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#1 2009-03-10 23:25:17

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Prove fixed point


This is an adaptation of a problem I came across on another forum. The problem was originally stated more generally as follows: if

is a compact subset of
with the usual topology and
is continuous with
, then
has a fixed point. Unfortunately this is not quite true:
needs to be connected as well as compact. Since, by the Heine–Borel theorem, any compact and connected subset of
is a closed and bounded interval, I have chosen the interval
for convenience. smile

Note also a slight difference in this problem from the originally stated one. In the original problem, the domain of

is a subset of the range of
. In my adaptation of the problem, the reverse is the case.

Last edited by JaneFairfax (2009-03-25 16:56:17)


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#2 2009-03-11 02:40:17

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Prove fixed point

Last edited by mathsyperson (2009-03-11 04:24:31)


Why did the vector cross the road?
It wanted to be normal.

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#3 2009-03-11 04:14:21

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Prove fixed point


Apart from the fact that you had
amd
the wrong way round, you are correct.

Applause.gif


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A: Click here for answer.

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#4 2009-03-11 04:23:24

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Prove fixed point

Ah, so I have. Fixed now, thanks.
I wrote g(x) differently to how I meant it, which is what caused those intervals to be the wrong way round.


Why did the vector cross the road?
It wanted to be normal.

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