On the convergence of fuzzy sets

Convergence of fuzzy sets with respect to the supremum metric

“…The antecedents of our approach may be found in [14], where the authors gave a characterization of the d ∞ -convergence (or Dconvergence) of upper semicontinuous fuzzy sets f with nonempty compact support and [f ] 1 ∅ (see Corollaries 3 and 6). Under these assumptions, several authors have studied the relationship between this convergence and the hypo-convergence (or Γ-convergence) [14], Hausdorff convergence of supported endographs (or Hconvergence) [20] and Hausdorff convergence of superlevel sets at a fixed height (also named L-convergence or levelwise convergence) [17]. Nevertheless, we cannot find results which do not impose conditions on the fuzzy sets.…”

“…In the literature, we can find several convergences for fuzzy sets whose properties and relationships have been studied by several authors [14,17,25]. One of the most used is the convergence in the supremum metric.…”

“…The supremum metric has its origins in the theory of convergence of fuzzy sets [10,12,13,14,15,16,17,18,22,23,26]. Recall that a fuzzy set f on a topological space (X, τ ), as originally defined by Zadeh, is a function f from X to the closed unit interval I.…”

On the hyperspace of bounded closed sets under a generalized Hausdorff stationary fuzzy metric

“…We also have discussed several important properties as completeness, completion and precompactness for this hyperspace. In fuzzy functional analysis, many researchers have been working on the fixed point theory in the space of compact fuzzy sets equipped with the supremum metric [1,5,10,11,21,22,30,40]. The approach of this paper, however, enables us to use fixed point theory to study the fuzzy mappings from the new perspective.…”

Bounded solutions for fuzzy integral equations

“…This in turn led Seikkala [30] to introduce the notion of fuzzy derivative, which is a generalization of the Hukuhara derivative and the fuzzy integral, which is the same as that proposed by Dubois and Prade [7,8]. A natural consequence of the above was the study of fuzzy differential and integral equations, see [5,9,10,11,12,18,19,20,21,23,24,25,26,28,29,30,31,32,33].…”