scholarly journals The fixed point property of the infinite K-sphere in the set Con*((Z2)*)

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4027-4042
Author(s):  
Sang-Eon Hana
Keyword(s):  
Fixed Point ◽  
Fixed Point Property ◽  
Point Property ◽  
Topological Plane ◽  
The One ◽  
One Point Compactification

In this paper the Alexandroff one point compactification of the 2-dimensional Khalimsky (K-, for brevity) plane (resp. the 1-dimensional Khalimsky line) is called the infinite K-sphere (resp. the infinite K-circle). The present paper initially proves that the infinite K-circle has the fixed point property (FPP, for short) in the set Con(Z*), where Con(Z*) means the set of all continuous self-maps f of the infinite K-circle. Next, we address the following query which remains open: Under what condition does the infinite K-sphere have the FPP? Regarding this issue, we prove that the infinite K-sphere has the FPP in the set Con*((Z2)*) (see Definition 1.1). Finally, we compare the FPP of the infinite K-sphere and that of the infinite M-sphere, where the infinite M-sphere means the one point compactification of the Marcus-Wyse topological plane.

scholarly journals The Fixed Point Property of the Infinite M-Sphere

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 599
Author(s):  
Sang-Eon Han ◽  
Selma Özçağ
Keyword(s):  
Applied Sciences ◽  
Fixed Point ◽  
Topological Space ◽  
Unsolved Problem ◽  
Fixed Point Property ◽  
Digital Geometry ◽  
Point Property ◽  
Open Set ◽  
One Point Compactification ◽  
Digital Or

The present paper is concerned with the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological space ( Z 2 , γ ) . This compactification is called the infinite M-topological sphere and denoted by ( ( Z 2 ) ∗ , γ ∗ ) , where ( Z 2 ) ∗ : = Z 2 ∪ { ∗ } , ∗ ∉ Z 2 and γ ∗ is the topology for ( Z 2 ) ∗ induced by the topology γ on Z 2 . With the topological space ( ( Z 2 ) ∗ , γ ∗ ) , since any open set containing the point “ ∗ ” has the cardinality ℵ 0 , we call ( ( Z 2 ) ∗ , γ ∗ ) the infinite M-topological sphere. Indeed, in the fields of digital or computational topology or applied analysis, there is an unsolved problem as follows: Under what category does ( ( Z 2 ) ∗ , γ ∗ ) have the fixed point property (FPP, for short)? The present paper proves that ( ( Z 2 ) ∗ , γ ∗ ) has the FPP in the category M o p ( γ ∗ ) whose object is the only ( ( Z 2 ) ∗ , γ ∗ ) and morphisms are all continuous self-maps g of ( ( Z 2 ) ∗ , γ ∗ ) such that | g ( ( Z 2 ) ∗ ) | = ℵ 0 with ∗ ∈ g ( ( Z 2 ) ∗ ) or g ( ( Z 2 ) ∗ ) is a singleton. Since ( ( Z 2 ) ∗ , γ ∗ ) can be a model for a digital sphere derived from the M-topological space ( Z 2 , γ ) , it can play a crucial role in topology, digital geometry and applied sciences.

scholarly journals The Fixed Point Property of Non-Retractable Topological Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 879 ◽  
Author(s):  
Jeong Kang ◽  
Sang-Eon Han ◽  
Sik Lee
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fixed Point Property ◽  
Topological Spaces ◽  
Point Property ◽  
Finite Planes ◽  
Topological Plane ◽  
Theoretic Foundations

Unlike the study of the fixed point property (FPP, for brevity) of retractable topological spaces, the research of the FPP of non-retractable topological spaces remains. The present paper deals with the issue. Based on order-theoretic foundations and fixed point theory for Khalimsky (K-, for short) topological spaces, the present paper studies the product property of the FPP for K-topological spaces. Furthermore, the paper investigates the FPP of various types of connected K-topological spaces such as non-K-retractable spaces and some points deleted K-topological (finite) planes, and so on. To be specific, after proving that not every one point deleted subspace of a finite K-topological plane X is a K-retract of X, we study the FPP of a non-retractable topological space Y, such as one point deleted space Y ∖ { p } .

scholarly journals New examples of subsets of c with the FPP and stability of the FPP in hyperconvex spaces

2021 ◽  
Vol 23 (3) ◽  
Author(s):  
Rafael Espínola-García ◽  
María Japón ◽  
Daniel Souza
Keyword(s):  
Fixed Point ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Normal Structure ◽  
Fixed Point Property ◽  
Point Property ◽  
Compact Sets ◽  
Weakly Compact ◽  
Weakly Compact Sets ◽  
The One

AbstractThe purpose of this work is two-fold. On the one side, we focus on the space of real convergent sequences c where we study non-weakly compact sets with the fixed point property. Our approach brings a positive answer to a recent question raised by Gallagher et al. in (J Math Anal Appl 431(1):471–481, 2015). On the other side, we introduce a new metric structure closely related to the notion of relative uniform normal structure, for which we show that it implies the fixed point property under adequate conditions. This will provide some stability fixed point results in the context of hyperconvex metric spaces. As a particular case, we will prove that the set $$M=[-1,1]^\mathbb {N}$$ M = [ - 1 , 1 ] N has the fixed point property for d-nonexpansive mappings where $$d(\cdot ,\cdot )$$ d ( · , · ) is a metric verifying certain restrictions. Applications to some Nakano-type norms are also given.

scholarly journals Brush spaces and the fixed point property

2011 ◽  
Vol 158 (8) ◽  
pp. 1085-1089 ◽  
Author(s):  
M.M. Marsh ◽  
J.R. Prajs
Keyword(s):  
Fixed Point ◽  
Fixed Point Property ◽  
Point Property

scholarly journals Towards the fixed point property for superreflexive spaces

2001 ◽  
Vol 64 (3) ◽  
pp. 435-444 ◽  
Author(s):  
Andrzej Wiśnicki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Unit Sphere ◽  
Convex Set ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Geometrical Condition

A Banach space X is said to have property (Sm) if every metrically convex set A ⊂ X which lies on the unit sphere and has diameter not greater than one can be (weakly) separated from zero by a functional. We show that this geometrical condition is closely connected with the fixed point property for nonexpansive mappings in superreflexive spaces.

scholarly journals A product space with the fixed point property

2012 ◽  
Vol 2012 (1) ◽  
Author(s):  
Helga Fetter Nathansky ◽  
Enrique Llorens-Fuster
Keyword(s):  
Fixed Point ◽  
Product Space ◽  
Fixed Point Property ◽  
Point Property
Sign in / Sign up

Export Citation Format

Share Document