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Volume 2012 |Article ID 765903 | https://doi.org/10.1155/2012/765903

Hans-Peter A. Künzi, Olivier Olela Otafudu, "q-Hyperconvexity in Quasipseudometric Spaces and Fixed Point Theorems", Journal of Function Spaces, vol. 2012, Article ID 765903, 18 pages, 2012. https://doi.org/10.1155/2012/765903

q-Hyperconvexity in Quasipseudometric Spaces and Fixed Point Theorems

Academic Editor: Salvador Romaguera
Received15 May 2012
Accepted04 Jul 2012
Published13 Sep 2012

Abstract

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,𝑦𝑋. (Here, 𝑑𝐻 denotes the usual (extended) Hausdorff quasipseudometric determined by 𝑑 on the set 𝒫0(𝑋) of nonempty subsets of X.)

1. Introduction

In a previous work, we started investigating a concept of hyperconvexity in quasipseudometric spaces, which we called 𝑞-hyperconvexity or Isbell-convexity (see [1], compare [2]). In this paper, we continue our studies of this concept by generalizing further known results about hyperconvexity from the metric setting to our theory. Among other things, in the present paper we consider subspaces of 𝑞-hyperconvex spaces and also present some fixed point theorems. In particular, we show 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,𝑦𝑋. (Here, 𝑑𝐻 denotes the usual (extended) Hausdorff quasipseudometric determined by 𝑑 on the set 𝒫0(𝑋) of nonempty subsets of 𝑋.)

Our investigations confirm the surprising fact that many classical results about hyperconvexity in metric spaces do not make essential use of the symmetry of the metric and, therefore, still hold—in a sometimes slightly modified form—for our concept of 𝑞-hyperconvexity in quasipseudometric spaces (see also [3] for a more general approach).

For the basic facts concerning quasipseudometrics and quasiuniformities we refer the reader to [4, 5]. Some recent work about quasipseudometric spaces can be found in the articles [69].

2. Preliminaries

In order to fix the terminology, we start with some basic concepts.

Definition 2.1. Let 𝑋 be a set and let 𝑑𝑋×𝑋[0,) be a function mapping into the set [0,) of the nonnegative reals. Then, 𝑑 is called a quasipseudometric on 𝑋 if(a)𝑑(𝑥,𝑥)=0 whenever 𝑥𝑋,(b)𝑑(𝑥,𝑧)𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧) whenever 𝑥,𝑦,𝑧𝑋.
We will say that 𝑑 is a 𝑇0-quasimetric provided that 𝑑 also satisfies the following condition: for each 𝑥,𝑦𝑋, 𝑑(𝑥,𝑦)=0=𝑑(𝑦,𝑥)impliesthat𝑥=𝑦.(2.1)

Remark 2.2. Let 𝑑 be a quasipseudometric on a set 𝑋, then 𝑑1𝑋×𝑋[0,) defined by 𝑑1(𝑥,𝑦)=𝑑(𝑦,𝑥) whenever 𝑥,𝑦𝑋 is also a quasipseudometric, called the conjugate quasipseudometric of 𝑑. As usual, a quasipseudometric 𝑑 on 𝑋 such that 𝑑=𝑑1 is called a pseudometric. Note that for any (𝑇0)-quasipseudometric 𝑑, the function 𝑑𝑠=max{𝑑,𝑑1}=𝑑𝑑1 is a pseudometric (metric).
For any 𝑎,𝑏[0,), we will set 𝑎̇𝑏=max{𝑎𝑏,0}.
Let (𝑋,𝑑) be a quasipseudometric space. For each 𝑥𝑋 and 𝜖>0, the set 𝐵𝑑(𝑥,𝜖)={𝑦𝑋𝑑(𝑥,𝑦)<𝜖} denotes the open 𝜖-ball at 𝑥. The collection of all “open” balls yields a base for a topology 𝜏(𝑑). It is called the topology induced by 𝑑 on 𝑋. Similarly, for each 𝑥𝑋 and 𝜖0, we define the ball 𝐶𝑑(𝑥,𝜖)={𝑦𝑋𝑑(𝑥,𝑦)𝜖}. Note that this latter set is 𝜏(𝑑1)-closed, but not 𝜏(𝑑)-closed in general. As usual, in the theory of quasiuniformities, for a subset 𝐴 of 𝑋 and 𝜖>0, we will also use notations like 𝐵𝑑(𝐴,𝜖)=𝑎𝐴𝐵𝑑(𝑎,𝜖) and similarly 𝐶𝑑(𝐴,𝜖)=𝑎𝐴𝐶𝑑(𝑎,𝜖).
A pair (𝐶𝑑(𝑥,𝑟);𝐶𝑑1(𝑥,𝑠)) with 𝑥𝑋 and nonnegative reals 𝑟,𝑠 will be called a double ball at 𝑥.
We shall also speak of a family [(𝐶𝑑(𝑥𝑖,𝑟𝑖))𝑖𝐼;(𝐶𝑑1(𝑥𝑖,𝑠𝑖))𝑖𝐼] of double balls, with 𝑥𝑖𝑋 and 𝑟𝑖,𝑠𝑖0 whenever 𝑖𝐼.
Let (𝑋,𝑑) be a quasipseudometric space and let 𝒫0(𝑋) be the set of all nonempty subsets of 𝑋. Given 𝐶𝒫0(𝑋), we will set dist(𝑥,𝐶)=inf{𝑑(𝑥,𝑐)𝑐𝐶} and dist(𝐶,𝑥)=inf{𝑑(𝑐,𝑥)𝑐𝐶} whenever 𝑥𝑋.
For any 𝐴,𝐵𝒫0(𝑋), we set 𝑑𝐻(𝐴,𝐵)=maxsup𝑏𝐵dist(𝐴,𝑏),sup𝑎𝐴dist(𝑎,𝐵)(2.2) (compare [10]).
Then 𝑑𝐻, is the so-called extended (as usual, a quasipseudometric that maps into [0,] (instead of [0,)) will be called extended)  Hausdorff(-Bourbaki) quasipseudometric on 𝒫0(𝑋). It is known that 𝑑𝐻 is an extended 𝑇0-quasimetric when restricted to the set of all the nonempty subsets 𝐴 of 𝑋 which satisfy 𝐴=cl𝜏(𝑑)𝐴cl𝜏(𝑑1)𝐴 (compare [11, page 164]).
A map 𝑓(𝑋,𝑑)(𝑌,𝑒) between two quasipseudometric spaces (𝑋,𝑑) and (𝑌,𝑒) is called an isometry or isometric map provided that 𝑒(𝑓(𝑥),𝑓(𝑦))=𝑑(𝑥,𝑦) whenever 𝑥,𝑦𝑋. Two quasipseudometric spaces (𝑋,𝑑) and (𝑌,𝑒) will be called isometric provided that there exists a bijective isometry 𝑓(𝑋,𝑑)(𝑌,𝑒). A map 𝑓(𝑋,𝑑)(𝑌,𝑒) between two quasipseudometric spaces (𝑋,𝑑) and (𝑌,𝑒) is called nonexpansive provided that 𝑒(𝑓(𝑥),𝑓(𝑦))𝑑(𝑥,𝑦) whenever 𝑥,𝑦𝑋.
The following definitions can be found in [1] (compare [12]).

