scholarly journals A note on compact sets in spaces of subsets

1988 ◽  
Vol 38 (3) ◽  
pp. 393-395 ◽  
Author(s):  
Phil Diamond ◽  
Peter Kloeden
Keyword(s):  
Metric Space ◽  
Complete Metric Space ◽  
Hausdorff Metric ◽  
Compact Sets ◽  
Selection Theorem ◽  
Starshaped Sets ◽  
Nonempty Compact ◽  
Compact Subsets

A simple characterisation is given of compact sets of the space K(X), of nonempty compact subsets of a complete metric space X, with the Hausdorff metric dH. It is used to give a new proof of the Blaschke selection theorem for compact starshaped sets.

scholarly journals Li-Yorke Sensitivity of Set-Valued Discrete Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Heng Liu ◽  
Fengchun Lei ◽  
Lidong Wang
Keyword(s):  
Metric Space ◽  
Short Proof ◽  
Hausdorff Metric ◽  
Discrete Systems ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Nonempty Compact ◽  
Compact Subsets

Consider the surjective, continuous mapf:X→Xand the continuous mapf¯of𝒦(X)induced byf, whereXis a compact metric space and𝒦(X)is the space of all nonempty compact subsets ofXendowed with the Hausdorff metric. In this paper, we give a short proof that iff¯is Li-Yoke sensitive, thenfis Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity offdoes not imply Li-Yorke sensitivity off¯.

Čebyšev Sets in Hyperspaces over Rn

2009 ◽  
Vol 61 (2) ◽  
pp. 299-314 ◽  
Author(s):  
Robert J. MacG. Dawson and ◽  
Maria Moszyńska
Keyword(s):  
Metric Space ◽  
Convex Sets ◽  
Compact Convex ◽  
Hausdorff Metric ◽  
Linear Structure ◽  
Convex Compact ◽  
Nearest Neighbour ◽  
Compact Sets ◽  
Strictly Convex ◽  
Convex Compact Sets

Abstract. A set in a metric space is called a Čebyšev set if it has a unique “nearest neighbour” to each point of the space. In this paper we generalize this notion, defining a set to be Čebyšev relative to another set if every point in the second set has a unique “nearest neighbour” in the first. We are interested in Čebyšev sets in some hyperspaces over Rn, endowed with the Hausdorff metric, mainly the hyperspaces of compact sets, compact convex sets, and strictly convex compact sets. We present some new classes of Čebyšev and relatively Čebyšev sets in various hyperspaces. In particular, we show that certain nested families of sets are Čebyšev. As these families are characterized purely in terms of containment,without reference to the semi-linear structure of the underlyingmetric space, their properties differ markedly from those of known Čebyšev sets.

EQUI-ATTRACTION AND THE CONTINUOUS DEPENDENCE OF PULLBACK ATTRACTORS ON PARAMETERS

2004 ◽  
Vol 04 (03) ◽  
pp. 373-384 ◽  
Author(s):  
DESHENG LI ◽  
P. E. KLOEDEN
Keyword(s):  
Metric Space ◽  
Continuous Dependence ◽  
Hausdorff Metric ◽  
Regularity Conditions ◽  
Pullback Attractors ◽  
Compact Metric Space ◽  
Nonautonomous Dynamical Systems ◽  
The Family ◽  
Nonempty Compact ◽  
Autonomous Dynamical System

The equi-attraction properties of uniform pullback attractors [Formula: see text] of nonautonomous dynamical systems (θ,ϕλ) with a parameter λ∈Λ, where Λ is a compact metric space, are investigated; here θ is an autonomous dynamical system on a compact metric space P which drives the cocycle ϕλon a complete metric state space X. In particular, under appropriate regularity conditions, it is shown that the equi-attraction of the family [Formula: see text] uniformly in p∈P is equivalent to the continuity of the setvalued mappings [Formula: see text] in λ with respect to the Hausdorff metric on the nonempty compact subsets of X.

scholarly journals Topologies on Superspaces of TVS-Cone Metric Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Xun Ge ◽  
Shou Lin
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Cone Metric Space ◽  
Vietoris Topology ◽  
Cone Metric Spaces ◽  
Nonempty Compact ◽  
Cone Metric ◽  
Compact Subsets

This paper investigates superspaces𝒫0(X)and𝒦0(X)of a tvs-cone metric space(X,d), where𝒫0(X)and𝒦0(X)are the space consisting of nonempty subsets ofXand the space consisting of nonempty compact subsets ofX, respectively. The purpose of this paper is to establish some relationships between the lower topology and the lower tvs-cone hemimetric topology (resp., the upper topology and the upper tvs-cone hemimetric topology to the Vietoris topology and the Hausdorff tvs-cone hemimetric topology) on𝒫0(X)and𝒦0(X), which makes it possible to generalize some results of superspaces from metric spaces to tvs-cone metric spaces.

scholarly journals Compact reduction in Lipschitz-free spaces

2021 ◽  
Vol 151 (6) ◽  
pp. 1683-1699
Author(s):  
Ramón J. Aliaga ◽  
Camille Noûs ◽  
Colin Petitjean ◽  
Antonín Procházka
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
General Principle ◽  
Metric Spaces ◽  
Approximation Property ◽  
Complete Metric Space ◽  
Infinite Dimensional ◽  
Schur Property ◽  
Free Spaces ◽  
Compact Subsets

