Convergence of fuzzy sets with respect to the supremum metric (original) (raw)
Related papers
Convexity and semicontinuity of fuzzy sets
Fuzzy Sets and Systems, 2004
In this paper we study a modiÿed concept called fuzzy convexity which was proposed by Ammar and Metz (Fuzzy Sets and Systems 49 (1992) 135). A criteria for convex fuzzy sets under lower semicontinuity is given. We prove in the upper semicontinuous case, that the class of semistrictly quasiconvex fuzzy sets lies between the quasiconvex and strictly quasiconvex classes. We also prove for both families of semistrictly quasiconvex and strictly quasiconvex fuzzy sets, that every local maximizer is also a global one. In addition, a characterization of quasimonotonic fuzzy sets in terms of level sets is given.
On the variational convergence of fuzzy sets in metric spaces
Annali Dell'universita' Di Ferrara, 1998
Sunto Kaleva ha investigato in [9] le relazioni esistenti tra due convergenze metriche, detteH eD, di sottoinsiemi fuzzy di spazi euclidei finito-dimensionali. In questo articolo le convergenzeH eD (la loro definizione dipende dalla distanza di Hausdorff tra insiemi compatti) sono confrontate con la convergenza variazionale, detta γ-convergenza, introdotta da De Giorgi and Franzoni in [3] nel contesto degli spazi topologici. Tale
Fuzzy Sets and Systems, 1999
Two classes of metrics are introduced for spaces of fuzzy sets. Their equivalence is discussed and basic properties established. A characterisation of compact and locally compact subsets is given in terms of boundedness and p-mean equileft-continuity, and the spaces shown to be locally compact, complete and separable metric spaces.
Study of Convexity in Sets and Fuzzy Sets
Fudma Journal of Sciences, 2022
In this paper, we present an overview of both classical and fuzzy convexity, particularly, in conjunction with continuity 5 and some topological concepts, and provide proofs of some of their algebraic properties along with suitable illustrations. We have extended the work of (Ammar E. E., 1999) by proving that if μ and ν are two convex fuzzy sets, then μ + ν, μ − ν, μ * ν, and μ/ν are also convex fuzzy sets. We have also shown that the convexity of a fuzzy set implies semistrict quasi-convexity.
Some Results in Fuzzy Metric Spaces
The problem of constructing a satisfactory theory of fuzzy metric spaces has been investigated by several authors from different points of view. In particular, and by modifying a definition of fuzzy metric space given by Kramosil and Michalek, George and Veeramani have introduced and studied the following interesting notion of a fuzzy metric space: A fuzzy metric space is an ordered triple (X , M, * ) such that X is a set, * is a continuous t-norm and M is a function defined on X × X ×]0, +∞[ with values in ]0, 1] satisfying certain axioms and M is called a fuzzy metric on X . It is proved that every fuzzy metric M on X generates a topology τ M on X which has as a base the family of open sets of the form B(x, ε, t) = {x ∈ X , 0 < ε < 1, t > 0} where B(x, ε, t) = {y ∈ X : M(x, y , t) > 1 − ε} for all ε ∈]0, 1[ and t > 0. The topological space (X , τ ) is said to be fuzzy metrizable if there is a fuzzy metric M on X such that τ = τ M . Then, it was proved that a topological space is fuzzy metrizable if and only if it is metrizable. From then, several fuzzy notions which are analogous to the corresponding ones in metric spaces have been given. Nevertheless, the theory of fuzzy metric completion is, in this context, very different to the classical theory of metric completion: indeed, there exist fuzzy metric spaces which are not completable. This class of fuzzy metrics can be easily included within fuzzy systems since the value given by them can be directly interpreted as a fuzzy certainty degree of nearness, and in particular, recently, they have been applied to colour image filtering, improving some filters when replacing classical metrics In this lecture we survey some results and questions obtained in recent years about this class of fuzzy metric spaces.
Information Sciences, 2009
One of the most important aspects of the (statistical) analysis of imprecise data is the usage of a suitable distance on the family of all compact, convex fuzzy sets, which is not too hard to calculate and which reflects the intuitive meaning of fuzzy sets. On the basis of expressing the metric of in terms of the mid points and spreads of the corresponding intervals we construct new families of metrics on the family of all d-dimensional compact convex sets as well as on the family of all d-dimensional compact convex fuzzy sets. It is shown that these metrics not only fulfill many good properties, but also that they are easy to calculate and easy to manage for statistical purposes, and therefore useful from the practical point of view.
Α-Optimal Best Proximity Point Result Involving Proximal Contraction Mappings in Fuzzy Metric Spaces
2017
In this paper, we introduce α-proximal fuzzy contraction of type−I and II in complete fuzzy metric space and obtain some fuzzy proximal and optimal coincidence point results. The obtained results further unify, extend and generalize some already existing results in literature. We also provide some examples which show the validity of obtained results and a comparison is also given which shows that contractive mappings and obtained results further generalizes already existing results in literature. c ©2017 all rights reserved.
Journal of Mathematics and Computer Science
In this paper we define complete fuzzy metric space and proved that a fuzzy topologically complete subset of a fuzzy metric space is a G set and prove that a converse of Sierpinsky theorem by showing that any G set in a complete metric space is a topologically complete fuzzy metrizable space (Alexandroff Theorem).
The Journal of Nonlinear Sciences and Applications, 2017
In this paper, we introduce α-proximal fuzzy contraction of type−I and II in complete fuzzy metric space and obtain some fuzzy proximal and optimal coincidence point results. The obtained results further unify, extend and generalize some already existing results in literature. We also provide some examples which show the validity of obtained results and a comparison is also given which shows that contractive mappings and obtained results further generalizes already existing results in literature.
Some Generalized IFS in Fuzzy Metric Spaces
2017
The intent of this paper is to study Suzuki type contraction (S-contraction) in fuzzy metric space (FMS) and obtain some existence and uniqueness results. Further, HutchinsonBarnsley theory is extended in this new setting and a collage theorem is derived as an application of it. The results obtain contain some of the recent results as special cases.