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A fixed point theorem for multivalued mappings with \(\delta\)-distance. (English) Zbl 1469.54033

Summary: We mainly study fixed point theorem for multivalued mappings with \(\delta\)-distance using Wardowski’s technique on complete metric space. Let \((X, d)\) be a metric space and let \(B(X)\) be a family of all nonempty bounded subsets of \(X\). Define \(\delta : B(X) \times B(X) \rightarrow \mathbb{R}\) by \(\delta(A, B) = \text{sup} \left\{d(a, b) : a \in A, b \in B\right\} \). Considering \(\delta\)-distance, it is proved that if \((X, d)\) is a complete metric space and \(T : X \rightarrow B(X)\) is a multivalued certain contraction, then \(T\) has a fixed point.

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54C60 Set-valued maps in general topology
54E40 Special maps on metric spaces
54E50 Complete metric spaces
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