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Fixed point theorems and characterizations of metric completeness. (English) Zbl 0902.47050
While discussing fixed point theorems on complete metric spaces, the authors focus their attention on characterization of metric completeness and derive the following theorems in this connection:
(1) Let \(X\) be a metric space. Then \(X\) is complete if and only if every weakly contractive mapping from \(X\) into itself has a fixed point in \(X\);
(2) Let \(X\) be a normed linear space and let \(D\) be a convex subset of \(X\). Then \(D\) is complete if and only if every contractive mapping from \(D\) into itself has a fixed point in \(D\).

MSC:
47H10 Fixed-point theorems
54E50 Complete metric spaces
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