en>RjwilmsiBot: fixing page range dashes using AWB

2013-02-25T02:45:40Z

fixing page range dashes using AWB

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	~~Name~~: ~~Leonard Ingham~~<br>~~My age~~: 38<br>~~Country~~: ~~Great Britain~~<br>~~Town~~: ~~Pitagowan~~ <br>~~Postal code~~: ~~Ph18 7as~~<br>~~Address~~: ~~42 Tadcaster Rd~~<br>~~[http~~://~~www~~.~~restaurantcalcuta~~.~~com~~/~~outlet~~/~~ugg~~.~~asp?p~~=~~231 Ugg Boots Uggs Outlet Store~~],[http://~~peterlongogolfshow~~.com/~~coach~~/~~?key~~=~~coach~~-~~factory~~-~~outlet~~-~~online~~-~~store~~-~~sale-5 Coach Factory Online Store]~~.~~<br><br>Feel free to visit my page [~~http://www.~~bendtrapclub~~.com/~~cheap~~/~~ugg.asp Cheap Uggs Boots~~]		In [[mathematics]], the '''Earle–Hamilton fixed point theorem''' is a result in [[geometric function theory]] giving sufficient conditions for a [[holomorphic mapping]] of an open domain in a complex [[Banach space]] into itself to have a fixed point. The result was proved in 1968 by Clifford Earle and [[Richard Hamilton (mathematician)\|Richard Hamilton]] by showing that, with respect to the [[Carathéodory metric]] on the domain, the holomorphic mapping becomes a [[contraction mapping]] to which the [[Banach fixed-point theorem]] can be applied.

			==Statement==
			Let ''D'' be a connected open subset of a complex [[Banach space]] ''X'' and let ''f'' be a holomorphic mapping of ''D'' into itself such that:

			*the image ''f''(''D'') is bounded in norm;
			*the distance between points ''f''(''D'') and points in the exterior of ''D'' is bounded below by a positive constant.

			Then the mapping ''f'' has a unique fixed point ''x'' in ''D'' and if ''y'' is any point in ''D'', the iterates ''f''<sup>''n''</sup>(''y'') converge to ''x''.

			==Proof==
			Replacing ''D'' by an ε-neighbourhood of ''f''(''D''), it can be assumed that ''D'' is itself bounded in norm.

			For ''z'' in ''D'' and ''v'' in ''X'', set

			:<math>\displaystyle{\alpha(z,v)=\sup_g \|g^\prime(z)v\|,}</math>

			where the supremum is taken over all holomorphic functions ''g'' on ''D'' with \|''g''(''z'')\| < 1.

			Define the α-length of a piecewise differentiable curve γ:[0,1] <math>\rightarrow</math> ''D'' by

			:<math>\displaystyle{\ell(\gamma)=\int_0^1 \alpha(\gamma(t),\dot{\gamma}(t))\, dt.}</math>

			The Carathéodory metric is defined by

			:<math>\displaystyle{d(x,y)=\inf_{\gamma(0)=x,\gamma(1)=y} \ell(\gamma)}</math>

			for ''x'' and ''y'' in ''D''. It is a continuous function on ''D'' x ''D'' for the norm topology.

			If the diameter of ''D'' is less than ''R'' then, by taking suitable holomorphic functions ''g'' of the form

			:<math>\displaystyle{g(z)=a(z) + b}</math>

			with ''a'' in ''X''* and ''b'' in '''C''', it follows that

			:<math>\displaystyle{\alpha(z,v)\ge \\|v\\|/R,}</math>

			and hence that

			:<math>\displaystyle{d(x,y)\ge \\|x-y\\|/R.}</math>

			In particular ''d'' defines a metric on ''D''.

