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 Equivalence of two BV classes of functions in metric spaces, and existence of a Semmes family of curves under a 1-Poincaré inequality

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Estibalitz Durand-Cartagena ◽  
Sylvester Eriksson-Bique ◽  
Riikka Korte ◽  
Nageswari Shanmugalingam
Keyword(s):  
Bounded Variation ◽  
Metric Space ◽  
Metric Spaces ◽  
Poincaré Inequality ◽  
The Other ◽  
Functions Of Bounded Variation ◽  
Metric Measure Spaces ◽  
Poincare Inequality ◽  
Family Of Curves ◽  
Measure Spaces

Abstract We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda Jr. We show that these two notions coincide if the measure is doubling and supports a 1-Poincaré inequality. In doing so, we also prove that if the measure is doubling and supports a 1-Poincaré inequality, then the metric space supports a Semmes family of curves structure.

The variational 1-capacity and BV functions with zero boundary values on doubling metric spaces

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Panu Lahti
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Poincaré Inequality ◽  
Doubling Measure ◽  
Boundary Values ◽  
Poincare Inequality ◽  
Compactly Supported ◽  
Bv Functions

AbstractIn the setting of a metric space that is equipped with a doubling measure and supports a Poincaré inequality, we define and study a class of{\mathrm{BV}}functions with zero boundary values. In particular, we show that the class is the closure of compactly supported{\mathrm{BV}}functions in the{\mathrm{BV}}norm. Utilizing this theory, we then study the variational 1-capacity and its Lipschitz and{\mathrm{BV}}analogs. We show that each of these is an outer capacity, and that the different capacities are equal for certain sets.

scholarly journals Semiregular and strongly irregular boundary points for p-harmonic functions on unbounded sets in metric spaces

2021 ◽  
Author(s):  
Anders Björn ◽  
Daniel Hansevi
Keyword(s):  
Harmonic Functions ◽  
Metric Spaces ◽  
Poincaré Inequality ◽  
Doubling Measure ◽  
Poincare Inequality ◽  
Irregular Boundary ◽  
Complete Metric Spaces ◽  
Boundary Points ◽  
Local Properties ◽  
Harmonic Measures

AbstractThe trichotomy between regular, semiregular, and strongly irregular boundary points for $$p$$ p -harmonic functions is obtained for unbounded open sets in complete metric spaces with a doubling measure supporting a $$p$$ p -Poincaré inequality, $$1<p<\infty $$ 1 < p < ∞ . We show that these are local properties. We also deduce several characterizations of semiregular points and strongly irregular points. In particular, semiregular points are characterized by means of capacity, $$p$$ p -harmonic measures, removability, and semibarriers.

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.

scholarly journals Suzuki ψF-contractions and some fixed point results

2018 ◽  
Vol 34 (1) ◽  
pp. 93-102
Author(s):  
NICOLAE-ADRIAN SECELEAN ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Continuity Point ◽  
Picard Operator ◽  
Nonlinear Anal ◽  
New Type

The purpose of this paper is to combine and extend some recent fixed point results of Suzuki, T., [A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313–5317] and Secelean, N. A. & Wardowski, D., [ψF-contractions: not necessarily nonexpansive Picard operators, Results Math., 70 (2016), 415–431]. The continuity and the completeness conditions are replaced by orbitally continuity and orbitally completeness respectively. It is given an illustrative example of a Picard operator on a non complete metric space which is neither nonexpansive nor expansive and has a unique continuity point.

Convergence of the Ishikawa Iterates for Multi-Valued Mappings in Convex Metric Spaces

2008 ◽  
Vol 15 (1) ◽  
pp. 39-43
Author(s):  
Ljubomir B. Ćirić ◽  
Nebojša T. Nikolić
Keyword(s):  
Metric Space ◽  
Convex Subset ◽  
Metric Spaces ◽  
Satisfy Condition ◽  
Convex Metric Space ◽  
Convex Metric Spaces ◽  
The Family

Abstract Let (𝑋, 𝑑) be a convex metric space, 𝐶 be a closed and convex subset of 𝑋 and let 𝐵(𝐶) be the family of all nonempty bounded subsets of 𝐶. In this paper some results are obtained on the convergence of the Ishikawa iterates associated with a pair of multi-valued mappings 𝑆,𝑇 : 𝐶 → 𝐵(𝐶) which satisfy condition (2.1) below.

On a theorem of Brian Fisher in the framework of w-distance

2017 ◽  
Vol 33 (2) ◽  
pp. 199-205
Author(s):  
DARKO KOCEV ◽  
◽  
VLADIMIR RAKOCEVIC ◽  
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Complete Metric Space ◽  
Point Theory ◽  
Common Fixed Points ◽  
Complete Metric Spaces ◽  
Condition C ◽  
Relaxing Condition

In 1980. Fisher in [Fisher, B., Results on common fixed points on complete metric spaces, Glasgow Math. J., 21 (1980), 165–167] proved very interesting fixed point result for the pair of maps. In 1996. Kada, Suzuki and Takahashi introduced and studied the concept of w–distance in fixed point theory. In this paper, we generalize Fisher’s result for pair of mappings on metric space to complete metric space with w–distance. The obtained results do not require the continuity of maps, but more relaxing condition (C; k). As a corollary we obtain a result of Chatterjea.

scholarly journals Minimal arcs in metric spaces

1975 ◽  
Vol 19 (4) ◽  
pp. 426-430 ◽  
Author(s):  
S. K. Hildebrand ◽  
Harold Willis Milnes
Keyword(s):  
Fixed Points ◽  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Minimal Length

In this paper we discuss the existence of an are of minimal length joining two arbitrary, yet fixed points in a complete metric space, where the metric is restricted only by the properties (A) and (B) given below. It is shown that under these conditions an arc of least length joining any two fixed points exists, and is unique. In addition, its length is shown to be equal to the metrie distance between the points.

scholarly journals A Universal Separable Diversity

2017 ◽  
Vol 5 (1) ◽  
pp. 138-151 ◽  
Author(s):  
David Bryant ◽  
André Nies ◽  
Paul Tupper
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Urysohn Space ◽  
Fundamental Properties ◽  
Finite Set ◽  
Whole Space ◽  
Set Of Points ◽  
Separable Metric Space

AbstractThe Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points.We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.

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