Eigenvalue and oscillation theorems for time scale symplectic systems

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Authors	KRATZ Werner ŠIMON HILSCHER Roman ZEIDAN Vera Michel
Year of publication	2011
Type	Article in Periodical
Magazine / Source	International Journal of Dynamical Systems and Differential Equations
MU Faculty or unit	Faculty of Science
Citation
Field	General mathematics
Keywords	Time scale; Time scale symplectic system; Linear Hamiltonian system; Discrete symplectic system; Finite eigenvalue; Proper focal point; Generalized focal point; Oscillation theorem; Conjoined basis; Controllability; Normality; Quadratic functional
Attached files	ijdsde_oscssts.pdf
Description	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 (or generalized) 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.
Related projects:	Mathematical structures and their physical applications Asymptotika, oscilace a kvadratické funkcionály v teorii dynamických rovnic Second order optimality conditions for optimization problems Oscillation and spectral theory of differential and difference systems