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 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 Towards the fixed point property for superreflexive spaces

2001 ◽  
Vol 64 (3) ◽  
pp. 435-444 ◽  
Author(s):  
Andrzej Wiśnicki
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Unit Sphere ◽  
Convex Set ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Geometrical Condition

A Banach space X is said to have property (Sm) if every metrically convex set A ⊂ X which lies on the unit sphere and has diameter not greater than one can be (weakly) separated from zero by a functional. We show that this geometrical condition is closely connected with the fixed point property for nonexpansive mappings in superreflexive spaces.

The fixed point property for some uniformly nonoctahedral Banach spaces

1999 ◽  
Vol 59 (3) ◽  
pp. 361-367 ◽  
Author(s):  
A. Jiménez-Melado
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property

Roughly speaking, we show that a Banach space X has the fixed point property for nonexpansive mappings whenever X has the WORTH property and the unit sphere of X does not contain a triangle with sides of length larger than 2.

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 The fixed-point property in Banach spaces containing a copy ofc0

2003 ◽  
Vol 2003 (3) ◽  
pp. 183-192
Author(s):  
Maria A. Japón Pineda
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Banach Spaces ◽  
Weak Topology ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
Asymptotically Nonexpansive ◽  
Locally Convex Topology ◽  
Locally Convex

We prove that every Banach space containing an isomorphic copy ofc0fails to have the fixed-point property for asymptotically nonexpansive mappings with respect to some locally convex topology which is coarser than the weak topology. If the copy ofc0is asymptotically isometric, this result can be improved, because we can prove the failure of the fixed-point property for nonexpansive mappings.

scholarly journals c0 can be renormed to have the fixed point property for affine nonexpansive mappings

Filomat ◽  
2018 ◽  
Vol 32 (16) ◽  
pp. 5645-5663 ◽  
Author(s):  
Veysel Nezir ◽  
Nizami Mustafa
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Reflexive Banach Space ◽  
Nonexpansive Mappings ◽  
Equivalent Norm ◽  
Positive Answer ◽  
Fixed Point Property ◽  
Null Sequence ◽  
Point Property ◽  
Usual Norm

P.K. Lin gave the first example of a non-reflexive Banach space (X,||?||) with the fixed point property (FPP) for nonexpansive mappings and showed this fact for (l1,||?||1) with the equivalent norm ||?|| given by ||x|| = sup k?N 8k/1+8k ?1,n=k |xn|, for all x = (xn)n?N ? l1. We wonder (c0, ||?||1) analogue of P.K. Lin?s work and we give positive answer if functions are affine nonexpansive. In our work, for x = (?k)k ? c0, we define |||x||| := lim p?? sup ?k?N ?k (?1,j=k |?j|p/2j)1/p where ?k ?k 3, k is strictly increasing with ?k > 2, ?k ? N, then we prove that (c0,|||?|||) has the fixed point property for affine |||?|||-nonexpansive self-mappings. Next, we generalize this result and show that if ?(?) is an equivalent norm to the usual norm on c0 such that lim sup n ?(1/n ?n,m=1 xm + x) = lim sup n ?(1/n ?n,m=1 xm) + ?(x) for every weakly null sequence (xn)n and for all x ? c0, then for every ? > 0, c0 with the norm ||?||? = ?(?)+?|||?||| has the FPP for affine ||?||?-nonexpansive self-mappings.

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 Theτ-fixed point property for nonexpansive mappings

1998 ◽  
Vol 3 (3-4) ◽  
pp. 343-362 ◽  
Author(s):  
Tomás Domínguez Benavides ◽  
jesús García Falset ◽  
Maria A. Japón Pineda
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Normal Structure ◽  
Fixed Point Property ◽  
Regularity Conditions ◽  
Point Property ◽  
Geometrical Properties ◽  
Bounded Convex ◽  
Geometric Constants

LetXbe a Banach space andτa topology onX. We say thatXhas theτ-fixed point property (τ-FPP) if every nonexpansive mappingTdefined from a bounded convexτ-sequentially compact subsetCofXintoChas a fixed point. Whenτsatisfies certain regularity conditions, we show that normal structure assures theτ-FPP and Goebel-Karlovitz's Lemma still holds. We use this results to study two geometrical properties which imply theτ-FPP: theτ-GGLD andM(τ)properties. We show several examples of spaces and topologies where these results can be applied, specially the topology of convergence locally in measure in Lebesgue spaces. In the second part we study the preservence of theτ-FPP under isomorphisms. In order to do that we study some geometric constants for a Banach spaceXsuch that theτ-FPP is shared by any isomorphic Banach spaceYsatisfying that the Banach-Mazur distance betweenXandYis less than some of these constants.

scholarly journals A remark on the approximate fixed-point property

2003 ◽  
Vol 2003 (2) ◽  
pp. 93-99 ◽  
Author(s):  
Tadeusz Kuczumow
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Orthogonal Basis ◽  
Fixed Point Property ◽  
Point Property ◽  
Closed Set ◽  
Approximate Fixed Point ◽  
Approximate Fixed Point Property ◽  
Unbounded Convex

We give an example of an unbounded, convex, and closed setCin the Hilbert spacel2with the following two properties: (i)Chas the approximate fixed-point property for nonexpansive mappings, (ii)Cis not contained in a block for every orthogonal basis inl2.

scholarly journals On some Banach space properties sufficient for normal structure

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1305-1315 ◽  
Author(s):  
Mina Dinarvand
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Recent Literature ◽  
Nonexpansive Mappings ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
Fixed Point Property ◽  
Point Property ◽  
Von Neumann

In this paper, we present some sufficient conditions for which a Banach space has normal structure and therefore the fixed point property for nonexpansive mappings in terms of the generalized James, von Neumann-Jordan, Zb?ganu constants, the Ptolemy constant and the Dom?nguez-Benavides coefficient. Our main results extend and improve some known results in the recent literature.

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