Contributions to Generalized Oscillation Theory of Linear
Hamiltonian Systems

J 2024

Contributions to Generalized Oscillation Theory of Linear Hamiltonian Systems

ŠEPITKA, Peter a Roman ŠIMON HILSCHER

Základní údaje

Originální název

Contributions to Generalized Oscillation Theory of Linear Hamiltonian Systems

Autoři

ŠEPITKA, Peter a Roman ŠIMON HILSCHER

Vydání

Results in Mathematics, Springer Nature, 2024, 1422-6383

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10101 Pure mathematics

Stát vydavatele

Švédsko

Utajení

není předmětem státního či obchodního tajemství

Odkazy

URL

Impakt faktor

Impact factor: 1.200

Označené pro přenos do RIV

Ano

Kód RIV

RIV/00216224:14310/24:00139566

Organizační jednotka

Přírodovědecká fakulta

Klíčová slova anglicky

Linear Hamiltonian systém; generalized right focal point; generalized left focal point; sturmian separation theorem; legendre condition; comparative index; dual comparative index

Štítky

14311010, rivok

Příznaky

Mezinárodní význam, Recenzováno

Změněno: 10. 12. 2024 15:44, Mgr. Marie Novosadová Šípková, DiS.

Anotace

V originále

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 (Discrete Contin. Dyn. Syst. 43(12):4139–4173, 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 (Analysis 23(2):163–183, 2003) and Wahrheit (Int. J. Differ. Equ. 2(2):221–244, 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.

Návaznosti

GA23-05242S, projekt VaV

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

Investor: Grantová agentura ČR, Oscilační teorie na hybridních časových doménách s aplikacemi ve spektrální a maticové analýze

Citovat

ŠEPITKA, Peter a Roman ŠIMON HILSCHER. Contributions to Generalized Oscillation Theory of Linear Hamiltonian Systems. Results in Mathematics. Springer Nature, 2024, roč. 79, č. 8, s. 1-47. ISSN 1422-6383. Dostupné z: https://doi.org/10.1007/s00025-024-02280-7.

@article{2445780,
   author = {Šepitka, Peter and Šimon Hilscher, Roman},
   article_number = {8},
   doi = {https://doi.org/10.1007/s00025-024-02280-7},
   keywords = {Linear Hamiltonian systém; generalized right focal point; generalized left focal point; sturmian separation theorem; legendre condition; comparative index; dual comparative index},
   language = {eng},
   issn = {1422-6383},
   journal = {Results in Mathematics},
   title = {Contributions to Generalized Oscillation Theory of Linear Hamiltonian Systems},
   url = {https://link.springer.com/article/10.1007/s00025-024-02280-7},
   volume = {79},
   year = {2024}
}

TY  - JOUR
ID  - 2445780
AU  - Šepitka, Peter - Šimon Hilscher, Roman
PY  - 2024
TI  - Contributions to Generalized Oscillation Theory of Linear Hamiltonian Systems
JF  - Results in Mathematics
VL  - 79
IS  - 8
SP  - 1-47
EP  - 1-47
PB  - Springer Nature
SN  - 14226383
KW  - Linear Hamiltonian systém
KW  - generalized right focal point
KW  - generalized left focal point
KW  - sturmian separation theorem
KW  - legendre condition
KW  - comparative index
KW  - dual comparative index
UR  - https://link.springer.com/article/10.1007/s00025-024-02280-7
N2  - 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 (Discrete Contin. Dyn. Syst. 43(12):4139–4173, 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 (Analysis 23(2):163–183, 2003) and Wahrheit (Int. J. Differ. Equ. 2(2):221–244, 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.
ER  -

ŠEPITKA, Peter a Roman ŠIMON HILSCHER. Contributions to Generalized Oscillation Theory of Linear Hamiltonian Systems. \textit{Results in Mathematics}. Springer Nature, 2024, roč.~79, č.~8, s.~1-47. ISSN~1422-6383. Dostupné z: https://doi.org/10.1007/s00025-024-02280-7.

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