Generalized focal points and local Sturmian theory for linear Hamiltonian systems

In this paper we present a new approach for the study of oscillation properties linear differential equations, in particular Hamiltonian systems. We introduce notion generalized left focal point as well its multiplicity, which do not depend on validity traditionally assumed Legendre condition. Based are able to develop local (or pointwise) version Sturmian separation theorem, provides lower bound and an upper multiplicity any conjoined basis system. apply knowledge several directions, such (ⅰ) explanation exact role condition theory, (ⅱ) second order optimality conditions variational problems, (ⅲ) analysis isolated non-isolated points, (ⅳ) so-called anti-Legendre As main tool use comparative index properties. The results even completely controllable systems, including Sturm–Liouville equations arbitrary order.