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CONVERGENCE THEOREMS FOR SET-VALUED DENJOY-PETTIS INTEGRABLE MAPPINGS

Park, Chun-Kee

  • Published : 2009.04.30

Abstract

In this paper, we introduce the Denjoy-Pettis integral of set-valued mappings and investigate some properties of the set-valued Denjoy-Pettis integral. Finally we obtain the Dominated Convergence Theorem and Monotone Convergence Theorem for set-valued Denjoy-Pettis integrable mappings.

Keywords

set-valued mapping;Denjoy integral;Denjoy-Pettis integral

References

  1. R. J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1–12 https://doi.org/10.1016/0022-247X(65)90049-1
  2. B. Cascales and J. Rodriguez, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), 540–560 https://doi.org/10.1016/j.jmaa.2004.03.026
  3. G. Debreu, Integration of Correspondences, Univ. California Press, Berkeley, CA, 1967
  4. L. Di Piazza and K. Musial, A decomposition theorem for compact-valued Henstock integral, Monatsh. Math. 148 (2006), 119–126 https://doi.org/10.1007/s00605-005-0376-2
  5. K. El Amri and C. Hess, On the Pettis integral of closed valued multifunctions, Set-Valued Analysis 8 (2000), 329–360 https://doi.org/10.1023/A:1026547222209
  6. J. L. Gamez and J. Mendoza, On Denjoy-Dunford and Denjoy-Pettis integrals, Studia Math. 130 (1998), 115–133
  7. R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron and Henstock, Grad. Stud. Math. 4, Amer. Math. Soc. Providence, R.I., 1994
  8. N. Papageoriou, On the theory of Banach space valued multifunctions, J. Multivariate Anal. 17 (1985), 185–206 https://doi.org/10.1016/0047-259X(85)90078-8
  9. S. Saks, Theory of the Integral, Dover, New York, 1964
  10. J. Wu and C. Wu, The w-derivatives of fuzzy mappings in Banach spaces, Fuzzy Sets and Systems 119 (2001), 375–381 https://doi.org/10.1016/S0165-0114(98)00468-0
  11. W. Zhang, Z. Wang, and Y. Gao, Set-Valued Stochastic Process, Academic Press, Beijing, 1996
  12. L. Di Piazza and K. Musial, Set-valued Kurzweil-Henstock-Pettis integral, Set-Valued Analysis 13 (2005), 167–179 https://doi.org/10.1007/s11228-004-0934-0
  13. R. A. Gordon, The Denjoy extension of the Bochner, Pettis and Dunford integrals, Studia Math. 92 (1989), 73–91