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Brouwer's fixed-point theorem: Under a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017670/b0176701.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017670/b0176702.png" />-dimensional simplex into itself there exists at least one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017670/b0176703.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017670/b0176704.png" />; this theorem was proved by L.E.J. Brouwer [[#References|[1]]]. An equivalent theorem had been proved by P.G. Bohl [[#References|[2]]] at a somewhat earlier date. Brouwer's theorem can be extended to continuous mappings of closed convex bodies in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017670/b0176705.png" />-dimensional topological vector space and is extensively employed in proofs of theorems on the existence of solutions of various equations. Brouwer's theorem can be generalized to infinite-dimensional topological vector spaces.
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==Brouwer's fixed-point theorem==
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Under a continuous mapping $f : S \rightarrow S$ of an $n$-dimensional simplex $S$ into itself there exists at least one point $x \in S$ such that $f(x) = x$; this theorem was proved by L.E.J. Brouwer [[#References|[1]]]. An equivalent theorem had been proved by P.G. Bohl [[#References|[2]]] at a somewhat earlier date. Brouwer's theorem can be extended to continuous mappings of closed convex bodies in an $n$-dimensional topological vector space and is extensively employed in proofs of theorems on the existence of solutions of various equations. Brouwer's theorem can be generalized to infinite-dimensional topological vector spaces.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.E.J. Brouwer,  "Ueber eineindeutige, stetige Transformationen von Flächen in sich"  ''Math. Ann.'' , '''69'''  (1910)  pp. 176–180</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Bohl,  "Ueber die Beweging eines mechanischen Systems in der Nähe einer Gleichgewichtslage"  ''J. Reine Angew. Math.'' , '''127'''  (1904)  pp. 179–276</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  L.E.J. Brouwer,  "Ueber eineindeutige, stetige Transformationen von Flächen in sich"  ''Math. Ann.'' , '''69'''  (1910)  pp. 176–180</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  P. Bohl,  "Ueber die Beweging eines mechanischen Systems in der Nähe einer Gleichgewichtslage"  ''J. Reine Angew. Math.'' , '''127'''  (1904)  pp. 179–276</TD></TR>
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</table>
  
  
  
 
====Comments====
 
====Comments====
There are many different proofs of the Brouwer fixed-point theorem. The shortest and conceptually easiest, however, use algebraic topology. Completely-elementary proofs also exist. Cf. e.g. [[#References|[a1]]], Chapt. 4. In 1886, H. Poincaré proved a fixed-point result on continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017670/b0176706.png" /> which is now known to be equivalent to the Brouwer fixed-point theorem, [[#References|[a2]]]. There are effective ways to calculate (approximate) Brouwer fixed points and these techniques are important in a multitude of applications including the calculation of economic equilibria, [[#References|[a1]]]. The first such algorithm was proposed by H. Scarf, [[#References|[a3]]]. Such algorithms later developed in the so-called homotopy or continuation methods for calculating zeros of functions, cf. e.g. [[#References|[a4]]], [[#References|[a5]]].
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There are many different proofs of the Brouwer fixed-point theorem. The shortest and conceptually easiest, however, use algebraic topology. Completely-elementary proofs also exist. Cf. e.g. [[#References|[a1]]], Chapt. 4. In 1886, H. Poincaré proved a fixed-point result on continuous mappings $f : \mathbf{E}^n \rightarrow \mathbf{E}^n$ which is now known to be equivalent to the Brouwer fixed-point theorem, [[#References|[a2]]]. There are effective ways to calculate (approximate) Brouwer fixed points and these techniques are important in a multitude of applications including the calculation of economic equilibria, [[#References|[a1]]]. The first such algorithm was proposed by H. Scarf, [[#References|[a3]]]. Such algorithms later developed in the so-called homotopy or continuation methods for calculating zeros of functions, cf. e.g. [[#References|[a4]]], [[#References|[a5]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Istrăţescu,  "Fixed point theory" , Reidel  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Poincaré,  "Sur les courbes definies par les équations différentielles"  ''J. de Math.'' , '''2'''  (1886)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Scarf,  "The approximation of fixed points of continuous mappings"  ''SIAM J. Appl. Math.'' , '''15'''  (1967)  pp. 1328–1343</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Karamadian (ed.) , ''Fixed points. Algorithms and applications'' , Acad. Press  (1977)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Allgower,  K. Georg,  "Simplicial and continuation methods for approximating fixed points and solutions to systems of equations"  ''SIAM Rev.'' , '''22'''  (1980)  pp. 28–85</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Istrăţescu,  "Fixed point theory" , Reidel  (1981)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Poincaré,  "Sur les courbes definies par les équations différentielles"  ''J. de Math.'' , '''2'''  (1886) {{ZBL|18.0314.01}}</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Scarf,  "The approximation of fixed points of continuous mappings"  ''SIAM J. Appl. Math.'' , '''15'''  (1967)  pp. 1328–1343</TD></TR>
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<TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Karamadian (ed.) , ''Fixed points. Algorithms and applications'' , Acad. Press  (1977)</TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Allgower,  K. Georg,  "Simplicial and continuation methods for approximating fixed points and solutions to systems of equations"  ''SIAM Rev.'' , '''22'''  (1980)  pp. 28–85</TD></TR>
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</table>
  
