Skip to main content
SpringerLink
Log in

An Extension of the Birkhoff Integrability for Multifunctions

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

A comparison between a set-valued Gould type and simple Birkhoff integrals of bf(X)-valued multifunctions with respect to a non-negative set function is given. Relationships among them and Mc Shane multivalued integrability is given under suitable assumptions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Angeloni L., Martellotti A.: Non-coalitional Core–Walras equivalence in finitely additive economies with extremely desirable commodities. Mediterr. J. Math. 4(1), 87–107 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balder E.J., Sambucini A.R.: On simple compactness and lower closure results for Pettis integrable (multi)functions. Bull. Pol. Acad. Sci. Math. 52(1), 53–61 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Balder E.J., Sambucini A.R.: Fatou’s lemma for multifunctions with unbounded values in a dual space. J. Convex Anal. 12(2), 383–395 (2005)

    MathSciNet  MATH  Google Scholar 

  4. Boccuto, A., Candeloro, D., Sambucini, A.R.: Henstock multivalued integrability in Banach lattices with respect to pointwise non atomic measures. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26(4) (2015). arXiv:1503.08285v1 [math.FA]

  5. Boccuto A., Minotti A.M., Sambucini A.R.: Set-valued Kurzweil–Henstock integral in Riesz space setting. PanAm. Math. J. 23(1), 57–74 (2013)

    MathSciNet  MATH  Google Scholar 

  6. Boccuto A., Sambucini A.R.: A Mc Shane integral for multifunctions. J. Concrete Appl. Math. 2(4), 307–325 (2004)

    MathSciNet  MATH  Google Scholar 

  7. Boccuto A., Sambucini A.R.: A note on comparison between Birkhoff and Mc Shane integrals for multifunctions. Real Anal. Exchange 37(2), 3–15 (2012)

    MathSciNet  Google Scholar 

  8. Candeloro, D., Croitoru, A., Gavrilut, A., Sambucini, A.R.: Radon Nikodym theorem for a pair of multimeasures with respect to the Gould integrability. arXiv:1504.04110v1 [math.FA] (2015) (submitted for publication)

  9. Candeloro, D., Sambucini, A.R.: Order-type Henstock and Mc Shane integrals in Banach lattice setting. In: Proceedings of the SISY 2014—IEEE 12th International Symposium on Intelligent Systems and Informatics, pp. 55–59 (2014) (ISBN 978-1-4799-5995-2)

  10. Candeloro, D., Sambucini, A.R.: Comparison between some norm and order gauge integrals in Banach lattices. PanAm. Math. J. 25(3) (2015). arXiv:1503.04968 [math.FA]

  11. Cascales B., Rodríguez J.: Birkhoff integral for multi-valued functions. J. Math. Anal. Appl. 297, 540–560 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cascales B., Rodríguez J.: The Birkhoff integral and the property of Bourgain. Math. Ann. 331(2), 259–279 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cascales B., Rodríguez J.: Birkhoff integral for multi-valued functions. J. Math. Anal. Appl. 297, 540–560 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Castaing, C., Valadier, M.: Convex analysis and measurable multifunctions. In: Lecture Notes in Mathematics, vol. 580. Springer, New York (1977)

  15. Cichoń, K., Cichoń, M.: Some applications of nonabsolute integrals in the theory of differential inclusions in Banach spaces. In: Curbera, G.P., Mockenhaupt, G., Ricker, W.J. (eds.) Vector Measures, Integration and Related Topics. Operator Theory: Advances and Applications, vol. 201, pp. 115–124. Birkhauser, New York (2010) (ISBN 978-3-0346-0210-5)

  16. Croitoru, A., Gavriluţ, A.: Comparison between Birkhoff integral and Gould integral. Mediterr. J. Math. 12, 329–347 (2015). doi:10.1007/50009-014-0410-5

  17. Croitoru, A., Gavriluţ, A., Iosif, A.E.: Birkhoff weak integrability of multifunctions. Int. J. Pure Math. 2, 47–54 (2015)

  18. Croitoru, A.: Strong integral of multifunctions relative to a fuzzy measure, Fuzzy Sets Syst. 244, 20–33 (2014). doi:10.1016/j.fss.2013.10.004

  19. Di Piazza L., Marraffa V.: The Mc Shane, PU and Henstock integrals of Banach valued functions. Czechoslov. Math. J. 52, 609–633 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Di Piazza, L., Marraffa, V., Satco, B.: Set valued integrability in non separable Frechet spaces and applications. Z. Anal. Anwend. (J. Anal. Appl.). arXiv:1507.07349v1

  21. Di Piazza L., Musiał K.: A characterization of variationally Mc Shane integrable Banach-space valued functions. Ill. J. Math. 45(1), 279–289 (2001)