Definition 2.3 (see [1, Definition 2]). A quasipseudometric space (𝑋,𝑑) is called 𝑞-hyperconvex (or Isbell-convex) provided that for each family (𝑥𝑖)𝑖𝐼 of points in 𝑋 and families (𝑟𝑖)𝑖𝐼 and (𝑠𝑖)𝑖𝐼 of nonnegative real numbers satisfying 𝑑(𝑥𝑖,𝑥𝑗)𝑟𝑖+𝑠𝑗 whenever 𝑖,𝑗𝐼, the following condition holds: 𝑖𝐼𝐶𝑑𝑥𝑖,𝑟𝑖𝐶𝑑1𝑥𝑖,𝑠𝑖.(2.3)

Definition 2.4 (see [1, Definition 5]). Let (𝑋,𝑑) be a quasipseudometric space. A family of double balls [(𝐶𝑑(𝑥𝑖,𝑟𝑖))𝑖𝐼;(𝐶𝑑1(𝑥𝑖,𝑠𝑖))𝑖𝐼] with 𝑟𝑖,𝑠𝑖[0,) and 𝑥𝑖𝑋 whenever 𝑖𝐼 is said to have the mixed binary intersection property if for all indices 𝑖,𝑗𝐼,𝐶𝑑(𝑥𝑖,𝑟𝑖)𝐶𝑑1(𝑥𝑗,𝑠𝑗).

Definition 2.5 (see [1, Definition 6]). A quasipseudometric space (𝑋,𝑑) is called 𝑞-hypercomplete (or Isbell-complete) if every family 𝐶𝑑𝑥𝑖,𝑟𝑖𝑖𝐼;𝐶𝑑1𝑥𝑖,𝑠𝑖𝑖𝐼(2.4) of double balls, where 𝑟𝑖,𝑠𝑖0 and 𝑥𝑖𝑋 whenever 𝑖𝐼, having the mixed binary intersection property satisfies 𝑖𝐼(𝐶𝑑(𝑥𝑖,𝑟𝑖)𝐶𝑑1(𝑥𝑖,𝑠𝑖)).

Definition 2.6 (see [1, Definition 4]). Let (𝑋,𝑑) be a quasipseudometric space. We say that 𝑋 is metrically convex if for any points 𝑥,𝑦𝑋 and nonnegative real numbers 𝑟 and 𝑠 such that 𝑑(𝑥,𝑦)𝑟+𝑠, there exists 𝑧𝑋 such that 𝑑(𝑥,𝑧)𝑟 and 𝑑(𝑧,𝑦)𝑠.
The following useful result was established in [1, Proposition 1]. A quasipseudometric space (𝑋,𝑑) is 𝑞-hyperconvex if and only if it is metrically convex and 𝑞-hypercomplete.
As usual, a subset 𝐴 of a quasipseudometric space (𝑋,𝑑) will be called bounded provided that there is a positive real constant 𝑀 such that 𝑑(𝑥,𝑦)<𝑀 whenever 𝑥,𝑦𝐴. Note that a subset 𝐴 of (𝑋,𝑑) is bounded if and only if there are 𝑥𝑋 and 𝑟,𝑠0 such that 𝐴𝐶𝑑(𝑥,𝑟)𝐶𝑑1(𝑥,𝑠).

3. Some First Results

Proposition 3.1 (compare [13, Proposition 4.5]). Let (𝑋,𝑑) be a 𝑞-hyperconvex quasipseudometric space. Let (𝑥𝑖)𝑖𝐼 be a nonempty family of points in 𝑋 and let (𝑟𝑖)𝑖𝐼 and (𝑠𝑖)𝑖𝐼 be two families of nonnegative reals such that 𝑑(𝑥𝑖,𝑥𝑗)𝑟𝑖+𝑠𝑗. Set 𝐷=𝑖𝐼(𝐶𝑑(𝑥𝑖,𝑟𝑖)𝐶𝑑1(𝑥𝑖,𝑠𝑖)). Then 𝐷 is nonempty and 𝑞-hyperconvex.

Proof. Note first that 𝐷 by 𝑞-hyperconvexity of 𝑋. For each 𝛼𝑆, let 𝑥𝛼𝐷 and let 𝑟𝛼,𝑠𝛼 be nonnegative reals such that 𝑑(𝑥𝛼,𝑥𝛽)𝑟𝛼+𝑠𝛽 whenever 𝛼,𝛽𝑆.
We show that the family 𝐶𝑑𝑥𝛼,𝑟𝛼𝛼𝑆,𝐶𝑑𝑥𝑖,𝑟𝑖𝑖𝐼;𝐶𝑑1𝑥𝛼,𝑠𝛼𝛼𝑆,𝐶𝑑1𝑥𝑖,𝑠𝑖𝑖𝐼(3.1) satisfies the hypothesis of 𝑞-hyperconvexity. Indeed, in particular, for each 𝛼𝑆 and 𝑖𝐼, we have that 𝑑(𝑥𝛼,𝑥𝑖)𝑠𝑖𝑟𝛼+𝑠𝑖 and 𝑑(𝑥𝑖,𝑥𝛼)𝑟𝑖𝑟𝑖+𝑠𝛼.
Hence, by 𝑞-hyperconvexity of 𝑋, we have that 𝑖𝐼𝐶𝑑𝑥𝑖,𝑟𝑖𝐶𝑑1𝑥𝑖,𝑠𝑖𝛼𝑆𝐶𝑑𝑥𝛼,𝑟𝛼𝐶𝑑1𝑥𝛼,𝑠𝛼=𝐷𝛼𝑆𝐶𝑑𝑥𝛼,𝑟𝛼𝐶𝑑1𝑥𝛼,𝑠𝛼.(3.2) Hence, the subspace 𝐷 of 𝑋 is 𝑞-hyperconvex.

Let (𝑋,𝑑) be a quasipseudometric space. For a nonempty bounded subset 𝐴 of 𝑋, we set bicov(𝐴)+=𝐶𝑑(𝑥,𝑟)𝐴𝐶𝑑,(𝑥,𝑟),𝑥𝑋,𝑟0bicov(𝐴)=𝐶𝑑1(𝑥,𝑠)𝐴𝐶𝑑1.(𝑥,𝑠),𝑥𝑋,𝑠0(3.3) Furthermore, we define the bicover of 𝐴 by bicov(𝐴)=bicov(𝐴)+bicov(𝐴).

A nonempty bounded set 𝐴 in a quasipseudometric space (𝑋,𝑑) that can be written as the intersection of a nonempty family of sets of the form 𝐶𝑑(𝑥,𝜖1)𝐶𝑑1(𝑥,𝜖2) where 𝜖1,𝜖20 and 𝑥𝑋, that is, 𝐴=bicov𝐴, will be called 𝑞-admissible in the following. By 𝒜𝑞(𝑋), we will denote the set of 𝑞-admissible subsets of 𝑋. Note that if (𝑋,𝑑) is 𝑞-hyperconvex, then any member of 𝒜𝑞(𝑋) is 𝑞-hyperconvex in the light of Proposition 3.1.

Let (𝑋,𝑑) be a quasipseudometric space and let 𝐴 be a nonempty bounded subset in (𝑋,𝑑). Then, in accordance with [13, page 79], we can define the cover cov𝐴 of 𝐴 as follows: cov𝐴={𝐶𝑑𝑠(𝑥)𝐴𝐶𝑑𝑠(𝑥),𝑥𝑋}. Obviously, we have 𝐴bicov(𝐴)cov(𝐴). The latter inclusion can be strict, as our first example shows.