Brouwer's theorem on the invariance of domain: Under any homeomorphic mapping of a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017670/b0176707.png" /> of a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017670/b0176708.png" /> into a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017670/b0176709.png" /> of that space any interior point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017670/b01767010.png" /> (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017670/b01767011.png" />) is mapped to an interior point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017670/b01767012.png" /> (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017670/b01767013.png" />), and any non-interior point is mapped to a non-interior point. It was proved by L.E.J. Brouwer [[#References|[1]]].
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==Brouwer's theorem on the invariance of domain==
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Under any homeomorphic mapping of a subset $A$ of a Euclidean space $\mathbf{E}^n$ into a subset $B$ of that space any interior point of $A$ (with respect to $\mathbf{E}^n$) is mapped to an interior point of $B$ (with respect to $\mathbf{E}^n$), and any non-interior point is mapped to a non-interior point. It was proved by L.E.J. Brouwer [[#References|[1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.E.J. Brouwer,  "Ueber Abbildungen von Mannigfaltigkeiten"  ''Math. Ann.'' , '''71'''  (1912)  pp. 97–115</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  L.E.J. Brouwer,  "Ueber Abbildungen von Mannigfaltigkeiten"  ''Math. Ann.'' , '''71'''  (1912)  pp. 97–115</TD></TR>
 +
</table>
  
 
''M.I. Voitsekhovskii''
 
''M.I. Voitsekhovskii''
  
 
====Comments====
 
====Comments====
For a modern account of the Brouwer invariance-of-domain theorem cf. [[#References|[a1]]], Chapt. 7, Sect. 3. The result is important for the idea of the topological dimension (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017670/b01767014.png" />).
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For a modern account of the Brouwer invariance-of-domain theorem cf. [[#References|[a1]]], Chapt. 7, Sect. 3. The result is important for the idea of the topological dimension ($\dim \mathbf{E}^n = n$).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,  "Topology" , Allyn &amp; Bacon  (1966)  (Theorem 8.4)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,  "Topology" , Allyn &amp; Bacon  (1966)  (Theorem 8.4) {{ZBL|0144.21501}}</TD></TR>
 +
</table>

Latest revision as of 11:27, 17 March 2023


Brouwer's fixed-point theorem

Under a continuous mapping $f : S \rightarrow S$ of an $n$-dimensional simplex $S$ into itself there exists at least one point $x \in S$ such that $f(x) = x$; this theorem was proved by L.E.J. Brouwer [1]. An equivalent theorem had been proved by P.G. Bohl [2] at a somewhat earlier date. Brouwer's theorem can be extended to continuous mappings of closed convex bodies in an $n$-dimensional topological vector space and is extensively employed in proofs of theorems on the existence of solutions of various equations. Brouwer's theorem can be generalized to infinite-dimensional topological vector spaces.

References

[1] L.E.J. Brouwer, "Ueber eineindeutige, stetige Transformationen von Flächen in sich" Math. Ann. , 69 (1910) pp. 176–180
[2] P. Bohl, "Ueber die Beweging eines mechanischen Systems in der Nähe einer Gleichgewichtslage" J. Reine Angew. Math. , 127 (1904) pp. 179–276


Comments

There are many different proofs of the Brouwer fixed-point theorem. The shortest and conceptually easiest, however, use algebraic topology. Completely-elementary proofs also exist. Cf. e.g. [a1], Chapt. 4. In 1886, H. Poincaré proved a fixed-point result on continuous mappings $f : \mathbf{E}^n \rightarrow \mathbf{E}^n$ which is now known to be equivalent to the Brouwer fixed-point theorem, [a2]. There are effective ways to calculate (approximate) Brouwer fixed points and these techniques are important in a multitude of applications including the calculation of economic equilibria, [a1]. The first such algorithm was proposed by H. Scarf, [a3]. Such algorithms later developed in the so-called homotopy or continuation methods for calculating zeros of functions, cf. e.g. [a4], [a5].

References

[a1] V.I. Istrăţescu, "Fixed point theory" , Reidel (1981)
[a2] H. Poincaré, "Sur les courbes definies par les équations différentielles" J. de Math. , 2 (1886) Zbl 18.0314.01
[a3] H. Scarf, "The approximation of fixed points of continuous mappings" SIAM J. Appl. Math. , 15 (1967) pp. 1328–1343
[a4] S. Karamadian (ed.) , Fixed points. Algorithms and applications , Acad. Press (1977)
[a5] E. Allgower, K. Georg, "Simplicial and continuation methods for approximating fixed points and solutions to systems of equations" SIAM Rev. , 22 (1980) pp. 28–85

Brouwer's theorem on the invariance of domain

Under any homeomorphic mapping of a subset $A$ of a Euclidean space $\mathbf{E}^n$ into a subset $B$ of that space any interior point of $A$ (with respect to $\mathbf{E}^n$) is mapped to an interior point of $B$ (with respect to $\mathbf{E}^n$), and any non-interior point is mapped to a non-interior point. It was proved by L.E.J. Brouwer [1].

References

[1] L.E.J. Brouwer, "Ueber Abbildungen von Mannigfaltigkeiten" Math. Ann. , 71 (1912) pp. 97–115

M.I. Voitsekhovskii

Comments

For a modern account of the Brouwer invariance-of-domain theorem cf. [a1], Chapt. 7, Sect. 3. The result is important for the idea of the topological dimension ($\dim \mathbf{E}^n = n$).

References

[a1] J. Dugundji, "Topology" , Allyn & Bacon (1966) (Theorem 8.4) Zbl 0144.21501
How to Cite This Entry:
Brouwer theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brouwer_theorem&oldid=17262
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article