en>Schmock: Template:Stochastic processes added

2012-09-29T12:18:45Z

← Older revision		Revision as of 14:18, 29 September 2012
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	~~Hi there~~. ~~Let me begin by introducing~~ the ~~writer~~, ~~her title~~ is ~~Sophia Boon but she~~ by ~~no means truly favored that name~~. It'~~s not~~ a ~~typical factor but what I like doing is to climb but I don~~'~~t have~~ the ~~time lately~~. ~~Her family life~~ in ~~Ohio~~. ~~My working day occupation is an invoicing officer but I~~'~~ve already applied for an additional one~~.<br><br>~~Look at~~ [~~http://www~~.~~taehyuna~~.~~net~~/xe/~~?document_srl~~=~~78721 free online tarot card readings] my webpage~~ ... ~~tarot card~~ [~~http://jplusfn~~.~~gaplus~~.~~kr/xe/qna/78647 accurate psychic readings~~]~~, [http://www~~.~~familysurvivalgroup.com~~/~~easy~~-~~methods~~-~~planting~~-~~looking~~-~~backyard~~/ ~~http~~:~~//www.familysurvivalgroup.com~~],		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]].

			==Statement of the theorem==
			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'',

			:<math>d \big( x, T(x) \big) \leq f(x) - f \big( T(x) \big).</math>

			Then ''T'' has a fixed point in ''X'', i.e. a point ''x''<sub>0</sub> such that ''T''(''x''<sub>0</sub>) = ''x''<sub>0</sub>.

			==References==

			* {{cite journal
			\| last = Caristi
			\| 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 = 241–251
			\| 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 = 324–353
			\| 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 = 443–474
			\| 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 = 186–188
			\| issn = 0002-9939
			\| doi = 10.2307/2041008
			\| jstor = 2041008
			}}

			[[Category:Fixed-point theorems]]
			[[Category:Theorems in real analysis]]
			[[Category:Metric geometry]]

66.159.228.210: /* See also */

2012-03-16T06:23:17Z

Telegraph process - Revision history

en>Schmock: Template:Stochastic processes added

66.159.228.210: /* See also */