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Re: Fixed point theorem
Posted:
Jan 6, 2004 12:04 PM
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Joona I Palaste wrote:
> W. Dale Hall <wd_hall@pacbell.net> scribbled the following:
>
>>Charlie Johnson wrote:
>>
>>>Hi all,
>>>
>>>I have just read about the fixed point theorem, but I can't seem to figure
>>>out how to apply it yet.
>>>
>
>
>>Just an exercise in nitpicking:
>
>
>> There are several theorems that are fixed point theorems. Here
>> are the ones I came up with after a quick Google-aided search:
>
>
>> Brouwer Fixed Point Theorem
>> Lefshetz Fixed Point Theorem
>> Nielsen Fixed Point Theorem
>> (Leray-)Schauder Fixed Point Theorem
>> Banach Fixed Point Theorem
>> Kakutani Fixed Point Theorem
>>
>>I believe you are probably working with Brouwer's theorem, and could be
>>unaware of the existence of other fixed point theorems.
>
>
> I've only heard of Brouwer and Banach. Where could I find out more
> information about the others?
>
The fixed point theorem I know about is the Picard theorem - my guess is that
this is another name for the Banach theorem, but I am not sure. The Picard
fixed point theorem is the easiest of the lot, the one which applies to
contractions on complete metric spaces, and is probably the appropriate one for
this problem. In general, I think that the Picard theorem is the most useful,
for example, for proving existence and uniqueness of solutions to ODE which
satisfy a Lipschitz comdition.
But I can see that the Brouwer fixed point theorem could also be useful for our
problem - in one dimension it is very easy to prove (basically it is the
intermediate value theorem), and that is all one needs to prove existence (but
not uniquness) for the problem given here. It says that continuous functions on
the closed ball in n dimensions must have a fixed point.
The Schauder fixed point theorem is just an infinite dimensional version of the
Brouwer fixed point theorem - it applies to convex compact subsets of Banach
spaces (or maybe even complete locally convex spaces), and I think is even
proved by approximating to the Brouwer fixed point theorem.
I have seen the Picard fixed point theorem used to prove existence of local
solutions to the Navier-Stokes equation, and the Schauder fixed point theorem
used to prove existence of global solutions to the 2D Euler equation. I think
that both of these were done in seperate papers by Kato, although perhaps he
wasn't the first to do this.
I did study the Lefshetz fixed point theorem in college - as I recall it
requires knowing some homological properties of the mapping, related to the
Euler characteristic. The only use I know for it is to prove that the sphere
S^n cannot be given a topological group structure if n is even, but that is
because I am ignorant of its other applications.
I don't know the other fixed point theorems - perhaps someone who is
knowledgable about them could summarize.
Stephen
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