scholarly journals Transitivity in Fuzzy Hyperspaces

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1862
Author(s):  
Daniel Jardón ◽  
Iván Sánchez ◽  
Manuel Sanchis
Keyword(s):  
Fuzzy Sets ◽  
Metric Space ◽  
Complete Metric Space ◽  
Discrete Dynamical System ◽  
Natural Extension ◽  
Classical Problem ◽  
The Family ◽  
Weakly Mixing ◽  
System F ◽  
Sendograph Metric

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 f^ of 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∞),f^) is transitive, (c) ((F(X),d0),f^) is transitive and (d) ((F(X),dS)),f^) is transitive, (2) if f:(X,d)→(X,d) is a continuous function, then the following hold: (a) if ((F(X),dS),f^) is transitive, then ((F(X),dE),f^) is transitive, (b) if ((F(X),dS),f^) 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.

scholarly journals Duality of Moduli and Quasiconformal Mappings in Metric Spaces

2020 ◽  
Vol 8 (1) ◽  
pp. 166-181
Author(s):  
Rebekah Jones ◽  
Panu Lahti
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Poincaré Inequality ◽  
Quasiconformal Mappings ◽  
Duality Relation ◽  
Doubling Measure ◽  
Poincare Inequality ◽  
Family Of Curves ◽  
The Family

AbstractWe prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.

scholarly journals Countable contraction mappings in metric spaces: invariant sets and measure

2014 ◽  
Vol 12 (4) ◽  
Author(s):  
María Barrozo ◽  
Ursula Molter
Keyword(s):  
Invariant Measure ◽  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Invariant Set ◽  
Invariant Sets ◽  
Countable Number ◽  
Contraction Mappings ◽  
Classic Result ◽  
System F

AbstractWe consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {F i: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps F i are of the form F i(x) = r i x + b i on X = ℝd, we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supi r i is strictly smaller than 1.Further, if ρ = {ρ k}k∈ℕ is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set.

scholarly journals IFSs consisting of generalized convex contractions

2017 ◽  
Vol 25 (1) ◽  
pp. 77-86 ◽  
Author(s):  
Flavian Georgescu
Keyword(s):  
Metric Space ◽  
Iterated Function System ◽  
Complete Metric Space ◽  
Continuous Functions ◽  
Finite Family ◽  
Function System ◽  
Picard Operator ◽  
Iterated Function ◽  
The Family ◽  
Convex Contractions

Abstract In this paper we introduce the concept of iterated function system consisting of generalized convex contractions. More precisely, given n ∈ ℕ*, an iterated function system consisting of generalized convex contractions on a complete metric space (X; d) is given by a finite family of continuous functions (fi)i ∈I , fi : X → X, having the property that for every ω ∈ λn(I) there exists a family of positive numbers (aω;υ)υ∈Vn(I) such that: x; y ∈ X. Here λn(I) represents the family of words with n letters from I, Vn(I) designates the family of words having at most n - 1 letters from I, while, if ω1 = ω1ω2 ... ωp, by fω we mean fω1 ⃘fω2 ⃘... ⃘ fωp. Denoting such a system by S = ((X; d); n; (fi)i∈I), one can consider the function FS : K(X) → K(X) described by , for all B ∈ K(X), where K(X) means the set of non-empty compact subsets of X. Our main result states that FS is a Picard operator for every iterated function system consisting of generalized convex contractions S.

scholarly journals Global asymptotic stability of inhomogeneous iterates

2002 ◽  
Vol 29 (3) ◽  
pp. 133-142 ◽  
Author(s):  
Yong-Zhuo Chen
Keyword(s):  
Dynamical System ◽  
Asymptotic Stability ◽  
Metric Space ◽  
Complete Metric Space ◽  
Discrete Dynamical System ◽  
Closed Convex Cone ◽  
Nonlinear Difference Equations ◽  
Finite Dimensional ◽  
Uniformly Convergent ◽  
The Stability

Let(M,d)be a finite-dimensional complete metric space, and{Tn}a sequence of uniformly convergent operators onM. We study the non-autonomous discrete dynamical systemxn+1=Tnxnand the globally asymptotic stability of the inhomogeneous iterates of{Tn}. Then we apply the results to investigate the stability of equilibrium ofTwhen it satisfies certain type of sublinear conditions with respect to the partial order defined by a closed convex cone. The examples of application to nonlinear difference equations are also given.

scholarly journals Residually many BV homeomorphisms map a null set onto a set of full measure

2018 ◽  
Vol 149 (04) ◽  
pp. 1047-1059
Author(s):  
Andrea Marchese
Keyword(s):  
Metric Space ◽  
Complete Metric Space ◽  
Full Measure ◽  
Open Unit ◽  
Unit Square ◽  
First Category ◽  
The Family ◽  
Baire Categories ◽  
Null Set

AbstractLet Q be the open unit square in ℝ2. We prove that in a natural complete metric space of BV homeomorphisms f : Q → Q with f|∂Q = Id, residually many homeomorphisms (in the sense of Baire categories) map a null set onto a set of full measure, and vice versa. Moreover, we observe that for 1 ⩽ p < 2, the family of W1,p homemomorphisms satisfying the above property is of the first category.

