In this paper we present several new contributions to the oscillation theory of linear differential equations, in particular of linear Hamiltonian systems, when the traditional Legendre condition is absent. Following our recent work (2023), we introduce the multiplicity of a generalized right focal point and derive the corresponding local Sturmian separation theorem. We also examine the relation between the existence of finitely many generalized right focal points, or in the special case the nonexistence of generalized right focal points, with the Legendre condition. As the main tools we use new notions of the minimal multiplicities at a given point and the dual comparative index - an object from matrix analysis or differential geometry (Maslov index theory). Furthermore, we study local limit properties of the dual comparative index and the comparative index and apply them for deriving new oscillation results phrased in terms of the generalized right and left focal points. The investigation of the interplay between generalized right and left focal points leads to conditions characterizing the situation, when in the local Sturmian separation theorem the corresponding multiplicities attain the minimal possible value. This also provides a generalization of the concepts of the right and left proper focal point defined by Kratz (2003) and Wahrheit (2007) to the setting, which does not impose the Legendre condition. The results are new even for completely controllable linear Hamiltonian systems, including the Sturm-Liouville differential equations of arbitrary even order.

Last change: 06.09.2024. (c) Roman Simon Hilscher