| Title:
|
Substitution formulas for the Kurzweil and Henstock vector integrals (English) |
| Author:
|
Federson, M. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
127 |
| Issue:
|
1 |
| Year:
|
2002 |
| Pages:
|
15-26 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Kurzweil-Henstock integrals |
| Keyword:
|
integration by parts |
| Keyword:
|
integration by substitution |
| MSC:
|
26A39 |
| MSC:
|
28B05 |
| MSC:
|
46G10 |
| idZBL:
|
Zbl 1002.28012 |
| idMR:
|
MR1895242 |
| DOI:
|
10.21136/MB.2002.133979 |
| . |
| Date available:
|
2009-09-24T21:57:17Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133979 |
| . |
| Reference:
|
[1] Federson, M.: The Fundamental Theorem of Calculus for multidimensional Banach space-valued Henstock vector integrals.Real Anal. Exchange 25 (2000), 469–480. Zbl 1015.28012, MR 1758903 |
| Reference:
|
[2] Federson, M., Bianconi, R.: Linear integral equations of Volterra concerning the integral of Henstock.Real Anal. Exchange 25 (2000), 389–417. MR 1758896 |
| Reference:
|
[3] Henstock, R.: The General Theory of Integration.Oxford Math. Monog., Clarendon Press, Oxford, 1991. Zbl 0745.26006, MR 1134656 |
| Reference:
|
[4] Hönig, C. S.: Volterra-Stieltjes Integral Equations.Math. Studies, 16, North-Holland Publ. Comp., Amsterdam, 1975. MR 0499969 |
| Reference:
|
[5] Hönig, C. S.: 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. |
| Reference:
|
[6] Hönig, C. S.: A Riemaniann characterization of the Bochner-Lebesgue integral.Seminário Brasileiro de Análise 35 (1992), 351–358. |
| Reference:
|
[7] Lee, P. Y.: Lanzhou Lectures on Henstock Integration.World Sci., Singapore, 1989. Zbl 0699.26004, MR 1050957 |
| Reference:
|
[8] McShane, E. J.: A unified theory of integration.Amer. Math. Monthly 80 (1973), 349–359. Zbl 0266.26008, MR 0318434, 10.1080/00029890.1973.11993291 |
| Reference:
|
[9] Schwabik, Š.: Abstract Perron-Stieltjes integral.Math. Bohem. 121 (1996), 425–447. Zbl 0879.28021, MR 1428144 |
| . |
