Publication details

Oscillation theorems and Rayleigh principle for linear Hamiltonian and symplectic systems with general boundary conditions



Year of publication 2012
Type Article in Periodical
Magazine / Source Applied Mathematics and Computation
MU Faculty or unit

Faculty of Science

Field General mathematics
Keywords Oscillation theorem; Rayleigh principle; Linear Hamiltonian system; Time scale symplectic system; Discrete symplectic system; Finite eigenvalue; Finite eigenfunction; Quadratic functional; Positivity; Selfadjoint eigenvalue problem
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Description The aim of this paper is to establish the oscillation theorems, Rayleigh principle, and coercivity results for linear Hamiltonian and symplectic systems with general boundary conditions, i.e., for the case of separated and jointly varying endpoints, and with no controllability (normality) and strong observability assumptions. Our method is to consider the time interval as a time scale and apply suitable time scales techniques to reduce the problem with separated endpoints into a problem with Dirichlet boundary conditions, and the problem with jointly varying endpoints into a problem with separated endpoints. These more general results on time scales then provide new results for the continuous time linear Hamiltonian systems as well as for the discrete symplectic systems. This paper also solves an open problem of deriving the oscillation theorem for problems with periodic boundary conditions. Furthermore, the present work demonstrates the utility and power of the analysis on time scales in obtaining new results especially in the classical continuous and discrete time theories.
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