Eigenvalues of discrete linear second-order periodic and antiperiodic eigenvalue problems with sign-changing weight

By virtue of the eigenvalues of discrete linear second-order Neumann eigenvalue problems, we study the eigenvalues of discrete linear second-order periodic and antiperiodic eigenvalue problems with sign-changing weight. We find that these two problems have T real eigenvalues (including the multiplicity) respectively. Furthermore, the numbers of positive eigenvalues are equal to the numbers of positive elements in the weight function, and the numbers of negative eigenvalues are equal to the numbers of negative elements in the weight function. Furthermore, these eigenvalues, including the eigenvalues of Neumann problem, satisfy the order relation.