Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Kurzweil integral; Young integral; Dushnik integral; Kurzweil-Stieltjes integral; Young-Stieltjes integral; Dushnik-Stieltjes integral; convergence theorem
Kurzweil integral; Young integral; Dushnik integral; Kurzweil-Stieltjes integral; Young-Stieltjes integral; Dushnik-Stieltjes integral; convergence theorem
Summary:
In this paper we explain the relationship between Stieltjes type integrals of Young, Dushnik and Kurzweil for functions with values in Banach spaces. To this aim also several new convergence theorems will be stated and proved.
References:
[1] Ashordia, M.: On the Opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations. Math. Bohem. 141 (2016), 183-215. DOI 10.21136/mb.2016.15 | MR 3499784 | Zbl 06587862
[2] Dushnik, B.: On the Stieltjes Integral. Thesis. University of Michigan, Ann Arbor (1931). MR 2936614
[3] Federson, M., Bianconi, R., Barbanti, L.: Linear Volterra integral equations as the limit of discrete systems. Acta Math. Appl. Sin., Engl. Ser. 20 (2004), 623-640. DOI 10.1007/s10255-004-0200-0 | MR 2173638 | Zbl 1067.45005
[4] Federson, M., Táboas, P.: Topological dynamics of retarded functional differential equations. J. Differ. Equations 195 (2003), 313-331. DOI 10.1016/s0022-0396(03)00061-5 | MR 2016815 | Zbl 1054.34102
[5] Hildebrandt, T. H.: On systems of linear differentio-Stieltjes-integral equations. Ill. J. Math. 3 (1959), 352-373. DOI 10.1215/ijm/1255455257 | MR 0105600 | Zbl 0088.31101
[6] Hildebrandt, T. H.: Introduction to the Theory of Integration. Pure and Applied Mathematics 13. Academic Press, New York (1963). DOI 10.1016/s0079-8169(08)x6120-3 | MR 0154957 | Zbl 0112.28302
[7] Hönig, C. S.: Volterra Stieltjes-Integral Equations. Functional Analytic Methods; Linear Constraints. North-Holland Mathematics Studies 16. Notas de Mathematica (56). North-Holland Publishing Company, Amsterdam; American Elsevier Publishing Company, New York (1975). MR 0499969 | Zbl 0307.45002
[8] Hönig, C. S.: Volterra-Stieltjes integral equations. Functional Differential Equations and Bifurcation. Proc. Conf., Inst. Ciênc. Mat. São Carlos, 1979 Lecture Notes in Math. 799. Springer, Berlin (1980), 173-216 A. F. Izé. DOI 10.1007/bfb0089315 | MR 0585488 | Zbl 0436.45021
[9] Kurzweil, J.: Generalized ordinary differential equations and continuous dependence on a parameter. Czech. Math. J. 7 (1957), 418-449. DOI 10.21136/CMJ.1957.100258 | MR 0111875 | Zbl 0090.30002
[10] Kurzweil, J.: Generalized Ordinary Differential Equations. Not Absolutely Continuous Solutions. Series in Real Analysis 11. World Scientific, Hackensack (2012). DOI 10.1142/9789814324038 | MR 2906899 | Zbl 1248.34001
[11] MacNerney, J. S.: An integration-by-parts formula. Bull. Am. Math. Soc. 69 (1963), 803-805. DOI 10.1090/s0002-9904-1963-11039-3 | MR 0153809 | Zbl 0129.27004
[12] Meng, G., Zhang, M.: Dependence of solutions and eigenvalues of measure differential equations on measures. J. Differ. Equations 254 (2013), 2196-2232. DOI 10.1016/j.jde.2012.12.001 | MR 3007109 | Zbl 1267.34014
[13] Monteiro, G. A., Slavík, A., Tvrdý, M.: Kurzweil-Stieltjes Integral. Theory and Applications. Series in Real Analysis 15. World Scientific, Hackensack (2018). DOI 10.1142/9432 | MR 3839599 | Zbl 06758513
[14] Monteiro, G. A., Tvrdý, M.: On Kurzweil-Stieltjes integral in a Banach space. Math. Bohem. 137 (2012), 365-381. DOI 10.21136/MB.2012.142992 | MR 3058269 | Zbl 1274.26014
[15] 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. DOI 10.3934/dcds.2013.33.283 | MR 2972960 | Zbl 1268.45009
[16] Saks, S.: Theory of the Integral. With two additional notes by Stefan Banach. Dover Publications, Mineola (1964). MR 0167578 | Zbl 1196.28001
[17] Schwabik, Š.: On a modified sum integral of Stieltjes type. Čas. Pěst. Mat. 98 (1973), 274-277. MR 0322114 | Zbl 0266.26007
[18] Schwabik, Š.: On the relation between Young's and Kurzweil's concept of Stieltjes integral. Čas. Pěst. Mat. 98 (1973), 237-251. MR 0322113 | Zbl 0266.26006
[19] Schwabik, Š.: Generalized Ordinary Differential Equations. Series in Real Analysis 5. World Scientific, Singapore (1992). DOI 10.1142/1875 | MR 1200241 | Zbl 0781.34003
[20] Schwabik, Š.: Abstract Perron-Stieltjes integral. Math. Bohem. 121 (1996), 425-447. DOI 10.21136/MB.1996.126036 | MR 1428144 | Zbl 0879.28021
[21] Schwabik, Š.: Linear Stieltjes integral equations in Banach spaces. Math. Bohem. 124 (1999), 433-457. DOI 10.21136/MB.1999.125994 | MR 1722877 | Zbl 0937.34047
[22] Schwabik, Š.: Linear Stieltjes integral equations in Banach spaces. II. Operator valued solutions. Math. Bohem. 125 (2000), 431-454. DOI 10.21136/MB.2000.126273 | MR 1802292 | Zbl 0974.34057
[23] Schwabik, Š.: A note on integration by parts for abstract Perron-Stieltjes integrals. Math. Bohem. 126 (2001), 613-629. DOI 10.21136/MB.2001.134198 | MR 1970264 | Zbl 0980.26005
[24] Schwabik, Š., Tvrdý, M., Vejvoda, O.: Differential and Integral Equations. Boundary Value Problems and Adjoints. D. Reidel Publishing Company, Dordrecht; Academia (Publishing House of the Czechoslovak Academy of Sciences), Praha (1979). MR 0542283 | Zbl 0417.45001
[25] Schwabik, Š., Ye, G.: Topics in Banach Space Integration. Series in Real Analysis 10. World Scientific, Hackensack (2005). DOI 10.1142/5905 | MR 2167754 | Zbl 1088.28008
[26] Young, W. H.: On integration with respect to a function of bounded variation. Proc. Lond. Math. Soc. (2) 13 (1914), 109-150. DOI 10.1112/plms/s2-13.1.109 | MR 1577495 | Zbl 45.0446.04
Search
Partner of
