scholarly journals Markov-Kakutani Theorem on Hyperspace of a Banach Space

2020 ◽  
Vol 52 (1) ◽  
Author(s):  
Shueh-Inn Hu ◽  
Thakyin Hu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Common Fixed Point ◽  
Convex Subset ◽  
Compact Convex ◽  
Hausdorff Metric ◽  
Compact Convex Subset ◽  
Affine Mappings ◽  
Kakutani Theorem ◽  
Commutative Family

Suppose $X$ is a Banach space and $K$ is a compact convex subset of $X$. Let $\mathcal{F}$ be a commutative family of continuous affine mappings of $K$ into $K$. It follows from Markov-Kakutani Theorem that $\mathcal{F}$ has a common fixed point in $K$. Suppose now $(CC(X), h)$ is the corresponding hyperspace of $X$ containing all compact, convex subsets of $X$ endowed with Hausdorff metric $h$. We shall prove the above version of Markov-Kakutani Theorem is valid on the hyperspace $(CC(X), h)$.

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 On a fixed point theorem of Greguš

1986 ◽  
Vol 9 (1) ◽  
pp. 23-28 ◽  
Author(s):  
Brian Fisher ◽  
Salvatore Sessa
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Common Fixed Point ◽  
Point Theorem ◽  
Convex Subset ◽  
Unique Common Fixed Point

We consider two selfmapsTandIof a closed convex subsetCof a Banach spaceXwhich are weakly commuting inX, i.e.‖TIx−ITx‖≤‖Ix−Tx‖   for   any   x   in   X,and satisfy the inequality‖Tx−Ty‖≤a‖Ix−Iy‖+(1−a)max{‖Tx−Ix‖,‖Ty−Iy‖}for allx,yinC, where0<a<1. It is proved that ifIis linear and non-expansive inCand such thatICcontainsTC, thenTandIhave a unique common fixed point inC.

Fixed point theorems for uniformly generalized Kannan type semigroup of self-mappings

2017 ◽  
Vol 26 (2) ◽  
pp. 231-240
Author(s):  
AHMED H. SOLIMAN ◽  
MOHAMMAD IMDAD ◽  
MD AHMADULLAH
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Common Fixed Point ◽  
Convex Subset ◽  
Fixed Point Theorems ◽  
Real Banach Space ◽  
Normal Structure ◽  
Uniform Normal Structure

In this paper, we consider a new uniformly generalized Kannan type semigroup of self-mappings defined on a closed convex subset of a real Banach space equipped with uniform normal structure and employ the same to show that such semigroup of self-mappings admits a common fixed point provided the underlying semigroup of self-mappings has a bounded orbit.

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 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 On Fixed Point Property under Lipschitz and Uniform Embeddings

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Jichao Zhang ◽  
Lingxin Bao ◽  
Lili Su
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Convex Sets ◽  
Fixed Point Property ◽  
Point Property ◽  
Lipschitz Mappings ◽  
Affine Mappings ◽  
Reflexive Space ◽  
Gateaux Differentiability

We first present a generalization of ω⁎-Gâteaux differentiability theorems of Lipschitz mappings from open sets to those closed convex sets admitting nonsupport points and then show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for isometries if it Lipschitz embeds into a super reflexive space. With the application of Baudier-Lancien-Schlumprecht’s theorem, we finally show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for continuous affine mappings if it uniformly embeds into the Tsirelson space T⁎.

scholarly journals Strong Convergence for Hybrid ImplicitS-Iteration Scheme of Nonexpansive and Strongly Pseudocontractive Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Shin Min Kang ◽  
Arif Rafiq ◽  
Faisal Ali ◽  
Young Chel Kwun
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Convex Subset ◽  
Real Banach Space ◽  
Iteration Scheme ◽  
Nonempty Closed Convex Subset ◽  
Pseudocontractive Mappings

LetKbe a nonempty closed convex subset of a real Banach spaceE, letS:K→Kbe nonexpansive, and let  T:K→Kbe Lipschitz strongly pseudocontractive mappings such thatp∈FS∩FT=x∈K:Sx=Tx=xandx-Sy≤Sx-Sy and x-Ty≤Tx-Tyfor allx, y∈K. Letβnbe a sequence in0, 1satisfying (i)∑n=1∞βn=∞; (ii)limn→∞⁡βn=0.For arbitraryx0∈K, letxnbe a sequence iteratively defined byxn=Syn, yn=1-βnxn-1+βnTxn, n≥1.Then the sequencexnconverges strongly to a common fixed pointpofSandT.

scholarly journals Fixed point properties for semigroups of nonexpansive mappings on convex sets in dual Banach spaces

2018 ◽  
Vol 17 (1) ◽  
pp. 67-87
Author(s):  
Anthony To-Ming Lau ◽  
Yong Zhang
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Convex Sets ◽  
Compact Convex ◽  
Nonexpansive Mappings ◽  
Point Property ◽  
Fixed Point Properties ◽  
Dual Banach Space ◽  
Standing Problem

Abstract It has been a long-standing problem posed by the first author in a conference in Marseille in 1990 to characterize semitopological semigroups which have common fixed point property when acting on a nonempty weak* compact convex subset of a dual Banach space as weak* continuous and norm nonexpansive mappings. Our investigation in the paper centers around this problem. Our main results rely on the well-known Ky Fan’s inequality for convex functions.

scholarly journals Existence of common fixed points of non-linear semigroups of Enriched Kannan contractive mappings

2021 ◽  
Vol 38 (1) ◽  
pp. 169-178
Author(s):  
SAYANTAN PANJA ◽  
◽  
MANTU SAHA ◽  
RAVINDRA K. BISHT ◽  
◽  
...  
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Common Fixed Point ◽  
Convex Subset ◽  
Real Banach Space ◽  
Normal Structure ◽  
Contractive Mapping ◽  
Common Fixed Points ◽  
Contractive Mappings ◽  
Non Linear

In this article, we consider the non-linear semigroup of \textit{enriched Kannan} contractive mapping and prove the existence of common fixed point on a non-empty closed convex subset $\mathcal C$ of a real Banach space $\mathscr X$, having uniformly normal structure.

scholarly journals Revisiting Cauty's proof of the Schauder conjecture

2003 ◽  
Vol 2003 (7) ◽  
pp. 407-433 ◽  
Author(s):  
Tadeusz Dobrowolski
Keyword(s):  
Fixed Point ◽  
Detailed Analysis ◽  
Linear Space ◽  
Convex Subset ◽  
Approximation Property ◽  
Compact Convex ◽  
Fixed Point Property ◽  
Compact Convex Subset ◽  
Point Property ◽  
Linear Spaces

The Schauder conjecture that every compact convex subset of a metric linear space has the fixed-point property was recently established by Cauty (2001). This paper elaborates on Cauty's proof in order to make it more detailed, and therefore more accessible. Such a detailed analysis allows us to show that the convex compacta in metric linear spaces possess the simplicial approximation property introduced by Kalton, Peck, and Roberts. The latter demonstrates that the original Schauder approach to solve the conjecture is in some sense “correctable.”

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