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Difference between revisions of "Schauder theorem"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Schauder,  "Der Fixpunktsatz in Funktionalräumen"  ''Stud. Math.'' , '''2'''  (1930)  pp. 171–180</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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</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)</TD></TR></table>
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<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>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,  A. Granas,  "Fixed-point theory" , '''I''' , PWN  (1982)</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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Istrăţescu,  "Fixed point theory" , Reidel  (1981)</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</TD></TR></table>
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<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>

Revision as of 12:12, 27 September 2012

One of the fixed point theorems: If a completely-continuous operator maps a bounded closed convex set of a Banach space into itself, then there exists at least one point such that . 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 be a locally convex topological space whose topology is defined by a family of continuous semi-norms. Let be compact and convex and a continuous mapping. Then has a fixed point in ([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 be a Banach space and a non-empty weakly compact subset. Let be a semi-group of mappings from to which is non-contracting, then there is a fixed point of . Here, a family of mappings is said to have a fixed point if for every , , [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=18281
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article