Eigenvalue theory for time scale symplectic systems depending nonlinearly on spectral parameter

Abstract

In this paper we develop the eigenvalue theory for general time scale symplectic systems, in which the dependence on the spectral parameter λ is allowed to be nonlinear. At the same time we do not impose any controllability or strict normality assumptions. We prove the oscillation theorems for eigenvalue problems with Dirichlet, separated, and jointly varying endpoints, including the periodic boundary conditions. We also allow the boundary conditions depending on the spectral parameter. Our new theory generalizes and unifies recently published results on continuous-time linear Hamiltonian systems and discrete symplectic systems with nonlinear dependence on λ and on time scale symplectic systems with linear dependence on λ. The results of this paper are also new in the special case of linear Hamiltonian systems with variable endpoints, as well as for Sturm–Liouville dynamic equations.