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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 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 q (H) be a mapping. There exists then a mapping T:XH which selects T * [i.e.: T(x)T * (x), for all xX], such that d(Tx,Ty)d H (T * (x),T * (y)), for all x,yX.

Theorem 2. Let T * :H q (H) be a nonexpansive map with Fix(T * ). There exists then a nonexpansive mapping T:HH which selects T * , such that Fix(T)=Fix(T * ).

The obtained facts complete the investigation of these concepts started in E. Kemajou, H.-P. Künzi and O. O. Otafudu [Topology Appl. 159, No. 9, 2463–2475 (2012; Zbl 1245.54023)].

MSC:
54H25Fixed-point and coincidence theorems in topological spaces
54C65Continuous selections