# Contraction Mapping Theorem

We have seen that solving an equation with iteratively by time stepping or by Newton’s method, can be formulated as the iteration

- , with a given initial value,

where

- or .

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 intial value,

is referred to as* fixed point iteration *because will converge to a* fixed point* satisfying as 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 .

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.

Next: Generalized Fundamental Theorem Previous: From Residual to Root Error

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