Reid roundabout theorem for symplectic dynamic systems on time scales

The principal aim of this paper is to state and prove the so-called Reid ro undabout theorem for the symplectic dynamic system (S) z(Delta) = S(l)z on an arbitrary time scale T, so that the well known case of differential line ar Hamiltonian systems (T = R) and the recently developed case of discrete symplectic systems (T = Z) are unified. We list conditions which are equiva lent to the positivity of the quadratic functional associated with (S), e.g , disconjugacy tin terms of no focal points of a conjoined basis) of (S), n o generalized zeros for vector solutions of (S), and the existence of a sol ution to the corresponding Riccati matrix equation. A certain normality ass umption is employed. The result requires treatment of the quadratic functio nals both with general and separated boundary conditions.