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One of the [[Fixed point|fixed point]] theorems: If a completely-continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s0833201.png" /> maps a bounded closed convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s0833202.png" /> of a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s0833203.png" /> into itself, then there exists at least one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s0833204.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s0833205.png" />. Proved by J. Schauder [[#References|[1]]] as a generalization of the [[Brouwer theorem|Brouwer theorem]].
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One of the [[Fixed point|fixed point]] theorems: If a [[Completely-continuous_operator|completely-continuous operator]] $A$ maps a bounded closed convex set $K$ of a Banach space $X$ into itself, then there exists at least one point $x\in K$ such that $Ax=x$. Proved by J. Schauder [[#References|[1]]] as a generalization of the [[Brouwer theorem|Brouwer theorem]].
  
 
There exist different generalizations of Schauder's theorem: the Markov–Kakutani theorem, Tikhonov's principle, etc.
 
There exist different generalizations of Schauder's theorem: the Markov–Kakutani theorem, Tikhonov's principle, etc.
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====Comments====
 
====Comments====
The Tikhonov fixed-point theorem (also spelled Tychonoff's fixed-point theorem) states the following. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s0833206.png" /> be a locally convex topological space whose topology is defined by a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s0833207.png" /> of continuous semi-norms. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s0833208.png" /> be compact and convex and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s0833209.png" /> a continuous mapping. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s08332010.png" /> has a fixed point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s08332011.png" /> ([[#References|[a2]]]; [[#References|[a3]]], p. 175). Both the Kakutani fixed-point theorem and the Markov fixed-point theorem are generalized in the Ryll-Nardzewski fixed-point theorem, which states: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s08332012.png" /> be a Banach space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s08332013.png" /> a non-empty weakly compact subset. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s08332014.png" /> be a semi-group of mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s08332015.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s08332016.png" /> which is non-contracting, then there is a fixed point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s08332017.png" />. Here, a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s08332018.png" /> of mappings is said to have a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s08332019.png" /> if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s08332020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083320/s08332021.png" />, [[#References|[a4]]]; cf. [[#References|[a3]]], Chapt. 9, for a discussion of the Ryll-Nardzewski fixed-point theorem in relation to the Kakutani and Markov ones and other fixed-point theorems for families of mappings.
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The Tikhonov fixed-point theorem (also spelled Tychonoff's fixed-point theorem) states the following. Let $  X $
 +
be a locally convex topological space whose topology is defined by a family $  \{ p _{i} \} $
 +
of continuous semi-norms. Let $  C \subset X $
 +
be compact and convex and $  f : \  C \rightarrow C $
 +
a continuous mapping. Then $  f $
 +
has a fixed point in $  C $([[#References|[a2]]]; [[#References|[a3]]], p. 175). Both the Kakutani fixed-point theorem and the Markov fixed-point theorem are generalized in the Ryll-Nardzewski fixed-point theorem, which states: Let $  X $
 +
be a Banach space and $  Q $
 +
a non-empty weakly compact subset. Let $  S $
 +
be a semi-group of mappings from $  Q $
 +
to $  Q $
 +
which is non-contracting, then there is a fixed point of $  S $.  
 +
Here, a family $  S $
 +
of mappings is said to have a fixed point $  p $
 +
if for every $  f \in S $,  
 +
$  f(p) = p $,  
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[[#References|[a4]]]; cf. [[#References|[a3]]], Chapt. 9, for a discussion of the Ryll-Nardzewski fixed-point theorem in relation to the Kakutani and Markov ones and other fixed-point theorems for families of mappings.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,  A. Granas,  "Fixed-point theory" , '''I''' , PWN  (1982)  {{MR|0660439}} {{ZBL|0483.47038}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tychonoff,  "Ein Fixpunktsatz"  ''Math. Ann.'' , '''111'''  (1935)  pp. 767–776  {{MR|1513031}} {{ZBL|0012.30803}}  {{ZBL|61.1195.01}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Istrăţescu,  "Fixed point theory" , Reidel  (1981)  {{MR|0620639}} {{ZBL|0465.47035}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Ryll-Nardzewski,  "On fixed points of semi-groups of endomorphisms of linear spaces" , ''Proc. 5-th Berkeley Symp. Probab. Math. Stat.'' , '''2: 1''' , Univ. California Press  (1967)  pp. 55–61  {{MR|}} {{ZBL|}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,  A. Granas,  "Fixed-point theory" , '''I''' , PWN  (1982)  {{MR|0660439}} {{ZBL|0483.47038}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.N. [A.N. Tikhonov] Tychonoff,  "Ein Fixpunktsatz"  ''Math. Ann.'' , '''111'''  (1935)  pp. 767–776  {{MR|1513031}} {{ZBL|0012.30803}}  {{ZBL|61.1195.01}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Istrăţescu,  "Fixed point theory" , Reidel  (1981)  {{MR|0620639}} {{ZBL|0465.47035}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. Ryll-Nardzewski,  "On fixed points of semi-groups of endomorphisms of linear spaces" , ''Proc. 5-th Berkeley Symp. Probab. Math. Stat.'' , '''2: 1''' , Univ. California Press  (1967)  pp. 55–61  {{MR|}} {{ZBL|}} </TD></TR></table>

