Abstract
The celebrated Brouwer’s Fixed Point Theorem is dated in 1912. Its extension to compact set setting in Banach spaces due to Schauder appeared in 1930. Immediately it raised the question whether the Theorem can be extended to noncompact setting. The works of Kakutani, Klee, Benyamini and Sterfeld, Sternfeld and Lim solved the qualitative part of the problem. Lack of compactness makes the statement of the theorem false. However, there are some quantitative aspects of the question. The two basic are called minimal displacement problem, and optimal retraction problem. The aim of this article is to present the historical back ground and possibly, up to date state of investigations in this field. A list of open problems with comments will be discussed.
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References
M. Annoni, Retrazioni e minimo spostamento negli spazi di Banach, Degree Thesis, (2005).
Annoni M., Casini E.: An upper bound for the Lipschitz retraction constant in l 1. Studia Math. 180, 73–76 (2007)
J. Appell, N. Erzakova, S. Falcon Santana, M. Väth, On some Banach space constants arising in nonlinear fixed point and eigenvalue theory, Fixed Point Theory Appl. (2004), no. 4, 317–336.
Baronti M., Casini E., Franchetti C.: The retraction constant in some Banach spaces. J. Approx. Theory 120(2), 296–308 (2003)
Benyamini Y., Sternfeld Y.: Spheres in infinite dimensional normed spaces are Lipschitz contractible. Proc. Amer. Math. Soc. 88, 439–445 (1983)
Bohl P.: Über die Bewengung eines mechanishen Systems in der Nähe einer Gleichgewichtslage. J. Reine Angew. Math. 127, 179–276 (1904)
K. Bolibok, Minimal displacement and retraction problems for balls in Banach spaces, Mariae Curie-Sklodowska University, Thesis (1999)
Bolibok K.: Minimal displacement and retraction problems in infinite-dimensional Hilbert spaces. Proc. Amer. Math. Soc. 132(4), 1103–111 (2004)
Brouwer L.E.J.: Über Abbildungen von Mannigfaltigkeiten. Math. Ann. 71, 97–115 (1912)
K. Goebel, Concise course on fixed point theorems, Yokohama Publishers, (2002).
Goebel K.: On the minimal displacement of points under lipschitzian mappings. Pacific J. Math. 48, 151–163 (1973)
Goebel K., Kaczor W.: Remarks on failure of Schauder’s theorem in noncompact settings. Ann. Univ. Mariae Curie Skłodowska Sect. A 51, 99–108 (1997)
K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge University Press, (1990).
Goebel K., Marino G., Muglia L., Volpe R.: The retraction constant and minimal displacement characteristic of some Banach spaces. Nonlinear Anal. 67, 735–744 (2007)
K.Goebel, Ł. Piasecki, A new estimate for the optimal retraction constant, Banach and Function Spaces II, Yokohama Publishers (2008), 77–83.
Goebel K., Wośko J.: Making a hole in the space. Proc. Amer. Math. Soc. 114(2), 475–476 (1992)
Kakutani S.: Topological properties of the unit sphere of a Hilbert space. Proc. Imp. Acad. Tokyo 19, 269–271 (1943)
Kannai Y.: An elementary proof of no retraction theorem. Amer. Math. Monthly 88, 264–268 (1981)
W. A. Kirk, B. Sims, Hanbook on metric fixed point theory, Kluwer Academic Publishers, (2001).
Klee V.: Convex bodies and periodic homeomorphisms in Hilbert spaces. Trans. Amer. Math. Soc. 74, 10–43 (1953)
Klee V.: Some topological properties of convex sets. Trans. Amer. Math. Soc. 78, 30–45 (1955)
Komorowski T., Wośko J.: A remark on retracting of a ball ontoa sphere in an infinite dimensional Hilbert spase. Math. Scan. 67, 223–226 (1990)
Knaster B., Kuratowski K., Mazurkiewicz S.: Ein Beweis des Fixpunktsatz für n-dimensionale Simplexe. Fund. Math. 14, 132–137 (1929)
Lax P.D.: Change of variables in multiple integrals. Amer. Math. Monthly 106, 497–501 (1999)
Lin P.K., Sternfeld Y.: Convex sets with the Lipschitz fixed point property are compact. Proc. Amer. Math. Soc. 93, 633–639 (1985)
Mauldin D.: The Scottish Book: mathematical problems from the Scottish Cafe. Birkhäuser, Boston (1981)
Milnor J.: Analytic proof of “Hairy Ball Theorem” and Brouwer Fixed Point Theorem. Amer. Math. Monthly 85, 521–524 (1978)
Nowak B.: On the Lipschitz retraction of the unit ball in infinite dimensional Banach space onto boundary. Bull. Acad. Polon. Sci. 27, 861–864 (1979)
Ł. Piasecki, Retracting ball onto sphere in some Banach spaces, (preprint).
Piasecki Ł.: Retracting ball onto sphere in \({{BC_0}\, (\mathbb{R})}\) . Topol. Methods Nonlinear Anal. 33(2), 307–314 (2009)
J.J. Schaffer, Geometry of Spheres in Normed Spaces, Marcel Dekker (1976).
Wośko J.: An example related to the retraction problem. Ann. Univ. Mariae Curie-Skłodowska, Sect. A 45, 127–130 (1991)
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Lecture held by K. Goebel in the Seminario Matematico e Fisico on November 16, 2007.
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Casini, E., Goebel, K. Why and How Much Brouwer’s Fixed Point Theorem Fails in Noncompact Setting?. Milan J. Math. 78, 371–394 (2010). https://doi.org/10.1007/s00032-010-0135-2
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DOI: https://doi.org/10.1007/s00032-010-0135-2