Fixed Point Theorem
Submitted by Structure on Sun, 11/11/2007 - 19:09.
Let (X,d) a complete metric space and f:X--->X a contraction (i.e. a function for which there is a real c, such as
Then there is a unique
such as
. Such a point
is called a fixed point for f.
Proof.
Unicity. Let two different fixed points.Then
. We have
contradiction!
Existence.
Let an arbitrary point and
If
then
is the unique fixed point.
Otherwise, let We shall prove that sequence
is a Cauchy sequence and as X is a complete metric space , the sequence
will be a convergent sequence.
We have
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Then we also have
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.
Now let .
if n is big enough as
As any contraction is a continuous function from we have
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From
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taking limit as p tends to we have also
which gives an estimation of the error made by replacing
by
.