Generalized focal points and local Sturmian theory for linear
Hamiltonian systems

J 2023

Generalized focal points and local Sturmian theory for linear Hamiltonian systems

ŠEPITKA, Peter a Roman ŠIMON HILSCHER

Základní údaje

Originální název

Generalized focal points and local Sturmian theory for linear Hamiltonian systems

Autoři

ŠEPITKA, Peter a Roman ŠIMON HILSCHER

Vydání

Discrete and Continuous Dynamical Systems, American Institute of Mathematical Sciences, 2023, 1078-0947

Další údaje

Jazyk

angličtina

Typ výsledku

Článek v odborném periodiku

Obor

10101 Pure mathematics

Stát vydavatele

Spojené státy

Utajení

není předmětem státního či obchodního tajemství

Odkazy

URL

Impakt faktor

Impact factor: 1.100

Označené pro přenos do RIV

Ano

Kód RIV

RIV/00216224:14310/23:00134179

Organizační jednotka

Přírodovědecká fakulta

Klíčová slova anglicky

Linear Hamiltonian system; generalized left focal point; Sturmian separation theorem; Legendre condition; comparative index; principal solution; Anti-Legendre condition

Štítky

14311010, rivok

Příznaky

Mezinárodní význam, Recenzováno

Změněno: 12. 1. 2024 14:30, Mgr. Marie Novosadová Šípková, DiS.

Anotace

V originále

In this paper we present a new approach for the study of the oscillation properties of linear differential equations, in particular of linear Hamiltonian systems. We introduce a new notion of a generalized left focal point as well as its multiplicity, which do not depend on the validity of the traditionally assumed Legendre condition. Based on this notion we are able to develop a local (or pointwise) version of the Sturmian separation theorem, which provides a lower bound and an upper bound for the multiplicity of a generalized left focal point for any conjoined basis of the system. We apply this knowledge in several directions, such as (ⅰ) in the explanation of the exact role of the Legendre condition in the Sturmian theory, (ⅱ) in the second order optimality conditions for variational problems, (ⅲ) in the analysis of isolated and non-isolated generalized left focal points, and (ⅳ) in the study of the so-called anti-Legendre condition. As a main tool we use the comparative index and its properties. 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. Generalized focal points and local Sturmian theory for linear Hamiltonian systems. Discrete and Continuous Dynamical Systems. American Institute of Mathematical Sciences, 2023, roč. 43, č. 12, s. 4139-4173. ISSN 1078-0947. Dostupné z: https://doi.org/10.3934/dcds.2023082.

@article{2301326,
   author = {Šepitka, Peter and Šimon Hilscher, Roman},
   article_number = {12},
   doi = {https://doi.org/10.3934/dcds.2023082},
   keywords = {Linear Hamiltonian system; generalized left focal point; Sturmian separation theorem; Legendre condition; comparative index; principal solution; Anti-Legendre condition},
   language = {eng},
   issn = {1078-0947},
   journal = {Discrete and Continuous Dynamical Systems},
   title = {Generalized focal points and local Sturmian theory for linear Hamiltonian systems},
   url = {https://doi.org/10.3934/dcds.2023082},
   volume = {43},
   year = {2023}
}

TY  - JOUR
ID  - 2301326
AU  - Šepitka, Peter - Šimon Hilscher, Roman
PY  - 2023
TI  - Generalized focal points and local Sturmian theory for linear Hamiltonian systems
JF  - Discrete and Continuous Dynamical Systems
VL  - 43
IS  - 12
SP  - 4139-4173
EP  - 4139-4173
PB  - American Institute of Mathematical Sciences
SN  - 10780947
KW  - Linear Hamiltonian system
KW  - generalized left focal point
KW  - Sturmian separation theorem
KW  - Legendre condition
KW  - comparative index
KW  - principal solution
KW  - Anti-Legendre condition
UR  - https://doi.org/10.3934/dcds.2023082
N2  - In this paper we present a new approach for the study of the oscillation properties of linear differential equations, in particular of linear Hamiltonian systems. We introduce a new notion of a generalized left focal point as well as its multiplicity, which do not depend on the validity of the traditionally assumed Legendre condition. Based on this notion we are able to develop a local (or pointwise) version of the Sturmian separation theorem, which provides a lower bound and an upper bound for the multiplicity of a generalized left focal point for any conjoined basis of the system. We apply this knowledge in several directions, such as (ⅰ) in the explanation of the exact role of the Legendre condition in the Sturmian theory, (ⅱ) in the second order optimality conditions for variational problems, (ⅲ) in the analysis of isolated and non-isolated generalized left focal points, and (ⅳ) in the study of the so-called anti-Legendre condition. As a main tool we use the comparative index and its properties. 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. Generalized focal points and local Sturmian theory for linear Hamiltonian systems. \textit{Discrete and Continuous Dynamical Systems}. American Institute of Mathematical Sciences, 2023, roč.~43, č.~12, s.~4139-4173. ISSN~1078-0947. Dostupné z: https://doi.org/10.3934/dcds.2023082.

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