This paper is concerned with coupled boundary value problems for self-adjoint second-order difference equations. Existence of eigenvalues is proved, numbers of eigenvalues are calculated, and relationships between the eigenvalues of a self-adjoint second-order difference equation with three different coupled boundary conditions are established. These results extend the relevant existing results of periodic and antiperiodic boundary value problems.

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 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.