# Linear Stieltjes integral equations in Banach spaces

Mathematica Bohemica (1999)

• Volume: 124, Issue: 4, page 433-457
• ISSN: 0862-7959

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## Abstract

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Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces have been presented in [5]. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. [3]). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. Here basic results concerning equations of the form x(t) = x(a) +at [A(s)]x(s) +f(t) - f(a) are presented on the basis of the Kurzweil type Stieltjes integration. We are looking for generally discontinuous solutions which belong to the space of Banach space-valued regulated functions in the case that $A$ is a suitable operator-valued function and $f$ is regulated.

## How to cite

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Schwabik, Štefan. "Linear Stieltjes integral equations in Banach spaces." Mathematica Bohemica 124.4 (1999): 433-457. <http://eudml.org/doc/248452>.

@article{Schwabik1999,
abstract = {Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces have been presented in [5]. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. [3]). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. Here basic results concerning equations of the form x(t) = x(a) +at [A(s)]x(s) +f(t) - f(a) are presented on the basis of the Kurzweil type Stieltjes integration. We are looking for generally discontinuous solutions which belong to the space of Banach space-valued regulated functions in the case that $A$ is a suitable operator-valued function and $f$ is regulated.},
author = {Schwabik, Štefan},
journal = {Mathematica Bohemica},
keywords = {linear Stieltjes integral equations; generalized linear differential equation; Banach space; equation in Banach space; linear Stieltjes integral equations; generalized linear differential equation; Banach space},
language = {eng},
number = {4},
pages = {433-457},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Linear Stieltjes integral equations in Banach spaces},
url = {http://eudml.org/doc/248452},
volume = {124},
year = {1999},
}

TY - JOUR
AU - Schwabik, Štefan
TI - Linear Stieltjes integral equations in Banach spaces
JO - Mathematica Bohemica
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 124
IS - 4
SP - 433
EP - 457
AB - Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces have been presented in [5]. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. [3]). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. Here basic results concerning equations of the form x(t) = x(a) +at [A(s)]x(s) +f(t) - f(a) are presented on the basis of the Kurzweil type Stieltjes integration. We are looking for generally discontinuous solutions which belong to the space of Banach space-valued regulated functions in the case that $A$ is a suitable operator-valued function and $f$ is regulated.
LA - eng
KW - linear Stieltjes integral equations; generalized linear differential equation; Banach space; equation in Banach space; linear Stieltjes integral equations; generalized linear differential equation; Banach space
UR - http://eudml.org/doc/248452
ER -

## References

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1. Dunford N., Schwartz J. T., Linear Operators I., Interscience Publishers, New York, London, 1958. (1958) Zbl0084.10402MR0117523
2. Hönig, Ch. S., Volterra-Stieltjes Integral Equations, North-Holland Publ. Comp., Amsterdam, 1975. (1975) MR0499969
3. Kurzweil J., Nichtabsolut konvergente Integrale, B. G.Teubner Verlagsgesellschaft, Leipzig, 1980. (1980) Zbl0441.28001MR0597703
4. Rudin W., Functional Analysis, McGraw-Hill Book Company, New York, 1973. (1973) Zbl0253.46001MR0365062
5. Schwabik Š., Abstract Perron-Stieltjes integral, Math. Bohem. 121 (1996), 425-447. (1996) Zbl0879.28021MR1428144
6. Schwabik Š., Generalized Ordinary Differential Equations, World Scientific, Singapore, 1992. (1992) Zbl0781.34003MR1200241
7. Schwabik Š., Tvrdý M., Vejvoda O., Differential and Integral Equations, Academia & Reidel, Praha & Dordrecht, 1979. (1979) MR0542283

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