Generalized oscillation theorems for symplectic difference systems with nonlinear dependence on spectral parameter

Abstract

In this paper we generalize oscillation theorems for discrete symplectic eigenvalue problems with nonlinear dependence on spectral parameter recently proved by R. Šimon Hilscher and W. Kratz under the assumption that the block Bk(λ) of the symplectic coefficient matrices located in the right upper corner has a constant image for all λ∈R. In our version of the discrete oscillation theorems we avoid this assumption admitting that rankBk(λ) is a piecewise constant function of the spectral parameter λ. Assuming a monotonicity condition for the symplectic coefficient matrices we show that spectrum of symplectic eigenvalue problems with the Dirichlet boundary conditions is bounded from below iff so is the number of jump discontinuities of rankBk(λ).