From: yiou@asterix.saclay.cea.fr (Pascal Yiou 77.28)
Newsgroups: sci.math
Subject: Re: fixed pt. thm?
Date: 13 Feb 1995 10:44:27 GMT
(* Mark Mak *) (s925959@hp9000.csc.cuhk.hk) wrote:
: Hi all,
: i heard that there is a so-called fixed point theorem.
: i.e. stating the necessary and sufficient conditions for
: a function whether has a fixed pt. or not?
: i want to know what is it about and it's proof as well if possible.p
: and what is it's significance?
: thanks.
: -- Mark
:
Hi!
I don't know about *necessary* conditions, but generally speaking, there
is Schauder's theorem that states that:
if B is a Banach space (possibly infinite dimentional), if C is convex
and compact (and nonempty), if T is a continuous function from C
to C, then T has a fixed point.
You can find a proof of this theorem in the book of Gilbarg and Trudinger,
Partial differential equations of second order, Springer (1983).
In fact, this theorem is used to prove the existence of nonlinear partial
differential equations (in a rather non constructive way): the above
T is then a differential operator on some functional space. And note
that it doen not say anything about the uniqueness (i.e. where the
fixed point is located).
I hope this helps. Cheers,
[sig deleted - djr]
==============================================================================
From: indmat1@convex.edvz.uni-linz.ac.at (Helmut Zeisel)
Newsgroups: sci.math
Subject: Re: FIXED POINT THEOREM
Date: 15 Feb 1995 08:40:54 GMT
In article <3hjm7t$bm5@news.iastate.edu>, abian@iastate.edu (Alexander Abian) writes:
|>
|> In the literature there is the following ABIAN-BROWN Fixed Point Theorem
|> generalizing Brouwer's fixed point theorem.
|>
|> Let B be a closed n-Ball in R^n . Let f be a continuous
|> mapping from B into R^n such that f maps the boundary of B into
|> B. Then f has a fixed point.
This is the finite dimensional case of the ROTHE Fixed Point Theorem (1937)
(Theorem 4.2.3 in
D. R. Smart: Fixed point theorems
Cambridge University Press 1974, ISBN 0 521 20289 2),
generalizing Schauder's Fixed Point Theorem
(a few years before Abian, I think):
Let B be a normed space, M the closed unit ball in B
and dM the unit sphere in B. Let T be a continuous compact mapping
of M into B such that T(dM) \subseteq M.
Then T has a fixed point.
Helmut Zeisel
==============================================================================
Newsgroups: sci.math
From: gordon@atria.com (Gordon McLean Jr.)
Subject: Re: fixed pt. thm?
Date: Fri, 17 Feb 1995 15:55:09 GMT
Rob De Villiers (rdv@planet.bt.co.uk) wrote:
: >
: >>(* Mark Mak *) (s925959@hp9000.csc.cuhk.hk) wrote:
: >>: Hi all,
: >>: i heard that there is a so-called fixed point theorem.
: >>: i.e. stating the necessary and sufficient conditions for
: >>: a function whether has a fixed pt. or not?
: >>: i want to know what is it about and it's proof as well if possible.p
: >>: and what is it's significance?
: >>: thanks.
: >>
: >>: -- Mark
: >>:
: >>Hi!
: >>I don't know about *necessary* conditions, but generally speaking, there
: >>is Schauder's theorem that states that:
: >>if B is a Banach space (possibly infinite dimentional), if C is convex
: >>and compact (and nonempty), if T is a continuous function from C
: >>to C, then T has a fixed point.
: >
: >I suspect the questioner hs in mind Brouwer's fixed point theorem:
: >if F maps a closed, connected, compact(? I'm pretty rusty) set into (or onto)
: >itself the F has a fixed pt. F(a) = a. Try any elementary point set
: >topology book. Apparently applications to the many body problem etc...
: >
: >Rob.
: I just got the following mail from Dave Rusin:
: > The domain and range for Brouwer's FPT have to be (homeomorphic to)
: > the closed unit disc in R^n. The conditions you posted are
: > insufficient, as is shown by a rotation of the circle.
: Absolutely right, i posted on the fly, giving it insufficient thought ( hence
: the "?" i included).
Actually, didn't Lefschetz generalize this, so that the FPT applies
whenever the domain (and co-domain) is a compact acyclic triangulable
space? The case where the domain is the "closed unit disc in R^n",
i.e. Brouwer's theorem, is a special case of this.
==============================================================================
From: mmurray@spam.maths.adelaide.edu.au (Michael Murray)
Newsgroups: sci.math
Subject: Re: fixed pt. thm?
Date: Fri, 17 Feb 1995 21:19:38 +1030
In article <3hg8ti$5r9@hpg30a.csc.cuhk.hk>, s925959@hp9000.csc.cuhk.hk ((*
Mark Mak *)) wrote:
> Hi all,
> i heard that there is a so-called fixed point theorem.
> i.e. stating the necessary and sufficient conditions for
> a function whether has a fixed pt. or not?
> i want to know what is it about and it's proof as well if possible.p
> and what is it's significance?
> thanks.
>
> -- Mark
>
There are lots of fixed point theorems but one of the most
useful relates to contractions. A map T : X \to X
where X is a metric space is a contraction if
d(T(x), T(y)) < K d(x, y) where 0 < K < 1. Then the
theorem says that if T is a contraction and X is a
complete metric space then T has a unique fixed point.
The proof is quite neat you start with any x \in X and
consider the sequence x_1 = T(x) , x_2 = T(x_1) etc. You can
show that this is Cauchy and hence ( by completeness)
converges to something which is a fixed point.
Uniqueness is easy as if T(x) = x and T(y) = y then
d(x,y) = d(T(x), T(y)) < K d(x,y) therefore d(x,y) = 0.
Its usually called the contraction mapping theorem or
Banachs fixed point theorem. It should be in most
functional analysis books. Its used to prove
Picards theorem on existence of ode's. Find solutions
to integral equations. Prove the inverse function
theorem etc etc.
Michael
% Michael Murray %
% Department of Pure Mathematics %
% University of Adelaide Fax: 61+ 8 232 5670 %
% Adelaide SA 5005 Phone: 61+ 8 303 4174 %
% Australia Email: mmurray@maths.adelaide.edu.au %