuously differentiable map near a fixed point In an n-dimensional real space and the eigenvalues of the Jacobi matrix of the map at the fixed point.

This paper is concerned with chaos of discrete dynamical systems in complete metric spaces. Discrete dynamical systems governed by continuous maps in general complete metric spaces are first discussed, and two criteria of chaos are then established. As a special case, two corresponding criteria of chaos for discrete dynamical systems in compact subsets of metric spaces are obtained. These results have extended and improved the existing relevant results of chaos in finite-dimensional Euclidean spaces.

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 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.