On Kurzweil-Stieltjes integral in a Banach space

Monteiro, Giselle A., and Tvrdý, Milan. "On Kurzweil-Stieltjes integral in a Banach space." Mathematica Bohemica 137.4 (2012): 365-381. <http://eudml.org/doc/246217>.

@article{Monteiro2012,
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 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 $F\colon [a,b]\rightarrow L(X)$ has a bounded semi-variation on $[a,b]$ and $g\colon [a,b]\rightarrow X$ 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.},
author = {Monteiro, Giselle A., Tvrdý, Milan},
journal = {Mathematica Bohemica},
keywords = {Kurzweil-Stieltjes integral; substitution formula; integration-by-parts; Kurzweil-Stieltjes integral; substitution formula; integration-by-parts},
language = {eng},
number = {4},
pages = {365-381},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On Kurzweil-Stieltjes integral in a Banach space},
url = {http://eudml.org/doc/246217},
volume = {137},
year = {2012},
}

TY - JOUR
AU - Monteiro, Giselle A.
AU - Tvrdý, Milan
TI - On Kurzweil-Stieltjes integral in a Banach space
JO - Mathematica Bohemica
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 137
IS - 4
SP - 365
EP - 381
AB - 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 $F\colon [a,b]\rightarrow L(X)$ has a bounded semi-variation on $[a,b]$ and $g\colon [a,b]\rightarrow X$ 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.
LA - eng
KW - Kurzweil-Stieltjes integral; substitution formula; integration-by-parts; Kurzweil-Stieltjes integral; substitution formula; integration-by-parts
UR - http://eudml.org/doc/246217
ER -