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.

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.

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 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.

Consistency conditions for a finite set of projections of a function

1979 ◽  
Vol 85 (1) ◽  
pp. 61-68 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Lebesgue Measure ◽  
Convex Subset ◽  
Compact Convex ◽  
Unit Vector ◽  
Compact Convex Subset ◽  
Finite Collection ◽  
Finite Set ◽  
Almost All ◽  
Dimensional Lebesgue Measure ◽  
Unit Vectors

Let H(μ, θ) be the hyperplane in Rn (n ≥ 2) that is perpendicular to the unit vector 6 and perpendicular distance μ from the origin; that is, H(μ, θ) = (x ∈ Rn: x. θ = μ). (Note that H(μ, θ) and H(−μ, −θ) are the same hyperplanes.) Let X be a proper compact convex subset of Rm. If f(x) ∈ L1(X) we will denote by F(μ, θ) the projection of f perpendicular to θ; that is, the integral of f(x) over H(μ, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ∈ L1(X), F(μ, θ) exists for almost all μ for every θ. Our aim in this paper is, given a finite collection of unit vectors θ1, …, θN, to characterize the F(μ, θi) that are the projections of some function f(x) with support in X for 1 ≤ i ≤ N.

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 Fixed point theorems for nonexpansive mappings on nonconvex sets in UCED Banach spaces

2002 ◽  
Vol 31 (4) ◽  
pp. 251-257 ◽  
Author(s):  
Wei-Shih Du ◽  
Young-Ye Huang ◽  
Chi-Lin Yen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Compact Convex ◽  
Nonexpansive Mappings ◽  
Finite Union ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Nonconvex Sets ◽  
Firmly Nonexpansive ◽  
Asymptotically Regular

It is shown that every asymptotically regular orλ-firmly nonexpansive mappingT:C→Chas a fixed point wheneverCis a finite union of nonempty weakly compact convex subsets of a Banach spaceXwhich is uniformly convex in every direction. Furthermore, if{T i}i∈Iis any compatible family of strongly nonexpansive self-mappings on such aCand the graphs ofT i,i∈I, have a nonempty intersection, thenT i,i∈I, have a common fixed point inC.

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

Some Inequalities Related to Planar Convex Sets

1982 ◽  
Vol 25 (3) ◽  
pp. 302-310 ◽  
Author(s):  
R. J. Gardner ◽  
S. Kwapien ◽  
D. P. Laurie
Keyword(s):  
Convex Subset ◽  
Convex Sets ◽  
Compact Convex ◽  
Compact Convex Subset ◽  
Planar Convex

AbstractB. Grünbaum and J. N. Lillington have considered inequalities defined by three lines meeting in a compact convex subset of the plane. We prove a conjecture of Lillington and propose some conjectures of our own.

scholarly journals On Bare Points

1969 ◽  
Vol 9 (1-2) ◽  
pp. 25-28 ◽  
Author(s):  
S. J. Bernau
Keyword(s):  
Linear Space ◽  
Convex Subset ◽  
Normed Linear Space ◽  
Compact Convex ◽  
Proper Subset ◽  
Compact Convex Subset

This note shows that the set of bare points of a compact convex subset of a normed linear space is, in general, a proper subset of its set of exposed points.

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