# Henstock-Kurzweil and McShane product integration; descriptive definitions

Antonín Slavík; Štefan Schwabik

Czechoslovak Mathematical Journal (2008)

- Volume: 58, Issue: 1, page 241-269
- ISSN: 0011-4642

## Access Full Article

top## Abstract

top## How to cite

topSlavík, Antonín, and Schwabik, Štefan. "Henstock-Kurzweil and McShane product integration; descriptive definitions." Czechoslovak Mathematical Journal 58.1 (2008): 241-269. <http://eudml.org/doc/31209>.

@article{Slavík2008,

abstract = {The Henstock-Kurzweil and McShane product integrals generalize the notion of the Riemann product integral. We study properties of the corresponding indefinite integrals (i.e. product integrals considered as functions of the upper bound of integration). It is shown that the indefinite McShane product integral of a matrix-valued function $A$ is absolutely continuous. As a consequence we obtain that the McShane product integral of $A$ over $[a,b]$ exists and is invertible if and only if $A$ is Bochner integrable on $[a,b]$.},

author = {Slavík, Antonín, Schwabik, Štefan},

journal = {Czechoslovak Mathematical Journal},

keywords = {Henstock-Kurzweil product integral; McShane product integral; Bochner product integral; Henstock-Kurzweil product integral; McShane product integral; Bochner product integral},

language = {eng},

number = {1},

pages = {241-269},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Henstock-Kurzweil and McShane product integration; descriptive definitions},

url = {http://eudml.org/doc/31209},

volume = {58},

year = {2008},

}

TY - JOUR

AU - Slavík, Antonín

AU - Schwabik, Štefan

TI - Henstock-Kurzweil and McShane product integration; descriptive definitions

JO - Czechoslovak Mathematical Journal

PY - 2008

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 58

IS - 1

SP - 241

EP - 269

AB - The Henstock-Kurzweil and McShane product integrals generalize the notion of the Riemann product integral. We study properties of the corresponding indefinite integrals (i.e. product integrals considered as functions of the upper bound of integration). It is shown that the indefinite McShane product integral of a matrix-valued function $A$ is absolutely continuous. As a consequence we obtain that the McShane product integral of $A$ over $[a,b]$ exists and is invertible if and only if $A$ is Bochner integrable on $[a,b]$.

LA - eng

KW - Henstock-Kurzweil product integral; McShane product integral; Bochner product integral; Henstock-Kurzweil product integral; McShane product integral; Bochner product integral

UR - http://eudml.org/doc/31209

ER -

## References

top- Product Integration with Applications to Differential Equations, Addison-Wesley Publ. Company, Reading, Massachusetts, 1979. (1979) MR0552941
- A general form of the product integral and linear ordinary differential equations, Czech. Math. J. 37 (1987), 642–659. (1987) MR0913996
- 10.1090/S0002-9947-1947-0018719-6, Trans. Am. Math. Soc. 61 (1947), 147–192. (1947) Zbl0037.03802MR0018719DOI10.1090/S0002-9947-1947-0018719-6
- Bochner product integration, Math. Bohem. 119 (1994), 305–335. (1994) Zbl0830.28006MR1305532
- The Perron product integral and generalized linear differential equations, Časopis pěst. mat. 115 (1990), 368–404. (1990) Zbl0724.26006MR1090861
- Topics in Banach Space Integration, World Scientific, Singapore, 2005. (2005) MR2167754

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.