Weyl-Titchmarsh theory for time scale symplectic systems on half line. (English) Zbl 1217.34144

Šimon Hilscher, Roman; Zemánek, Petr

Abstr. Appl. Anal. 2011, Article ID 738520, 41 p. (2011).

Summary: We develop the Weyl-Titchmarsh theory for time scale symplectic systems. We introduce the \(M(\lambda)\)-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 [Appl. Math. Comput. 217, No. 7, 2952–2976 (2010; Zbl 1214.37034)]. It also unifies the results in many other papers on 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 second-order Sturm-Liouville equations on time scales.

MSC:

34N05	Dynamic equations on time scales or measure chains
34B20	Weyl theory and its generalizations for ordinary differential equations

Citations:

Zbl 1214.37034

References:

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