			The chain rule

			:<math>\displaystyle{(g\circ f)^\prime(z)=g^\prime(f(z))f^\prime(z)}</math>

			implies that

			:<math>\displaystyle{\alpha(f(z),f^\prime(z)v)\le \alpha(z,v)}</math>

			and hence ''f'' satisfies the following generalization of the [[Schwarz lemma#Schwarz–Pick theorem\|Schwarz-Pick inequality]]:

			:<math>\displaystyle{d(f(x),f(y))\le d(x,y).}</math>

			For δ sufficiently small and ''y'' fixed in ''D'', the same inequality can be applied to the holomorphic mapping

			:<math>\displaystyle{f_{\delta,y}(z)=f(z) +\delta (f(z)-f(y))}</math>

			and yields the improved estimate:

			:<math>\displaystyle{d(f(x),f(y))\le (1+\delta)^{-1} d(x,y).}</math>

			The Banach fixed-point theorem can be applied to the restriction of ''f'' to the closure of ''f''(''D'') on which ''d'' defines a complete metric, defining the same
			topology as the norm.

			==Other holomorphic fixed point theorems==
			In finite dimensions the existence of a fixed point can often be deduced from the [[Brouwer fixed point theorem]] without any appeal to holomorphicity of the mapping. In the case of [[bounded symmetric domain]]s with the [[Bergman metric]], {{harvtxt\|Neretin\|1996}} and {{harvtxt\|Clerc\|1999}} showed that the same scheme of proof as that used in the Earle-Hamilton theorem applies. The bounded symmetric domain ''D'' = ''G'' / ''K'' is a complete metric space for the Bergman metric. The open semigroup of the complexification ''G''<sub>''c''</sub> taking the closure of ''D'' into ''D'' acts by [[contraction mapping]]s, so again the Banach fixed-point theorem can be applied. Neretin extended this argument by continuity to some infinite-dimensional bounded symmetric domains, in particular the Siegel generalized disk of symmetric Hilbert-Schmidt operators with operator norm less than 1. The Earle-Hamilton theorem applies equally well in this case.

			==References==
			*{{citation\|last=Earle\|first= Clifford J.\|last2=Hamilton\|first2= Richard S.\|
			title=A fixed point theorem for holomorphic mappings\|year= 1970\|series=Proc. Sympos. Pure Math.\|volume= XVI\|pages= 61–65\|publisher =American Mathemetical Society}}
			*{{citation\|last=Harris\|first= Lawrence A.\|title=Fixed points of holomorphic mappings for domains in Banach spaces\|journal=Abstr. Appl. Anal.\|year= 2003\|volume= 5\| pages=261–274\|url=http://www.hindawi.com/journals/aaa/2003/121329/abs/}}
			*{{citation\|last=Neretin\|first= Y. A.\|title=Categories of symmetries and infinite-dimensional groups\|series= London Mathematical Society Monographs\|volume=16\|publisher= Oxford University Press\|year= 1996\|isbn=0-19-851186-8}}
			*{{citation\|last=Clerc\|first=Jean-Louis\|title=Compressions and contractions of Hermitian symmetric spaces\|journal=Math. Z.\|volume= 229\|year=1998\|pages=1–8\|url=http://www.springerlink.com/content/29yxnqe2ph9y4hhk/}}

			{{DEFAULTSORT:Earle-Hamilton fixed-point theorem}}
			[[Category:Functional analysis]]
			[[Category:Complex analysis]]
			[[Category:Mathematical theorems]]
			[[Category:Fixed-point theorems]]

en>Michael Hardy at 14:04, 16 June 2012

2012-06-16T14:04:49Z

New page

Name: Leonard Ingham My age: 38 Country: Great Britain Town: Pitagowan Postal code: Ph18 7as Address: 42 Tadcaster Rd [http://www.restaurantcalcuta.com/outlet/ugg.asp?p=231 Ugg Boots Uggs Outlet Store],[http://peterlongogolfshow.com/coach/?key=coach-factory-outlet-online-store-sale-5 Coach Factory Online Store]. Feel free to visit my page [http://www.bendtrapclub.com/cheap/ugg.asp Cheap Uggs Boots]

Triangular network coding - Revision history

en>RjwilmsiBot: fixing page range dashes using AWB

en>Michael Hardy at 14:04, 16 June 2012