History of the Brouwer Fixed Point Theorem

2020 ◽  
pp. 391-412
Author(s):  
George Dinca ◽  
Jean Mawhin
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Brouwer Fixed Point Theorem ◽  
History Of

scholarly journals Connected choice and the Brouwer fixed point theorem

2019 ◽  
Vol 19 (01) ◽  
pp. 1950004 ◽  
Author(s):  
Vasco Brattka ◽  
Stéphane Le Roux ◽  
Joseph S. Miller ◽  
Arno Pauly
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Unit Cube ◽  
Closed Set ◽  
Brouwer Fixed Point Theorem ◽  
Fixed Dimension ◽  
Lipschitz Constants ◽  
Weak König’S Lemma ◽  
König’S Lemma

We study the computational content of the Brouwer Fixed Point Theorem in the Weihrauch lattice. Connected choice is the operation that finds a point in a non-empty connected closed set given by negative information. One of our main results is that for any fixed dimension the Brouwer Fixed Point Theorem of that dimension is computably equivalent to connected choice of the Euclidean unit cube of the same dimension. Another main result is that connected choice is complete for dimension greater than or equal to two in the sense that it is computably equivalent to Weak Kőnig’s Lemma. While we can present two independent proofs for dimension three and upward that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upward. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater than or equal to one is equivalent to Weak Kőnig’s Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes.

scholarly journals Spherical Designs via Brouwer Fixed Point Theorem

2010 ◽  
Vol 24 (1) ◽  
pp. 207-217 ◽  
Author(s):  
Andriy V. Bondarenko ◽  
Maryna S. Viazovska
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Spherical Designs ◽  
Brouwer Fixed Point Theorem

Equivalent forms of the Brouwer fixed point theorem

2016 ◽  
Vol 44 (1) ◽  
pp. 263
Author(s):  
Adam Idzik ◽  
Władysław Kulpa ◽  
Piotr Maćkowiak
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Brouwer Fixed Point Theorem

scholarly journals On the Brouwer fixed point theorem

2002 ◽  
Vol 119 (1) ◽  
pp. 53-64 ◽  
Author(s):  
Dariusz Miklaszewski
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Brouwer Fixed Point Theorem
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