q-Hyperconvexity in Quasipseudometric Spaces and Fixed Point Theorems

Abstract EN

In a previous work, we started investigating the concept of hyperconvexity in quasipseudometric spaces which we called q-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 of q-hyperconvex spaces and also present some fixed point theorems for nonexpansive self-maps on a bounded q-hyperconvex quasipseudometric space.

In analogy with a metric result, we show among other things that a set-valued mapping T∗ on a q-hyperconvex T0-quasimetric space (X, d) which takes values in the space of nonempty externally q-hyperconvex subsets of (X, d) always has a single-valued selection T which satisfies d(T(x),T(y))≤dH(T∗(x),T∗(y)) whenever x,y∈X.

(Here, dH denotes the usual (extended) Hausdorff quasipseudometric determined by d on the set ?0(X) of nonempty subsets of X.)