Example 3.2. Let 𝑋=[0,1]×[1/4,3/4] be equipped with the 𝑇0-quasimetric 𝑑 defined by 𝑑((𝛼,𝛽),(𝛼,𝛽))=(𝛼̇𝛼)(𝛽̇𝛽) whenever (𝛼,𝛽),(𝛼,𝛽)𝑋.
Consider 𝐴={(0,1/2),(1,1/2)}𝑋. Then, bicov(𝐴) is equal to the line segment in 𝑋 from 𝑥=(0,1/2) to 𝑦=(1,1/2). This follows from the fact that, for each 𝜖[0,1/4], we have 𝑥[0,1]×[1/4,(1/2)+𝜖]=𝐶𝑑1(𝑦,𝜖) and 𝑦[0,1]×[(1/2)𝜖,3/4]=𝐶𝑑(𝑥,𝜖), and that the line segment is a subset of any set of the form 𝐶𝑑(𝑎,𝑟)𝐶𝑑1(𝑏,𝑠) for which {𝑥,𝑦}𝐶𝑑(𝑎,𝑟)𝐶𝑑1(𝑏,𝑠). Indeed, assume that the point 𝑧 belongs to this segment. Then 𝑑(𝑧,𝑦)=0=𝑑(𝑥,𝑧) and, therefore, 𝑧𝐶𝑑(𝑎,𝑟)𝐶𝑑1(𝑏,𝑠) by the triangle inequality.
On the other hand, cov(𝐴)=𝑋, since {𝑥,𝑦}𝐶𝑑𝑠(𝑧,𝜖) with 𝑧𝑋 implies that 𝜖1/2. Indeed, assume that 𝑧=(𝑎,𝑏)𝑋. Then, 𝑎𝑑𝑠((𝑎,𝑏),(0,1/2))𝜖 and 1𝑎𝑑𝑠((𝑎,𝑏),(1,1/2))𝜖. Thus, 𝜖max{𝑎,1𝑎}1/2 with 𝑎[0,1]. In the light that the interval [1/4,3/4] has length 1/2, it follows that 𝑋𝐶𝑑𝑠(𝑧,𝜖). Therefore, cov(𝐴)=𝑋.
By the results of [1, Example 1], (bicov(𝐴),𝑑) is 𝑞-hyperconvex, while the metric space (bicov(𝐴),𝑑𝑠) is hyperconvex [1, Proposition 2], but not 𝑞-hyperconvex (see [1, Example 2]).
The following result gives a quasipseudometric variant of a well-known result usually attibuted to Sine [14] (compare also [15]).

Theorem 3.3. If (𝑋,𝑑) is a bounded 𝑞-hyperconvex 𝑇0-quasimetric space and if  𝑇(𝑋,𝑑)(𝑋,𝑑) is a nonexpansive map, then the fixed point set Fix(𝑇) of   𝑇 in (𝑋,𝑑) is nonempty and 𝑞-hyperconvex.

Proof. We first show that Fix(𝑇). Note that 𝑇(𝑋,𝑑𝑠)(𝑋,𝑑𝑠) is nonexpansive, since for any 𝑥,𝑦𝑋, we have 𝑑(𝑇𝑥,𝑇𝑦)𝑑(𝑥,𝑦) and 𝑑(𝑇𝑦,𝑇𝑥)𝑑(𝑦,𝑥), and thus 𝑑𝑠(𝑇𝑥,𝑇𝑦)𝑑𝑠(𝑥,𝑦). By assumption (𝑋,𝑑𝑠) is bounded. Furthermore, (𝑋,𝑑𝑠) is a hyperconvex space according to [1, Proposition 2]. So, by [13, Theorem 4.8], we know that 𝑇 has a fixed point and Fix(𝑇) is hyperconvex in (𝑋,𝑑𝑠).
We need to show that Fix(𝑇) is indeed 𝑞-hyperconvex. Let (𝐶𝑑(𝑥𝑖,𝑟𝑖))𝑖𝐼;(𝐶𝑑1(𝑥𝑖,𝑠𝑖))𝑖𝐼(3.4) be a nonempty family of double balls, where 𝑥𝑖Fix(𝑇) and (𝑟𝑖)𝑖𝐼 and (𝑠𝑖)𝑖𝐼 are two families of nonnegative reals such that 𝑑(𝑥𝑖,𝑥𝑗)𝑟𝑖+𝑠𝑗 whenever 𝑖,𝑗𝐼. Since 𝑋 is a 𝑞-hyperconvex 𝑇0-quasimetric space, the set 𝑋0=𝑖𝐼𝐶𝑑𝑥𝑖,𝑟𝑖𝐶𝑑1𝑥𝑖,𝑠𝑖.(3.5) Let 𝑥𝑋0. Then, 𝑑(𝑇(𝑥),𝑥𝑖)=𝑑(𝑇(𝑥),𝑇(𝑥𝑖))𝑑(𝑥,𝑥𝑖)𝑠𝑖 and 𝑑𝑥𝑖𝑇𝑥,𝑇(𝑥)=𝑑𝑖𝑥,𝑇(𝑥)𝑑𝑖,𝑥𝑟𝑖(3.6) whenever 𝑖𝐼. Thus, 𝑇(𝑥)𝑋0 and we have 𝑇𝑋0𝑋0.
Moreover, 𝑋0 is a bounded 𝑞-hyperconvex 𝑇0-quasimetric space by Proposition 3.1. So the first part of the proof implies that 𝑇 has a fixed point in 𝑋0, which implies that Fix(T)[𝑖𝐼(𝐶𝑑(𝑥𝑖,𝑟𝑖)𝐶𝑑1(𝑥𝑖,𝑠𝑖)]. We have shown that Fix(𝑇) is 𝑞-hyperconvex.

4. Chains of 𝑞-Hyperconvex Subspaces

In this section, we will prove the analogue of a famous theorem due to Baillon [16].

Theorem 4.1. Let (𝑋,𝑑) be a bounded 𝑇0-quasimetric space and let (𝐻𝑖)𝑖𝐼 be a descending family of nonempty 𝑞-hyperconvex subsets of 𝑋, where one assumes that 𝐼 is totally ordered such that 𝑖1,𝑖2𝐼 and 𝑖1𝑖2 hold if and only if 𝐻𝑖2𝐻𝑖1. Then, 𝑖𝐼𝐻𝑖 is nonempty and 𝑞-hyperconvex.

Proof. We begin by showing that 𝑖𝐼𝐻𝑖. We first note that (𝑋,𝑑𝑠) is a bounded metric space and (𝐻𝑖)𝑖𝐼 is a descending chain of hyperconvex sets in (𝑋,𝑑𝑠) by [1, Proposition 2]. By the well-known result of Baillon [16, Theorem 7], we conclude that 𝑖𝐼𝐻𝑖 is nonempty and hyperconvex in (𝑋,𝑑𝑠).
In order to complete the proof, we need to show that 𝐻=𝑖𝐼𝐻𝑖 is 𝑞-hyperconvex. Let a nonempty family (𝑥𝛼)𝛼𝑆 of points in 𝐻 and families of nonnegative real numbers (𝑟𝛼)𝛼𝑆 and (𝑠𝛼)𝛼𝑆 be given such that 𝑑(𝑥𝛼,𝑥𝛽)𝑟𝛼+𝑠𝛽 whenever 𝛼,𝛽𝑆. Fix 𝑖𝐼. Since 𝐻𝑖 is a 𝑞-hyperconvex space and since 𝑥𝛼𝐻𝑖 whenever 𝛼𝑆, therefore, 𝒟𝑖=𝛼𝑆(𝐶𝑑(𝑥𝛼,𝑟𝛼)𝐶𝑑1(𝑥𝛼,𝑠𝛼))𝐻𝑖 is nonempty and 𝑞-hyperconvex by the proof of Proposition 3.1 and thus a hyperconvex subset of (𝑋,𝑑𝑠) by [1, Proposition 2].
Thus by the first part of our present proof, 𝑖𝐼𝒟𝑖=𝑖𝐼𝛼𝑆𝐶𝑑𝑥𝛼,𝑟𝛼𝐶𝑑1𝑥𝛼,𝑠𝛼𝐻𝑖=𝛼𝑆𝐶𝑑𝑥𝛼,𝑟𝛼𝐶𝑑1𝑥𝛼,𝑠𝛼𝑖𝐼𝐻𝑖,(4.1) since (𝒟𝑖)𝑖𝐼 is descending. This proves that 𝐻=𝑖𝐼𝐻𝑖 is 𝑞-hyperconvex.

