Oscillation theorems for linear Hamiltonian systems with nonlinear dependence on the spectral parameter and separated boundary conditions

Abstract

In this paper, we consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with separated boundary conditions. In our consideration we do not impose any controllability and strict normality assumptions and omit the Legendre condition for the Hamiltonian. The main results generalize our previous investigations for the Hamiltonian spectral problems with Dirichlet boundary conditions. We prove the local and global oscillation theorems relating the number of left finite eigenvalues of the problem in the given interval with the values of the oscillation numbers at the end points of this interval.