Contraction Mapping Principle

Contraction Mapping Principle

Definition

Banach Fixed Point Theorem

Proof Outline

1. Cauchy sequence: , so converges, making Cauchy
2. Convergence: Complete space
3. Fixed point:
4. Uniqueness: If , then , so

Example

Solve :

Let . Then on .

Starting from : , , ,

Applications to Differential Equations

Applications to Linear Systems

Convergence Rate

Linear convergence: error decreases by factor each step.

Generalizations

Applications

Banach's Lemma

This Neumann series converges and gives the inverse.

Error Bounds