48

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.

YCor
  • 60,149
57319
  • 603
  • 1
    Wow, how old is this? This conjecture should have more attention; so simple to state, and clearly has a lot of impact... – Per Alexandersson May 11 '14 at 21:18
  • 1
    @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. – Nate Eldredge May 12 '14 at 06:33
  • 3

    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).

     – Mohammad Golshani May 18 '14 at 13:52
  • The present author uses this result to obtain some more or less obvious generalizations to the case of set-valued mappings. Here, he reports on rumors that both Cauty's and Dobrowolski's proofs might contain gaps (without giving any evidence for this claim) and he reconsiders his results under the assumption that the rumors should prove correct. – Mohammad Golshani May 18 '14 at 13:52
  • I have taken the following from the review of the following paper "Schuader'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→E has a fixed point.

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

     – Mohammad Golshani May 19 '14 at 07:12
  • 3
    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." – 57319 May 19 '14 at 15:09
  • 5
    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 – Johannes Hahn Mar 23 '15 at 14:58
  • Sehie Park also has a more recent publication, Remarks on fixed point and generalized vector equilibrium problems, Nonlinear Anal. Forum 20 (2015), 33–41; the summary, as found at MR3444052, goes, "In the first half of this paper, we introduce the contents of some of our previous papers on fixed point problems related to the Schauder conjecture. Some of them contain incorrect statements. The second half devotes to improve or correct the results in certain papers of other authors based on one of our incorrect statements related to a fixed point conjecture. [continued] – Gerry Myerson Jul 25 '16 at 23:13
  • [continued from previous comment] These results are mainly concerned with generalized vector equilibrium problems.'' – Gerry Myerson Jul 25 '16 at 23:14

4 Answers4

16

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.

Martin Sleziak
  • 4,608
Mohammad Golshani
  • 31,966
  • 3
    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." – François G. Dorais May 18 '14 at 09:58
  • You are right. In fact it seems the problem is still open. See my new answer. – Mohammad Golshani May 18 '14 at 15:12
12

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.

Martin Sleziak
  • 4,608
Jochen Wengenroth
  • 15,438
  • 2
    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. – Taras Banakh Aug 12 '16 at 19:24
  • Could you please indicate which result in the cited paper gives Schauder's conjecture? I am not able to locate it. – Doug Liu Apr 06 '22 at 13:04
  • Sorry, me neither. – Jochen Wengenroth Apr 06 '22 at 15:52
  • 3
    @TarasBanakh it might be useful if you reply to the above comments – YCor Oct 14 '22 at 15:24
  • 1
    @DougLiu The Schauder conjecture follows from Theoreme 8 combined with Theoreme 6 or Theoreme 7 in Cauty's paper. – Taras Banakh Oct 14 '22 at 19:23
6

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

YCor
  • 60,149
jaco
  • 161
  • 2
    The Cauty paper was published online in January 2015, available at https://www.degruyter.com/view/j/crll.ahead-of-print/crelle-2014-0134/crelle-2014-0134.xml A footnote says, "After submitting this work Robert Cauty passed away. The invaluable help of H. Steinlein (steinl@math.lmu.de) and T. Banakh (t.o.banakh@gmail.com) in going through the manuscript and making minor revisions is gratefully acknowledged." – Gerry Myerson Nov 03 '16 at 01:37
  • 1
    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. – Gerry Myerson Nov 03 '16 at 01:47
4

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."

Doug Liu
  • 433