scholarly journals Universal and proximately universal limits

1996 ◽  
Vol 61 (1) ◽  
pp. 96-105
Author(s):  
Zvonko Čerin ◽  
Jóse M. R. Sanjurjo
Keyword(s):  
Fixed Point ◽  
Sufficient Conditions ◽  
Fixed Point Property ◽  
Inverse System ◽  
Point Property ◽  
Approximate Inverse ◽  
Similar Theorem ◽  
Inverse Systems

AbstractWe present sufficient conditions on an approximate mapping F: X → Y of approximate inverse systems in order that the limit f: X → Y of F is a universal map in the sense of Holsztyński. A similar theorem holds for a more restrictive concept of a proximately universal map introduced recently by the second author. We get as corollaries some sufficient conditions on an approximate inverse system implying that the its limit has the (proximate) fixed point property. In particular, every chainable compact Hausdorif space has the proximate fixed point property.

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.

scholarly journals The fixed point property and normal structure for some B-convex Banach spaces

2001 ◽  
Vol 63 (1) ◽  
pp. 75-81 ◽  
Author(s):  
Jesús García-Falset ◽  
Enrique Llorens-Fuster ◽  
Eva M. Mazcuñán-Navarro
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Normal Structure ◽  
Fixed Point Property ◽  
Sufficient Condition ◽  
Point Property

We give a sufficient condition for normal structure more general than the well known ɛ0(X) < 1. Moreover we obtain sufficient conditions for the fixed point property for some B-convex Banach spaces.

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 Brush spaces and the fixed point property

2011 ◽  
Vol 158 (8) ◽  
pp. 1085-1089 ◽  
Author(s):  
M.M. Marsh ◽  
J.R. Prajs
Keyword(s):  
Fixed Point ◽  
Fixed Point Property ◽  
Point Property

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 A product space with the fixed point property

2012 ◽  
Vol 2012 (1) ◽  
Author(s):  
Helga Fetter Nathansky ◽  
Enrique Llorens-Fuster
Keyword(s):  
Fixed Point ◽  
Product Space ◽  
Fixed Point Property ◽  
Point Property
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