scholarly journals Characterizing the metric compactification of $$L_{p}$$ spaces by random measures

2020 ◽  
Vol 11 (2) ◽  
pp. 227-243
Author(s):  
Armando W. Gutiérrez
Keyword(s):  
Fixed Points ◽  
Ergodic Theorem ◽  
Convex Subset ◽  
Polish Space ◽  
Compact Convex ◽  
Random Measure ◽  
Complete Characterization ◽  
Random Measures ◽  
Weakly Compact

AbstractWe present a complete characterization of the metric compactification of $$L_{p}$$Lp spaces for $$1\le p < \infty $$1≤p<∞. Each element of the metric compactification of $$L_{p}$$Lp is represented by a random measure on a certain Polish space. By way of illustration, we revisit the $$L_{p}$$Lp-mean ergodic theorem for $$1< p < \infty $$1<p<∞, and Alspach’s example of an isometry on a weakly compact convex subset of $$L_{1}$$L1 with no fixed points.

scholarly journals Some remarks on the location of fixed points

1984 ◽  
Vol 37 (3) ◽  
pp. 358-365
Author(s):  
Michael Edelstein ◽  
Daryl Tingley
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Convex Subset ◽  
Compact Convex ◽  
Compact Convex Subset ◽  
Chebyshev Center ◽  
Weakly Compact ◽  
Asymptotic Center

AbstractSeveral procedures for locating fixed points of nonexpansive selfmaps of a weakly compact convex subset of a Banach space are presented. Some of the results involve the notion of an asymptotic center or a Chebyshev center.

A Uniformly Asymptotically Regular Mapping Without Fixed Points

1987 ◽  
Vol 30 (4) ◽  
pp. 481-483 ◽  
Author(s):  
Pei-kee Lin
Keyword(s):  
Fixed Points ◽  
Convex Subset ◽  
Compact Convex ◽  
Compact Convex Subset ◽  
Lipschitzian Mapping ◽  
Weakly Compact ◽  
Regular Mapping ◽  
Asymptotically Regular

AbstractWe construct a uniformly asymptotically regular, Lipschitzian mapping acting on a weakly compact convex subset of l2 which has no fixed points.

A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi

1976 ◽  
Vol 19 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Joseph Bogin
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Compact Convex ◽  
Continuous Mapping ◽  
Normal Structure ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Image Position

In [7], Goebel, Kirk and Shimi proved the following:Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:(1)where ai≥0 and Then F has a fixed point in K.In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).

scholarly journals Decompositions of Weakly Compact Valued Integrable Multifunctions

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 863 ◽  
Author(s):  
Luisa Di Piazza ◽  
Kazimierz Musiał
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Compact Convex ◽  
Weakly Compact ◽  
Seminal Paper ◽  
Decomposition Property ◽  
Short Overview ◽  
Decomposition Theorems ◽  
State Of Art

We give a short overview on the decomposition property for integrable multifunctions, i.e., when an “integrable in a certain sense” multifunction can be represented as a sum of one of its integrable selections and a multifunction integrable in a narrower sense. The decomposition theorems are important tools of the theory of multivalued integration since they allow us to see an integrable multifunction as a translation of a multifunction with better properties. Consequently, they provide better characterization of integrable multifunctions under consideration. There is a large literature on it starting from the seminal paper of the authors in 2006, where the property was proved for Henstock integrable multifunctions taking compact convex values in a separable Banach space X. In this paper, we summarize the earlier results, we prove further results and present tables which show the state of art in this topic.

scholarly journals Fixed points of isometries on weakly compact convex sets

2003 ◽  
Vol 282 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Teck-Cheong Lim ◽  
Pei-Kee Lin ◽  
C. Petalas ◽  
T. Vidalis
Keyword(s):  
Fixed Points ◽  
Convex Sets ◽  
Compact Convex ◽  
Weakly Compact

scholarly journals Remarks on Asymptotic Centers and Fixed Points

2010 ◽  
Vol 2010 ◽  
pp. 1-5
Author(s):  
A. Kaewkhao ◽  
K. Sokhuma
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
Compact Convex ◽  
Bounded Sequence ◽  
Point Property ◽  
Weakly Compact ◽  
Asymptotic Center ◽  
Continuous Mappings ◽  
Weak Fixed Point Property ◽  
Asymptotic Centers

We introduce a class of nonlinear continuous mappings defined on a bounded closed convex subset of a Banach spaceX. We characterize the Banach spaces in which every asymptotic center of each bounded sequence in any weakly compact convex subset is compact as those spaces having the weak fixed point property for this type of mappings.

