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Keywords:
Henstock-Kurzweil product integral; McShane product integral; Bochner product integral
Henstock-Kurzweil product integral; McShane product integral; Bochner product integral
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 $[a,b]$.
References:
[1] J. D. Dollard and C. N. Friedman: Product Integration with Applications to Differential Equations. Addison-Wesley Publ. Company, Reading, Massachusetts, 1979. MR 0552941
[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
[3] P. R. Masani: Multiplicative Riemann integration in normed rings. Trans. Am. Math. Soc. 61 (1947), 147–192. DOI 10.1090/S0002-9947-1947-0018719-6 | MR 0018719 | Zbl 0037.03802
[4] Š. Schwabik: Bochner product integration. Math. Bohem. 119 (1994), 305–335. MR 1305532 | Zbl 0830.28006
[5] Š. Schwabik: The Perron product integral and generalized linear differential equations. Časopis pěst. mat. 115 (1990), 368–404. MR 1090861 | Zbl 0724.26006
[6] Š. Schwabik and Ye Guoju: Topics in Banach Space Integration. World Scientific, Singapore, 2005. MR 2167754
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