Abstract and Applied Analysis
Volume 2011 (2011), Article ID 738520, 41 pages
doi:10.1155/2011/738520
Research Article

Weyl-Titchmarsh Theory for Time Scale Symplectic Systems on Half Line

Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic

Received 8 October 2010; Accepted 3 January 2011

Academic Editor: Miroslava Růžičková

Copyright © 2011 Roman Šimon Hilscher and Petr Zemánek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We develop the Weyl-Titchmarsh theory for time scale symplectic systems. We introduce the 𝑀 ( 𝜆 ) -function, study its properties, construct the corresponding Weyl disk and Weyl circle, and establish their geometric structure including the formulas for their center and matrix radii. Similar properties are then derived for the limiting Weyl disk. We discuss the notions of the system being in the limit point or limit circle case and prove several characterizations of the system in the limit point case and one condition for the limit circle case. We also define the Green function for the associated nonhomogeneous system and use its properties for deriving further results for the original system in the limit point or limit circle case. Our work directly generalizes the corresponding discrete time theory obtained recently by S. Clark and P. Zemánek (2010). It also unifies the results in many other papers on the Weyl-Titchmarsh theory for linear Hamiltonian differential, difference, and dynamic systems when the spectral parameter appears in the second equation. Some of our results are new even in the case of the second-order Sturm-Liouville equations on time scales.