    MATH  Google Scholar 

  22. Di Piazza L., Musiał K.: Set-valued Kurzweil–Henstock–Pettis integral. Set-Valued Anal. 3, 167–179 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Di Piazza L., Musiał K.: A decomposition theorem for compact-valued Henstock integral. Monatsh. Math. 148(2), 119–126 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Di Piazza, L., Musiał, K.: A decomposition of Denjoy–Hintchine–Pettis and Henstock–Kurzweil–Pettis integrable multifunctions. In: Curbera, G.P., Mockenhaupt, G., Ricker, W.J. (eds.) Vector Measures, Integration and Related Topics. Operator Theory: Advances and Applications, vol. 201, pp. 171–182. BirHauser, New York (2010) (ISBN 978-3-0346-0210-5)

  25. Di Piazza L., Musiał K.: Henstock–Kurzweil–Pettis integrability of compact valued multifunctions with values in an arbitrary Banach space. J. Math. Anal. Appl. 408, 452–464 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Di Piazza, L., Musiał, K.: Relations among Henstock, McShane and Pettis integrals for multifunctions with compact convex values. Monatsh. Math. 173, 459–470 (2014). doi:10.1007/s00605-013-0594

  27. Di Piazza L., Satco B.: A new result on impulsive differential equations involving non-absolutely convergent integrals. J. Math. Anal. Appl. 352, 954–963 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Fremlin, D.H.: The Mc Shane and Birkhoff integrals of vector-valued functions. In: University of Essex Mathematics Department Research Report 92-10, version 13.10.04. http://www.essex.ac.uk/maths/staff/fremlin/preprints.htm

  29. Fremlin D.H.: The generalized Mc Shane integral. Ill. J. Math. 39, 39–67 (1995)

    MathSciNet  MATH  Google Scholar 

  30. Gavriluţ, A.: A Gould type integral with respect to a multisubmeasure. Math. Slovaca 58 (1), 43–62(2008). doi:10.2478/s12175-007-0054-z

  31. Gavriluţ A., Croitoru A.: Classical thorems for a Gould type fuzzy integral. WSEAS Trans. Math. 11, 421–433 (2012)

    Google Scholar 

  32. Gould G.G.: On integration of vector-valued measures. Proc. Lond. Math. Soc. 15, 193–225 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hu, S., Papageorgiou, N.S.: Handbook of multivalued analysis, vol. I. Theory. Mathematics and Its Applications, vol. 419. Kluwer Academic, Dordrecht (1997)

  34. Labuschagne, C.C.A., Pinchuck, A.L., van Alten, C.J.: A vector lattice version of Rådstrőm’s embedding theorem. Quaest. Math. 30(3), 285–308 (2007)

  35. Marraffa V.: A Birkhoff type integral and the Bourgain property in a locally convex space. Real Anal. Exchange 32(2), 409–428 (2006)

    MathSciNet  MATH  Google Scholar 

  36. Martellotti A., Sambucini A.R.: On the comparison between Aumann and Bochner integral. J. Math. Anal. Appl. 260(1), 6–17 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Martellotti A., Sambucini A.R.: The finitely additive integral of multifunctions with closed and convex values. Z. Anal. Anwend. (J. Anal. Appl.) 21(45), 851–864 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Pap, E., Gavriluţ, A., Croitoru, A.: Gould type integral of multifunctions relative to non negative set functions (2015) (preprint, submitted for publication)

  39. Potyrala M.M.: The Birkhoff and variational Mc Shane integrals of vector valued functions. Folia Math. Acta Univ. Lodz. 13, 31–40 (2006)

    MathSciNet  MATH  Google Scholar 

  40. Potyrala M.M.: Some remarks about Birkhoff and Riemann–Lebesgue integrability of vector valued functions. Tatra Mt. Math. Publ. 35, 97–106 (2007)

    MathSciNet  MATH  Google Scholar 

  41. Riečan B.: On the Kurzweil integral in compact topological spaces. Rad. Mat. 2, 151–163 (1986)

    MathSciNet  MATH  Google Scholar 

  42. Rodríguez J.: On the existence of Pettis integrable functions which are not Birkhoff integrable. Proc. Am. Math. Soc. 133(4), 1157–1163 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Rodríguez J.: On the equivalence of Mc Shane and Pettis integrability in non-separable Banach spaces. J. Math. Anal. Appl. 350, 514–524 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  44. Rodríguez J.: Some examples in vector integration. Bull. Aust. Math. Soc. 80(03), 384–392 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, University of Perugia, Via Vanvitelli, 1, 06123, Perugia, Italy

    Domenico Candeloro & Anna Rita Sambucini

  2. Faculty of Mathematics, “Al.I. Cuza” University, Bd. Carol I, no 11, 700506, Iasi, Romania

    Anca Croitoru & Alina Gavriluţ

Authors
  1. Domenico Candeloro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anca Croitoru
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alina Gavriluţ
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Anna Rita Sambucini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Domenico Candeloro.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Candeloro, D., Croitoru, A., Gavriluţ, A. et al. An Extension of the Birkhoff Integrability for Multifunctions. Mediterr. J. Math. 13, 2551–2575 (2016). https://doi.org/10.1007/s00009-015-0639-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00009-015-0639-7

Mathematics Subject Classification

Keywords

Search

Navigation