# Linear Stieltjes integral equations in Banach spaces

Mathematica Bohemica (1999)

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

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topSchwabik, 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

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

## Citations in EuDML Documents

top- Dana Fraňková, Regulated functions with values in Banach space
- Štefan Schwabik, Linear Stieltjes integral equations in Banach spaces. II. Operator valued solutions
- Štefan Schwabik, Operator-valued functions of bounded semivariation and convolutions
- Štefan Schwabik, A note on integration by parts for abstract Perron-Stieltjes integrals
- Giselle A. Monteiro, Milan Tvrdý, On Kurzweil-Stieltjes integral in a Banach space
- Umi Mahnuna Hanung, Milan Tvrdý, On the relationships between Stieltjes type integrals of Young, Dushnik and Kurzweil
- Rodolfo Collegari, Márcia Federson, Miguel Frasson, Linear FDEs in the frame of generalized ODEs: variation-of-constants formula

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