Appl Math Optim 43:129–146 (2001)
2001 Springer-Verlag New York Inc.
Reid Roundabout Theorem for Symplectic Dynamic Systems
on Time Scales
Department of Mathematics, Masaryk University Brno,
Jan´aˇckovo n´am. 2a, CZ-66295 Brno, Czech Republic
Abstract. The principal aim of this paper is to state and prove the so-called Reid
roundabout theorem for the symplectic dynamic system (S) z
z on an arbitrary
time scale T, so that the well known case of differential linear Hamiltonian systems
(T = R) and the 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), 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.
KeyWords. Timescale,Symplectic system, LinearHamiltoniansystem,Quadratic
functional, Disconjugacy, Focal point, Principal solution, Riccati equation, Jacobi
condition, Legendre condition.
AMS Classiﬁcation. 34C10, 39A10, 93C70
The main purpose of this paper is to state and prove the so-called Reid roundabout
theorem for symplectic dynamic systems on time scales. Such a result not only uniﬁes
the continuous (= differential) and discrete (= difference) roundabout theorems, but
also explicitly explains the essential discrepancies between them.
This research was supported by Grants 201/98/0677 and 201/99/0295 (GA