A Fixed-Point Theorem for Commuting Monotone Functions

1969 ◽  
Vol 21 ◽  
pp. 502-504 ◽  
Author(s):  
William J. Gray
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Fixed Point Property ◽  
Point Property ◽  
Monotone Functions ◽  
Abelian Semigroup

Hamilton (1) proved that a hereditarily unicoherent, hereditarily decomposable metric continuum has the fixed-point property for homeomorphisms. In this paper we shall generalize this result by showing that if X is a hereditarily unicoherent, hereditarily decomposable Hausdorff continuum and 5 is an abelian semigroup of continuous monotone functions from X into X, then S leaves a point of X fixed.Let X be a Hausdorff continuum. X is unicoherent if, whenever X = A ∪ B, where A and B are subcontinua of X, A ∩ B is a continuum. If each subcontinuum of X is unicoherent, X is hereditarily unicoherent. X is decomposable if X is the union of two of its proper subcontinua. If each subcontinuum of X which contains more than one point is decomposable, X is hereditarily decomposable.

A note on Banach fixed point property

2021 ◽  
Vol 1 (1) ◽  
pp. 47-52
Author(s):  
Vlasta Matijević
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Space ◽  
Closed Subspace ◽  
Short Note ◽  
Fixed Point Property ◽  
Banach Fixed Point Theorem ◽  
Point Property

In this short note we consider a sort of converse of the Banach fixed point theorem and prove that a metric space X is complete if and only if, for each closed subspace Y ⊆ X, any contraction f : Y → Y has a fixed point y ∈ Y.

scholarly journals A Fixed Point Theorem and the Existence of a Haar Measure for Hypergroups Satisfying Conditions Related to Amenability

2015 ◽  
Vol 58 (2) ◽  
pp. 415-422 ◽  
Author(s):  
Benjamin Willson
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Haar Measure ◽  
Continuous Functions ◽  
Fixed Point Property ◽  
Invariant Mean ◽  
Point Property ◽  
Uniformly Continuous

AbstractIn this paperwe present a fixed point property for amenable hypergroups that is analogous to Rickert’s fixed point theorem for semigroups. It equates the existence of a left invariant mean on the space of weakly right uniformly continuous functions to the existence of a fixed point for any action of the hypergroup. Using this fixed point property, certain hypergroups are shown to have a left Haar measure.

scholarly journals New Results and Generalizations for Approximate Fixed Point Property and Their Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Wei-Shih Du ◽  
Farshid Khojasteh
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Fixed Point Theorems ◽  
Existence Theorems ◽  
Fixed Point Property ◽  
Point Property ◽  
Approximate Fixed Point ◽  
Nadler’S Fixed Point Theorem ◽  
Approximate Fixed Point Property

We first introduce the concept of manageable functions and then prove some new existence theorems related to approximate fixed point property for manageable functions andα-admissible multivalued maps. As applications of our results, some new fixed point theorems which generalize and improve Du's fixed point theorem, Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, and Nadler's fixed point theorem and some well-known results in the literature are given.

scholarly journals Generalized trend constants of Lipschitz mappings

2018 ◽  
Vol 72 (2) ◽  
pp. 71
Author(s):  
Mariusz Szczepanik
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Lipschitz Mappings ◽  
Trend Coefficient

In 2015, Goebel and Bolibok defined the initial trend coefficient of a mapping and the class of initially nonexpansive mappings. They proved that the fixed point property for nonexpansive mappings implies the fixed point property for initially nonexpansive mappings. We generalize the above concepts and prove an analogous fixed point theorem. We also study the initial trend coefficient more deeply.

scholarly journals General Higher-Order Lipschitz Mappings

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Joseph Frank Gordon
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Nonexpansive Mappings ◽  
Higher Order ◽  
Fixed Point Property ◽  
Point Property ◽  
New Class ◽  
Lipschitz Mappings ◽  
Approximate Fixed Point Property

In this paper, we introduce a new class of mappings and investigate their fixed point property. In the first direction, we prove a fixed point theorem for general higher-order contraction mappings in a given metric space and finally prove an approximate fixed point property for general higher-order nonexpansive mappings in a Banach space.

scholarly journals Coarse dynamics and fixed-point theorem

2011 ◽  
Vol 202 ◽  
pp. 1-13
Author(s):  
Tomohiro Fukaya
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Point Of View ◽  
Abelian Semigroup ◽  
Semigroup Actions ◽  
Brouwer’S Fixed Point Theorem ◽  
Brouwer's Fixed Point Theorem ◽  
Higson Corona ◽  
Coarse Space

AbstractWe study semigroup actions on a coarse space and the induced actions on the Higson corona from a dynamical point of view. Our main theorem states that if an action of an abelian semigroup on a proper coarse space satisfies certain conditions, the induced action has a fixed point in the Higson corona. As a corollary, we deduce a coarse version of Brouwer’s fixed-point theorem.

scholarly journals New Existence Results and Generalizations for Coincidence Points and Fixed Points without Global Completeness

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Wei-Shih Du
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Spaces ◽  
Coincidence Point ◽  
Existence Theorems ◽  
Coupled Fixed Point ◽  
Point Property ◽  
Coincidence Points ◽  
Approximate Fixed Point Property

Some new existence theorems concerning approximate coincidence point property and approximate fixed point property for nonlinear maps in metric spaces without global completeness are established in this paper. By exploiting these results, we prove some new coincidence point and fixed point theorems which generalize and improve Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, Kikkawa-Suzuki's fixed point theorem, and some well known results in the literature. Moreover, some applications of our results to the existence of coupled coincidence point and coupled fixed point are also presented.

scholarly journals On a probabilistic version of Meir-Keeler type fixed point theorem for a family of discontinuous operators

2021 ◽  
Vol 22 (2) ◽  
pp. 435
Author(s):  
Ravindra K. Bisht ◽  
Vladimir Rakocević
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Space ◽  
Point Property ◽  
Probabilistic Metric Space ◽  
Discontinuous Mappings ◽  
Probabilistic Metric ◽  
Menger Probabilistic Metric Space ◽  
Probabilistic Version

<p>A Meir-Keeler type fixed point theorem for a family of mappings is proved in Menger probabilistic metric space (Menger PM-space). We establish that completeness of the space is equivalent to fixed point property for a larger class of mappings that includes continuous as well as discontinuous mappings. In addition to it, a probabilistic fixed point theorem satisfying (ϵ - δ) type non-expansive mappings is established.</p>

scholarly journals Coarse dynamics and fixed-point theorem

2011 ◽  
Vol 202 ◽  
pp. 1-13
Author(s):  
Tomohiro Fukaya
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Point Of View ◽  
Abelian Semigroup ◽  
Semigroup Actions ◽  
Brouwer’S Fixed Point Theorem ◽  
Brouwer's Fixed Point Theorem ◽  
Higson Corona ◽  
Coarse Space

AbstractWe study semigroup actions on a coarse space and the induced actions on the Higson corona from a dynamical point of view. Our main theorem states that if an action of an abelian semigroup on a proper coarse space satisfies certain conditions, the induced action has a fixed point in the Higson corona. As a corollary, we deduce a coarse version of Brouwer’s fixed-point theorem.

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