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If is a contraction of a non-empty closed set , i.e., if

- ,
- ,

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

(iv) Again, using the Lipschitz condition for ,

The fixed point is unique since

(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 |