| Title:
|
Role of the Harnack extension principle in the Kurzweil-Stieltjes integral (English) |
| Author:
|
Hanung, Umi Mahnuna |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
149 |
| Issue:
|
3 |
| Year:
|
2024 |
| Pages:
|
337-363 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the theories of integration and of ordinary differential and integral equations, convergence theorems provide one of the most widely used tools. Since the values of the Kurzweil-Stieltjes integrals over various kinds of bounded intervals having the same infimum and supremum need not coincide, the Harnack extension principle in the Kurzweil-Henstock integral, which is a key step to supply convergence theorems, cannot be easily extended to the Kurzweil-type Stieltjes integrals with discontinuous integrators. Moreover, in general, the existence of integral over an elementary set $E$ does not always imply the existence of integral over every subset $T$ of $E.$ The goal of this paper is to construct the Harnack extension principle for the Kurzweil-Stieltjes integral with values in Banach spaces and then to demonstrate its role in guaranteeing the integrability over arbitrary subsets of elementary sets. New concepts of equiintegrability and equiregulatedness involving elementary sets are pivotal to the notion of the Harnack extension principle for the Kurzweil-Stieltjes integration. (English) |
| Keyword:
|
Kurzweil-Stieltjes integral |
| Keyword:
|
integral over arbitrary bounded sets |
| Keyword:
|
equiintegrability |
| Keyword:
|
equiregulatedness |
| Keyword:
|
convergence theorem |
| Keyword:
|
Harnack extension principle |
| MSC:
|
26A36 |
| MSC:
|
26A39 |
| MSC:
|
26A42 |
| MSC:
|
28B05 |
| MSC:
|
28C20 |
| DOI:
|
10.21136/MB.2023.0162-22 |
| . |
| Date available:
|
2024-09-11T13:46:26Z |
| Last updated:
|
2024-09-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152538 |
| . |
| Reference:
|
[1] Arredondo, J. H., Bernal, M., Morales, M. G.: Fourier analysis with generalized integration.Mathematics 8 (2020), Article ID 1199, 16 pages. 10.3390/math8071199 |
| Reference:
|
[2] Bartle, R. G.: A convergence theorem for generalized Riemann integrals.Real Anal. Exch. 20 (1994/1995), 119-124. Zbl 0828.26006, MR 1313676, 10.2307/44152472 |
| Reference:
|
[3] Bartle, R. G.: A Modern Theory of Integration.Graduate Studies in Mathematics 32. AMS, Providence (2001). Zbl 0968.26001, MR 1817647, 10.1090/gsm/032 |
| Reference:
|
[4] Bongiorno, B., Piazza, L. Di: Convergence theorems for generalized Riemann-Stieltjes integrals.Real Anal. Exch. 17 (1991/1992), 339-361. Zbl 0758.26006, MR 1147373, 10.2307/44152212 |
| Reference:
|
[5] Bonotto, E. M., Federson, M., (eds.), J. G. Mesquita: Generalized Ordinary Differential Equations in Abstract Spaces and Applications.John Wiley & Sons, Hoboken (2021). Zbl 1475.34001, MR 4485099, 10.1002/9781119655022 |
| Reference:
|
[6] Cao, S. S.: The Henstock integral for Banach-valued functions.Southeast Asian Bull. Math. 16 (1992), 35-40. Zbl 0749.28007, MR 1173605 |
| Reference:
|
[7] Caponetti, D., Cichoń, M., Marraffa, V.: On a step method and a propagation of discontinuity.Comput. Appl. Math. 38 (2019), Article ID 172, 20 pages. Zbl 1438.34219, MR 4017897, 10.1007/s40314-019-0927-0 |
| Reference:
|
[8] Carl, S., Heikkilä, S.: Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory.Springer, New York (2011). Zbl 1209.47001, MR 2760654, 10.1007/978-1-4419-7585-0 |
| Reference:
|
[9] Cousin, P.: Sur les fonctions de $n$ variables complexes.Acta Math. 19 (1895), 1-62 French \99999JFM99999 26.0456.02. MR 1554861, 10.1007/BF02402869 |
| Reference:
|
[10] Federson, M., Mawhin, J., Mesquita, C.: Existence of periodic solutions and bifurcation points for generalized ordinary differential equations.Bull. Sci. Math. 169 (2021), Article ID 102991, 31 pages. Zbl 1471.34011, MR 4253257, 10.1016/j.bulsci.2021.102991 |
