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

Abstract

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.