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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]] | |||
Latest revision as of 03:45, 25 February 2013
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 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 fn(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
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] D by
The Carathéodory metric is defined by
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
with a in X* and b in C, it follows that
and hence that
In particular d defines a metric on D.
The chain rule
implies that
and hence f satisfies the following generalization of the Schwarz-Pick inequality:
For δ sufficiently small and y fixed in D, the same inequality can be applied to the holomorphic mapping
and yields the improved estimate:
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 domains with the Bergman metric, Template:Harvtxt and Template:Harvtxt 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 Gc taking the closure of D into D acts by contraction mappings, 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
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In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010