Mathematica Bohemica, Vol. 144, No. 2, pp. 137-148, 2019
New extension of the variational McShane integral of vector-valued functions
Sokol Bush Kaliaj
Received October 8, 2017. Published online June 20, 2018.
Abstract: We define the Hake-variational McShane integral of Banach space valued functions defined on an open and bounded subset $G$ of $m$-dimensional Euclidean space $\mathbb{R}^m$. It is a "natural" extension of the variational McShane integral (the strong McShane integral) from $m$-dimensional closed non-degenerate intervals to open and bounded subsets of $\mathbb{R}^m$. We will show a theorem that characterizes the Hake-variational McShane integral in terms of the variational McShane integral. This theorem reduces the study of our integral to the study of the variational McShane integral. As an application, a full descriptive characterization of the Hake-variational McShane integral is presented in terms of the cubic derivative.
References: [1] L. Di Piazza: Variational measures in the theory of the integration in $\mathbb{R}^m$. Czech. Math. J. 51 (2001), 95-110. DOI 10.1023/A:1013705821657 | MR 1814635 | Zbl 1079.28500
[2] L. Di Piazza, K. Musial: A characterization of variationally McShane integrable Banach-space valued functions. Ill. J. Math. 45 (2001), 279-289. MR 1849999 | Zbl 0999.28006
[3] G. B. Folland: Real Analysis. Modern Techniques and Their Applications. Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts. Wiley, New York (1999). MR 1681462 | Zbl 0924.28001
[4] D. H. Fremlin: The generalized McShane integral. Ill. J. Math. 39 (1995), 39-67. MR 1299648 | Zbl 0810.28006
[5] R. A. Gordon: The Denjoy extension of the Bochner, Pettis, and Dunford integrals. Stud. Math. 92 (1989), 73-91. DOI 10.4064/sm-92-1-73-91 | MR 0984851 | Zbl 0681.28006
[6] R. A. Gordon: The McShane integral of Banach-valued functions. Ill. J. Math. 34 (1990), 557-567. MR 1053562 | Zbl 0685.28003
[7] R. A. Gordon: The Integrals of Lebesgue, Denjoy, Perron, and Henstock. Graduate Studies in Mathematics 4. AMS, Providence (1994). DOI 10.1090/gsm/004 | MR 1288751 | Zbl 0807.26004
[8] S. B. Kaliaj: The new extensions of the Henstock-Kurzweil and the McShane integrals of vector-valued functions. Mediterr. J. Math. 15 (2018), Article ID 22, 16 pages. DOI 10.1007/s00009-018-1067-2 | MR 3746986 | Zbl 06860542
[9] S. B. Kaliaj: Some remarks about descriptive characterizations of the strong McShane integral. To appear in Math. Bohem.
[10] J. Kurzweil, Š. Schwabik: On the McShane integrability of Banach space-valued functions. Real Anal. Exchange 29 (2003-2004), 763-780. DOI 10.14321/realanalexch.29.2.0763 | MR 2083811 | Zbl 1078.28007
[11] E. J. McShane: Unifed Integration. Pure and Applied Mathematics 107. Academic Press, Orlando (1983). MR 0740710 | Zbl 0551.28001
[12] W. F. Pfeffer: Derivation and Integration. Cambridge Tracts in Mathematics 140. Cambridge University Press, Cambridge (2001). DOI 10.1017/CBO9780511574764 | MR 1816996 | Zbl 0980.26008
[13] Š. Schwabik, Y. Guoju: Topics in Banach Space Integration. Series in Real Analysis 10. World Scientific, Hackensack (2005). DOI 10.1142/9789812703286 | MR 2167754 | Zbl 1088.28008
[14] V. A. Skvortsov, A. P. Solodov: A variational integral for Banach-valued functions. Real Anal. Exch. 24 (1999), 799-805. MR 1704751 | Zbl 0967.28007
[15] B. S. Thomson: Derivates of interval functions. Mem. Am. Math. Soc. 93 (1991), 96 pages. DOI 10.1090/memo/0452 | MR 1078198 | Zbl 0734.26003
[16] B. S. Thomson: Differentiation. Handbook of Measure Theory. Vol. I. and II. (E. Pap, ed.). North-Holland, Amsterdam (2002), 179-247. DOI /10.1016/B978-044450263-6/50006-3 | MR 1954615 | Zbl 1028.28001
[17] C. Wu, Y. Xiaobo: A Riemann-type definition of the Bochner integral. J. Math. Study 27 (1994), 32-36. MR 1318255 | Zbl 0947.28010
Affiliations: Sokol Bush Kaliaj, Department of Mathematics, Faculty of Natural Sciences, Aleksander Xhuvani University, Rruga Rinia, Elbasan, Albania, e-mail: sokol_bush@yahoo.co.uk
