Czechoslovak Mathematical Journal

, Volume 56, Issue 3, pp 1029–1047 | Cite as

A Komlós-type theorem for the set-valued Henstock-Kurzweil-Pettis integral and applications

  • B. Satco


This paper presents a Komlós theorem that extends to the case of the set-valued Henstock-Kurzweil-Pettis integral a result obtained by Balder and Hess (in the integrably bounded case) and also a result of Hess and Ziat (in the Pettis integrability setting). As applications, a solution to a best approximation problem is given, weak compactness results are deduced and, finally, an existence theorem for an integral inclusion involving the Henstock-Kurzweil-Pettis set-valued integral is obtained.


Komlós convergence Henstock-Kurzweil integral Henstock-Kurzweil-Pettis set-valued integral selection 


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  1. [1]
    E. Balder: New sequential compactness results for spaces of scalarly integrable functions. J. Math. Anal. Appl. 151 (1990), 1–16.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    E. Balder, C. Hess: Two generalizations of Komlós theorem with lower closure-type applications. J. Convex Anal. 3 (1996), 25–44.MATHMathSciNetGoogle Scholar
  3. [3]
    E. Balder, A. R. Sambucini: On weak compactness and lower closure results for Pettis integrable (multi)functions. Bull. Pol. Acad. Sci. Math. 52 (2004), 53–61.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    C. Castaing: Weak compactness and convergences in Bochner and Pettis integration. Vietnam J. Math. 24 (1996), 241–286.MathSciNetGoogle Scholar
  5. [5]
    C. Castaing, P. Clauzure: Compacité faible dans l’espace L E1 et dans l’espace des multifonctions intégrablement bornées, et minimisation. Ann. Mat. Pura Appl. 140 (1985), 345–364.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions. Lect. Notes Math. Vol. 580. Springer-Verlag, Berlin, 1977.MATHGoogle Scholar
  7. [7]
    T. S. Chew, F. Flordeliza: On x′ = f(t, x) and Henstock-Kurzweil integrals. Differential Integral Equations 4 (1991), 861–868.MATHMathSciNetGoogle Scholar
  8. [8]
    M. Cichón, I. Kubiaczyk, A. Sikorska: The Henstock-Kurzweil-Pettis integrals and existence theorems for the Cauchy problem. Czechoslovak Math. J. 54 (2004), 279–289.MATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    K. El Amri, C. Hess: On the Pettis integral of closed valued multifunctions. Set-Valued Analysis 8 (2000), 329–360.MATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    M. Federson, R. Bianconi: Linear integral equations of Volterra concerning Henstock integrals. Real Anal. Exchange 25 (1999/00), 389–417.MathSciNetGoogle Scholar
  11. [11]
    M. Federson, P. Táboas: Impulsive retarded differential equations in Banach spaces via Bochner-Lebesgue and Henstock integrals. Nonlinear Anal. Ser. A: Theory Methods 50 (2002), 389–407.MATHCrossRefGoogle Scholar
  12. [12]
    J. L. Gamez, J. Mendoza: On Denjoy-Dunford and Denjoy-Pettis integrals. Studia Math. 130 (1998), 115–133.MATHMathSciNetGoogle Scholar
  13. [13]
    R. A. Gordon: The Denjoy extension of the Bochner, Pettis and Dunford integrals]. Studia Math. 92 (1989), 73–91.MATHMathSciNetGoogle Scholar
  14. [14]
    R. A. Gordon: The Integrals of Lebesgue, Denjoy, Perron and Henstock. Grad. Stud. Math. Vol 4. AMS, Providence, 1994.MATHGoogle Scholar
  15. [15]
    C. Hess: On multivalued martingales whose values may be unbounded: martingale selectors and Mosco convergence. J. Multivariate Anal. 39 (1991), 175–201.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    C. Hess, H. Ziat: Théorème de Komlós pour des multifonctions intégrables au sens de Pettis et applications. Ann. Sci. Math. Québec 26 (2002), 181–198.MATHMathSciNetGoogle Scholar
  17. [17]
    J. Komlós: A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18 (1967), 217–229.MATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    K. Musial: Topics in the theory of Pettis integration. In: School of Measure theory and Real Analysis, Grado, Italy, May 1992. Rend. Ist. Mat. Univ. Trieste 23 (1991), 177–262.Google Scholar
  19. [19]
    L. Di Piazza, K. Musial: Set-valued Kurzweil-Henstock-Pettis integral. Set-Valued Analysis 13 (2005), 167–179.MATHMathSciNetCrossRefGoogle Scholar
  20. [20]
    S. Schwabik: The Perron integral in ordinary differential equations. Differential Integral Equations 6 (1993), 863–882.MATHMathSciNetGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2006

Authors and Affiliations

  • B. Satco
    • 1
  1. 1.UFR Sciences et Techniques, Laboratoire de Mathématiques CNRS-UMR 6205Université de Bretagne OccidentaleBrest Cedex 3France

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