Definition 4.2. Let (𝑋,𝑑) be a 𝑇0-quasimetric space and let a family of nonexpansive maps (𝑇𝑖)𝑖𝐼, with 𝑇𝑖(𝑋,𝑑)(𝑋,𝑑), be given. We say that (𝑇𝑖)𝑖𝐼 is a commuting family if 𝑇𝑖𝑇𝑗=𝑇𝑗𝑇𝑖 whenever 𝑖,𝑗𝐼.
Our next lemma is motivated by [16, Corollary 8].

Lemma 4.3. If (𝐻𝛼)𝛼𝑆 is a family of bounded 𝑞-hyperconvex subsets of a 𝑇0-quasimetric space 𝑋 such that 𝛼𝐹𝐻𝛼 is nonempty and 𝑞-hyperconvex whenever 𝐹𝑆 is finite, then the intersection 𝛼𝑆𝐻𝛼 is nonempty and 𝑞-hyperconvex.

Proof. Consider ={𝐼𝑆 for all 𝐽 finite, 𝐽𝑆,   𝐼𝐽𝐻𝛼 is nonempty and 𝑞-hyperconvex}.
Obviously and satisfies the hypothesis of Zorn's lemma because of Theorem 4.1. Let 𝐼 be maximal in . Then, 𝐼{𝛼} whenever 𝛼𝑆. Because of the maximality of 𝐼, we, therefore, have 𝛼𝐼 whenever 𝛼𝑆.

The next result is a consequence of Theorems 3.3 and 4.1. It is analogous to [17, Theorem 6.2].

Theorem 4.4. Let (𝑋,𝑑) be a bounded 𝑞-hyperconvex 𝑇0-quasimetric space. Any commuting family of nonexpansive maps (𝑇𝑖)𝑖𝐼, with 𝑇𝑖(𝑋,𝑑)(𝑋,𝑑), has a common fixed point. Moreover, the common fixed point set 𝑖𝐼𝑇Fix𝑖,(4.2) is 𝑞-hyperconvex.

Proof. We observe that (𝑋,𝑑𝑠) is a bounded hyperconvex metric space by [1, Proposition 2], and for each 𝑖𝐼, the map 𝑇𝑖(𝑋,𝑑𝑠)(𝑋,𝑑𝑠) is nonexpansive, as we noted before (see proof of Theorem 3.3). By Theorem 3.3, each 𝑇𝑖 has a fixed point. Hence, there is 𝑥𝑋 such that 𝑇𝑖(𝑥)=𝑥. We now show that, given any 𝑗𝐼, we have that 𝑇𝑗(Fix(𝑇𝑖))Fix(𝑇𝑖) Indeed, if for some 𝑥𝑋, we have 𝑥=𝑇𝑖(𝑥), then 𝑇𝑗(𝑥)=𝑇𝑗(𝑇𝑖(𝑥))=𝑇𝑖(𝑇𝑗(𝑥)). So 𝑇𝑗(𝑥)Fix(𝑇𝑖).
By Theorem 3.3, we conclude that 𝑇𝑗Fix(𝑇𝑖)Fix(𝑇𝑖) has a fixed point 𝑦Fix(𝑇𝑖), which then is a fixed point of 𝑇𝑖 and 𝑇𝑗. Therefore, the set of common fixed points of 𝑇𝑖 and 𝑇𝑗 is 𝑞-hyperconvex by Theorem 3.3. Hence, by induction for each finite family (𝑇𝑖)𝑖𝐹 of nonexpansive self-maps on 𝑋 the set of common fixed points is nonempty and 𝑞-hyperconvex.
Since 𝑖𝐹Fix(𝑇𝑖) is nonempty subset and 𝑞-hyperconvex whenever 𝐹 is a finite subset of 𝐼, by Lemma 4.3 we conclude that 𝑖𝐼Fix(𝑇𝑖) is nonempty and 𝑞-hyperconvex.

5. Approximate Fixed Points

In this section, we investigate the approximation of fixed points by generalizing some results from [13] (compare [18]). We first define an 𝜖1,𝜖2-parallel set of a subset in a quasipseudometric space similarly to [13, page 89].

Definition 5.1. Let (𝑋,𝑑) be a quasipseudometric space. Given a subset 𝐴 of 𝑋, we define for 𝜖1,𝜖20 the 𝜖1,𝜖2-parallel set of 𝐴 as 𝑁𝜖1,𝜖2(𝐴)=𝑎𝐴𝐶𝑑𝑎,𝜖2𝐶𝑑1𝑎,𝜖1.(5.1)
(Note that for each 𝜖>0 in particular 𝑁𝜖,𝜖(𝐴)=𝑎𝐴𝐶𝑑𝑠(𝑎,𝜖).)
Thus, 𝑥𝑁𝜖1,𝜖2(𝐴) if and only if there exists 𝑎𝐴 such that 𝑑(𝑎,𝑥)𝜖2 and 𝑑1(𝑎,𝑥)𝜖1.
We next give a characterization of 𝑁𝜖1,𝜖2(𝐴) if A is a 𝑞-admissible set in a 𝑞-hyperconvex quasipseudometric space (compare [13, Lemma 4.2]).

Lemma 5.2. Let (𝑋,𝑑) be a 𝑞-hyperconvex quasipseudometric space and let 𝐴 be a 𝑞-admissible subset of 𝑋, say 𝐴=𝑖𝐼(𝐶𝑑(𝑥𝑖,𝑟𝑖)𝐶𝑑1(𝑥𝑖,𝑠𝑖)) with 𝑥𝑖𝑋 and 𝑟𝑖,𝑠𝑖 nonnegative reals whenever 𝑖𝐼. Then, for each 𝜖1,𝜖20, 𝑁𝜖1,𝜖2(𝐴)=𝑖𝐼𝐶𝑑𝑥𝑖,𝑟𝑖+𝜖2𝐶𝑑1𝑥𝑖,𝑠𝑖+𝜖1.(5.2)