| Reference:
|
[11] Fraňková, D.: Regulated functions.Math. Bohem. 116 (1991), 20-59. Zbl 0724.26009, MR 1100424, 10.21136/MB.1991.126195 |
| Reference:
|
[12] Gordon, R. A.: Another look at a convergence theorem for the Henstock integral.Real Anal. Exch. 15 (1989/1990), 724-728. Zbl 0708.26005, MR 1059433, 10.2307/44152048 |
| Reference:
|
[13] 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:
|
[14] Hanung, U. M., Tvrdý, M.: On the relationships between Stieltjes type integrals of Young, Dushnik and Kurzweil.Math. Bohem. 144 (2019), 357-372. Zbl 07217260, MR 4047342, 10.21136/MB.2019.0015-19 |
| Reference:
|
[15] Henstock, R.: Lectures on the Theory of Integration.Series in Real Analysis 1. World Scientific, Singapore (1988). Zbl 0668.28001, MR 0963249, 10.1142/0510 |
| Reference:
|
[16] Hildebrandt, T. H.: Introduction to the Theory of Integration.Pure and Applied Mathematics 13. Academic Press, New York (1963). Zbl 0112.28302, MR 0154957 |
| Reference:
|
[17] Hönig, C. S.: Volterra Stieltjes-Integral Equations: Functional Analytic Methods, Linear Constraints.North-Holland Mathematics Studies 16. Notas de Mathematica 56. North-Holland, Amsterdam (1975). Zbl 0307.45002, MR 0499969 |
| Reference:
|
[18] Krejčí, P., Lamba, H., Monteiro, G. A., Rachinskii, D.: The Kurzweil integral in financial market modeling.Math. Bohem. 141 (2016), 261-286. Zbl 1389.34140, MR 3499787, 10.21136/MB.2016.18 |
| Reference:
|
[19] Kubota, Y.: The Cauchy property of the generalized approximately continuous Perron integral.Tohoku Math. J., II. Ser. 12 (1960), 171-174. Zbl 0109.28002, MR 121461, 10.2748/tmj/1178244433 |
| Reference:
|
[20] Kurtz, D. S., Swartz, C. W.: Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock-Kurzweil, and McShane.Series in Real Analysis 13. World Scientific, Hackensack (2012). Zbl 1263.26019, MR 2894455, 10.1142/8291 |
| Reference:
|
[21] Kurzweil, J.: Generalized ordinary differential equations and continuous dependence on a parameter.Czech. Math. J. 7 (1957), 418-449. Zbl 0090.30002, MR 0111875, 10.21136/CMJ.1957.100258 |
| Reference:
|
[22] Kurzweil, J.: Nichtabsolut konvergente Integrale.Teubner-Texte zur Mathematik 26. B. G. Teubner, Leipzig (1980), German. Zbl 0441.28001, MR 0597703 |
| Reference:
|
[23] Kurzweil, J.: Henstock-Kurzweil Integration: Its Relation to Topological Vector Spaces.Series in Real Analysis 7. World Scientific, Singapore (2000). Zbl 0954.28001, MR 1763305, 10.1142/4333 |
| Reference:
|
[24] Kurzweil, J.: Generalized Ordinary Differential Equations: Not Absolutely Continuous Solutions.Series in Real Analysis 11. World Scientific, Hackensack (2012). Zbl 1248.34001, MR 2906899, 10.1142/7907 |
| Reference:
|
[25] Kurzweil, J., Jarník, J.: Equiintegrability and controlled convergence of Perron-type integrable functions.Real Anal. Exch. 17 (1991/1992), 110-139. Zbl 0754.26003, MR 1147361, 10.2307/44152200 |
| Reference:
|
[26] 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:
|
[27] Lee, P. Y.: Harnack Extension for the Henstock integral in the Euclidean space.J. Math. Study 27 (1994), 5-8. Zbl 0927.26014, MR 1318250 |
| Reference:
|
[28] Lee, T. Y.: Henstock-Kurzweil Integration on Euclidean Spaces.Series in Real Analysis 12. World Scientific, Hackensack (2011). Zbl 1246.26002, MR 2789724, 10.1142/7933 |
| Reference:
|
[29] Liu, W., Krejčí, P., Ye, G.: Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions.Discrete Contin. Dyn. Syst., Ser. B 22 (2017), 3783-3795. Zbl 1375.34073, MR 3693841, 10.3934/dcdsb.2017190 |
| Reference:
|
[30] Albés, I. Márquez, Slavík, A., Tvrdý, M.: Duality for Stieltjes differential and integral equations.J. Math. Anal. Appl. 519 (2023), Article ID 126789, 52 pages. Zbl 07616180, MR 4499373, 10.1016/j.jmaa.2022.126789 |
| Reference:
|
