^{}"Al.I. Cuza" University, Faculty of Mathematics, Bd. Carol I, No.
11, Iasi, 700506, Romania

Abstract

We study a fuzzy type integral for measurable multifunctions with respect to a fuzzy measure. Some classical properties and convergence theorems are presented.

[1] R. J. Aumann, Integrals of set-valued maps, J. Math. Anal. Appl., 12 (1965), 1-12. [2] G. Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble), 5 (1953-1954), 131-292. [3] A. Croitoru, Set-norm continuity of set multifunctions, ROMAI Journal, 6 (2010), 47-56 . [4] A. Croitoru, A. Gavrilut, N. E. Mastorakis and G. Gavrilut, On dierent types of non-additive set multifunctions, WSEAS Transactions on Mathematics, 8 (2009), 246-257. [5] G. Debreu, Integration of correspondences, Proc. 5th Berkely Symposium on Math. Stat. Prob. II, Part. I, (1967), 351-372. [6] L. Drewnowski, Topological rings of sets, continuous set functions. Integration, I, II, III, Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys., 20 (1972), 269-286. [7] D. Dubois and H. Prade, Fuzzy sets and systems: theory and applications, Academic Press, New York, 1980.

[8] N. Dunford and J. Schwartz, Linear operators I. general theory, Interscience, New York, 1958. [9] A. Gavrilut, A Gould type integral with respect to a multisubmeasure, Math. Slovaca, 58 (2008), 1-20. [10] A. Gavrilut, The general Gould type integral with respect to a multisubmeasure, Math. Slovaca, 60(3) (2010), 289-318. [11] A. Gavrilut and A. Croitoru, Non-atomicity for fuzzy and non-fuzzy set multifunctions, Fuzzy Sets and Systems, 160 (2009), 2106-2116. [12] C. Guo and D. Zhang, On set-valued fuzzy measures, Information Sciences, 160 (2004), 13-25. [13] S. Hu and N. S. Papageorgiou, Handbook of multivalued analysis, Kluwer Acad. Publ., Dordrecht, I (1997). [14] L. C. Jang and J. S. Kwon, On the representation of Choquet integrals of set-valued functions and null sets, Fuzzy Sets and Systems, 112 (2000), 233-239. [15] F. Merghadi and A. Aliouche, A related xed point theorem in n fuzzy metric spaces, Iranian Journal of Fuzzy Systems, 7(3) (2010), 73-86. [16] E. Pap, Null-additive set functions, Kluwer Academic Publishers, Dordrecht, 1995. [17] A. M. Precupanu, On the set valued additive and subadditive set functions, An. St. Univ. "Al.I. Cuza" Iasi, 29 (1984), 41-48. [18] A. Precupanu and A. Croitoru, A Gould type integral with respect to a multimeasure. I, An. St. Univ. "Al.I. Cuza" Iasi, 48 (2002), 165-200. [19] A. Precupanu, A. Gavrilut and A. Croitoru, A fuzzy Gould type integral, Fuzzy Sets and Systems, 161 (2010), 661-680. [20] H. Radstrom, An embedding theorem for spaces of convex sets, Proc. A.M.S., 3 (1952), 151- 158. [21] D. A. Ralescu and M. Sugeno, Fuzzy integral representation, Fuzzy Sets and Systems, 84 (1996), 127-133. [22] H. Roman-Flores, A. Flores-Franulic and Y. Chalco-Cano, The fuzzy integral for monotone functions, Applied Mathematics and Computation, 185 (2007), 492-498. [23] C. Stamate, Vector fuzzy integral, Recent Advances in Neural Network, Fuzzy Systems and Evolutionary Computing, Proceeding of the 11th WSEAS International Conference on Fuzzy Systems (FS'10), Iasi, Romania, June 13-15, (2010), 221-224. [24] M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. Thesis, Tokyo Institute of Technology, 1974. [25] H. Suzuki, Atoms of fuzzy measures and fuzzy integrals, Fuzzy Sets and Systems, 41 (1991), 329-342. [26] S. M. Vaezpour and F. Karini, t-Best approximation in fuzzy normed spaces, Iranian Journal of Fuzzy Systems, 5(2) (2008), 93-99. [27] Z. Wang and G. J. Klir, Fuzzy measure theory, Plenum Press, New York, 1992. [28] G. F. Wen, F. G. Shi and H. Y. Li, Almost S-compactness in L-topological spaces, Iranian Journal of Fuzzy Systems, 5(3) (2008), 31-44. [29] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. [30] D. Zhang and C. Guo, Fuzzy integrals of set-valued mappings and fuzzy mappings, Fuzzy Sets and Systems, 75 (1995), 103-109.