q-Hyperconvexity in Quasipseudometric Spaces and Fixed Point Theorems

In a previous work, we started investigating the concept of hyperconvexity in quasipseudometric spaces which we calledq-hyperconvexity or Isbell-convexity. In this paper, we continue our studies of this concept, generalizing further known results about hyperconvexity from the metric setting to our theory. In particular, in the present paper, we consider subspaces ofq-hyperconvex spaces and also present some fixed point theorems for nonexpansive self-maps on a boundedq-hyperconvex quasipseudometric space. In analogy with a metric result, we show among other things that a set-valued mappingT∗on aq-hyperconvexT0-quasimetric space (X, d) which takes values in the space of nonempty externallyq-hyperconvex subsets of (X, d) always has a single-valued selectionTwhich satisfiesd(T(x),T(y))≤dH(T∗(x),T∗(y))wheneverx,y∈X. (Here,dHdenotes the usual (extended) Hausdorff quasipseudometric determined bydon the set𝒫0(X)of nonempty subsets ofX.)