scholarly journals On \(\ell_1\)-preduals distant by 1

2018 ◽  
Vol 72 (2) ◽  
pp. 41
Author(s):  
Łukasz Piasecki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Wide Class ◽  
Fixed Point Theory ◽  
General Setting ◽  
Point Theory ◽  
Fixed Point Property ◽  
Sufficient Condition ◽  
Point Property ◽  
Weak Fixed Point Property

For every predual \(X\) of \(\ell_1\) such that the standard basis in \(\ell_1\) is weak\(^*\) convergent, we give explicit models of all Banach spaces \(Y\) for which the Banach-Mazur distance \(d(X,Y)=1\). As a by-product of our considerations, we obtain some new results in metric fixed point theory. First, we show that the space \(\ell_1\), with a predual \(X\) as above, has the stable weak\(^*\) fixed point property if and only if it has almost stable weak\(^*\) fixed point property, i.e. the dual \(Y^*\) of every Banach space \(Y\) has the weak\(^*\) fixed point property (briefly, \(\sigma(Y^*,Y)\)-FPP) whenever \(d(X,Y)=1\). Then, we construct a predual \(X\) of \(\ell_1\) for which \(\ell_1\) lacks the stable \(\sigma(\ell_1,X)\)-FPP but it has almost stable \(\sigma(\ell_1,X)\)-FPP, which in turn is a strictly stronger property than the \(\sigma(\ell_1,X)\)-FPP. Finally, in the general setting of preduals of \(\ell_1\), we give a sufficient condition for almost stable weak\(^*\) fixed point property in \(\ell_1\) and we prove that for a wide class of spaces this condition is also necessary.

scholarly journals Fixed Point Theory for Digital k-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1244 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Fixed Point ◽  
Euler Characteristic ◽  
Fixed Point Theory ◽  
Geometric Realization ◽  
Point Theory ◽  
Fixed Point Property ◽  
Point Property ◽  
Euler Characteristics ◽  
Surface Theory ◽  
Closed Surfaces

The present paper studies the fixed point property (FPP) for closed k-surfaces. We also intensively study Euler characteristics of a closed k-surface and a connected sum of closed k-surfaces. Furthermore, we explore some relationships between the FPP and Euler characteristics of closed k-surfaces. After explaining how to define the Euler characteristic of a closed k-surface more precisely, we confirm a certain consistency of the Euler characteristic of a closed k-surface and a continuous analog of it. In proceeding with this work, for a simple closed k-surface in Z 3 , say S k , we can see that both the minimal 26-adjacency neighborhood of a point x ∈ S k , denoted by M k ( x ) , and the geometric realization of it in R 3 , denoted by D k ( x ) , play important roles in both digital surface theory and fixed point theory. Moreover, we prove that the simple closed 18-surfaces M S S 18 and M S S 18 ′ do not have the almost fixed point property (AFPP). Consequently, we conclude that the triviality or the non-triviality of the Euler characteristics of simple closed k-surfaces have no relationships with the FPP in digital topology. Using this fact, we correct many errors in many papers written by L. Boxer et al.

scholarly journals On the modulus ofu-convexity of Ji Gao

2003 ◽  
Vol 2003 (1) ◽  
pp. 49-54 ◽  
Author(s):  
Eva María Mazcuñán-Navarro
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Normal Structure ◽  
Fixed Point Property ◽  
Sufficient Condition ◽  
Point Property

We consider the modulus ofu-convexity of a Banach space introduced by Ji Gao (1996) and we improve a sufficient condition for the fixed-point property (FPP) given by this author. We also give a sufficient condition for normal structure in terms of the modulus ofu-convexity.

scholarly journals A Counterexample in Finite Fixed Point Theory

1979 ◽  
Vol 22 (1) ◽  
pp. 99-100
Author(s):  
H. C. Enos
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fixed Point Property ◽  
Point Property ◽  
Closed Sets

