| Title:
|
On Kurzweil-Stieltjes equiintegrability and generalized BV functions (English) |
| Author:
|
Monteiro, Giselle A. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
144 |
| Issue:
|
4 |
| Year:
|
2019 |
| Pages:
|
423-436 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We present sufficient conditions ensuring Kurzweil-Stieltjes equiintegrability in the case when integrators belong to the class of functions of generalized bounded variation. (English) |
| Keyword:
|
Kurzweil-Stieltjes integral |
| Keyword:
|
generalized bounded variation |
| Keyword:
|
variational measure |
| Keyword:
|
Stieltjes derivative |
| MSC:
|
26A24 |
| MSC:
|
26A39 |
| MSC:
|
26A42 |
| MSC:
|
26A45 |
| idZBL:
|
07217263 |
| idMR:
|
MR4047345 |
| DOI:
|
10.21136/MB.2019.0041-19 |
| . |
| Date available:
|
2019-12-09T11:54:13Z |
| Last updated:
|
2020-08-14 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147936 |
| . |
| Reference:
|
[1] Bongiorno, B., Piazza, L. Di: Convergence theorems for generalized Riemann-Stieltjes integrals.Real Anal. Exch. 17 (1991-92), 339-361. Zbl 0758.26006, MR 1147373, 10.2307/44152212 |
| Reference:
|
[2] Bongiorno, B., Piazza, L. Di, Skvortsov, V.: A new full descriptive characterization of \hbox{Denjoy-Perron} integral.Real Anal. Exch. 21 (1995-96), 656-663. Zbl 0879.26026, MR 1407278, 10.2307/44152676 |
| Reference:
|
[3] Faure, C.-A.: A descriptive definition of the KH-Stieltjes integral.Real Anal. Exch. 23 (1998-99), 113-124. Zbl 0944.26014, MR 1609775, 10.2307/44152834 |
| Reference:
|
[4] Fraňková, D.: Regulated functions.Math. Bohem. 116 (1991), 20-59. Zbl 0724.26009, MR 1100424 |
| Reference:
|
[5] Frigon, M., Pouso, R. L.: Theory and applications of first-order systems of Stieltjes differential equations.Adv. Nonlinear Anal. 6 (2017), 13-36. Zbl 1361.34010, MR 3604936, 10.1515/anona-2015-0158 |
| Reference:
|
[6] Gordon, R. A.: Another look at a convergence theorem for the Henstock integral.Real Anal. Exch. 15 (1989-90), 724-728. Zbl 0708.26005, MR 1059433, 10.2307/44152048 |
| Reference:
|
[7] Gordon, R. A.: The Integrals of Lebesgue, Denjoy, Perron, and Henstock.Graduate Studies in Mathematics 4. AMS, Providence (1994). Zbl 0807.26004, MR 1288751, 10.1090/gsm/004 |
| Reference:
|
[8] Hoffmann, H.: Descriptive Characterisation of the Variational Henstock-Kurzweil-Stieltjes Integral and Applications.PhD thesis. Karlsruher Institue of Technology, Karlsruhe. Available at https://publikationen.bibliothek.kit.edu/1000046600 (2014). |
| Reference:
|
[9] Kurzweil, J., Jarník, J.: Equiintegrability and controlled convergence of Perron-type integrable functions.Real Anal. Exch. 17 (1991-92), 110-139. Zbl 0754.26003, MR 1147361, 10.2307/44152200 |
| Reference:
|
[10] Lee, P. Y.: Lanzhou Lectures on Henstock Integration.Series in Real Analysis 2. World Scientific, London (1989). Zbl 0699.26004, MR 1050957, 10.1142/0845 |
| Reference:
|
[11] Monteiro, G. A., Satco, B.: Distributional, differential and integral problems: equivalence and existence results.Electron. J. Qual. Theory Differ. Equ. 2017 (2017), Paper No. 7, 26 pages. Zbl 06931238, MR 3606985, 10.14232/ejqtde.2017.1.7 |
| Reference:
|
[12] Monteiro, G. A., Slavík, A., Tvrdý, M.: Kurzweil-Stieltjes Integral. Theory and Applications.Series in Real Analysis 15. World Scientific, Hackensack (2019). Zbl 06758513, MR 3839599, 10.1142/9432 |
| Reference:
|
[13] Pouso, R. L., Rodríguez, A.: A new unification of continuous, discrete, and impulsive calculus through Stieltjes derivatives.Real Anal. Exch. 40 (2015), 319-354. Zbl 1384.26024, MR 3499768, 10.14321/realanalexch.40.2.0319 |
| Reference:
|
[14] Saks, S.: Theory of the Integral. With two additional notes by Stefan Banach.Monografie Matematyczne Tom. 7. G. E. Stechert & Co., New York (1937). Zbl 0017.30004, MR 0167578 |
| Reference:
|
[15] Satco, B.-R.: Measure integral inclusions with fast oscillating data.Electron. J. Differ. Equ. 2015 (2015), Paper No. 107, 13 pages. Zbl 1314.45005, MR 3358479 |
| Reference:
|
[16] Schwabik, Š.: Variational measures and the Kurzweil-Henstock integral.Math. Slovaca 59 (2009), 731-752. Zbl 1212.26014, MR 2564330, 10.2478/s12175-009-0160-1 |
| Reference:
|
[17] Schwabik, Š.: General integration and extensions I.Czech. Math. J. 60 (2010), 961-981. Zbl 1224.26030, MR 2738960, 10.1007/s10587-010-0087-2 |
| Reference:
|
[18] Schwabik, Š.: General integration and extensions II.Czech. Math. J. 60 (2010), 983-1005. Zbl 1224.26031, MR 2738961, 10.1007/s10587-010-0088-1 |
| Reference:
|
[19] Schwabik, Š., Vrkoč, I.: On Kurzweil-Henstock equiintegrable sequences.Math. Bohem. 121 (1996), 189-207. Zbl 0863.26009, MR 1400612, 10.21136/MB.1996.126102 |
| Reference:
|
[20] Schwabik, Š., Ye, G.: Topics in Banach Space Integration.Series in Real Analysis 10. World Scientific, Hackensack (2005). Zbl 1088.28008, MR 2167754, 10.1142/5905 |
| Reference:
|
[21] Thomson, B. S.: Real Functions.Lecture Notes in Mathematics 1170. Springer, Berlin (1985). Zbl 0581.26001, MR 0818744, 10.1007/BFb0074380 |
| Reference:
|
[22] Ward, A. J.: The Perron-Stieltjes integral.Math. Z. 41 (1936), 578-604. Zbl 0014.39702, MR 1545641, 10.1007/BF01180442 |
| . |