Proof. Suppose that 𝑦𝑁𝜖1,𝜖2(𝐴). Then, 𝑑(𝑎,𝑦)𝜖2 and 𝑑(𝑦,𝑎)𝜖1 for some 𝑎𝐴. But for each 𝑖𝐼, 𝑑𝑥𝑖𝑥,𝑦𝑑𝑖,𝑎+𝑑(𝑎,𝑦)𝑟𝑖+𝜖2,𝑑𝑦,𝑥𝑖𝑑(𝑦,𝑎)+𝑑𝑎,𝑥𝑖𝜖1+𝑠𝑖.(5.3) Then, for each 𝑖𝐼, we have 𝑦𝐶𝑑(𝑥𝑖,𝑟𝑖+𝜖2) and 𝑦𝐶𝑑1(𝑥𝑖,𝑠𝑖+𝜖1). This proves that 𝑁𝜖1,𝜖2(𝐴)𝑖𝐼(𝐶𝑑(𝑥𝑖,𝑟𝑖+𝜖2)𝐶𝑑1(𝑥𝑖,𝑠𝑖+𝜖1)).
Now suppose that 𝑦𝑖𝐼(𝐶𝑑(𝑥𝑖,𝑟𝑖+𝜖2)𝐶𝑑1(𝑥𝑖,𝑠𝑖+𝜖1)) and let 𝑖𝐼.
Hence, 𝑑𝑥𝑖,𝑦𝑟𝑖+𝜖2,𝑑𝑦,𝑥𝑖𝜖1+𝑠𝑖.(5.4) By definition of 𝐴 and the triangle inequality, for any 𝑎𝐴 and any 𝑖,𝑗𝐼 we must have that 𝑑𝑥𝑖,𝑥𝑗𝑥𝑑𝑖,𝑎+𝑑𝑎,𝑥𝑗𝑟𝑖+𝑠𝑗.(5.5) Hence, [(𝐶𝑑(𝑥𝑖,𝑟𝑖))𝑖𝐼,𝐶𝑑(𝑦,𝜖1);(𝐶𝑑1(𝑥𝑖,𝑠𝑖))𝑖𝐼,𝐶𝑑1(𝑦,𝜖2)] satisfies the hypothesis in the definition of 𝑞-hyperconvexity of (𝑋,𝑑).
So, by 𝑞-hyperconvexity of 𝑋, 𝑖𝐼𝐶𝑑𝑥𝑖,𝑟𝑖𝐶𝑑𝑦,𝜖1𝑖𝐼𝐶𝑑1𝑥𝑖,𝑠𝑖𝐶𝑑1𝑦,𝜖2=𝑖𝐼𝐶𝑑𝑥𝑖,𝑟𝑖𝐶𝑑1𝑥𝑖,𝑠𝑖𝐶𝑑𝑦,𝜖1𝐶𝑑1𝑦,𝜖2𝐶=𝐴𝑑𝑦,𝜖1𝐶𝑑1𝑦,𝜖2.(5.6)
Therefore, there is 𝑎𝐴 such that 𝑑(𝑦,𝑎)𝜖1 and 𝑑(𝑎,𝑦)𝜖2. Hence, 𝑦𝑁𝜖1,𝜖2(𝐴) and the proof is complete.

The following lemma will be needed in our discussion below of approximate fixed point sets.

Lemma 5.3 (compare [13, Lemma 4.3]). Suppose that (𝑋,𝑑) is a 𝑞-hyperconvex 𝑇0-quasimetric space and let 𝐴 be a 𝑞-admissible subset of  𝑋. Then, for each 𝜖1,𝜖20 there is a nonexpansive retraction 𝑅 of 𝑁𝜖1,𝜖2(𝐴) onto 𝐴 which has the property that 𝑑(𝑥,𝑅(𝑥))𝜖1 and 𝑑(𝑅(𝑥),𝑥)𝜖2 whenever 𝑥𝑁𝜖1,𝜖2(𝐴).

Proof. Assume 𝐴=𝑖𝐼(𝐶𝑑(𝑥𝑖,𝑟𝑖)𝐶𝑑1(𝑥𝑖,𝑠𝑖)) with 𝐼. By Lemma 5.2, we know that 𝑁𝜖1,𝜖2(𝐴) is 𝑞-admissible in (𝑋,𝑑) and so 𝑁𝜖1,𝜖2(𝐴) is itself 𝑞-hyperconvex by Proposition 3.1. Consider the family ={(𝐷,𝑅𝐷)𝐴𝐷𝑁𝜖1,𝜖2(𝐴) and 𝑅𝐷𝐷𝐴 is a nonexpansive retraction such that 𝑑(𝑥,𝑅(𝑥))𝜖1 and 𝑑(𝑅(𝑥),𝑥)𝜖2 whenever 𝑥𝐷}.
Let Id denote the identity map on 𝐴. Note that (𝐴,Id). So . If one partially orders in the usual way ((𝐷,𝑅𝐷)(𝐻,𝑅𝐻) if and only if 𝐷𝐻 and 𝑅𝐻 is an extension of 𝑅𝐷), then each chain in (,) is bounded above. So by Zorn's lemma has a maximal element which we denote by (𝐷,𝑅𝐷). We need to show that 𝐷=𝑁𝜖1,𝜖2(𝐴). Suppose that there exists 𝑥𝑁𝜖1,𝜖2(𝐴) such that 𝑥𝐷, and consider the set 𝐶=𝑤𝐷𝐶𝑑𝑅𝐷(𝑤),𝑑(𝑤,𝑥)𝐶𝑑1𝑅𝐷(𝑤),𝑑(𝑥,𝑤)𝑖𝐼𝐶𝑑𝑥𝑖,𝑟𝑖𝐶𝑑1𝑥𝑖,𝑠𝑖𝐶𝑑𝑥,𝜖1𝐶𝑑1𝑥,𝜖2.(5.7) First, we want to show that 𝐶, and in order to do this by [1, Proposition 1], we need only to show that the family 𝐶𝑑𝑅𝐷(𝑤),𝑑(𝑤,𝑥)𝑤𝐷,𝐶𝑑𝑥𝑖,𝑟𝑖𝑖𝐼,𝐶𝑑𝑥,𝜖1;𝐶𝑑1𝑅𝐷(𝑤),𝑑(𝑥,𝑤)𝑤𝐷,𝐶𝑑1𝑥𝑖,𝑠𝑖𝑖𝐼,𝐶𝑑1𝑥,𝜖2(5.8) of double balls has the mixed binary intersection property.
First note that if 𝑤1,𝑤2𝐷, then 𝑑𝑅𝐷𝑤1,𝑅𝐷𝑤2𝑤𝑑1,𝑤2𝑤𝑑1,𝑥+𝑑𝑥,𝑤2.(5.9) Therefore, 𝐶𝑑(𝑅𝐷(𝑤1),𝑑(𝑤1,𝑥)) and 𝐶𝑑1(𝑅𝐷(𝑤2),𝑑(𝑥,𝑤2)) intersect by metric convexity of (𝑋,𝑑).
Furthermore, by the definition of 𝐴, for each 𝑖,𝑗𝐼, we see that 𝐶𝑑(𝑥𝑖,𝑟𝑖) and 𝐶𝑑1(𝑥𝑗,𝑠𝑗) intersect.
Also for each 𝑤𝐷, 𝑅𝐷(𝑤)𝐴=𝑖𝐼(𝐶𝑑(𝑥𝑖,𝑟𝑖)𝐶𝑑1(𝑥𝑖,𝑠𝑖)). Hence, for any 𝑤𝐷 and 𝑖𝐼,𝐶𝑑(𝑅𝐷(𝑤),𝑑(𝑤,𝑥)) and 𝐶𝑑1(𝑥𝑖,𝑠𝑖) intersect, as well as for any 𝑤𝐷 and 𝑖𝐼,𝐶𝑑1(𝑅𝐷(𝑤),𝑑(𝑥,𝑤)) and 𝐶𝑑(𝑥𝑖,𝑟𝑖) intersect.
Since 𝑥𝑁𝜖1,𝜖2(𝐴)=𝑖𝐼𝐶𝑑𝑥𝑖,𝑟𝑖+𝜖2𝐶𝑑1𝑥𝑖,𝑠𝑖+𝜖1,(5.10) by Lemma 5.2 there is 𝑎𝐴 such that 𝑥𝐶𝑑(𝑎,𝜖2)𝐶𝑑1(𝑎,𝜖1) and, therefore, (𝐶𝑑(𝑥,𝜖1)𝐶𝑑1(𝑥,𝜖2))(𝐶𝑑(𝑥𝑖,𝑟𝑖)𝐶𝑑1(𝑥𝑖,𝑠𝑖)) whenever 𝑖𝐼.
Finally, if 𝑤𝐷, then by assumption on 𝑅𝐷, 𝑑𝑅𝐷𝑅(𝑤),𝑥𝑑𝐷(𝑤),𝑤+𝑑(𝑤,𝑥)𝜖2𝑑+𝑑(𝑤,𝑥),𝑥,𝑅𝐷(𝑤)𝑑(𝑥,𝑤)+𝑑𝑤,𝑅𝐷(𝑤)𝑑(𝑥,𝑤)+𝜖1.(5.11) Thus, by metric convexity of (𝑋,𝑑), we have that 𝐶𝑑(𝑅𝐷(𝑤),𝑑(𝑤,𝑥)) and 𝐶𝑑1(𝑥,𝜖2) intersect as well as 𝐶𝑑1(𝑅𝐷(𝑤),𝑑(𝑥,𝑤)) and 𝐶𝑑(𝑥,𝜖1) intersect.
Of course, 𝐶𝑑(𝑥,𝜖1) and 𝐶𝑑1(𝑥,𝜖2) intersect.
We have shown that the family 𝐶𝑑𝑅𝐷(𝑤),𝑑(𝑤,𝑥)𝑤𝐷,𝐶𝑑𝑥𝑖,𝑟𝑖𝑖𝐼,𝐶𝑑𝑥,𝜖1;𝐶𝑑1𝑅𝐷(𝑤),𝑑(𝑥,𝑤)𝑤𝐷,𝐶𝑑1𝑥𝑖,𝑠𝑖𝑖𝐼,𝐶𝑑1𝑥,𝜖2(5.12) of double balls has the mixed binary intersection property.
Hence, 𝐶𝐴. Now, let 𝑢𝐶 and define 𝑅𝐷{𝑥}𝐴 by setting 𝑅(𝑤)=𝑅𝐷(𝑤) if 𝑤𝐷 and 𝑅(𝑥)=𝑢. Then, for each 𝑤𝐷, we have 𝑑𝑅(𝑥),𝑅(𝑤)=𝑑𝑢,𝑅𝐷𝑑𝑅(𝑤)𝑑(𝑥,𝑤),(𝑤),𝑅(𝑅𝑥)=𝑑𝐷(𝑤),𝑢𝑑(𝑤,𝑥),(5.13) so that 𝑅 is nonexpansive. Also, 𝑑(𝑅(𝑥),𝑥)=𝑑(𝑢,𝑥)𝜖2 and 𝑑(𝑥,𝑅(𝑥))=𝑑(𝑥,𝑢)𝜖1. Therefore, we conclude that the pair (𝐷{𝑥},𝑅) contradicts the maximality of (𝐷,𝑅𝐷) in (,). Consequently, 𝐷=𝑁𝜖1,𝜖2(𝐴) and we are done.

