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In [[mathematics]], the '''Caristi fixed-point theorem''' (also known as the '''Caristi–Kirk fixed-point theorem''') generalizes the [[Banach fixed point theorem]] for maps of a [[complete space|complete]] [[metric space]] into itself. Caristi's fixed-point theorem is a variation of the ''ε''-[[Ekeland's variational principle|variational principle of]] [[Ivar Ekeland|Ekeland]] (1974, 1979). Moreover, the conclusion of Caristi's theorem is equivalent to metric completeness, as proved by Weston (1977). The original result is due to the mathematicians [[James Caristi]] and [[William Arthur Kirk]].
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==Statement of the theorem==
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Let (''X'', ''d'') be a complete metric space. Let ''T'' : ''X'' → ''X'' and ''f'' : ''X'' → [0, +∞) be a [[lower semicontinuous]] function from ''X'' into the non-negative [[real numbers]]. Suppose that, for all points ''x'' in ''X'',
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:<math>d \big( x, T(x) \big) \leq f(x) - f \big( T(x) \big).</math>
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Then ''T'' has a fixed point in ''X'', i.e. a point ''x''<sub>0</sub> such that ''T''(''x''<sub>0</sub>)&nbsp;=&nbsp;''x''<sub>0</sub>.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


==References==
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


* {{cite journal
'''source'''
|    last = Caristi
:<math forcemathmode="source">E=mc^2</math> -->
|    first = James
|    title = Fixed point theorems for mappings satisfying inwardness conditions
|  journal = [[Transactions of the American Mathematical Society|Trans. Amer. Math. Soc.]]
|  volume = 215
|    year = 1976
|    pages = 241&ndash;251
|    issn = 0002-9947
|      doi = 10.2307/1999724
|      jstor = 1999724
}}
* {{cite journal
|    doi = 10.1016/0022-247X(74)90025-0
|    last = Ekeland
|    first = Ivar
|    title = On the variational principle
|  journal = J. Math. Anal. Appl.
|  volume = 47
|    year = 1974
|    pages = 324&ndash;353
|    issn = 0022-247x
|    issue = 2
}}
* {{cite journal
|    last = Ekeland
|    first = Ivar
|    title = Nonconvex minimization problems
|  journal = [[Bulletin of the American Mathematical Society|Bull. Amer. Math. Soc. (N.S.)]]
|  volume = 1
|    year = 1979
|    issue = 3
|    pages = 443&ndash;474
|    issn = 0002-9904
|      doi = 10.1090/S0273-0979-1979-14595-6
}}
* {{cite journal
|    last = Weston
|    first = J. D.
|    title = A characterization of metric completeness
|  journal = [[Proceedings of the American Mathematical Society|Proc. Amer. Math. Soc.]]
|  volume = 64
|    year = 1977
|    issue = 1
|    pages = 186&ndash;188
|    issn = 0002-9939
|      doi = 10.2307/2041008
|      jstor = 2041008
}}


[[Category:Fixed-point theorems]]
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
[[Category:Theorems in real analysis]]
[[Category:Metric geometry]]


[[it:Teorema di Caristi]]
==Demos==
[[nl:Dekpuntstelling van Caristi]]
 
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
 
 
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
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** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.
 
==Test pages ==
 
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
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Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

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Test pages

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Bug reporting

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