Substitution formulas for the Kurzweil and Henstock vector integrals

Márcia Federson

Mathematica Bohemica (2002)

  • Volume: 127, Issue: 1, page 15-26
  • ISSN: 0862-7959

Abstract

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Results on integration by parts and integration by substitution for the variational integral of Henstock are well-known. When real-valued functions are considered, such results also hold for the Generalized Riemann Integral defined by Kurzweil since, in this case, the integrals of Kurzweil and Henstock coincide. However, in a Banach-space valued context, the Kurzweil integral properly contains that of Henstock. In the present paper, we consider abstract vector integrals of Kurzweil and prove Substitution Formulas by functional analytic methods. In general, Substitution Formulas need not hold for Kurzweil vector integrals even if they are defined.

How to cite

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Federson, Márcia. "Substitution formulas for the Kurzweil and Henstock vector integrals." Mathematica Bohemica 127.1 (2002): 15-26. <http://eudml.org/doc/249019>.

@article{Federson2002,
abstract = {Results on integration by parts and integration by substitution for the variational integral of Henstock are well-known. When real-valued functions are considered, such results also hold for the Generalized Riemann Integral defined by Kurzweil since, in this case, the integrals of Kurzweil and Henstock coincide. However, in a Banach-space valued context, the Kurzweil integral properly contains that of Henstock. In the present paper, we consider abstract vector integrals of Kurzweil and prove Substitution Formulas by functional analytic methods. In general, Substitution Formulas need not hold for Kurzweil vector integrals even if they are defined.},
author = {Federson, Márcia},
journal = {Mathematica Bohemica},
keywords = {Kurzweil-Henstock integrals; integration by parts; integration by substitution; Kurzweil-Henstock integrals; integration by parts; integration by substitution; Banach space-valued functions; vector integrals},
language = {eng},
number = {1},
pages = {15-26},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Substitution formulas for the Kurzweil and Henstock vector integrals},
url = {http://eudml.org/doc/249019},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Federson, Márcia
TI - Substitution formulas for the Kurzweil and Henstock vector integrals
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 1
SP - 15
EP - 26
AB - Results on integration by parts and integration by substitution for the variational integral of Henstock are well-known. When real-valued functions are considered, such results also hold for the Generalized Riemann Integral defined by Kurzweil since, in this case, the integrals of Kurzweil and Henstock coincide. However, in a Banach-space valued context, the Kurzweil integral properly contains that of Henstock. In the present paper, we consider abstract vector integrals of Kurzweil and prove Substitution Formulas by functional analytic methods. In general, Substitution Formulas need not hold for Kurzweil vector integrals even if they are defined.
LA - eng
KW - Kurzweil-Henstock integrals; integration by parts; integration by substitution; Kurzweil-Henstock integrals; integration by parts; integration by substitution; Banach space-valued functions; vector integrals
UR - http://eudml.org/doc/249019
ER -

References

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  1. The Fundamental Theorem of Calculus for multidimensional Banach space-valued Henstock vector integrals, Real Anal. Exchange 25 (2000), 469–480. (2000) Zbl1015.28012MR1758903
  2. Linear integral equations of Volterra concerning the integral of Henstock, Real Anal. Exchange 25 (2000), 389–417. (2000) MR1758896
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  5. There is no natural Banach space norm on the space of Kurzweil-Henstock-Denjoy-Perron integrable functions, Seminário Brasileiro de Análise 30 (1989), 387–397. (1989) 
  6. A Riemaniann characterization of the Bochner-Lebesgue integral, Seminário Brasileiro de Análise 35 (1992), 351–358. (1992) 
  7. Lanzhou Lectures on Henstock Integration, World Sci., Singapore, 1989. (1989) Zbl0699.26004MR1050957
  8. 10.1080/00029890.1973.11993291, Amer. Math. Monthly 80 (1973), 349–359. (1973) Zbl0266.26008MR0318434DOI10.1080/00029890.1973.11993291
  9. Abstract Perron-Stieltjes integral, Math. Bohem. 121 (1996), 425–447. (1996) Zbl0879.28021MR1428144

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