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On Kurzweil-Stieltjes integral in a Banach space. (English) Zbl 1274.26014
The authors deal with the Kurzweil-Stieltjes integration of functions having values in a Banach space \(X\). This article extends results obtained by Š. Schwabik and completes the theory such that it will be well applicable to prove results on the continuous dependence of solutions to generalized linear differential equations in a Banach space. By Š. Schwabik, the integral \(\int _a^b d[F]g\) exists if \(F\: [a, b]\rightarrow L(X)\) has a bounded semi-variation on \([a,b]\) and \(g\: [a, b]\rightarrow X\) is regulated on \([a, b]\). The authors prove that this integral makes sense also if \(F\) is regulated on \([a, b]\) and \(g\) has a bounded semi-variation on \([a, b]\). Furthermore, the integration-by-parts theorem is presented under assumptions not covered by Š. Schwabik [Math. Bohem. 126, No. 3, 613–629 (2001; Zbl 0980.26005)] and K. M. Naralenkov [Real Anal. Exch. 30, No. 1, 235–260 (2004–2005; Zbl 1069.28008)]. Further, a substitution formula is proved.
Reviewer: Guoju Ye (Nanjing)

26A39 Denjoy and Perron integrals, other special integrals
28B05 Vector-valued set functions, measures and integrals
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