Discrete oscillation theorems and weighted focal points for Hamiltonian difference systems with nonlinear dependence on a spectral parameter

In this paper we generalize oscillation theorems for discrete Hamiltonian eigenvalue problems with nonlinear dependence on the spectral parameter λλ. In our version of the discrete oscillation theorems, we incorporate the case when the block Bk(λ)Bk(λ) of the discrete Hamiltonian Hk(λ)Hk(λ) has nonconstant rank with respect to λλ. We introduce a new notion of weighted focal points for conjoined bases of the Hamiltonian difference systems and we show that the number of weighted focal points plays the role of the classical number of focal points in the discrete oscillation theorems for the Hamiltonian spectral problems with the nonconstant rank of Bk(λ)Bk(λ).