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. The principal aim of this paper is to state and prove the so-called Reid roundabout theorem for the symplectic dynamic system (S) zΔ= \cal S tzon an arbitrary time scale \Bbb T , so that the well known case of differential linear Hamiltonian systems ( \Bbb T = \Bbb R ) and the recently developed case of discrete symplectic systems ( \Bbb T = \Bbb 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), and 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.
Applied Mathematics & Optimization – Springer Journals
Published: Jan 1, 2001
Keywords: Time scale; Symplectic system; Linear Hamiltonian system; Quadratic functional; Disconjugacy; Focal point; Principal solution; Riccati equation; Jacobi condition; Legendre condition; AMS Classification. 34C10, 39A10, 93C70
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