Home Mathematics A New fuzzy McShane integrability
Article
Licensed
Unlicensed Requires Authentication

A New fuzzy McShane integrability

  • Redouane Sayyad EMAIL logo
Published/Copyright: June 8, 2021
Become an author with De Gruyter Brill

Abstract

We introduce the notion of the fuzzy McShane integral in the linear topology sense and we discuse its relation with the fuzzy Pettis integral introduced recently by Chun-Kee Park in [On the Pettis integral of fuzzy mappings in Banach spaces, Commun. Korean Math. Soc. 22 (2007), 535–545].

  1. (Communicated by Anatolij Dvurečenskij)

Acknowledgement

The author is grateful to the anonymous reviewers for their valuable remarks.

References

[1] Aliprantis, C. D.—Border, K. C.: Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd ed., Springer, 2006.Search in Google Scholar

[2] Aliprantis, C. D.—Burkinshow, O.: Positive operators, Springer, 2006.10.1007/978-1-4020-5008-4Search in Google Scholar

[3] El Amri, K.—Hess, C.: On The Pettis integral of closed valued multifunctions, Set-Valued Anal. 8 (2000), 329–360.10.1023/A:1026547222209Search in Google Scholar

[4] Avilés, A.—Plebanek, G.—Rodríguez, J.: The McShane integral in weakly compactly generated spaces, J. Funct. Anal. 259(11) (2010), 2776–2792.10.1016/j.jfa.2010.08.007Search in Google Scholar

[5] Beer, G.: Topologies on Closed and Closed Convex Sets, Kluwer Academic Publischers, Dordrech, Boston, London, 1993.10.1007/978-94-015-8149-3Search in Google Scholar

[6] Bongiorno, B.—Di Piazza, L.—Musiał, K.: A decomposition theorem for the fuzzy Henstock integral, Fuzzy Sets and Systems 200 (2012), 36–47.10.1016/j.fss.2011.12.006Search in Google Scholar

[7] Candeloro, D.—Di Piazza, L.—Musiał, K.—Sambucini, A. R.: Gauge integrals and selections of weakly compact valued multifunctions, J. Math. Anal. Appl. 441 (2016), 293–308.10.1016/j.jmaa.2016.04.009Search in Google Scholar

[8] Candeloro, D.—Di Piazza, L.—Musiał, K.—Sambucini, A. R.: Relations among Gauge and Pettis integrals for multifunctions with compact convex values, Ann. Mat. Pura Appl. 197 (2018), 171–183.10.1007/s10231-017-0674-zSearch in Google Scholar

[9] Caponetti, D.—Marraffa, V.—Naralenkov, K.: On the integration of Riemann-measurable vectoe-valued functions, Monatsh. Math. 182 (2017), 513–536.10.1007/s00605-016-0923-zSearch in Google Scholar

[10] Cascales, C.—Kadets, V.—Rodríguez, J.: The Pettis integral for multi-valued functions via single ones, J. Math. Anal. Appl. 332(1) (2007), 1–10.10.1016/j.jmaa.2006.10.003Search in Google Scholar

[11] Castaing, C.: Quelques résultats de convergence des suites adaptées, Acta Math. Vietnam. 14(1) (1989), 51–66.Search in Google Scholar

[12] Castaing, C.—Valadier, M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Math. 580, Springer, 1977.10.1007/BFb0087685Search in Google Scholar

[13] Congxin, W.—Zengtai, G.: On the Henstock integrals of fuzzy-valued functions (I), Fuzzy Sets and Systems 120 (2001), 523–532.10.1016/S0165-0114(99)00057-3Search in Google Scholar

[14] Deville, R.—Rodríguez, J.: Integration in Hilbert Generated Banach spaces, Israel J. Math. 177 (2010), 285–306.10.1007/s11856-010-0047-4Search in Google Scholar

[15] Diestel, J.—Uhl Jr., J. J.: Vector Measures. Math. Surveys Monogr. 15, AMS, 1977.10.1090/surv/015Search in Google Scholar

[16] Di Piazza, L.—Musiał, K.: Set-valued Kurzweil-Henstock-Pettis integral, Set-Valued Anal. 13 (2005), 167–179.10.1007/s11228-004-0934-0Search in Google Scholar

[17] Di Piazza, L.—Musiał, K.: A decomposition theorem for compact-valued Henstock integral, Monatsh. Math. 1482 (2006), 119–126.10.1007/s00605-005-0376-2Search in Google Scholar

[18] Di Piazza, L.—Musiał, K.: Decomposition of weakly compact valued integrable multifunctions, ∑ Mathematics 8 (2020), Art. 836.10.3390/math8060863Search in Google Scholar

[19] Di Piazza, L.—Marraffa, V.: Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces, Math. Slovaca 67(6) (2017), 1359–1370.10.1515/ms-2017-0057Search in Google Scholar

[20] Di Piazza, L.—Preiss, D.: When do McShane and Pettis integrals coincide, Illinois J. Math. 47(4) (2003), 1177–1187.10.1215/ijm/1258138098Search in Google Scholar

[21] Fremlin, D. H.: The generalzed McShane integral, Illinois J. Math. 39(1) (1995), 39–67.10.1215/ijm/1255986628Search in Google Scholar

[22] Fremlin, D. H.: The Henstock and McShane integrals of vector-valued function, Illinois J. Math. 38(3) (1994), 471–479.10.1215/ijm/1255986726Search in Google Scholar

