Publication details
Oscillation theory on hybrid time domains: Global oscillation properties
| Authors | |
|---|---|
| Year of publication | 2026 |
| Type | Peer-reviewed scientific article |
| Magazine / Source | Journal of Differential Equations |
| MU Faculty or unit | |
| Citation | |
| web | https://www.sciencedirect.com/science/article/pii/S0022039626003311 |
| Doi | https://doi.org/10.1016/j.jde.2026.114424 |
| Keywords | Canonical system; Time scale; Generalized focal point; Sturm separation theorem; Comparative index; Matrix Jacobi system |
| Description | In this paper we investigate global oscillation properties of solutions to canonical differential systems on arbitrary hybrid time domains (time scales). We use the concepts of generalized left and right focal points for solutions defined on hybrid time domains, which we introduced and developed in our previous paper, in order to derive the Sturm separation theorems on a given time scale interval. These separation theorems are new even for matrix Jacobi systems arising in the optimal control problems on time scales or for the second order Sturm–Liouville equations on time scales. We utilize the comparative index theory of two Lagrangian planes as a key tool to study the global nonexistence of generalized left and right focal points, the monotonicity properties of the comparative index involving two solutions of the system, and the mutual relations between the numbers of generalized left and right focal points. We also consider the minimal multiplicities at left-dense and right-dense points and the strengthened Legendre condition to establish additional new results in this subject. The presented theory unifies and extends the global oscillation (Sturm–type) results known in the special cases of continuous time linear Hamiltonian differential systems and discrete time symplectic difference systems. |
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