Definition 5.4 (compare [19] and [20]). Let (𝑋,𝑑) be a 𝑇0-quasimetric space. We say that a map 𝑇(𝑋,𝑑)(𝑋,𝑑) has approximate fixed points if inf𝑥𝑋𝑑𝑠(𝑥,𝑇(𝑥))=0.

Definition 5.5. Let (𝑋,𝑑) be a 𝑇0-quasimetric space. For a map 𝑇(𝑋,𝑑)(𝑋,𝑑) and for any 𝜖1,𝜖20, we use 𝐹𝜖1,𝜖2(𝑇) to denote the set of 𝜖1,𝜖2-approximate fixed points of 𝑇; that is, 𝐹𝜖1,𝜖2(𝑇)={𝑥𝑋𝑑(𝑥,𝑇(𝑥))𝜖2 and 𝑑(𝑇(𝑥),𝑥)𝜖1}.

Theorem 5.6 (compare [13, Theorem 4.11]). Suppose that (𝑋,𝑑) is a 𝑞-hyperconvex 𝑇0-quasimetric space and that the map 𝑇(𝑋,𝑑)(𝑋,𝑑) is nonexpansive. Furthermore suppose that for some 𝜖1,𝜖20 one has that 𝐹𝜖1,𝜖2(𝑇) is nonempty. Then, the set 𝐹𝜖1,𝜖2(𝑇) is 𝑞-hyperconvex.

Proof. For each 𝑖 in some nonempty index set 𝐼, let 𝑥𝑖𝐹𝜖1,𝜖2(𝑇), and let 𝑟𝑖0 and 𝑠𝑖0 satisfy 𝑑𝑥𝑖,𝑥𝑗𝑟𝑖+𝑠𝑗.(5.14) We need to show that 𝑖𝐼𝐶𝑑𝑥𝑖,𝑟𝑖𝐶𝑑1𝑥𝑖,𝑠𝑖𝐹𝜖1,𝜖2(𝑇).(5.15) We know that 𝐽=𝑖𝐼(𝐶𝑑(𝑥𝑖,𝑟𝑖)𝐶𝑑1(𝑥𝑖,𝑠𝑖)) is 𝑞-hyperconvex according to Proposition 3.1, since (𝑋,𝑑) is 𝑞-hyperconvex. Furthermore, 𝐽 is obviously bounded in (𝑋,𝑑).
Also, if 𝑥𝐽, then for each 𝑖𝐼, 𝑑𝑥𝑖𝑥,𝑇(𝑥)𝑑𝑖𝑥,𝑇𝑖𝑇𝑥+𝑑𝑖,𝑇(𝑥)𝜖2𝑥+𝑑𝑖,𝑥𝜖2+𝑟𝑖,𝑑𝑇(𝑥),𝑥𝑖𝑥𝑑𝑇(𝑥),𝑇𝑖𝑇𝑥+𝑑𝑖,𝑥𝑖𝑑𝑥,𝑥𝑖+𝜖1𝑠𝑖+𝜖1.(5.16)
This proves that 𝑇(𝑥)𝑁𝜖1,𝜖2(𝐽) by Lemma 5.2. Now, by Lemma 5.3, there is a nonexpansive retraction 𝑅 of 𝑁𝜖1,𝜖2(𝐽) onto 𝐽 for which 𝑑(𝑅(𝑥),𝑥)𝜖2 and 𝑑(𝑥,𝑅(𝑥))𝜖1 whenever 𝑥𝑁𝜖1,𝜖2(𝐽). Also since 𝑅𝑇 is a nonexpansive map of 𝐽 into 𝐽, it must have a fixed point by Theorem 3.3.
Suppose that (𝑅𝑇)(𝑥0)=𝑥0 for some 𝑥0𝐽. Then, 𝑑𝑥0𝑥,𝑇0𝑥=𝑑(𝑅𝑇)0𝑥,𝑇0𝜖2,𝑑𝑇𝑥0,𝑥0𝑇𝑥=𝑑0𝑥,(𝑅𝑇)0𝜖1.(5.17) Thus, the proof is complete, since 𝑥0𝐽𝐹𝜖1,𝜖2(𝑇).

