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Supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings. (English) Zbl 1151.54010

Let \((X,d)\) be a metric space. Let \({\mathcal C}{\mathcal B}(X)\) be the collection of fuzzy sets \(\mu\), such that the \(\alpha\)-cut of \(\mu\) is a nonempty bounded closed set in \(X\), and let \(d_\infty\) be the metric for \({\mathcal C}{\mathcal B}(X)\) where \(d_\infty\) is induced by the Hausdorff metric. The authors establish that if \((X,d)\) is complete, then \(({\mathcal C}{\mathcal B}(X))\) is also a complete metric space. Some common fixed point theorems for fuzzy mappings are proved.

MSC:

54A40 Fuzzy topology
54E35 Metric spaces, metrizability
03E72 Theory of fuzzy sets, etc.
54H25 Fixed-point and coincidence theorems (topological aspects)
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