MATHEMATICA BOHEMICA, Vol. 137, No. 4, pp. 365-381, 2012

# On Kurzweil-Stieltjes integral in a Banach space

## Giselle A. Monteiro, Milan Tvrdý

Giselle Antunes Monteiro, Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos, SP, Brasil, e-mail: giantunesmonteiro@gmail.com; Milan Tvrdý, Institute of Mathematics, Academy of Sciences of the Czech Republic, CZ 115 67 Praha 1, Žitná 25, Czech Republic, e-mail: tvrdy@math.cas.cz

Abstract: In the paper we deal with the Kurzweil-Stieltjes integration of functions having values in a Banach space $X.$ We extend results obtained by Stefan 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 $\int_a^b d[F]g$ exists if $F[a,b]\to L(X)$ has a bounded semi-variation on $[a,b]$ and $g [a,b]\to X$ 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.

Keywords: Kurzweil-Stieltjes integral, substitution formula, integration-by-parts

Classification (MSC 2010): 26A39, 28B05

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