View source for Schauder 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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Schauder,   "Der Fixpunktsatz in Funktionalräumen"  ''Stud. Math.'' , '''2'''  (1930)  pp. 171–180  {{MR|}}  {{ZBL|56.0355.01}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.A. Lyusternik,   V.I. Sobolev,   "Elements of functional analysis" , Hindushtan Publ. Comp.  (1974)  (Translated from Russian)  {{MR|0539144}} {{MR|0048693}} {{ZBL|0141.11601}} {{ZBL|0096.07802}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. General theory" , '''1''' , Interscience  (1958)  {{MR|0117523}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.E. Edwards,   "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)  {{MR|0221256}} {{ZBL|0182.16101}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L. Nirenberg,   "Topics in nonlinear functional analysis" , New York Univ. Inst. Math. Mech.  (1974)  {{MR|0488102}} {{ZBL|0286.47037}} </TD></TR></table>



====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 $([[#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 $, 
[[#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====
<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>