6. External 𝑞-Hyperconvexity

We next define an externally 𝑞-hyperconvex subset of a quasipseudometric space (𝑋,𝑑) in analogy to [17, Definition 3.5]. Note that this definition strengthens the concept of a 𝑞-hyperconvex subset of (𝑋,𝑑) (compare also [21, Definition 3]).

Definition 6.1. Let (𝑋,𝑑) be a quasipseudometric space. A subspace 𝐸 of (𝑋,𝑑) is said to be externally 𝑞-hyperconvex (relative to 𝑋) if given any family (𝑥𝑖)𝑖𝐼 of points in 𝑋 and families of nonnegative real numbers (𝑟𝑖)𝑖𝐼 and (𝑠𝑖)𝑖𝐼 the following condition holds:if 𝑑(𝑥𝑖,𝑥𝑗)𝑟𝑖+𝑠𝑗 whenever 𝑖,𝑗𝐼, dist(𝑥𝑖,𝐸)𝑟𝑖 and dist(𝐸,𝑥𝑖)𝑠𝑖 whenever 𝑖𝐼, then 𝑖𝐼(𝐶𝑑(𝑥𝑖,𝑟𝑖)𝐶𝑑1(𝑥𝑖,𝑠𝑖))𝐸.
In the following, 𝑞(𝑋) will denote the set of nonempty externally 𝑞-hyperconvex subsets of (𝑋,𝑑).

Example 6.2 (compare [21, Theorem 7]). Let 𝐸 be a nonempty externally 𝑞-hyperconvex subset in a quasipseudometric space (𝑋,𝑑) and let 𝑥 be any point of 𝑋. Set dist(𝑥,𝐸)=𝑟 and dist(𝐸,𝑥)=𝑠. Then, by applying external 𝑞-hyperconvexity of 𝐸 to the double ball (𝐶𝑑(𝑥,𝑟);𝐶𝑑1(𝑥,𝑠)), we conclude that there is 𝑝𝐶𝑑(𝑥,𝑟)𝐶𝑑1(𝑥,𝑠)𝐸. Thus, 𝑑(𝑥,𝑝)=dist(𝑥,𝐸) and 𝑑(𝑝,𝑥)=dist(𝐸,𝑥).

Lemma 6.3 (compare [17, Lemma 3.8]). Let (𝑋,𝑑) be a 𝑞-hyperconvex space and let 𝑥𝑋. Furthermore, let 𝐴=𝑖𝐼(𝐶𝑑(𝑥𝑖,𝑟𝑖)𝐶𝑑1(𝑥𝑖,𝑠𝑖)) where (𝑥𝑖)𝑖𝐼 is a nonempty family of points in 𝑋 and (𝑟𝑖)𝑖𝐼 and (𝑠𝑖)𝑖𝐼 are families of nonnegative reals. Then, there is 𝑝𝐴 such that dist(𝑥,𝐴)=𝑑(𝑥,𝑝) and dist(𝐴,𝑥)=𝑑(𝑝,𝑥).

Proof. Evidently, 𝐶𝑑𝑥𝑖,𝑟𝑖𝑖𝐼,𝐶𝑑(𝑥,dist(𝑥,𝐴)+𝜖)𝜖>0;(𝐶𝑑1(𝑥𝑖,𝑠𝑖))𝑖𝐼,(𝐶𝑑1(𝑥,dist(𝐴,𝑥)+𝜖))𝜖>0(6.1) satisfies the mixed binary intersection property. Thus, there is 𝑝𝐴𝐶𝑑(𝑥,dist(𝑥,𝐴))𝐶𝑑1(𝑥,dist(𝐴,𝑥))(6.2) by 𝑞-hyperconvexity of (𝑋,𝑑). Obviously, 𝑝 then satisfies the stated condition.

The following lemma, which makes use of Lemma 6.3, will be useful in the proof of Theorem 6.5. Considering the case that 𝐸=𝑋, we see that Lemma 6.4 improves on Proposition 3.1.

Lemma 6.4 (compare [18, Lemma 2]). Let (𝑋,𝑑) be a 𝑞-hyperconvex quasipseudometric space. Suppose that 𝐸𝑋 is externally 𝑞-hyperconvex relative to 𝑋 and suppose that 𝐴 is a 𝑞-admissible subset of (𝑋,𝑑) such that 𝐸𝐴. Then 𝐸𝐴 is externally 𝑞-hyperconvex relative to 𝑋.

Proof. Assume that a given nonempty family (𝑥𝛼)𝛼𝑆 of points in 𝑋 and families of nonnegative real numbers (𝑟𝛼)𝛼𝑆 and (𝑠𝛼)𝛼𝑆 satisfy 𝑑(𝑥𝛼,𝑥𝛽)𝑟𝛼+𝑠𝛽, dist(𝑥𝛼,𝐴𝐸)𝑟𝛼, and dist(𝐴𝐸,𝑥𝛼)𝑠𝛼 whenever 𝛼,𝛽𝑆.
Since 𝐴 is 𝑞-admissible, 𝐴=𝑖𝐼(𝐶𝑑(𝑥𝑖,𝑟𝑖)𝐶𝑑1(𝑥𝑖,𝑠𝑖)) with 𝑥𝑖𝑋 and 𝑟𝑖,𝑠𝑖0 whenever 𝑖𝐼. Because dist(𝑥𝛼,𝐴)dist(𝑥𝛼,𝐴𝐸)𝑟𝛼 and dist(𝐴,𝑥𝛼)dist(𝐴𝐸,𝑥𝛼)𝑠𝛼 whenever 𝛼𝑆, it follows that for each  𝛼𝑆, 𝑖𝐼 and for 𝑝𝐴 chosen according to Lemma 6.3 we have 𝑑𝑥𝛼,𝑥𝑖𝑥𝑑𝛼,𝑝+𝑑𝑝,𝑥𝑖𝑟𝛼+𝑠𝑖,𝑑𝑥𝑖,𝑥𝛼𝑥𝑑𝑖,𝑝+𝑑𝑝,𝑥𝛼𝑟𝑖+𝑠𝛼.(6.3) Also, since for each 𝑖𝐼, 𝐴𝐶𝑑(𝑥𝑖,𝑟𝑖)𝐶𝑑1(𝑥𝑖,𝑠𝑖), and since 𝐴𝐸, it follows that 𝑥dist𝑖,𝐸𝑟𝑖,dist𝐸,𝑥𝑖𝑠𝑖,(6.4) and that 𝑑(𝑥𝑖,𝑥𝑗)𝑟𝑖+𝑠𝑗 whenever 𝑖,𝑗𝐼. Trivially, we have dist(𝑥𝛼,𝐸)𝑟𝛼 and dist(𝐸,𝑥𝛼)𝑠𝛼 whenever 𝛼𝑆.
Therefore, by external 𝑞-hyperconvexity of 𝐸, we conclude that 𝑖𝐼𝐶𝑑𝑥𝑖,𝑟𝑖𝐶𝑑1𝑥𝑖,𝑠𝑖𝛼𝑆𝐶𝑑𝑥𝛼,𝑟𝛼𝐶𝑑1𝑥𝛼,𝑠𝛼=𝐸𝛼𝑆𝐶𝑑𝑥𝛼,𝑟𝛼𝐶𝑑1𝑥𝛼,𝑠𝛼(𝐸𝐴).(6.5) Thus, the proof is complete.

