Linear Stieltjes integral equations in Banach spaces

Štefan Schwabik

Mathematica Bohemica (1999)

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

Abstract

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Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces have been presented in \cite5. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. \cite3). 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) +\int_a^t \dd[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 \cite5. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. \cite3). 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) +\int\_a^t \dd[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},
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 \cite5. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. \cite3). 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) +\int_a^t \dd[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
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

Citations in EuDML Documents

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

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