On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate

2013 ◽  
Vol 56 (2) ◽  
pp. 272-282 ◽  
Author(s):  
Lixin Cheng ◽  
Zhenghua Luo ◽  
Yu Zhou
Keyword(s):  
Banach Space ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Lower Semicontinuous ◽  
Weakly Compact ◽  
Bounded Set ◽  
Fréchet Differentiable ◽  
Bounded Convex

AbstractIn this note, we first give a characterization of super weakly compact convex sets of a Banach space X: a closed bounded convex set K ⊂ X is super weakly compact if and only if there exists a w* lower semicontinuous seminorm p with p ≥ σK ≌ supxєK 〈.,x〉 such that p2 is uniformly Fréchet differentiable on each bounded set of X*. Then we present a representation theoremfor the dual of the semigroup swcc(X) consisting of all the nonempty super weakly compact convex sets of the space X.

A Note on Asymptotic Normal Structure and Close-to-Normal Structure

1982 ◽  
Vol 25 (3) ◽  
pp. 339-343 ◽  
Author(s):  
Kok-Keong Tan
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
Convex Set ◽  
Reflexive Banach Space ◽  
Compact Convex ◽  
Normal Structure ◽  
Weakly Compact ◽  
Compact Convex Set ◽  
Asymptotic Normal ◽  
Open Question

AbstractA closed convex subset X of a Banach space E is said to have (i) asymptotic normal structure if for each bounded closed convex subset C of X containing more than one point and for each sequence in C satisfying ‖xn − xn + 1‖ → 0 as n → ∞, there is a point x ∈ C such that ; (ii) close-to-normal structure if for each bounded closed convex subset C of X containing more than one point, there is a point x ∈ C such that ‖x − y‖ < diam‖ ‖(C) for all y ∈ C While asymptotic normal structure and close-to-normal structure are both implied by normal structure, they are not related. The example that a reflexive Banach space which has asymptotic normal structure but not close-to normal structure provides us a non-empty weakly compact convex set which does not have close-to-normal structure. This answers an open question posed by Wong in [9] and hence also provides us a Kannan map defined on a weakly compact convex set which does not have a fixed point.

scholarly journals Fixed-Point Theorem for Isometric Self-Mappings

2020 ◽  
Vol 2020 ◽  
pp. 1-4
Author(s):  
Joseph Frank Gordon
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Convex Subset ◽  
Compact Convex ◽  
Convex Banach Space ◽  
Compact Convex Subset ◽  
Weakly Compact ◽  
Strictly Convex Banach Space

In this paper, we derive a fixed-point theorem for self-mappings. That is, it is shown that every isometric self-mapping on a weakly compact convex subset of a strictly convex Banach space has a fixed point.

scholarly journals The range of an o-weakly compact mapping

1985 ◽  
Vol 39 (3) ◽  
pp. 391-399 ◽  
Author(s):  
P. G. Dodds
Keyword(s):  
Convex Set ◽  
Vector Measure ◽  
Compact Convex ◽  
Locally Convex Space ◽  
Weakly Compact ◽  
Compact Mapping ◽  
Compact Convex Set ◽  
Locally Convex ◽  
Order Continuous

AbstractIt is shown that a weakly compact convex set in a locally convex space is a zonoform if and only if it is the order continuous image of an order interval in a Dedekind complete Riesz space. While this result implies the Kluv´nek characterization of the range of a vector measure, the techniques of the present paper are purely order theoretic.

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