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

scholarly journals Some Chaotic Properties of Discrete Fuzzy Dynamical Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Yaoyao Lan ◽  
Qingguo Li ◽  
Chunlai Mu ◽  
Hua Huang
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Continuous Map ◽  
Weakly Mixing ◽  
Dynamical Properties ◽  
Fuzzy Dynamical Systems ◽  
Compact Subsets

Letting(X,d)be a metric space,f:X→Xa continuous map, and(ℱ(X),D)the space of nonempty fuzzy compact subsets ofXwith the Hausdorff metric, one may study the dynamical properties of the Zadeh's extensionf̂:ℱ(X)→ℱ(X):u↦f̂u. In this paper, we present, as a response to the question proposed by Román-Flores and Chalco-Cano 2008, some chaotic relations betweenfandf̂. More specifically, we study the transitivity, weakly mixing, periodic density in system(X,f), and its connections with the same ones in its fuzzified system.

On multi-sensitivity with respect to a vector

2018 ◽  
Vol 32 (15) ◽  
pp. 1850166 ◽  
Author(s):  
Lixin Jiao ◽  
Lidong Wang ◽  
Fengquan Li ◽  
Heng Liu
Keyword(s):  
Dynamical System ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Sufficient Conditions ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Continuous Map ◽  
Periodic Sets ◽  
Vector Formula ◽  
Compact Subsets

Consider the surjective continuous map [Formula: see text]: [Formula: see text] defined on a compact metric space X. Let [Formula: see text] be the space of all non-empty compact subsets of X equipped with the Hausdorff metric and define [Formula: see text]: [Formula: see text] by [Formula: see text] for any [Formula: see text]. In this paper, we introduce several stronger versions of sensitivities, such as multi-sensitivity with respect to a vector, [Formula: see text]-sensitivity, strong multi-sensitivity. We obtain some basic properties of the concepts of these sensitivities and discuss the relationships with other sensitivities for continuous self-map on [0,[Formula: see text]1]. Some sufficient conditions for a dynamical system to be [Formula: see text]-sensitive are presented. Also, it is shown that the strong multi-sensitivity of f implies that [Formula: see text] is [Formula: see text]-sensitive. In turn, the [Formula: see text]-sensitivity of [Formula: see text] implies that [Formula: see text] is [Formula: see text]-sensitive. In particular, it is proved that if [Formula: see text] is a multi-transitive map with dense periodic sets, then f is [Formula: see text]-sensitive. Finally, we give a multi-sensitive example which is not [Formula: see text]-sensitive.

scholarly journals Generalized fuzzy GV-Hausdorff distance in GFGV-fractal spaces with application in integral equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Alireza Alihajimohammad ◽  
Reza Saadati
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Space ◽  
Hausdorff Distance ◽  
Fuzzy Metric Space ◽  
Fuzzy Metric ◽  
Nonempty Compact ◽  
Compact Subsets

AbstractWe propose a method for constructing a generalized fuzzy Hausdorff distance on the set of the nonempty compact subsets of a given generalized fuzzy metric space in the sence of George–Veeramni and Tian–Ha–Tian. Next, we define the generalized fuzzy fractal spaces. Morever, we obtain a fixed point theorem of a class of generalized fuzzy contractions and present an application in integranl equation.

A new approach to fixed point theorems for multivalued contractive maps

2015 ◽  
Vol 31 (2) ◽  
pp. 241-248
Author(s):  
GULHAN MINAK ◽  
◽  
MURAT OLGUN ◽  
ISHAK ALTUN ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Complete Metric Space ◽  
Hausdorff Metric ◽  
Multivalued Maps ◽  
New Approach ◽  
Contractive Maps

In the present paper, considering the Wardowski’s technique we give many fixed point results for multivalued maps on complete metric space without using the Hausdorff metric. Our results are real generalization of some related fixed point theorems including the famous Feng and Liu’s result in the literature. We also give some examples to both illustrate and show that our results are proper generalizations of the mentioned theorems.

scholarly journals Fixed Points for Multivalued Convex Contractions on Nadler Sense Types in a Geodesic Metric Space

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 155 ◽  
Author(s):  
Amelia Bucur
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Metric Space ◽  
Complete Metric Space ◽  
Hausdorff Metric ◽  
Geodesic Metric Space ◽  
Geodesic Metric ◽  
Convex Contractions ◽  
The Right ◽  
Multivalued Contractions

In 1969, based on the concept of the Hausdorff metric, Nadler Jr. introduced the notion of multivalued contractions. He demonstrated that, in a complete metric space, a multivalued contraction possesses a fixed point. Later on, Nadler’s fixed point theorem was generalized by many authors in different ways. Using a method given by Angrisani, Clavelli in 1996 and Mureşan in 2002, we prove in this paper that, for a class of convex multivalued left A-contractions in the sense of Nadler and the right A-contractions with a convex metric, the fixed points set is non-empty and compact. In this paper we present the fixed point theorems for convex multivalued left A-contractions in the sense of Nadler and right A-contractions on the geodesic metric space. Our results are particular cases of some general theorems, to the multivalued left A-contractions in the sense of Nadler and right A-contractions, and particular cases of the results given by Rus (1979, 2008), Nadler (1969), Mureşan (2002, 2004), Bucur, Guran and Petruşel (2009), Petre and Bota (2013), etc., and are applicable in many fields, such as economy, management, society, biology, ecology, etc.

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