Title: Eigenvalue and oscillation theorems for time scale symplectic systems

Authors: Werner Kratz, Roman Simon Hilscher, Vera Zeidan

Addresses: Faculty of Mathematics and Economics, Department of Applied Analysis, University of Ulm, D-89069, Ulm, Germany. ' Faculty of Science, Department of Mathematics and Statistics, Masaryk University, Kotlarska 2, CZ-61137 Brno, Czech Republic. ' Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA

Abstract: In this paper we study eigenvalue and oscillation properties of time scale symplectic systems with Dirichlet boundary conditions. The focus is on deriving the so-called oscillation theorems for these systems, which relate the number of finite eigenvalues of the system with the number of proper focal points of the principal solution of the system. This amounts to defining and developing the central notions of finite eigenvalues and proper focal points for the time scale environment. We establish the traditional geometric properties of finite eigenvalues and eigenfunctions enjoyed by self-adjoint linear systems. We assume no controllability or normality of the system.

Keywords: time scales; time scale symplectic systems; linear Hamiltonian systems; discrete symplectic systems; finite eigenvalue; proper focal point; generalised focal point; oscillation theorem; onjoined basis; controllability; normality; quadratic functional.

DOI: 10.1504/IJDSDE.2011.038497

International Journal of Dynamical Systems and Differential Equations, 2011 Vol.3 No.1/2, pp.84 - 131

Published online: 09 Feb 2011 *

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