Abstract Perron-Stieltjes integral

Štefan Schwabik

Mathematica Bohemica (1996)

  • Volume: 121, Issue: 4, page 425-447
  • ISSN: 0862-7959

Abstract

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Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces are presented. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. [4]). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. In [3] Ch. S. Honig presented a Stieltjes integral for Banach space valued functions. For Honig’s integral the Dushnik interior integral presents the background. It should be mentioned that abstract Stieltjes integration was recently used by O. Diekmann, M. Gyllenberg and H. R. Thieme in [1] and [2] for describing the behaviour of some evolutionary systems originating in problems concerning structured population dynamics.

How to cite

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Schwabik, Štefan. "Abstract Perron-Stieltjes integral." Mathematica Bohemica 121.4 (1996): 425-447. <http://eudml.org/doc/247962>.

@article{Schwabik1996,
abstract = {Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces are presented. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. [4]). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. In [3] Ch. S. Honig presented a Stieltjes integral for Banach space valued functions. For Honig’s integral the Dushnik interior integral presents the background. It should be mentioned that abstract Stieltjes integration was recently used by O. Diekmann, M. Gyllenberg and H. R. Thieme in [1] and [2] for describing the behaviour of some evolutionary systems originating in problems concerning structured population dynamics.},
author = {Schwabik, Štefan},
journal = {Mathematica Bohemica},
keywords = {bilinear triple; Perron-Stieltjes integral; bilinear triple; Perron-Stieltjes integral},
language = {eng},
number = {4},
pages = {425-447},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Abstract Perron-Stieltjes integral},
url = {http://eudml.org/doc/247962},
volume = {121},
year = {1996},
}

TY - JOUR
AU - Schwabik, Štefan
TI - Abstract Perron-Stieltjes integral
JO - Mathematica Bohemica
PY - 1996
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 121
IS - 4
SP - 425
EP - 447
AB - Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces are presented. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. [4]). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. In [3] Ch. S. Honig presented a Stieltjes integral for Banach space valued functions. For Honig’s integral the Dushnik interior integral presents the background. It should be mentioned that abstract Stieltjes integration was recently used by O. Diekmann, M. Gyllenberg and H. R. Thieme in [1] and [2] for describing the behaviour of some evolutionary systems originating in problems concerning structured population dynamics.
LA - eng
KW - bilinear triple; Perron-Stieltjes integral; bilinear triple; Perron-Stieltjes integral
UR - http://eudml.org/doc/247962
ER -

References

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  1. O. Diekmann M. Gyllenberg H. R. Thieme, Perturbing semigroups by solving Stieltjes renewal equations, Differential Integral Equations 6 (1993), 155-181. (1993) MR1190171
  2. O. Diekmann M. Gyllenberg H. R. Thieme, Perturbing evolutionary systems by step responses on cumulative outputs, Differential Integral Equations 7 (1995). To appear. (1995) MR1325554
  3. Ch. S. Hönig, Volterra Stieltjes-Integral Equations, North-Holland Publ. Comp., Amsterdam, 1975. (1975) MR0499969
  4. J. Kurzweil, Nichtabsolut konvergente Integrate, Teubner Verlagsgesellschaft, Leipzig, Teubner-Texte zur Mathematik Bd. 26, 1980. (1980) MR0597703
  5. W. Rudin, Functional Analysis, McGraw-Hill Book Company, New York, 1973. (1973) Zbl0253.46001MR0365062

Citations in EuDML Documents

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  1. Márcia Federson, The monotone convergence theorem for multidimensional abstract Kurzweil vector integrals
  2. Márcia Federson, Substitution formulas for the Kurzweil and Henstock vector integrals
  3. Márcia Federson, Ricardo Bianconi, Linear Volterra-Stieltjes integral equations in the sense of the Kurzweil-Henstock integral
  4. Štefan Schwabik, Linear Stieltjes integral equations in Banach spaces
  5. Štefan Schwabik, Linear Stieltjes integral equations in Banach spaces. II. Operator valued solutions
  6. Štefan Schwabik, Operator-valued functions of bounded semivariation and convolutions
  7. Pavel Krejčí, The Kurzweil integral with exclusion of negligible sets
  8. Štefan Schwabik, A note on integration by parts for abstract Perron-Stieltjes integrals
  9. Giselle A. Monteiro, Milan Tvrdý, On Kurzweil-Stieltjes integral in a Banach space
  10. Umi Mahnuna Hanung, Milan Tvrdý, On the relationships between Stieltjes type integrals of Young, Dushnik and Kurzweil

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