On disconjugacy for vector linear Hamiltonian systems on time scales

摘要

In this work we present a unified treatment of continuous and discrete vector linear Hamiltonian systems on a general time scale T, with the matrix B_t not necessarily invertible. This contains as special cases the Sturm-Liouville differential and difference equations of higher order. We define generalized zeros for vector solutions (x, u) of the Hamiltonian system. From this we read off the corresponding definition of generalized zero points for solutions of the Sturm-Liouville equations, so that the well known continuous case (T = R) and recently developed discrete one (T = Z) are unified. We show that disconjugacy of the equation implies positivity of the corresponding quadratic functional.