Note on singular Sturm comparison theorem and strict majorant condition
| Authors | |
|---|---|
| Year of publication | 2024 |
| Type | Article in Periodical |
| Magazine / Source | Journal of Mathematical Analysis and Applications |
| MU Faculty or unit | |
| Citation | |
| Web | https://www.sciencedirect.com/science/article/pii/S0022247X24003135 |
| Doi | http://dx.doi.org/10.1016/j.jmaa.2024.128391 |
| Keywords | Linear Hamiltonian system; Sturm comparison theorem; Focal point; Principal solution; Strict majorant condition; Second order linear differential equation |
| Description | In this note we present a singular Sturm comparison theorem for two linear Hamiltonian systems satisfying a standard majorant condition and the identical normality assumption. Both endpoints of the considered interval may be singular. We identify the exact form of the strict majorant condition, which is necessary and sufficient for the property that every solution (conjoined basis) of the majorant system has more focal points than the solutions of the minorant system. We provide a formula for the exact number of focal points of any solution of the majorant system, depending on the number of focal points of solutions of the minorant system and on the number of right focal points of a solution of a certain transformed linear Hamiltonian system. This transformed system may be in general abnormal. Our result extends the previous Sturm comparison theorems for linear Hamiltonian systems by Kratz (1995) [18] on a compact interval and by the authors (2020) [35], [36] on an open or unbounded interval. The main result is also new for the second order differential equations, where it extends the singular comparison theorem by Aharonov and Elias (2010) [1]. |
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