\(q\)-hyperconvexity in quasipseudometric spaces and fixed point theorems
Let \((H,d)\) be a \(q\)-hyperconvex \(T_0\)-quasimetric space, and \({\mathcal E}_q(H)\) 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 \(T^*:X\to {\mathcal E}_q(H)\) 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 \(d(Tx,Ty)\leq d_H(T^*(x),T^*(y))\), for all \(x,y\in X\). Theorem 2. Let \(T^*:H\to {\mathcal E}_q(H)\) be a nonexpansive map with \(\text{Fix}(T^*)\neq \emptyset\). 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 \textit{E. Kemajou, H.-P. Künzi} and \textit{O. O. Otafudu} [Topology Appl. 159, No. 9, 2463--2475 (2012; Zbl 1245.54023)].
scientific article; zbMATH DE number 6102741
scientific article