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 chaos of discrete dynamical systems in complete metric spaces. Discrete dynamical systems governed by continuous maps in general complete metric spaces are first discussed, and two criteria of chaos are then established. As a special case, two corresponding criteria of chaos for discrete dynamical systems in compact subsets of metric spaces are obtained. These results have extended and improved the existing relevant results of chaos in finite-dimensional Euclidean spaces.

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.

In this paper, the notion of anti-control of chaos (or chaotification) is introduced, which means to make an originally non-chaotic dynamical system chaotic or enhance the existing chaos of a chaotic system. The main interest in this paper is to employ the classical feedback control techniques. Only the discrete case is discussed in detail, including both finite-dimensional and infinite-dimensional settings.

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.