Fixed Point Theorem
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.
Unicity. Let two different fixed points.Then . We have
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.
Then we also have
Now let .
if n is big enough as
As any contraction is a continuous function from we have
taking limit as p tends to we have also
which gives an estimation of the error made by replacing by .