Role of the Harnack extension principle in the Kurzweil-Stieltjes integral

Umi Mahnuna Hanung

Mathematica Bohemica (2024)

  • Volume: 149, Issue: 3, page 337-363
  • ISSN: 0862-7959

Abstract

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In the theories of integration and of ordinary differential and integral equations, convergence theorems provide one of the most widely used tools. Since the values of the Kurzweil-Stieltjes integrals over various kinds of bounded intervals having the same infimum and supremum need not coincide, the Harnack extension principle in the Kurzweil-Henstock integral, which is a key step to supply convergence theorems, cannot be easily extended to the Kurzweil-type Stieltjes integrals with discontinuous integrators. Moreover, in general, the existence of integral over an elementary set E does not always imply the existence of integral over every subset T of E . The goal of this paper is to construct the Harnack extension principle for the Kurzweil-Stieltjes integral with values in Banach spaces and then to demonstrate its role in guaranteeing the integrability over arbitrary subsets of elementary sets. New concepts of equiintegrability and equiregulatedness involving elementary sets are pivotal to the notion of the Harnack extension principle for the Kurzweil-Stieltjes integration.

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Hanung, Umi Mahnuna. "Role of the Harnack extension principle in the Kurzweil-Stieltjes integral." Mathematica Bohemica 149.3 (2024): 337-363. <http://eudml.org/doc/299481>.

@article{Hanung2024,
abstract = {In the theories of integration and of ordinary differential and integral equations, convergence theorems provide one of the most widely used tools. Since the values of the Kurzweil-Stieltjes integrals over various kinds of bounded intervals having the same infimum and supremum need not coincide, the Harnack extension principle in the Kurzweil-Henstock integral, which is a key step to supply convergence theorems, cannot be easily extended to the Kurzweil-type Stieltjes integrals with discontinuous integrators. Moreover, in general, the existence of integral over an elementary set $E$ does not always imply the existence of integral over every subset $T$ of $E.$ The goal of this paper is to construct the Harnack extension principle for the Kurzweil-Stieltjes integral with values in Banach spaces and then to demonstrate its role in guaranteeing the integrability over arbitrary subsets of elementary sets. New concepts of equiintegrability and equiregulatedness involving elementary sets are pivotal to the notion of the Harnack extension principle for the Kurzweil-Stieltjes integration.},
author = {Hanung, Umi Mahnuna},
journal = {Mathematica Bohemica},
keywords = {Kurzweil-Stieltjes integral; integral over arbitrary bounded sets; equiintegrability; equiregulatedness; convergence theorem; Harnack extension principle},
language = {eng},
number = {3},
pages = {337-363},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Role of the Harnack extension principle in the Kurzweil-Stieltjes integral},
url = {http://eudml.org/doc/299481},
volume = {149},
year = {2024},
}

TY - JOUR
AU - Hanung, Umi Mahnuna
TI - Role of the Harnack extension principle in the Kurzweil-Stieltjes integral
JO - Mathematica Bohemica
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 149
IS - 3
SP - 337
EP - 363
AB - In the theories of integration and of ordinary differential and integral equations, convergence theorems provide one of the most widely used tools. Since the values of the Kurzweil-Stieltjes integrals over various kinds of bounded intervals having the same infimum and supremum need not coincide, the Harnack extension principle in the Kurzweil-Henstock integral, which is a key step to supply convergence theorems, cannot be easily extended to the Kurzweil-type Stieltjes integrals with discontinuous integrators. Moreover, in general, the existence of integral over an elementary set $E$ does not always imply the existence of integral over every subset $T$ of $E.$ The goal of this paper is to construct the Harnack extension principle for the Kurzweil-Stieltjes integral with values in Banach spaces and then to demonstrate its role in guaranteeing the integrability over arbitrary subsets of elementary sets. New concepts of equiintegrability and equiregulatedness involving elementary sets are pivotal to the notion of the Harnack extension principle for the Kurzweil-Stieltjes integration.
LA - eng
KW - Kurzweil-Stieltjes integral; integral over arbitrary bounded sets; equiintegrability; equiregulatedness; convergence theorem; Harnack extension principle
UR - http://eudml.org/doc/299481
ER -

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