Fixed points of products and the strong fixed point property

Order ◽  
1987 ◽  
Vol 4 (3) ◽  
pp. 221-231 ◽  
Author(s):  
Dwight Duffus ◽  
Norbert Sauer
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Fixed Point Property ◽  
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 Jungck theorem for triangular maps and related results

2000 ◽  
Vol 1 (1) ◽  
pp. 83 ◽  
Author(s):  
M. Grinc ◽  
L. Snoha
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Fixed Point ◽  
Periodic Points ◽  
Fixed Point Property ◽  
Compact Interval ◽  
Invariance Property ◽  
Point Property ◽  
Dimensional Cube ◽  
Triangular Maps

<p>We prove that a continuous triangular map G of the n-dimensional cube I<sup>n</sup> has only fixed points and no other periodic points if and only if G has a common fixed point with every continuous triangular map F that is nontrivially compatible with G. This is an analog of Jungck theorem for maps of a real compact interval. We also discuss possible extensions of Jungck theorem, Jachymski theorem and some related results to more general spaces. In particular, the spaces with the fixed point property and the complete invariance property are considered.</p>

scholarly journals A NOTE ON A FIXED POINT PROPERTY FOR METRIC PROJECTIONS

1996 ◽  
Vol 27 (1) ◽  
pp. 55-57
Author(s):  
WAGDY GOMAA EL-SAYED
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Fixed Point Property ◽  
Partial Answer ◽  
Point Property ◽  
Metric Projections

The paper contains a partial answer to a question r邳 ied recently by S. P. Singh concerning the existence of fixed points of metric projections

scholarly journals Fixed point property of amenable planar vortexes

2021 ◽  
Vol 22 (2) ◽  
pp. 385
Author(s):  
James Francis Peters ◽  
Tane Vergili
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Free Group ◽  
Fixed Point Property ◽  
Fundamental Result ◽  
Point Property ◽  
Group Representations ◽  
Amenable Groups ◽  
Natural Outcome ◽  
The 1960S

This article introduces free group representations of planar vortexes in a CW space that are a natural outcome of results for amenable groups and fixed points found by M.M. Day during the 1960s and a fundamental result for fixed points given by L.E.J. Brouwer.

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
Sign in / Sign up

Export Citation Format

Share Document