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 the limit circle and limit point criteria of second-order linear difference equations. A sufficient and necessary condition, a sufficient and necessary condition subject to a certain restriction, and several sufficient conditions are established. These results improve and extend some previous results.

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 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 oscillation of self-adjoint second-order vector difference equations with respect to a parameter. Properties of zeros and monotonicity of matrix-valued so-lutions are studied. The oscillation of two consecutive polynomials for vector-valued solutions is discussed. A separation theorem for matrix-valued solutions is also obtained.