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On general Sturmian theory for abnormal linear Hamiltonian systems

Pages: 684 - 691, Issue Special, September 2011

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Roman Šimon Hilscher - Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, CZ-61137 Brno, Czech Republic (email)

Abstract: In this paper we discuss oscillation theory for linear Hamiltonian systems for which we do not impose the controllability (or equivalently normality) assumption. Based on the Sturmian separation and comparison theorems on a compact interval, derived earlier by the author for these systems, we classify them as oscillatory or nonoscillatory. Moreover, we provide comparison theorems for such oscillatory and nonoscillatory systems. One of the goals of this paper is to provide several examples illustrating this new theory.

Keywords:  Linear Hamiltonian system, Sturmian separation theorem, Sturmian comparison theorem, proper focal point, conjoined basis, controllability, normality
Mathematics Subject Classification:  Primary: 34C10; Secondary: 34B24

Received: July 2010;      Revised: February 2011;      Published: October 2011.