This paper is concerned with spectral problems of second-order vector difference equation with two-point boundary value conditions, where the matrix-valued coneffcient of the leading term may be singular. A concept of self-adjointness of the boundary value conditions is introduced. The self-adjointness of the corresponding difference operator is discussed on a suitable admissible function space, and fundamental spcetal results are obtained. The dual orthogonality of eigenfunc-tions is shown in a special case. Reyleigh's principles and the minimax theorems in two linear spaces are given. As an application, a comparison theorem for eigenval-ues of two Sturm-Liouville problems is presented.

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 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.