Note on singular Sturm comparison theorem and strict majorant
condition

J 2024

Note on singular Sturm comparison theorem and strict majorant condition

ŠEPITKA, Peter and Roman ŠIMON HILSCHER

Basic information

Original name

Note on singular Sturm comparison theorem and strict majorant condition

Authors

ŠEPITKA, Peter (703 Slovakia, belonging to the institution) and Roman ŠIMON HILSCHER (203 Czech Republic, guarantor, belonging to the institution)

Edition

Journal of Mathematical Analysis and Applications, Elsevier, 2024, 0022-247X

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

United States of America

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

Impact factor

Impact factor: 1.300 in 2022

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.1016/j.jmaa.2024.128391

UT WoS

001229936600001

Keywords in English

Linear Hamiltonian system; Sturm comparison theorem; Focal point; Principal solution; Strict majorant condition; Second order linear differential equation

Abstract

V originále

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].

Links

GA23-05242S, research and development project

Name: Oscilační teorie na hybridních časových doménách s aplikacemi ve spektrální teorii a maticové analýze

Investor: Czech Science Foundation, Oscillation theory on hybrid time domains with applications in spectral and matrix analysis

Citovat

ŠEPITKA, Peter and Roman ŠIMON HILSCHER. Note on singular Sturm comparison theorem and strict majorant condition. Journal of Mathematical Analysis and Applications. Elsevier, 2024, vol. 538, No 2, p. 1-16. ISSN 0022-247X. Available from: https://dx.doi.org/10.1016/j.jmaa.2024.128391.

@article{2394577,
   author = {Šepitka, Peter and Šimon Hilscher, Roman},
   article_number = {2},
   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},
   language = {eng},
   issn = {0022-247X},
   journal = {Journal of Mathematical Analysis and Applications},
   title = {Note on singular Sturm comparison theorem and strict majorant condition},
   url = {https://www.sciencedirect.com/science/article/pii/S0022247X24003135},
   volume = {538},
   year = {2024}
}

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AU  - Šepitka, Peter - Šimon Hilscher, Roman
PY  - 2024
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JF  - Journal of Mathematical Analysis and Applications
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PB  - Elsevier
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KW  - Linear Hamiltonian system
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UR  - https://www.sciencedirect.com/science/article/pii/S0022247X24003135
N2  - 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].
ER  -

ŠEPITKA, Peter and Roman ŠIMON HILSCHER. Note on singular Sturm comparison theorem and strict majorant condition. \textit{Journal of Mathematical Analysis and Applications}. Elsevier, 2024, vol.~538, No~2, p.~1-16. ISSN~0022-247X. Available from: https://dx.doi.org/10.1016/j.jmaa.2024.128391.

Detailed Information on Publication Record