This note answers a question raised by Lee Mohler in 1970, by exhibiting a finite topological space X which is the union of closed subspaces Y, Z, such that Y, Z, and Y ⋂ Z, but not X, have the fixed point property. The example is a triangulation △ of S3, the points of X being the simplices of Δ and the closed sets the subcomplexes of △.

scholarly journals A class of spaces with weak normal structure

1994 ◽  
Vol 49 (3) ◽  
pp. 523-528 ◽  
Author(s):  
Brailey Sims
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Normal Structure ◽  
Fixed Point Property ◽  
Point Property ◽  
Weak Fixed Point Property

It has recently been shown that a Banach space enjoys the weak fixed point property if it is ε0-inquadrate for some ε0 < 2 and has WORTH; that is, if then, ║xn — x║ — ║xn + x║ → 0, for all x. We establish the stronger conclusion of weak normal structure under the substantially weaker assumption that the space has WORTH and is ‘ε0-inquadrate in every direction’ for some ε0 < 2.

scholarly journals On Almost-Fixed-Point Theory

1978 ◽  
Vol 30 (4) ◽  
pp. 673-699 ◽  
Author(s):  
Michiel Hazewinkel ◽  
Marcel Van De Vel
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fixed Point Property ◽  
Point Property ◽  
Finite Covering ◽  
Continuous Maps

Let X be a topological space, a finite covering of X (the words ‘covering’ and ‘cover’ are used interchangeably). We say that has the almost fixed point property for a class of continuous maps f : X → X if for all there is an x ∈ X and such that x ∈ U and f(x) ∈ U, or, equivalently, if there is a such that .

Fixed Point in Minimal Spaces

2005 ◽  
Vol 10 (4) ◽  
pp. 305-314 ◽  
Author(s):  
M. Alimohammady ◽  
M. Roohi
Keyword(s):  
Fixed Point ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Open Cover ◽  
Fixed Point Property ◽  
Point Property

This paper deals with fixed point theory and fixed point property in minimal spaces. We will prove that under some conditions f : (X,M) → (X,M) has a fixed point if and only if for each m-open cover {Bα} for X there is at least one x ∈ X such that both x and f(x) belong to a common Bα. Further, it is shown that if (X,M) has the fixed point property, then its minimal retract subset enjoys this property.

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 A note on properties that imply the fixed point property

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
S. Dhompongsa ◽  
A. Kaewkhao
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Property ◽  
Point Property ◽  
Geometric Properties ◽  
Weak Fixed Point Property

We give relationships between some Banach-space geometric properties that guarantee the weak fixed point property. The results extend some known results of Dalby and Xu.

scholarly journals The generalized Day norm. Part II. Applications

2017 ◽  
Vol 71 (2) ◽  
pp. 51
Author(s):  
Monika Budzyńska ◽  
Aleksandra Grzesik ◽  
Mariola Kot
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Empty Interior ◽  
Weak Fixed Point Property ◽  
Complete Set ◽  
Alpha Beta

In this paper we prove that for each \(1&lt; p, \tilde{p} &lt; \infty\), the Banach space \((l^{\tilde{p}}, \left\|\cdot\right\|_{\tilde{p}})\) can be equivalently renormed in such a way that  the Banach space \((l^{\tilde{p}},\left\|\cdot\right\|_{L,\alpha,\beta,p,\tilde{p}})\) is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in \(l^2\) with the Day norm. We also show that the Banach space \((l^{\tilde{p}},\left\|\cdot\right\|_{L,\alpha,\beta,p,\tilde{p}})\) has the weak fixed point property for nonexpansive mappings.

scholarly journals The effect of the dual on a Banach space and the weak fixed point propery

2003 ◽  
Vol 67 (2) ◽  
pp. 177-185 ◽  
Author(s):  
Tim Dalby
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Property ◽  
Point Property ◽  
Original Space ◽  
Weak Fixed Point Property

A number of Banach space properties have been shown to imply the weak fixed point property. If the dual of a Banach space were to possess some of these properties then the original space can been shown to satisfy related conditions.

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