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Mathematics-Online lexicon: Annotation to
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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 .
(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 .
|automatisch erstellt am 22. 9. 2016|