Publication details
Oscillation theory on hybrid time domains: Local oscillation properties
| Authors | |
|---|---|
| Year of publication | 2026 |
| Type | Article in Periodical |
| Magazine / Source | Journal of Differential Equations |
| MU Faculty or unit | |
| Citation | |
| web | https://www.sciencedirect.com/science/article/pii/S0022039625011076 |
| Doi | https://doi.org/10.1016/j.jde.2025.114080 |
| Keywords | Canonical system; Symplectic system; Time scale; Generalized focal point; Sturm separation theorem; Comparative index |
| Description | In this paper we introduce a new approach suitable for studying the local oscillation properties of solutions to canonical systems defined on arbitrary hybrid time domains, also called general time scales. Such systems are known as symplectic or Hamiltonian systems on time scales. We define the notions of the local multiplicities of generalized left and right focal points for conjoined bases of the system and establish, among other results, a local version of the Sturm separation theorem. This result leads to a new concept in the oscillation theory on time scales, which we call the minimal multiplicity at the given point. We derive several properties of these minimal multiplicities with special focus on their zero value. Our analysis is based on the theory of comparative index and dual comparative index of two Lagrangian planes, which is introduced and applied for the first time in this paper to canonical systems on time scales. We also relate the local multiplicities of generalized focal points corresponding to two conjoined bases with the limit properties of the comparative index and the dual comparative index. This theory produces new results when also applied to matrix Jacobi systems arising in variational analysis over time scales or to second order Sturm–Liouville equations on time scales. |
| Related projects: |