Publication details

Rayleigh principle for time scale symplectic systems and applications

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Year of publication 2011
Type Article in Periodical
Magazine / Source Electronic Journal of Qualitative Theory of Differential Equations
MU Faculty or unit

Faculty of Science

Field General mathematics
Keywords Rayleigh principle; Time scale symplectic system; Linear Hamiltonian system; Discrete symplectic system; Finite eigenvalue; Finite eigenfunction; Sturmian separation theorem; Sturmian comparison theorem; Quadratic functional; Positivity
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Description In this paper we establish the Rayleigh principle, i.e., the variational characterization of the eigenvalues, for a general eigenvalue problem consisting of a time scale symplectic system and the Dirichlet boundary conditions. No normality or controllability assumption is imposed on the system. Applications of this result include the Sturmian comparison and separation theorems for time scale symplectic systems. This paper generalizes and unifies the corresponding results obtained recently for the discrete symplectic systems and continuous time linear Hamiltonian systems. The results are also new and particularly interesting for the case when the considered time scale is ``special'', that is, consisting of a union of finitely many disjoint compact real intervals and/or finitely many isolated points.
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