Some questions about Zadeh's extension on metric spaces

Jardón, Daniel; Sánchez, Iván; Sanchis, Manuel

doi:10.1016/j.fss.2018.10.019

Cited by 14 publications

(18 citation statements)

References 8 publications

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“…The family {ω α : α ∈ [0, 1]} ⊂ K(X) is a decreasing family satisfying the conditions of Proposition 4.9 in [18]; therefore, there exists a unique ω ∈ F ∞ (X) such that…”

Section: Periodic Points and Devaney Chaosmentioning

confidence: 99%

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Chaos on Fuzzy Dynamical Systems

Martínez-Giménez¹,

Peris²,

Ródenas³

2021

Mathematics

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Given a continuous map f:X→X on a metric space, it induces the maps f¯:K(X)→K(X), on the hyperspace of nonempty compact subspaces of X, and f^:F(X)→F(X), on the space of normal fuzzy sets, consisting of the upper semicontinuous functions u:X→[0,1] with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems (X,f), (K(X),f¯), and (F(X),f^). In particular, we considered several dynamical properties related to chaos: Devaney chaos, A-transitivity, Li–Yorke chaos, and distributional chaos, extending some results in work by Jardón, Sánchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451–463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics).

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“…The family {ω α : α ∈ [0, 1]} ⊂ K(X) is a decreasing family satisfying the conditions of Proposition 4.9 in [18]; therefore, there exists a unique ω ∈ F ∞ (X) such that…”

Section: Periodic Points and Devaney Chaosmentioning

confidence: 99%

“…In the next section, we use the following properties of fuzzy sets on the spaces F ∞ (X) and F 0 (X) (see [4,18,19] for the details). Proposition 1.…”

Section: Introduction and Basic Definitionsmentioning

confidence: 99%

Chaos on Fuzzy Dynamical Systems

Martínez-Giménez¹,

Peris²,

Ródenas³

2021

Mathematics

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“…Recently, Jardón et al [5] proved that both f and F are contractive if the previous dynamical systems are contractive under the level-wise metric d ∞ , and they proposed the following two questions. For more recent results on Zadeh's extension and g-fuzzification, refer to [2,3,7,8,9,10] and some references therein.…”

Section: Andmentioning

confidence: 99%

“…There is a close relation between this topology and fuzzy numbers. The Skorokhod metric, induced by the Skorokhod topology, is used in the study of fuzzy numbers [5,6].…”

mentioning

confidence: 99%

Answers to some questions about Zadeh's extension on metric spaces

Wu,

Zhang,

Chen

2018

Preprint

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“…These results are obtained on the set of normal, upper semi-continuous and compact fuzzy sets on R m . Recently, Jardón, Sánchez and Sanchis [8] discussed the Skorokhod metric on compact fuzzy sets on metric space rather than just on R m . Here, a compact fuzzy set is a fuzzy set whose support set is compact, and a noncompact fuzzy set is a general fuzzy set whose support set may be compact or non-compact.…”

Section: Introductionmentioning

confidence: 99%

Some properties of Skorokhod metric on fuzzy sets

Huang

2019

Preprint

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In this paper, we have our discussions on normal and upper semi-continuous fuzzy sets on metric spaces. The Skorokhod-type metric is stronger than the Skorokhod metric. It is found that the Skorokhod metric and the Skorokhodtype metric are equivalent on compact fuzzy sets. However, the Skorokhod metric and the Skorokhod-type metric need not be equivalent on L p -integrable fuzzy sets. Based on this, we investigate relations between these two metrics and the L p -type d p metric. It is found that the relations can be divided into three cases. On compact fuzzy sets, the Skorokhod metric is stronger than the d p metric. On L p -integrable fuzzy sets, which take compact fuzzy sets as special cases, the Skorokhod metric is not necessarily stronger than the d p metric, but the Skorokhod-type metric is still stronger than the d p metric. On general fuzzy sets, even the Skorokhod-type metric is not necessarily stronger than the d p metric. We also show that the Skorokhod metric is stronger than the sendograph metric.

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Some questions about Zadeh's extension on metric spaces

Cited by 14 publications

References 8 publications

Chaos on Fuzzy Dynamical Systems

Chaos on Fuzzy Dynamical Systems

Answers to some questions about Zadeh's extension on metric spaces

Some properties of Skorokhod metric on fuzzy sets

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