This paper is concerned with the symplectic structure of discrete nonlinear Hamiltonian systems. The results are related to an open problem that was first proposed by C. D. Ahlbrandt [J. Math. Anal. Appl. 180 (1993), 498-517] discussed elsewhere in the literature. But we give a different statement and different proof. Under a solvable condition, we show that the solution operator of a discrete nonlinear Halmiltonian system is symplectic. Then its phase flow is a discrete one-parameter family of symplectic transformations and preserves the phase volume.

This paper is concerned with spectral problems of higher-order vector difference equations with self-adjoint boundary conditions, where the coefficient of the leading term may be singular. A suitable admissible function space is constructed so that the corresponding difference operator is self-adjoint in it, and the fundamental pectral results are obtained. Rayleigh's principles and minimax theorems in two special linear spaces are given. As an application, comparison theorems for eigenvalues of two Sturm-Liouville problems are presented. Especially, the dual orthogonality and multiplicity of eigenvalues are discussed.

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.