scholarly journals A renorming ofℓ2, rare but with the fixed-point property

2003 ◽  
Vol 2003 (65) ◽  
pp. 4115-4129 ◽  
Author(s):  
Antonio Jiménez-Melado ◽  
Enrique Llorens-Fuster
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Sufficient Conditions ◽  
Fixed Point Property ◽  
Point Property

We give an example of a renorming ofℓ2with the fixed-point property (FPP) for nonexpansive mappings, but which seems to fall out of the scope of all the commonly known sufficient conditions for FPP.

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.

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.

scholarly journals Remarks on the stability of the fixed point property for nonexpansive mappings

2012 ◽  
Vol 1 (4) ◽  
pp. 417-430 ◽  
Author(s):  
Krzysztof Bolibok ◽  
Kazimierz Goebel ◽  
W. A. Kirk
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Point Property ◽  
The Stability

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 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 The fixed point property and a technique to harness double fixed point combinators

2019 ◽  
Vol 29 (5) ◽  
pp. 831-880
Author(s):  
Giulio Manzonetto ◽  
Andrew Polonsky ◽  
Alexis Saurin ◽  
Jakob Grue Simonsen
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Sufficient Conditions ◽  
Lambda Calculus ◽  
Extension Property ◽  
Fixed Point Property ◽  
Proof Technique ◽  
Point Property ◽  
Negative Solution ◽  
Conservative Extension

Abstract The ${\lambda }$-calculus enjoys the property that each ${\lambda }$-term has at least one fixed point, which is due to the existence of a fixed point combinator. It is unknown whether it enjoys the ‘fixed point property’ stating that each ${\lambda }$-term has either one or infinitely many pairwise distinct fixed points. We show that the fixed point property holds when considering possibly open fixed points. The problem of counting fixed points in the closed setting remains open, but we provide sufficient conditions for a ${\lambda }$-term to have either one or infinitely many fixed points. In the main result of this paper we prove that in every sensible ${\lambda }$-theory there exists a ${\lambda }$-term that violates the fixed point property. We then study the open problem concerning the existence of a double fixed point combinator and propose a proof technique that could lead towards a negative solution. We consider interpretations of the ${\lambda } {\mathtt{Y}}$-calculus into the ${\lambda }$-calculus together with two reduction extension properties, whose validity would entail the non-existence of any double fixed point combinators. We conjecture that both properties hold when typed ${\lambda } {\mathtt{Y}}$-terms are interpreted by arbitrary fixed point combinators. We prove reduction extension property I for a large class of fixed point combinators. Finally, we prove that the ${\lambda }{\mathtt{Y}}$-theory generated by the equation characterizing double fixed point combinators is a conservative extension of the ${\lambda }$-calculus.

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