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Schauder's conjecture: "Every continuous function, from a nonempty compact and convex set in a (Hausdorff) topological vector space into itself, has a fixed point." [Problem 54 in The Scottish Book]

I wonder whether this conjecture is resolved. I know R. Cauty [Solution du problème de point fixe de Schauder, Fund. Math. 170 (2001) 231–246] proposed an answer, but apparently in the international conference "Fixed Point Theory and its Applications" in 2005, T. Dobrowolski remarked that there is a gap in the proof.

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    $\begingroup$ Wow, how old is this? This conjecture should have more attention; so simple to state, and clearly has a lot of impact... $\endgroup$
    – Per Alexandersson
    Commented May 11, 2014 at 21:18
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    $\begingroup$ @PerAlexandersson: Wikipedia says 1930, which is when Schauder proved the theorem for Banach spaces (the well-known Schauder fixed point theorem). Tychonoff extended it to locally convex spaces soon after. Note that Wikipedia calls the above conjecture a theorem, apparently accepting Cauty's argument as a proof. $\endgroup$
    – Nate Eldredge
    Commented May 12, 2014 at 6:33
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    $\begingroup$ >Park, Sehie; Remarks on recent results in analytical fixed point theory. Nonlinear analysis and convex analysis, 517–525, Yokohama Publ., Yokohama, 2007 R. Cauty proved the Schauder fixed point theorem in topological vector spaces without assuming local convexity (see also T. Dobrowolski, [Revisiting Cauty's proof of the Schauder conjecture] for an expanded version which is more easily accessible). $\endgroup$
    – Mohammad Golshani
    Commented May 18, 2014 at 13:52
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    $\begingroup$ In 2005 paper, Park claimed this. But at that time, the gap was not discovered. It is discovered in 2005. And 3 years later, the same author, in "Compact Browder maps and equilibria of abstract economies, Journal of Applied Mathematics and Computing, 26, 555-564 (2008)" wrote: "In 2001, Cauty claimed to resolve the conjecture affirmatively, but later, it was known that his proof had a gap." $\endgroup$
    – 57319
    Commented May 19, 2014 at 15:09
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    $\begingroup$ This could have all been found out (and resolved?) more quickly if the paper wasn't a.) behind a paywall no german university that I've tried had access to at the time and b.) written in french $\endgroup$
    – Johannes Hahn
    Commented Mar 23, 2015 at 14:58

4 Answers 4

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I have taken the following from the review of the following paper "Schauder's conjecture on convex metric spaces" written in 2010 :

One of the most resistant open problems in the theory of nonlocally convex linear metric spaces is:

Schauder's Conjecture. Let $E$ be a compact convex subset in a topological vector space. Then any continuous mapping $f:E\to E$ has a fixed point.

In this paper, the authors prove that it holds for convex metric spaces and consequently compact convex subsets of a $CAT(0)$ space have the fixed point property for continuous mappings.

So it seems that the problem in its general form is still open.

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    $\begingroup$ How does this 2003 paper address this point of the question: "apparently in the international conference 'Fixed Point Theory and its Applications' in 2005, T. Dobrowolski remarked that there is a gap in the proof." $\endgroup$
    – François G. Dorais
    Commented May 18, 2014 at 9:58
  • $\begingroup$ You are right. In fact it seems the problem is still open. See my new answer. $\endgroup$
    – Mohammad Golshani
    Commented May 18, 2014 at 15:12
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In Points fixes des applications compactes dans les espaces ULC published in in the arXiv in 2010 Robert Cauty wrote

il y a d’ailleurs une erreur dans la demonstration du lemme 3 de [2], qu’il n’y a plus de raison de corriger, vu la superiorite de la nouvelle approche

(there is, by the way, an error in the proof of lemma 3 from [2] for which there is no need for correction in view of the superiority of the new approach)

It seems thus that Cauty still (or again) claims that the Schauder conjecture is settled.

[2] is: R. Cauty. Solution du problème de point fixe de Schauder. Fund. Math. 170, 2001, 231-246.

Edit (August 2016). I was quite surprised that apparently there is no version of the mentioned 2010 arXiv article published in an international journal. While searching the web I learned that Robert Cauty died in 2013.

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    $\begingroup$ The new (correct) proof of Schauder Conjecture was published by R.Cauty in [R.Cauty, Rétractes absolus de voisinage algébriques, Serdica Math. J. 31 (2005), no. 4, 309--354]. The older proofs of Cauty (2001) and Dobrowolski (2003) both contained gaps. $\endgroup$
    – Taras Banakh
    Commented Aug 12, 2016 at 19:24
  • $\begingroup$ Could you please indicate which result in the cited paper gives Schauder's conjecture? I am not able to locate it. $\endgroup$
    – Doug Liu
    Commented Apr 6, 2022 at 13:04
  • $\begingroup$ Sorry, me neither. $\endgroup$
    – Jochen Wengenroth
    Commented Apr 6, 2022 at 15:52
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    $\begingroup$ @TarasBanakh it might be useful if you reply to the above comments $\endgroup$
    – YCor
    Commented Oct 14, 2022 at 15:24
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    $\begingroup$ @DougLiu The Schauder conjecture follows from Theoreme 8 combined with Theoreme 6 or Theoreme 7 in Cauty's paper. $\endgroup$
    – Taras Banakh
    Commented Oct 14, 2022 at 19:23
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There is R. Cauty paper from 2012 titled 'Un theoreme de Lefschetz-Hopf pour les fonctions a iterees compactes' which from what I heard was reviewed to establish a correct proof of Schauder's conjecture ( or rather a generalization for iterates of f ), and will appear in an international journal.

Edit: published: R. Cauty, J. Reine Angew. Math. 729 (2017), 1–27 DOI link

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    $\begingroup$ The Cauty paper was published online in January 2015, available at degruyter.com/view/j/crll.ahead-of-print/crelle-2014-0134/… A footnote says, "After submitting this work Robert Cauty passed away. The invaluable help of H. Steinlein ([email protected]) and T. Banakh ([email protected]) in going through the manuscript and making minor revisions is gratefully acknowledged." $\endgroup$
    – Gerry Myerson
    Commented Nov 3, 2016 at 1:37
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    $\begingroup$ The Cauty 2015 result is also sketched in Steinlein, 70 years of asymptotic fixed point theory, J Fixed Point Theory Appl 17 (2015) 3-21. $\endgroup$
    – Gerry Myerson
    Commented Nov 3, 2016 at 1:47
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An old proof (by M. R. Taskovic) is here: Mathematica Moravica, volume 2, (1998).

Summary: " The most famous of many problems in nonlinear analysis is Schauder’s problem (Scottish book, problem 54) of the following form, that if $C$ is a nonempty convex compact subset of a linear topological space does every continuous mapping $f : C → C$ has a fixed point? The answer we give in this paper is yes."

A new proof (by Mohamed Ennassik & Mohamed Aziz Taoudi ) is here: Journal of Fixed Point Theory and Applications volume 23, 52 (2021).

Summary: "In this paper, we prove that every nonempty compact $s$-convex subset ($0<s\le 1$) of a Hausdorff topological vector space has the fixed point property. Our approach allows us to provide a simple alternative proof of Schauder's conjecture."

But see also this: The Journal of Analysis, 2023

Summary: "We explain why a recent proof of Schauder’s conjecture is not correct. A counterexample to a cited intermediate lemma is given."

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