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Monatshefte für Mathematik

, Volume 180, Issue 3, pp 409–434 | Cite as

Bounded convergence theorem for abstract Kurzweil–Stieltjes integral

  • Giselle Antunes Monteiro
  • Umi Mahnuna Hanung
  • Milan TvrdýEmail author
Article

Abstract

In the theories of Lebesgue integration and of ordinary differential equations, the Lebesgue Dominated Convergence Theorem provides one of the most widely used tools. Available analogy in the Riemann or Riemann–Stieltjes integration is the Bounded Convergence Theorem, sometimes called also the Arzelà or Arzelà–Osgood or Osgood Theorem. In the setting of the Kurzweil–Stieltjes integral for real valued functions its proof can be obtained by a slight modification of the proof given for the \(\sigma \)-Young–Stieltjes integral by T.H. Hildebrandt in his monograph from 1963. However, it is clear that the Hildebrandt’s proof cannot be extended to the case of Banach space-valued functions. Moreover, it essentially utilizes the Arzelà Lemma which does not fit too much into elementary text-books. In this paper, we present the proof of the Bounded Convergence Theorem for the abstract Kurzweil–Stieltjes integral in a setting elementary as much as possible.

Keywords

Kurzweil–Stieltjes integral Bounded convergence theorem Integral over elementary set 

Mathematics Subject Classification

26A39 28B05 

References

  1. 1.
    Apostol, T.M.: Mathematical Analysis, 2nd edn. Addison-Wesley, Reading (1974)zbMATHGoogle Scholar
  2. 2.
    Bartle, R.G., Sherbert, D.R.: Introduction to Real Analysis. Wiley, New York (1982)zbMATHGoogle Scholar
  3. 3.
    Chistyakov, V.V.: On the theory of set-valued maps of bounded variation of one real variable. Sb. Math. 189, 153–176 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Diestel, J.: Sequences and Series in Banach Spaces. Springer, New York (1984)CrossRefzbMATHGoogle Scholar
  5. 5.
    Eberlein, W.F.: Notes on integration I: the underlying convergence theorem. Comm. Pure Appl. Math. 10, 357–360 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gordon, R.A.: The integrals of Lebesgue, Denjoy, Perron, and Henstock. In: Graduate Studies in Mathematics. AMS, Providence (1994)Google Scholar
  7. 7.
    Hildebrandt, T.H.: Theory of Integration. Academic Press, New York (1963)zbMATHGoogle Scholar
  8. 8.
    Hönig, C.S.: The abstract Riemann–Stieltjes integral and its applications to linear differential equations with generalized boundary conditions. Notas do Instituto de Matemática e Estatística da Universidade de S. Paulo, Série de Matemática no. 1 (1973)Google Scholar
  9. 9.
    Hönig, C.S.: Volterra Stieltjes-Integral Equations. In: Mathematics Studies, vol. 16. North Holland and American Elsevier, Amsterdam, New York (1975)Google Scholar
  10. 10.
    Lewin, J.W.: A truly elementary approach to the bounded convergence theorem. Amer. Math. Monthly 93, 395–397 (1986)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Luxemburg, W.A.J.: Arzela’s dominated convergence theorem for the Riemann integral. Am. Math. Monthly 78, 970–979 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Monteiro, G.A., Tvrdý, M.: On Kurzweil–Stieltjes integral in Banach space. Math. Bohem. 137, 365–381 (2012)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Monteiro, G.A., Tvrdý, M.: Generalized linear differential equations in a Banach space: continuous dependence on a parameter. Discrete Contin. Dyn. Syst. 33(1), 283–303 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Rudin, W.: Principles of Mathematical Analysis. McGraw-Hill, New York (1964)zbMATHGoogle Scholar
  15. 15.
    Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1966)zbMATHGoogle Scholar
  16. 16.
    Saks, S.: Theory of the Integral, 2nd edn. Hafner Publishing Company, New York (1937)zbMATHGoogle Scholar
  17. 17.
    Schwabik, Š.: Generalized Ordinary Differential Equations. World Scientific, Singapore (1992)CrossRefzbMATHGoogle Scholar
  18. 18.
    Schwabik, Š.: Abstract Perron–Stieltjes integral. Math. Bohem. 121, 425–447 (1996)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Schwabik, Š.: Linear Stieltjes integral equations in Banach spaces. Math. Bohem. 124, 433–457 (1999)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Schwabik, Š.: Linear Stieltjes integral equations in Banach spaces II. Operator valued solutions. Math. Bohem. 125, 431–454 (2000)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Schwabik, Š., Tvrdý, M., Vejvoda, O.: Differential and Integral Equations: Boundary Value Problems and Adjoints. Academia and D. Reidel, Praha and Dordrecht (1979)zbMATHGoogle Scholar
  22. 22.
    Tvrdý, M.: Differential and integral equations in the space of regulated functions. Mem. Differ. Equ. Math. Phys. 25, 1–104 (2002)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2015

Authors and Affiliations

  • Giselle Antunes Monteiro
    • 1
  • Umi Mahnuna Hanung
    • 1
    • 2
    • 3
  • Milan Tvrdý
    • 1
    Email author
  1. 1.Institute of MathematicsCzech Academy of SciencesPraha 1Czech Republic
  2. 2.Department of MathematicsUniversitas Gadjah MadaYogyakartaIndonesia
  3. 3.KdV Institute for MathematicsUniversiteit van AmsterdamAmsterdamThe Netherlands

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