Bounded convergence theorem for abstract Kurzweil–Stieltjes integral

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F16%3A00460232" target="_blank" >RIV/67985840:_____/16:00460232 - isvavai.cz</a>

Result on the web

<a href="http://dx.doi.org/10.1007/s00605-015-0774-z" target="_blank" >http://dx.doi.org/10.1007/s00605-015-0774-z</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00605-015-0774-z" target="_blank" >10.1007/s00605-015-0774-z</a>

Result language

angličtina

Original language name

Bounded convergence theorem for abstract Kurzweil–Stieltjes integral

Original language description

In the theories of Lebesgue integration and of ordinary differential equations, the Lebesgue Dominated Convergence Theorem provides one of the most widely used tools. Available analogy in the Riemann or Riemann–Stieltjes integration is the Bounded Convergence Theorem, sometimes called also the Arzelà or Arzelà–Osgood or Osgood Theorem. In the setting of the Kurzweil–Stieltjes integral for real valued functions its proof can be obtained by a slight modification of the proof given for the ...-Young–Stieltjes integral by T.H. Hildebrandt in his monograph from 1963. However, it is clear that the Hildebrandt’s proof cannot be extended to the case of Banach space-valued functions. Moreover, it essentially utilizes the Arzelà Lemma which does not fit too much into elementary text-books. In this paper, we present the proof of the Bounded Convergence Theorem for the abstract Kurzweil–Stieltjes integral in a setting elementary as much as possible. In the theories of Lebesgue integration and of ordinary differential equations, the Lebesgue Dominated Convergence Theorem provides one of the most widely used tools. Available analogy in the Riemann or Riemann–Stieltjes integration is the Bounded Convergence Theorem, sometimes called also the Arzelà or Arzelà–Osgood or Osgood Theorem. In the setting of the Kurzweil–Stieltjes integral for real valued functions its proof can be obtained by a slight modification of the proof given for the ...-Young–Stieltjes integral by T.H. Hildebrandt in his monograph from 1963. However, it is clear that the Hildebrandt’s proof cannot be extended to the case of Banach space-valued functions. Moreover, it essentially utilizes the Arzelà Lemma which does not fit too much into elementary text-books. In this paper, we present the proof of the Bounded Convergence Theorem for the abstract Kurzweil–Stieltjes integral in a setting elementary as much as possible.

Czech name

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Czech description

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Type

J_x - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)

CEP classification

BA - General mathematics

OECD FORD branch

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Project

<a href="/en/project/GA14-06958S" target="_blank" >GA14-06958S: Singularities and impulses in boundary value problems for nonlinear ordinary differential equations</a><br>

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2016

Confidentiality

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

Name of the periodical

Monatshefte für Mathematik

ISSN

0026-9255

e-ISSN

—

Volume of the periodical

180

Issue of the periodical within the volume

3

Country of publishing house

AT - AUSTRIA

Number of pages

26

Pages from-to

409-434

UT code for WoS article

000378786900001

EID of the result in the Scopus database

2-s2.0-84931049667