[31] Monteiro, G. A.: On Kurzweil-Stieltjes equiintegrability and generalized BV functions.Math. Bohem. 144 (2019), 423-436. Zbl 1499.26024, MR 4047345, 10.21136/MB.2019.0041-19 |
| Reference:
|
[32] Monteiro, G. A., Hanung, U. M., Tvrdý, M.: Bounded convergence theorem for abstract Kurzweil-Stieltjes integral.Monatsh. Math. 180 (2016), 409-434. Zbl 1355.26008, MR 3513214, 10.1007/s00605-015-0774-z |
| Reference:
|
[33] Monteiro, G. A., Slavík, A., Tvrdý, M.: Kurzweil-Stieltjes Integral: Theory and Applications.Series in Real Analysis 15. World Scientific, Hackensack (2019). Zbl 1437.28001, MR 3839599, 10.1142/9432 |
| Reference:
|
[34] Monteiro, G. A., Tvrdý, M.: On Kurzweil-Stieltjes integral in a Banach space.Math. Bohem. 137 (2012), 365-381. Zbl 1274.26014, MR 3058269, 10.21136/MB.2012.142992 |
| Reference:
|
[35] Monteiro, G. A., Tvrdý, M.: Generalized linear differential equations in a Banach space: Continuous dependence on a parameter.Discrete Contin. Dyn. Syst. 33 (2013), 283-303. Zbl 1268.45009, MR 2972960, 10.3934/dcds.2013.33.283 |
| Reference:
|
[36] Ng, W. L.: Nonabsolute Integration on Measure Spaces.Series in Real Analysis 14. World Scientific, Hackensack (2018). Zbl 1392.28001, MR 3752602, 10.1142/10489 |
| Reference:
|
[37] Pfeffer, W. F.: The Riemann Approach to Integration: Local Geometric Theory.Cambridge Tracts in Mathematics 109. Cambridge University Press, Cambridge (1993). Zbl 0804.26005, MR 1268404 |
| Reference:
|
[38] Saks, S.: Theory of the Integral.Monografie Matematyczne 7. G. E. Stechert & Co., New York (1937). Zbl 0017.30004, MR 0167578 |
| Reference:
|
[39] Sánchez-Perales, S., Mendoza-Torres, F. J.: Boundary value problems for the Schrödinger equation involving the Henstock-Kurzweil integral.Czech. Math. J. 70 (2020), 519-537. Zbl 07217149, MR 4111857, 10.21136/CMJ.2019.0388-18 |
| Reference:
|
[40] Sánchez-Perales, S., Pérez-Becerra, T., Vázquez-Hipólito, V., Oliveros-Oliveros, J. J. O.: Sturm-Liouville differential equations involving Kurzweil-Henstock integrable functions.Mathematics 9 (2021), Article ID 1403, 20 pages. 10.3390/math9121403 |
| Reference:
|
[41] Schwabik, Š.: Generalized Ordinary Differential Equations.Series in Real Analysis 5. World Scientific, Singapore (1992). Zbl 0781.34003, MR 1200241, 10.1142/1875 |
| Reference:
|
[42] Schwabik, Š.: Abstract Perron-Stieltjes integral.Math. Bohem. 121 (1996), 425-447. Zbl 0879.28021, MR 1428144, 10.21136/MB.1996.126036 |
| Reference:
|
[43] Schwabik, Š.: Linear Stieltjes integral equations in Banach spaces.Math. Bohem. 124 (1999), 433-457. Zbl 0937.34047, MR 1722877, 10.21136/MB.1999.125994 |
| Reference:
|
[44] Schwabik, Š.: A note on integration by parts for abstract Perron-Stieltjes integrals.Math. Bohem. 126 (2001), 613-629. Zbl 0980.26005, MR 1970264, 10.21136/MB.2001.134198 |
| Reference:
|
[45] Schwabik, Š., Tvrdý, M., Vejvoda, O.: Differential and Integral Equations: Boundary Value Problems and Adjoints.Academia and D. Reidel, Praha and Dordrecht (1979). Zbl 0417.45001, MR 0542283 |
| Reference:
|
[46] 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:
|
[47] 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:
|
[48] Slavík, A.: Explicit solutions of linear Stieltjes integral equations.Result. Math. 78 (2023), Article ID 40, 28 pages. Zbl 1507.45004, MR 4523289, 10.1007/s00025-022-01816-z |
| Reference:
|
[49] Stieltjes, T. J.: Recherches sur les fractions continues.Ann. Fac. Sci. Toulouse, VI. Sér., Math. 4 (1995), J76--J122 French. Zbl 0861.01036, MR 1607517, 10.5802/afst.808 |
| Reference:
|
[50] Trench, W. F.: Introduction to Real Analysis.Prentice Hall, Upper Saddle River (2003). Zbl 1204.00023 |
| Reference:
|
[51] Tvrdý, M.: Regulated functions and the Perron-Stieltjes integral.Čas. Pěstování Mat. 114 (1989), 187-209. Zbl 0671.26006, MR 1063765, 10.21136/CPM.1989.108713 |
| . |
