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If is a contraction of a non-empty closed set , i.e., if
• ,
• ,
with , then has a unique fixed point . Starting with any point , can be approximated by the sequence

The error satisfies

i.e., the iteration converges linearly.

More generally, the fixed point theorem holds in complete metric spaces. Since the proof neither requires translation invariance nor homogeneity of the norm, can be replaced by a general distance function .

The proof is divided into several steps.

(i) Since , for all .

(ii) Iteration of the inequality

(iii) With the aid of the triangle inequality we obtain

establishing Cauchy covergence of the sequence to a limit .

(iv) Again, using the Lipschitz condition for ,

Passing to the limit for shows that is a fixed point of .

The fixed point is unique since

with .

(vi) Finally, we obtain the estimate for the error by letting tend to in the inequality (iii) for .

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 automatisch erstellt am 22.  9. 2016