Invariant means and fixed point properties on completely regular spaces

1977 ◽  
Vol 16 (2) ◽  
pp. 203-212
Author(s):  
Marvin W. Grossman
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Topological Semigroup ◽  
Related Result ◽  
Continuous Functions ◽  
Point Property ◽  
Completely Regular ◽  
Fixed Point Properties ◽  
Bounded Functions ◽  
Invariant Means

Two theorems are presented which characterize the existence of multiplicative left invariant means on a given algebra of unbounded continuous functions on a topological semigroup S in terms of certain common fixed point properties of actions of S on completely regular spaces. Also a lattice formulation of a related result of Theodore Mitchell for the case of bounded functions is shown to be equivalent to a certain common fixed point property on Bauer simplexes.

INVARIANT MEANS AND ACTIONS OF SEMITOPOLOGICAL SEMIGROUPS ON COMPLETELY REGULAR SPACES AND APPLICATIONS

2020 ◽  
pp. 1-12
Author(s):  
KHADIME SALAME
Keyword(s):  
Fixed Point ◽  
Topological Spaces ◽  
Compact Groups ◽  
Locally Convex Spaces ◽  
Locally Compact ◽  
Completely Regular ◽  
Continuous Mappings ◽  
Fixed Point Properties ◽  
Invariant Means ◽  
Locally Convex

In this paper, we extend the study of fixed point properties of semitopological semigroups of continuous mappings in locally convex spaces to the setting of completely regular topological spaces. As applications, we establish a general fixed point theorem, a convergence theorem and an application to amenable locally compact groups.

GEOMETRIC AND FIXED POINT PROPERTIES IN PRODUCTS OF NORMED SPACES

2019 ◽  
Vol 99 (2) ◽  
pp. 262-273 ◽  
Author(s):  
M. VEENA SANGEETHA
Keyword(s):  
Fixed Point ◽  
Dimensional Subspace ◽  
Point Property ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Nonexpansive Maps ◽  
Uniform Rotundity ◽  
Fixed Point Properties ◽  
Weak Fixed Point Property ◽  
Monotone Norm

Given two (real) normed (linear) spaces $X$ and $Y$, let $X\otimes _{1}Y=(X\otimes Y,\Vert \cdot \Vert )$, where $\Vert (x,y)\Vert =\Vert x\Vert +\Vert y\Vert$. It is known that $X\otimes _{1}Y$ is $2$-UR if and only if both $X$ and $Y$ are UR (where we use UR as an abbreviation for uniformly rotund). We prove that if $X$ is $m$-dimensional and $Y$ is $k$-UR, then $X\otimes _{1}Y$ is $(m+k)$-UR. In the other direction, we observe that if $X\otimes _{1}Y$ is $k$-UR, then both $X$ and $Y$ are $(k-1)$-UR. Given a monotone norm $\Vert \cdot \Vert _{E}$ on $\mathbb{R}^{2}$, we let $X\otimes _{E}Y=(X\otimes Y,\Vert \cdot \Vert )$ where $\Vert (x,y)\Vert =\Vert (\Vert x\Vert _{X},\Vert y\Vert _{Y})\Vert _{E}$. It is known that if $X$ is uniformly rotund in every direction, $Y$ has the weak fixed point property for nonexpansive maps (WFPP) and $\Vert \cdot \Vert _{E}$ is strictly monotone, then $X\otimes _{E}Y$ has WFPP. Using the notion of $k$-uniform rotundity relative to every $k$-dimensional subspace we show that this result holds with a weaker condition on $X$.

scholarly journals The cardinality of the set of left invariant means on topological semigroups

1988 ◽  
Vol 37 (2) ◽  
pp. 247-262 ◽  
Author(s):  
Heneri A.M. Dzinotyiweyi
Keyword(s):  
Large Class ◽  
Topological Semigroup ◽  
Continuous Functions ◽  
Upper Bounds ◽  
Compact Sets ◽  
Lower And Upper Bounds ◽  
Topological Semigroups ◽  
Invariant Means ◽  
Uniformly Continuous

For a very large class of topological semigroups, we establish lower and upper bounds for the cardinality of the set of left invariant means on the space of left uniformly continuous functions. In certain cases we show that such a cardinality is exactly , where b is the smallest cardinality of the covering of the underlying topological semigroup by compact sets.

scholarly journals On common fixed and periodic points of commuting functions

1998 ◽  
Vol 21 (2) ◽  
pp. 269-276 ◽  
Author(s):  
Aliasghar Alikhani-Koopaei
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Periodic Point ◽  
Periodic Points ◽  
Continuous Functions ◽  
Contractive Condition ◽  
Unique Common Fixed Point

It is known that two commuting continuous functions on an interval need not have a common fixed point. It is not known if such two functions have a common periodic point. In this paper we first give some results in this direction. We then define a new contractive condition, under which two continuous functions must have a unique common fixed point.

scholarly journals Digital fixed points, approximate fixed points, and universal functions

2016 ◽  
Vol 17 (2) ◽  
pp. 159 ◽  
Author(s):  
Laurence Boxer ◽  
Ozgur Ege ◽  
Ismet Karaca ◽  
Jonathan Lopez ◽  
Joel Louwsma
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Digital Images ◽  
Continuous Functions ◽  
Universal Functions ◽  
Approximate Fixed Point ◽  
Fixed Point Properties ◽  
Digitally Continuous ◽  
Approximate Fixed Point Property ◽  
The Relationship

A. Rosenfeld [23] introduced the notion of a digitally continuous function between digital images, and showed that although digital images need not have fixed point properties analogous to those of the Euclidean spaces modeled by the images, there often are approximate fixed point properties of such images. In the current paper, we obtain additional results concerning fixed points and approximate fixed points of digitally continuous functions. Among these are several results concerning the relationship between universal functions and the approximate fixed point property (AFPP).

scholarly journals Common fixed point properties and amenability of a class of Banach algebras

2013 ◽  
Vol 402 (2) ◽  
pp. 536-544 ◽  
Author(s):  
Shawn Desaulniers ◽  
Rasoul Nasr-Isfahani ◽  
Mehdi Nemati
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Banach Algebras ◽  
Fixed Point Properties
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