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On the hyperspace of bounded closed sets under a generalized Hausdorff stationary fuzzy metric. (English) Zbl 1309.54005
Summary: In this paper, we generalize the classical Hausdorff metric with t-norms and obtain its basic properties. Furthermore, for a given stationary fuzzy metric space with a t-norm without zero divisors, we propose a method for constructing a generalized Hausdorff fuzzy metric on the set of the nonempty bounded closed subsets. Finally we discuss several important properties as completeness, completion and precompactness.

MSC:
54A40 Fuzzy topology
54B20 Hyperspaces in general topology
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[1] Adibi, H., Cho, Y. J., O’Regan, D., Saadati, R.: Common fixed point theorems in \(L\)-fuzzy metric spaces. Appl. Math. Comput. 182 (2006), 820-828. · Zbl 1106.54016
[2] Aubin, J. P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York 1984. · Zbl 1115.47049
[3] Azé, D., Corvellec, J. N., Lucchetti, R. E.: Variational pairs and applications to stability in nonsmooth analysis. Nonlinear Anal. Theory Methods Appl. 49 (2002), 643-670. · Zbl 1035.49014
[4] Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer Academic Publishers, Dordrecht 1993. · Zbl 0792.54008
[5] Chang, S. S., Cho, Y. J., Lee, B. S., Jung, J. S., Kang, S. M.: Coincidence point and minimization theorems in fuzzy metric spaces. Fuzzy Sets and Systems 88 (1997), 119-128. · Zbl 0912.54013
[6] Cho, Y. J., Petrot, N.: Existence theorems for fixed fuzzy points with closed \(\alpha\)-cut sets in complete metric spaces. Fuzzy Sets and Systems 26 (2011), 115-124. · Zbl 1207.54055
[7] Deng, Z. K.: Fuzzy pseudo-metric spaces. J. Math. Anal. Appl. 86 (1982), 74-95. · Zbl 0589.54006
[8] Engelking, R.: General Topology. PWN-Polish Science Publishers, Warsaw 1977. · Zbl 1281.54001
[9] George, A., Veeramani, P.: On some results in fuzzy metric spaces. Fuzzy Sets and Systems 64 (1994), 395-399. · Zbl 0843.54014
[10] Ghil, B. M., Kim, Y. K.: Continuity of functions defined on the space of fuzzy sets. Inform. Sci. 157 (2003), 155-165. · Zbl 1051.54004
[11] Gregori, V., Sapena, A.: On fixed point theorem in fuzzy metric spaces. Fuzzy Sets and Systems 125 (2002), 245-252. · Zbl 0995.54046
[12] Gregori, V., Romaguera, S.: On completion of fuzzy metric spaces. Fuzzy Sets and Systems 130 (2002), 399-404. · Zbl 1010.54002
[13] Gregori, V., Romaguera, S.: Characterizing completable fuzzy metric spaces. Fuzzy Sets and Systems 144 (2004), 411-420. · Zbl 1057.54010
[14] Gregori, V., Morillas, S., Sapena, A.: Examples of fuzzy metrics and applications. Fuzzy Sets and Systems 170 (2011), 95-111. · Zbl 1210.94016
[15] Hausdorff, F.: Set Theory. Chelsea, New York 1957. · Zbl 1149.01022
[16] Hop, N.V.: Solving fuzzy (stochastic) linear programming problems using superiority and inferiority measures. Inform. Sci. 177 (2007), 1977-1991. · Zbl 1128.90061
[17] Hop, N. V.: Solving linear programming problems under fuzziness and randomness environment using attainment values. Inform. Sci. 177 (2007), 2971-2984. · Zbl 1178.90363
[18] Joo, S. Y., Kim, Y. K.: The Skorokhod topology on space of fuzzy numbers. Fuzzy Sets and Systems 111 (2000), 497-501. · Zbl 0961.54024
[19] Joo, S. Y., Kim, Y. K.: Topological properties on the space of fuzzy sets. J. Math. Anal. Appl. 246 (2000), 576-590. · Zbl 0986.54012
[20] Kaleva, O., Seikkala, S.: On fuzzy metric spaces. Fuzzy Sets and Systems 12 (1984), 215-229. · Zbl 0558.54003
[21] Kaleva, O.: On the convergence of fuzzy sets. Fuzzy Sets and Systems 17 (1985), 53-65. · Zbl 0584.54004
[22] Kaleva, O.: Fuzzy differential equations. Fuzzy Sets and Systems 24 (1987), 301-317. · Zbl 1100.34500
[23] Kim, Y. K.: Compactness and convexity on the space of fuzzy sets. J. Math. Anal. Appl. 264 (2001), 122-132. · Zbl 1065.54001
[24] Kim, Y. K.: Compactness and convexity on the space of fuzzy sets II. Nonlinear Anal. Theory Methods Appl. 57 (2004), 639-653. · Zbl 1065.54001
[25] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000. · Zbl 1087.20041
[26] Kramosil, I., Michálek, J.: Fuzzy metric and statistical metric spaces. Kybernetika 11 (1975), 326-334.
[27] Matheron, G.: Random Sets and Integral Geometry. Wiley, New York 1975. · Zbl 0321.60009
[28] Mordukhovich, B. S., Shao, Y.: Fuzzy calculus for coderivatives of multifunctions. Nonlinear Anal. Theory Methods Appl. 29 (1997), 605-626. · Zbl 0879.58006
[29] Jr, S. B. Nadler: Multi-valued contraction mappings. Pacific J. Math. 30 (1969), 475-487. · Zbl 0187.45002
[30] Puri, M. L., Ralescu, D. A.: Fuzzy random variables. J. Math. Anal. Appl. 114 (1986), 409-422. · Zbl 0605.60038
[31] Qiu, D., Zhang, W.: On Decomposable Measures Induced by Metrics. J. Appl. Math. Volume 2012, Article ID 701206, 8 pages. · Zbl 1252.28001
[32] Qiu, D., Zhang, W.: The strongest t-norm for fuzzy metric spaces. Kybernetika 49 (2013), 141-148. · Zbl 1264.54020
[33] Qiu, D., Zhang, W., Li, C.: On decomposable measures constructed by using stationary fuzzy pseudo-ultrametrics. Int. J. Gen. Syst. 42 (2013), 395-404. · Zbl 1280.28018
[34] Qiu, D., Zhang, W., Li, C.: Extension of a class of decomposable measures using fuzzy pseudometrics. Fuzzy Sets and Systems 222 (2013), 33-44. · Zbl 1284.28017
[35] Rodríguez-López, J., Romaguera, S.: The Hausdorff fuzzy metric on compact sets. Fuzzy Sets and Systems 147 (2004), 273-283. · Zbl 1069.54009
[36] Romaguera, S., Sanchis, M.: On fuzzy metric groups. Fuzzy Sets and Systems 124 (2001), 109-115. · Zbl 0994.54007
[37] Shi, F. G., Zheng, C. Y.: Metrization theorems in L-topological spaces. Fuzzy Sets and Systems 149 (2005), 455-471. · Zbl 1070.54007
[38] Trillas, E.: On the use of words and fuzzy sets. Inf. Sci. 176 (2006), 1463-1487. · Zbl 1098.03066
[39] Zadeh, L. A.: Fuzzy sets. Inform. Control 8 (1965), 338-353. · Zbl 0942.00007
[40] Zhang, W., Qiu, D., Li, Z., Xiong, G.: Common fixed point theorems in a new fuzzy metric space. J. Appl. Math. Volume 2012, Article ID 890678, 18 pages. · Zbl 1235.54057
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