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$q$-hyperconvexity in quasipseudometric spaces and fixed point theorems. (English) Zbl 1257.54040
Let $(H,d)$ be a $q$-hyperconvex $T_0$-quasimetric space, and ${\cal E}_q(H)$ stand for the set of all (nonempty) externally $q$-hyperconvex subsets of $(H,d)$. The following are the main results in the paper: Theorem 1. Let $X$ be a nonempty set, and $T^*:X\to {\cal E}_q(H)$ be a mapping. There exists then a mapping $T:X\to H$ which selects $T^*$ [i.e.: $T(x)\in T^*(x)$, for all $x\in X$], such that $d(Tx,Ty)\le d_H(T^*(x),T^*(y))$, for all $x,y\in X$. Theorem 2. Let $T^*:H\to {\cal E}_q(H)$ be a nonexpansive map with $\text{Fix}(T^*)\ne \emptyset$. There exists then a nonexpansive mapping $T:H\to H$ which selects $T^*$, such that $\text{Fix}(T)=\text{Fix}(T^*)$. The obtained facts complete the investigation of these concepts started in {\it E. Kemajou, H.-P. Künzi} and {\it O. O. Otafudu} [Topology Appl. 159, No. 9, 2463--2475 (2012; Zbl 1245.54023)].

54H25Fixed-point and coincidence theorems in topological spaces
54C65Continuous selections
Full Text: DOI
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