This paper is concerned with the rank of the matrix radius of the limiting set for a singular Hamiltonian system with one singular end point. The exact relationship between the rank of the matrix radius and the number of square integrable solutions is obtained and then the defect index of the corresponding minimal operator can be represented in terms of the rank of the matrix radius. So two results obtained by Allan M. Krall [SIAM J. Math. Anal. 20 (1989) 664] are improved. In addition, it is discussed that the rank of the matrix radius is independent of the spectral parameter and a certain matrix. Especially, the classification of singular linear Hamiltonian systems is present and several sufficient and necessary conditions for the limit point and limit circle cases are established.

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.

This paper is concerned with chaoti cation of discrete dynamical systems in nite-dimensional real spaces, via feedback control techniques. A chaoti cation theorem for one-dimensional discrete dynamical systems and a chaoti cation theorem for general higher-dimensional discrete dynamical systems are established, respectively. The controlled systems are proved to be chaotic in the sense of Devaney. In particular, the maps corresponding to the original systems and designed controllers are only required to satisfy some mild assumptions on two very small disjoint closed subsets in the domains of interest. This condition is weaker than those in the existing relevant literature.

This paper is concerned with periodic and antiperiodic boundary value problems for self-adjoint second-order difference equations. Existence of eigenvalues of these two different boundary value problems is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. In addition, a representation of solutions of a nonhomogeneous linear equation with initial conditions is given.

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.