Latest revision as of 20:00, 28 January 2020


One of the fixed point theorems: If a completely-continuous operator $A$ maps a bounded closed convex set $K$ of a Banach space $X$ into itself, then there exists at least one point $x\in K$ such that $Ax=x$. Proved by J. Schauder [1] as a generalization of the Brouwer theorem.

There exist different generalizations of Schauder's theorem: the Markov–Kakutani theorem, Tikhonov's principle, etc.

References

[1] J. Schauder, "Der Fixpunktsatz in Funktionalräumen" Stud. Math. , 2 (1930) pp. 171–180 Zbl 56.0355.01
[2] L.A. Lyusternik, V.I. Sobolev, "Elements of functional analysis" , Hindushtan Publ. Comp. (1974) (Translated from Russian) MR0539144 MR0048693 Zbl 0141.11601 Zbl 0096.07802
[3] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[4] R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965) MR0221256 Zbl 0182.16101
[5] L. Nirenberg, "Topics in nonlinear functional analysis" , New York Univ. Inst. Math. Mech. (1974) MR0488102 Zbl 0286.47037


Comments

The Tikhonov fixed-point theorem (also spelled Tychonoff's fixed-point theorem) states the following. Let $ X $ be a locally convex topological space whose topology is defined by a family $ \{ p _{i} \} $ of continuous semi-norms. Let $ C \subset X $ be compact and convex and $ f : \ C \rightarrow C $ a continuous mapping. Then $ f $ has a fixed point in $ C $([a2]; [a3], p. 175). Both the Kakutani fixed-point theorem and the Markov fixed-point theorem are generalized in the Ryll-Nardzewski fixed-point theorem, which states: Let $ X $ be a Banach space and $ Q $ a non-empty weakly compact subset. Let $ S $ be a semi-group of mappings from $ Q $ to $ Q $ which is non-contracting, then there is a fixed point of $ S $. Here, a family $ S $ of mappings is said to have a fixed point $ p $ if for every $ f \in S $, $ f(p) = p $, [a4]; cf. [a3], Chapt. 9, for a discussion of the Ryll-Nardzewski fixed-point theorem in relation to the Kakutani and Markov ones and other fixed-point theorems for families of mappings.

References

[a1] J. Dugundji, A. Granas, "Fixed-point theory" , I , PWN (1982) MR0660439 Zbl 0483.47038
[a2] A.N. [A.N. Tikhonov] Tychonoff, "Ein Fixpunktsatz" Math. Ann. , 111 (1935) pp. 767–776 MR1513031 Zbl 0012.30803 Zbl 61.1195.01
[a3] V.I. Istrăţescu, "Fixed point theory" , Reidel (1981) MR0620639 Zbl 0465.47035
[a4] C. Ryll-Nardzewski, "On fixed points of semi-groups of endomorphisms of linear spaces" , Proc. 5-th Berkeley Symp. Probab. Math. Stat. , 2: 1 , Univ. California Press (1967) pp. 55–61
How to Cite This Entry:
Schauder theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schauder_theorem&oldid=28264
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article