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Title: Henstock-Kurzweil and McShane product integration; descriptive definitions (English)
Author: Slavík, Antonín
Author: Schwabik, Štefan
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 58
Issue: 1
Year: 2008
Pages: 241-269
Summary lang: English
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Category: math
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Summary: 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 $[a,b]$. (English)
Keyword: Henstock-Kurzweil product integral
Keyword: McShane product integral
Keyword: Bochner product integral
MSC: 26A39
MSC: 28B05
MSC: 46G10
idZBL: Zbl 1174.28013
idMR: MR2402536
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Date available: 2009-09-24T11:54:41Z
Last updated: 2016-04-07
Stable URL: http://hdl.handle.net/10338.dmlcz/128256
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Reference: [2] J. Jarník and J. Kurzweil: A general form of the product integral and linear ordinary differential equations.Czech. Math. J. 37 (1987), 642–659. MR 0913996
Reference: [3] P. R. Masani: Multiplicative Riemann integration in normed rings.Trans. Am. Math. Soc. 61 (1947), 147–192. Zbl 0037.03802, MR 0018719, 10.1090/S0002-9947-1947-0018719-6
Reference: [4] Š. Schwabik: Bochner product integration.Math. Bohem. 119 (1994), 305–335. Zbl 0830.28006, MR 1305532
Reference: [5] Š. Schwabik: The Perron product integral and generalized linear differential equations.Časopis pěst. mat. 115 (1990), 368–404. Zbl 0724.26006, MR 1090861
Reference: [6] Š. Schwabik and Ye Guoju: Topics in Banach Space Integration.World Scientific, Singapore, 2005. MR 2167754
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