Multiplicities of focal points for discrete symplectic systems: revisited

Abstract

In this note, we define a notion of multiplicity of focal points for conjoined bases of discrete symplectic systems. We show that this definition is equivalent to the one given by Kratz in [Discrete oscillation, J. Difference Equ. Appl., 9(1), 135–147 (2003)] and, furthermore, it has a natural connection to the newly developed continuous time theory on linear Hamiltonian differential systems. Many results obtained recently by Bohner, Došlý, and Kratz regarding the non-negativity of the corresponding discrete quadratic functionals, Sturmian separation and comparison theorems, and oscillation theorems relating the number of focal points of a certain special conjoined basis with the number of eigenvalues of the associated discrete symplectic eigenvalue problem, are now formulated in terms of this alternative definition of multiplicities.