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Keywords:
Kurzweil-Stieltjes integral; substitution formula; integration-by-parts
Summary:
In the paper we deal with the Kurzweil-Stieltjes integration of functions having values in a Banach space $X.$ We extend results obtained by Štefan Schwabik and complete the theory so 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 Schwabik, the integral $\int _a^b {\rm d}[F]g$ exists if $F\colon [a,b]\to L(X)$ has a bounded semi-variation on $[a,b]$ and $g\colon [a,b]\to X$ is regulated on $[a,b].$ We prove that this integral has 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 the assumptions not covered by Schwabik (2001) and Naralenkov (2004), and the substitution formula is proved.
References:
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