[23] Fremlin, D. H.—Mendoza, J.: On the integration of vector-valued functions, Illinois J. Math. 38(1) (1994), 127–147.10.1215/ijm/1255986891Search in Google Scholar

[24] Gordon, R. A.: The McShane integral of Banach-valued functions, Illinois J. Math. 34(3) (1990), 557–567.10.1215/ijm/1255988170Search in Google Scholar

[25] Gordon, R. A.: The integrals of Lebesgue, Denjoy, Perron, and Henstock. Grad. Stud. Math. 4, AMS, Providence, R.I., 1994.10.1090/gsm/004Search in Google Scholar

[26] Hess, C.: Measurability and integrability of the weak upper limit of a sequence of multifunctions, J. Math. Anal. Appl. 153(1) (1990), 226–249.10.1016/0022-247X(90)90275-KSearch in Google Scholar

[27] Hess, C.—Ziat, H.: Théorème de Komlós pour les multifonctions intégrables au sens de Pettis et applications, Ann. Sci. Math. Québec 26 (2002), 181–198.Search in Google Scholar

[28] Kurzweil, J.—Schwabik, S.: On McShane integrability of Banach space-valued functions, Real. Anal. Exchange 92(2) (2003–2004), 763–780.10.14321/realanalexch.29.2.0763Search in Google Scholar

[29] Lamei, S.—Li, G.: Strong laws of large numbers for fuzzy set-valued Random variables in Gα space, Adv. Pure Math. 6 (2016), 583–592.10.4236/apm.2016.69047Search in Google Scholar

[30] Musiał, K.: Topics in the theory of Pettis integration. In: School of Measure Theory and Real Analysis, Italy, May 1992.Search in Google Scholar

[31] Musiał, K.: A decomposition theorem for Banach space valued fuzzy Henstock integral, Fuzzy Sets and Systems 259 (2015), 21–28.10.1016/j.fss.2014.03.012Search in Google Scholar

[32] Neveu, J.: Mathematical Foundations of the Calculus of Probability, Holden-Day, San Francisco, 1965.Search in Google Scholar

[33] Park, C. K: On the Pettis integral of fuzzy mappings in Banach spaces, Commun. Korean Math. Soc. 22 (2007), 535–545.10.4134/CKMS.2007.22.4.535Search in Google Scholar

[34] Reynolds, R.—Swartz, C: The Vitali convergence theorem for the vector-valued McShane integral, Math. Bohem. 129(2) (2004), 159–176.10.21136/MB.2004.133906Search in Google Scholar

[35] Rodríguez, J.: On the equivalence of McShane and Pettis integrability in non-separable Banach spaces, J. Math. Anal. Appl. 341 (2008), 80–90.10.1016/j.jmaa.2007.10.017Search in Google Scholar

[36] Schwabik, S.—Ye, G.: On the strong McShane integral of functions with values in a Banach space, Czechoslovak Math. J. 51(126) (2001), 819–830.10.1023/A:1013721114330Search in Google Scholar

[37] Sayyad, R.: The weak integral by partitions of unity, Real Anal. Exchange 44(1) (2019), 181–198.10.14321/realanalexch.44.1.0181Search in Google Scholar

[38] Sayyad, R.: The multiplier for the weak McShane integral, Math. Bohem. 144(1) (2019), 13–24.10.21136/MB.2018.0044-17Search in Google Scholar

[39] Sayyad, R.: The McShane integral in the limit, Real Anal. Exchange 42(2) (2017), 283–310.10.14321/realanalexch.42.2.0283Search in Google Scholar

[40] Sayyad, R.: Contribution à l’étude des Intégrales de McShane et de Henstock Vectorielles et Multivoques. Thèse de doctorat, Agadir, 2014.Search in Google Scholar

[41] Sonntag, Y.—Zălinescu, C.: Scalar convergence of convex sets, JMAA 164 (1992), 219–241.10.1016/0022-247X(92)90154-6Search in Google Scholar

[42] Wu, J.—Wu, C.: The w-derivatives of fuzzy mapping in Banach spaces, Fuzzy Sets and Systems 119 (2001), 375–381.10.1016/S0165-0114(98)00468-0Search in Google Scholar

[43] Xue, X.—Ma, M.: Random fuzzy number integrals in Banach spaces, Fuzzy Sets and Systems 66 (1994), 97–111.10.1016/0165-0114(94)90303-4Search in Google Scholar

[44] Xue, X.—Wu, C.: On the extension of fuzzy number measures in Banach spaces. Part I. Representation of fuzzy number measures, Fuzzy Sets and Systems 78 (1996), 347–354.10.1016/0165-0114(96)84616-1Search in Google Scholar

[45] Ziat, H.: Convergence theorems for Pettis integrable multifunctions, Bull. Pol. Acad. Sci. Math. 45 (1997), 123–137.Search in Google Scholar

[46] Ziat, H.: On characterisation of Pettis integrable multifunctions, Bull. Pol. Acad. Sci. Math. 44 (2000), 227–230.Search in Google Scholar

Received: 2020-06-22
Accepted: 2020-07-21
Published Online: 2021-06-08
Published in Print: 2021-06-25

© 2021 Mathematical Institute Slovak Academy of Sciences

Downloaded on 15.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2021-0009/pdf?lang=en
Scroll to top button