Transitivity in Fuzzy Hyperspaces

Given a metric space (X, d), we deal with a classical problem in the theory of hyperspaces: how some important dynamical properties (namely, weakly mixing, transitivity and point-transitivity) between a discrete dynamical system f : (X, d) → (X, d) and its natural extension to the hyperspace are related. In this context, we consider the Zadeh’s extension fbof f to F(X), the family of all normal fuzzy sets on X, i.e., the hyperspace F(X) of all upper semicontinuous fuzzy sets on X with compact supports and non-empty levels and we endow F(X) with different metrics: the supremum metric d∞, the Skorokhod metric d0, the sendograph metric dS and the endograph metric dE. Among other things, the following results are presented: (1) If (X, d) is a metric space, then the following conditions are equivalent: (a) (X, f) is weakly mixing, (b) ((F(X), d∞), fb) is transitive, (c) ((F(X), d0), fb) is transitive and (d) ((F(X), dS)), fb) is transitive, (2) if f : (X, d) → (X, d) is a continuous function, then the following hold: (a) if ((F(X), dS), fb) is transitive, then ((F(X), dE), fb) is transitive, (b) if ((F(X), dS), fb) is transitive, then (X, f) is transitive; and (3) if (X, d) be a complete metric space, then the following conditions are equivalent: (a) (X × X, f × f) is point-transitive and (b) ((F(X), d0) is point-transitive

Convergence of fuzzy sets with respect to the supremum metric

We characterize the convergence of fuzzy sets in the supremum metric given by the supremum of the Hausdorff distances of the alpha-cuts of the fuzzy sets. We do it by dividing this metric into its lower and upper quasi-pseudometric parts. This characterization is given in the more general context with no assumption on the fuzzy sets. Furthermore, motivated from the theory of Convex Analysis, we also provide some results about the behavior of the convergence in the supremum metric with respect to maximizers. (C) 2014 Elsevier B.V. All rights reserved.The second and third authors thank the support of the Ministry of Economy and Competitiveness of Spain under grant MTM2012-37894-C02-01.Pedraza Aguilera, T.; Rodríguez López, J.; Romaguera Bonilla, S. (2014). Convergence of fuzzy sets with respect to the supremum metric. Fuzzy Sets and Systems. 245:83-100. https://doi.org/10.1016/j.fss.2014.03.005S8310024

Properties of several metric spaces of fuzzy sets

This paper discusses the properties the spaces of fuzzy sets in a metric space equipped with the endograph metric and the sendograph metric, respectively. We first give some relations among the endograph metric, the sendograph metric and the $\Gamma$-convergence, and then investigate the level characterizations of the endograph metric and the $\Gamma$-convergence. By using the above results, we give some relations among the endograph metric, the sendograph metric, the supremum metric and the $d_p^*$ metric, $p\geq 1$. On the basis of the above results, we present the characterizations of total boundedness, relative compactness and compactness in the space of fuzzy sets whose $\alpha$-cuts are compact when $\alpha>0$ equipped with the endograph metric, and in the space of compact support fuzzy sets equipped with the sendograph metric, respectively. Furthermore, we give completions of these metric spaces, respectively

On The Variational Convergence Of Fuzzy Sets In Metric Spaces

Kaleva [9] has studied the relationships between the metric convergencesH andD of fuzzy convex sets on Euclidean spaces. The distanceH between two fuzzy set is given by Hausdorff distance of their sendographs, whileD is the supremum of the Hausdorff distances of the level sets corresponding to the fuzzy sets. The aim of this paper is to compareH andD with the variational convergence, called γ-convergence (see De Giorgi and Franzoni [3]). Our analysis which is carried out in the setting of metric spaces (not necessarily locally compact or vector spaces), improves Kaleva's results. © 1998 Università degli Studi di Ferrara.4412739Attouch, H., (1984) Variational Convergence for Functions and Operators, , London: PitmanDantzig, G.B., Folkman, J., Shapiro, N., On the Continuity of the Minimum Set of a Continuous Function (1967) J. Math. An. And Appl., 17, pp. 519-548de Giorgi, E., Franzoni, T., Su un tipo di convergenza variazionale (1975) Atti Acc. Naz. Lincei Rend. Cl. Sc. Mat. Fis. Nat. (8), 58, pp. 842-850De Giorgi, E., Franzoni, T., Su un tipo di convergenza variazionale (1979) Rend. Sem. Brescia, 3, pp. 63-101De Giorgi, E., Generalized Limits in Calculus of Variations (1981) Topics in functional analysis 1980-1981, Scuola Normale Superiore di Pisa», pp. 117-148De Giorgi, E., G-operators and γ-convergence (1983) Proc. Int. Congress of Math. Warsaw, pp. 1174-1191Goetschel, R., Voxman, W., A pseudometric for fuzzy sets and certain related results (1981) J. Math. Anal. Appl., 81, pp. 507-523Hausdorff, F., (1978) Set Theory, , 3rd Ed.th edn., Milan: Chelsea Publ. CompKaleva, O., On the convergence of fuzzy sets (1985) Fuzzy Sets and Systems, 17, pp. 53-65Kloeden, P.E., Compact supported endographs and fuzzy sets (1980) Fuzzy Sets and Systems, 4, pp. 193-201Kuratowski, C., (1958) Topologie, , Warsaw: P. W. NPuri, M.L., Ralescu, D.A., Différentielle d'une fonction floue (1981) C. R. Acad. Sc. Paris, 293, pp. 237-23