- , for a given initial value,
This our experience can be formalized as follows: A mapping given by a Lipschitz continuous function with Lipschitz constant , is said to be a contraction. This is because
expressing that the distance between the images is smaller than the distance between the arguments .
For a given contraction mapping , the iteration
- , with a given initial value,
is referred to as fixed point iteration because will converge to a fixed point satisfying as n gets large. More precisely, we have the following basic result:
Contraction Mapping Theorem: If is a contraction with Lipschitz constant , then fixed point iteration converges to a unique fixed point with residual
and corresponding fixed point error
Proof: To prove uniqueness assume and are two fixed points, and conclude that
which is only possible if (since ).
Next, we have
from which follows
- , (1)
as stated. More generally, we have for , by adding and subtracting and so on,
from which follows that
Recalling now the following formula for the sum of a geometric series:
we have finally the following generalization of (1)
- for $m=1, 2,…$.
This estimate shows that the sequence , determines the decimals of a unique vector which satisfies . This proves that the contraction mapping has a unique fixed point, and the proof is complete.