Generalized focal points and local Sturmian theory for linear Hamiltonian systems

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F23%3A00134179" target="_blank" >RIV/00216224:14310/23:00134179 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.3934/dcds.2023082" target="_blank" >https://doi.org/10.3934/dcds.2023082</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.3934/dcds.2023082" target="_blank" >10.3934/dcds.2023082</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Generalized focal points and local Sturmian theory for linear Hamiltonian systems

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Generalized focal points and local Sturmian theory for linear Hamiltonian systems

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA23-05242S" target="_blank" >GA23-05242S: Oscilační teorie na hybridních časových doménách s aplikacemi ve spektrální teorii a maticové analýze</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2023

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Discrete and Continuous Dynamical Systems

ISSN

1078-0947

e-ISSN

1553-5231

Svazek periodika

43

Číslo periodika v rámci svazku

12

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

35

Strana od-do

4139-4173

Kód UT WoS článku

001044344800001

EID výsledku v databázi Scopus

2-s2.0-85176553055