We next show that the intersection of a descending family of externally 𝑞-hyperconvex nonempty subspaces of a bounded 𝑞-hyperconvex 𝑇0-quasimetric space behaves as expected.

Theorem 6.5 (compare [18, Theorem 4]). Let (𝑋,𝑑) be a bounded 𝑞-hyperconvex 𝑇0-quasimetric space. Moreover, let (𝑋𝑖)𝑖𝐼 be a descending family of nonempty externally 𝑞-hyperconvex subsets of 𝑋, where 𝐼 is assumed to be totally ordered such that 𝑖1,𝑖2𝐼 and 𝑖1𝑖2 if and only if 𝑋𝑖2𝑋𝑖1. Then𝑖𝐼𝑋𝑖 is nonempty and externally 𝑞-hyperconvex relative to 𝑋.

Proof. Theorem 4.1 implies that 𝐷=𝑖𝐼𝑋𝑖. In order to show that 𝐷 is externally 𝑞-hyperconvex, let a nonempty family (𝑥𝛼)𝛼𝑆 of points in 𝑋 and families of nonnegative real numbers (𝑟𝛼)𝛼𝑆 and (𝑠𝛼)𝛼𝑆 be given such that 𝑑(𝑥𝛼,𝑥𝛽)𝑟𝛼+𝑠𝛽, and dist(𝑥𝛼,𝐷)𝑟𝛼 and dist(𝐷,𝑥𝛼)𝑠𝛼 whenever 𝛼,𝛽𝑆.
Since 𝑋 is 𝑞-hyperconvex, we know that 𝐴=𝛼𝑆(𝐶𝑑(𝑥𝛼,𝑟𝛼)𝐶𝑑1(𝑥𝛼,𝑠𝛼)). Also, since for each 𝛼𝑆, dist(𝑥𝛼,𝐷)𝑟𝛼 and dist(𝐷,𝑥𝛼)𝑠𝛼, we have dist(𝑥𝛼,𝑋𝑖)𝑟𝛼 and dist(𝑋𝑖,𝑥𝛼)𝑠𝛼 whenever 𝑖𝐼, so that, by external 𝑞-hyperconvexity of 𝑋𝑖, we conclude that 𝐴𝑋𝑖 whenever 𝑖𝐼.
By Lemma 6.4,  (𝐴𝑋𝑖)𝑖𝐼 is a descending chain of nonempty (externally) 𝑞-hyperconvex subsets of (𝑋,𝑑), so that again by Theorem 4.1  𝑖𝐼(𝐴𝑋𝑖)=𝐴𝐷.

Let us note that the result stated in our abstract is a consequence of our next theorem.

Theorem 6.6 (compare [18, Theorem 1]). Let (𝐻,𝑑) be a 𝑞-hyperconvex 𝑇0-quasimetric space, let 𝑋 be any set, and let a map 𝑇𝑋𝑞(𝐻) be given. Then, there exists a map 𝑇𝑋𝐻 for which 𝑇(𝑥)𝑇(𝑥) whenever 𝑥𝑋 and for which 𝑑(𝑇(𝑥),𝑇(𝑦))𝑑𝐻(𝑇(𝑥),𝑇(𝑦)) whenever 𝑥,𝑦𝑋.

Proof. Let denote the collection of all pairs (𝐷,𝑇), where 𝐷𝑋,𝑇𝐷𝐻,𝑇(𝑑)𝑇(𝑑) whenever 𝑑𝐷, and 𝑑(𝑇(𝑥),𝑇(𝑦))𝑑𝐻(𝑇(𝑥),𝑇(𝑦)) whenever 𝑥,𝑦𝐷. Note that , since ({𝑥0},𝑇) for any choice of 𝑥0𝑋 and 𝑇(𝑥0)𝑇(𝑥0). Define a partial order relation on by setting (𝐷1,𝑇1)(𝐷2,𝑇2) if and only if 𝐷1𝐷2, and, 𝑇2|𝐷1=𝑇1.
Let ((𝐷𝛼,𝑇𝛼))𝛼𝑆 be an increasing chain in (,). Then it follows that (𝛼𝑆𝐷𝛼,𝑇) where 𝑇|𝐷𝛼=𝑇𝛼. By Zorn's lemma, (,) has a maximal element, say (𝐷,𝑇). Assume that 𝐷𝑋 and select 𝑥0𝑋𝐷. Set 𝐷=𝐷{𝑥0} and consider the set 𝐽=𝑥𝐷𝐶𝑑𝑇(𝑥),𝑑𝐻𝑇(𝑥),𝑇𝑥0𝐶𝑑1𝑇(𝑥),𝑑𝐻𝑇𝑥0,𝑇(𝑥)𝑇𝑥0.(6.6) Since 𝑇(𝑥0)𝑞(𝐻), by definition of external 𝑞-hyperconvexity, 𝐽 if for each 𝑥𝐷, we have dist(𝑇(𝑥),𝑇(𝑥0))𝑑𝐻(𝑇(𝑥),𝑇(𝑥0)) and 𝑇dist𝑥0,𝑇(𝑥)𝑑𝐻𝑇𝑥0,𝑇,(𝑥)(6.7) and for any 𝑥,𝑦𝐷, also 𝑑(𝑇(𝑥),𝑇(𝑦))𝑑𝐻𝑇(𝑥),𝑇𝑥0+𝑑𝐻𝑇𝑥0,𝑇.(𝑦)(6.8) We are going to check that these conditions hold.
Let 𝑥𝐷. For each 𝜖>0, we have 𝑇(𝑥)𝐵𝑑1(𝑇(𝑥0),𝑑𝐻(𝑇(𝑥),𝑇(𝑥0))+𝜖) and 𝑇(𝑥)𝐵𝑑(𝑇(𝑥0),𝑑𝐻(𝑇(𝑥0),𝑇(𝑥))+𝜖) by definition of the Hausdorff quasipseudometric.
Since 𝑇(𝑥)𝑇(𝑥), for each 𝜖>0, there is 𝑎𝑇(𝑥0) such that 𝑑(𝑇(𝑥),𝑎)𝑑𝐻(𝑇(𝑥),𝑇(𝑥0))+𝜖, and there is 𝑏𝑇(𝑥0) such that 𝑑(𝑏,𝑇(𝑥))𝑑𝐻𝑇𝑥0,𝑇(𝑥)+𝜖.(6.9) Therefore, dist(𝑇(𝑥),𝑇(𝑥0))𝑑𝐻(𝑇(𝑥),𝑇(𝑥0)) and dist(𝑇(𝑥0),𝑇(𝑥))𝑑𝐻(𝑇(𝑥0),𝑇(𝑥)).
We finally also note that by assumption on 𝑇, for each 𝑥,𝑦𝐷 we have that 𝑑(𝑇(𝑥),𝑇(𝑦))𝑑𝐻𝑇(𝑥),𝑇(𝑦)𝑑𝐻𝑇(𝑥),𝑇𝑥0+𝑑𝐻𝑇𝑥0,𝑇.(𝑦)(6.10) Thus, we have shown that 𝐽. Choose 𝑦0𝐽 and define
𝑇(𝑥)=𝑦0 if 𝑥=𝑥0 and 𝑇(𝑥)=𝑇(𝑥) if 𝑥𝐷.
Since for each 𝑥𝐷,