Equivalence of completeness and contraction property. (English) Zbl 1121.54049
The author shows that in a metric space, the contraction property implies Lipschitz-completeness or arcwise completeness, and the contraction property does not imply the usual completeness. The author proves that a locally Lipschitz-connected metric space has the contraction property if and only if it is Lipschitz-complete, and that a locally arcwise-connected metric space \(X\) is arcwise-complete if and only if \(X\) has the strong contraction property.
Reviewer: Palaniappan Vijayaraju (Chennai)
MSC:
| 54E50 | Complete metric spaces |
| 54E40 | Special maps on metric spaces |
| 54E35 | Metric spaces, metrizability |
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 47H10 | Fixed-point theorems |
References:
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