This paper is concerned with 2-D discrete systems of the form xm-1, n=f(xm,n, xm,n+1), where f:R2→R is a function, m, n e N0={0, 1, 2, . . .}. Some sufficient conditions for this system to be stable and a verification of this system to be chaotic in the sense of Devaney, respectively, are derived.

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 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 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 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.