Oscillation theorems for discrete symplectic systems with nonlinear dependence in spectral parameter

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Authors

ŠIMON HILSCHER Roman

Year of publication 2012
Type Article in Periodical
Magazine / Source Linear Algebra and Its Applications
MU Faculty or unit

Faculty of Science

Citation
Doi http://dx.doi.org/10.1016/j.laa.2012.06.033
Field General mathematics
Keywords Discrete symplectic system; Oscillation theorem; Finite eigenvalue; Finite eigenfunction; Linear Hamiltonian system; Quadratic functional
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Description In this paper we open a new direction in the study of discrete symplectic systems and Sturm-Liouville difference equations by introducing nonlinear dependence in the spectral parameter. We develop the notions of (finite) eigenvalues and (finite) eigenfunctions and their multiplicities, and prove the corresponding oscillation theorem for Dirichlet boundary conditions. The present theory generalizes several known results for discrete symplectic systems which depend linearly on the spectral parameter. Our results are new even for special discrete symplectic systems, namely for Sturm-Liouville difference equations, symmetric three-term recurrence equations, and linear Hamiltonian difference systems.
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