R.Hilscher

Reid roundabout theorem for symplectic dynamic systems on time scales

Abstract

The principal aim of this paper is to state and prove the so called Reid roundabout theorem for symplectic dynamic system (S) z\Delta=Stz on an arbitrary time scale T, so that the well known case of differential linear Hamiltonian systems (T=R) and recently developed case of discrete symplectic systems (T=Z) are unified. We list conditions which are equivalent to the positivity of the quadratic functional associated with (S), e.g. disconjugacy (in terms of no focal points of a conjoined basis) of (S), no generalized zeros for vector solutions of (S), the existence of a solution to the corresponding Riccati matrix equation. A certain normality assumption is employed. The result requires treatment of the quadratic functionals both with general and separated boundary conditions.



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Last change: October 5, 2000. (c) Roman Hilscher