scholarly journals Dense and σ-Porous Subsets in Some Families of Darboux Functions

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 759
Author(s):  
Gertruda Ivanova ◽  
Irena Domnik
Keyword(s):  
Metric Space ◽  
Continuous Functions ◽  
Darboux Functions ◽  
The Family ◽  
Nowhere Dense Set ◽  
Supremum Metric ◽  
Dense Set

G. Ivanova and E. Wagner-Bojakowska shown that the set of Darboux quasi-continuous functions with nowhere dense set of discontinuity points is dense in the metric space of Darboux quasi-continuous functions with the supremum metric. We prove that this set also is σ-strongly porous in such space. We obtain the symmetrical result for the family of strong Świątkowski functions, i.e., that the family of strong Świątkowski functions with nowhere dense set of discontinuity points is dense (thus, “large”) and σ-strongly porous (thus, asymmetrically, “small”) in the family of strong Świątkowski functions.

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.

Computably Isometric Spaces

2013 ◽  
Vol 78 (4) ◽  
pp. 1055-1085 ◽  
Author(s):  
Alexander G. Melnikov
Keyword(s):  
Hilbert Space ◽  
Metric Space ◽  
Separable Hilbert Space ◽  
Continuous Functions ◽  
Unit Interval ◽  
Computable Analysis ◽  
Sufficient Condition ◽  
Cantor Space ◽  
Urysohn Space ◽  
Supremum Metric

AbstractWe say that an uncountable metric space is computably categorical if every two computable structures on this space are equivalent up to a computable isometry. We show that Cantor space, the Urysohn space, and every separable Hilbert space are computably categorical, but the space [0, 1] of continuous functions on the unit interval with the supremum metric is not. We also characterize computably categorical subspaces of ℝn, and give a sufficient condition for a space to be computably categorical. Our interest is motivated by classical and recent results in computable (countable) model theory and computable analysis.

scholarly journals Comparison of Some Subfamilies of Functions Having The Baire Property

2016 ◽  
Vol 65 (1) ◽  
pp. 151-159
Author(s):  
Gertruda Ivanova ◽  
Aleksandra Karasińska ◽  
Elżbieta Wagner-Bojakowska
Keyword(s):  
Continuous Functions ◽  
Baire Property ◽  
Porous Set ◽  
Darboux Property ◽  
Darboux Functions ◽  
The Family

Abstract We prove that the family Q of quasi-continuous functions is a strongly porous set in the space Ba of functions having the Baire property. Moreover, the family DQ of all Darboux quasi-continuous functions is a strongly porous set in the space DBa of Darboux functions having the Baire property. It implies that each family of all functions having the A-Darboux property is strongly porous in DBa if A has the (∗)-property.

scholarly journals Ideal and some applications of simply open sets

2017 ◽  
Vol 13 (3) ◽  
pp. 7264-7271
Author(s):  
Arafa A Nasefa ◽  
R Mareay
Keyword(s):  
Closed Subset ◽  
Continuous Functions ◽  
Logical Space ◽  
Fundamental Properties ◽  
New Type ◽  
Nowhere Dense Set ◽  
Dense Set ◽  
The Ideal

Recently there has been some interest in the notion of a locally closed subset of a topo- logical space. In this paper, we introduce a useful characterizations of simply open sets in terms of the ideal of nowhere dense set. Also, we study a new notion of functions in topo- logical spaces known as dual simply-continuous functions and some of their fundamental properties are investigated. Finally, a new type of simply open sets is introduced.

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 Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

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 Semi-smooth Points in Some Classical Function Spaces

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Beata Derȩgowska ◽  
Beata Gryszka ◽  
Karol Gryszka ◽  
Paweł Wójcik
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Metric Space ◽  
Spaces Of Continuous Functions ◽  
Classical Function ◽  
Necessary And Sufficient ◽  
Vector Valued

AbstractThe investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

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.

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