Relative oscillation theory for linear Hamiltonian systems with nonlinear dependence on the spectral parameter

Abstract

AbstractIn this paper, we consider two linear Hamiltonian differential systems that depend in general nonlinearly on the spectral parameter λ and with Dirichlet boundary conditions. For the Hamiltonian problems, we do not assume any controllability and strict normality assumptions and also omit the classical Legendre condition for their Hamiltonians. The main result of the paper, the relative oscillation theorem, relates the difference of the numbers of finite eigenvalues of the two problems in the intervals and , respectively, with the so‐called oscillation numbers associated with the Wronskian of the principal solutions of the systems evaluated for and . As a corollary to the main result, we prove the renormalized oscillation theorems presenting the number of finite eigenvalues of one single problem in . The consideration is based on the comparative index theory applied to the continuous case.