On the existence problem for a fixed point of a generalized contracting multivalued mapping

2021 ◽  
pp. 372-381
Author(s):  
Evgeny S. Zhukovskiy
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Complete Metric Space ◽  
Natural Extension ◽  
Multivalued Map ◽  
Existence Problem ◽  
Multivalued Maps ◽  
Generalized Contraction ◽  
Contraction Map ◽  
Definition Of

We discuss the still unresolved question, posed in [S. Reich, Some Fixed Point Problems, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 57:8 (1974), 194–198], of existence in a complete metric space X of a fixed point for a generalized contracting multivalued map Φ: X⇉X having closed values Φ(x)⊂X for all x∈X. Generalized contraction is understood as a natural extension of the Browder–Krasnoselsky definition of this property to multivalued maps: ∀x,u∈X h(φ(x),φ(u))≤ η(ρ(x,u)), where the function η:R_+→R_+ is increasing, right continuous, and for all d>0, η(d)<d (h(•,•) denotes the Hausdorff distance between sets in the space X). We give an outline of the statements obtained in the literature that solve the S. Reich problem with additional requirements on the generalized contraction Φ. In the simplest case, when the multivalued generalized contraction map Φ acts in R, without any additional conditions, we prove the existence of a fixed point for this map.

On Fixed Point Theorems in Complete Metric Space

2019 ◽  
Vol 10 (7) ◽  
pp. 1419-1425
Author(s):  
Jayashree Patil ◽  
Basel Hardan
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Complete Metric Space

Topological Completeness of Function Spaces Arising in the Hausdorff Approximation of Functions

1992 ◽  
Vol 35 (4) ◽  
pp. 439-448 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Approximation Theory ◽  
Metric Space ◽  
Polish Space ◽  
Complete Metric Space ◽  
Approximation Of Functions ◽  
Lower Envelopes ◽  
Constructive Approximation ◽  
Hausdorff Approximation ◽  
Set Of Points ◽  
Completely Metrizable

AbstractLet X be a complete metric space. Viewing continuous real functions on X as closed subsets of X × R, equipped with Hausdorff distance, we show that C(X, R) is completely metrizable provided X is complete and sigma compact. Following the Bulgarian school of constructive approximation theory, a bounded discontinuous function may be identified with its completed graph, the set of points between the upper and lower envelopes of the function. We show that the space of completed graphs, too, is completely metrizable, provided X is locally connected as well as sigma compact and complete. In the process, when X is a Polish space, we provide a simple answer to the following foundational question: which subsets of X × R arise as completed graphs?

scholarly journals Best Proximity Point for Generalized Proximal Weak Contractions in Complete Metric Space

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Erdal Karapınar ◽  
V. Pragadeeswarar ◽  
M. Marudai
Keyword(s):  
Approximate Solution ◽  
Metric Space ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Best Proximity Point ◽  
Complete Metric Space ◽  
Optimal Approximate Solution ◽  
Weak Contraction ◽  
Complete Metric Spaces ◽  
New Class

We introduce a new class of nonself-mappings, generalized proximal weak contraction mappings, and prove the existence and uniqueness of best proximity point for such mappings in the context of complete metric spaces. Moreover, we state an algorithm to determine such an optimal approximate solution designed as a best proximity point. We establish also an example to illustrate our main results. Our result provides an extension of the related results in the literature.

scholarly journals Fixed Point Results for the Family of Multivalued F-Contractive Mappings on Closed Ball in Complete Dislocated b-Metric Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 56 ◽  
Author(s):  
Qasim Mahmood ◽  
Abdullah Shoaib ◽  
Tahair Rasham ◽  
Muhammad Arshad
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Metric Space ◽  
Metric Spaces ◽  
Closed Ball ◽  
Multivalued Mappings ◽  
System Of Integral Equations ◽  
Contractive Mappings ◽  
The Family ◽  
Rational Type

The purpose of this paper is to find out fixed point results for the family of multivalued mappings fulfilling a generalized rational type F-contractive conditions on a closed ball in complete dislocated b-metric space. An application to the system of integral equations is presented to show the novelty of our results. Our results extend several comparable results in the existing literature.

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