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Hans-Peter A. Künzi, Olivier Olela Otafudu, "
-Hyperconvexity in Quasipseudometric Spaces and Fixed Point Theorems
In a previous work, we started investigating the concept of hyperconvexity in quasipseudometric spaces which we called -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 on a q-hyperconvex -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 whenever . (Here, denotes the usual (extended) Hausdorff quasipseudometric determined by on the set of nonempty subsets of X.)
In a previous work, we started investigating a concept of hyperconvexity in quasipseudometric spaces, which we called -hyperconvexity or Isbell-convexity (see , compare ). 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 -hyperconvex spaces and also present some fixed point theorems. In particular, we show that a set-valued mapping on a -hyperconvex -quasimetric space which takes values in the space of nonempty externally -hyperconvex subsets of always has a single-valued selection which satisfies whenever . (Here, denotes the usual (extended) Hausdorff quasipseudometric determined by on the set of nonempty subsets of .)
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 -hyperconvexity in quasipseudometric spaces (see also  for a more general approach).
In order to fix the terminology, we start with some basic concepts.
Definition 2.1. Let be a set and let be a function mapping into the set of the nonnegative reals. Then, is called a quasipseudometric on if(a) whenever ,(b) whenever .
We will say that is a -quasimetric provided that also satisfies the following condition: for each ,
Remark 2.2. Let be a quasipseudometric on a set , then defined by whenever is also a quasipseudometric, called the conjugate quasipseudometric of . As usual, a quasipseudometric on such that is called a pseudometric. Note that for any ()-quasipseudometric , the function is a pseudometric (metric).
For any , we will set .
Let be a quasipseudometric space. For each and , the set denotes the open -ball at . The collection of all “open” balls yields a base for a topology . It is called the topology induced by on . Similarly, for each and , we define the ball . Note that this latter set is -closed, but not -closed in general. As usual, in the theory of quasiuniformities, for a subset of and , we will also use notations like and similarly .
A pair with and nonnegative reals will be called a double ball at .
We shall also speak of a family of double balls, with and whenever .
Let be a quasipseudometric space and let be the set of all nonempty subsets of . Given , we will set and whenever .
For any , we set (compare ).
Then , is the so-called extended (as usual, a quasipseudometric that maps into (instead of will be called extended) Hausdorff(-Bourbaki) quasipseudometric on . It is known that is an extended -quasimetric when restricted to the set of all the nonempty subsets of which satisfy (compare [11, page 164]).
A map between two quasipseudometric spaces and is called an isometry or isometric map provided that whenever . Two quasipseudometric spaces and will be called isometric provided that there exists a bijective isometry . A map between two quasipseudometric spaces and is called nonexpansive provided that whenever .
The following definitions can be found in  (compare ).
Definition 2.3 (see [1, Definition 2]). A quasipseudometric space is called -hyperconvex (or Isbell-convex) provided that for each family of points in and families and of nonnegative real numbers satisfying whenever , the following condition holds:
Definition 2.4 (see [1, Definition 5]). Let be a quasipseudometric space. A family of double balls with and whenever is said to have the mixed binary intersection property if for all indices .
Definition 2.5 (see [1, Definition 6]). A quasipseudometric space is called -hypercomplete (or Isbell-complete) if every family of double balls, where and whenever , having the mixed binary intersection property satisfies .
Definition 2.6 (see [1, Definition 4]). Let be a quasipseudometric space. We say that is metrically convex if for any points and nonnegative real numbers and such that , there exists such that and .
The following useful result was established in [1, Proposition 1]. A quasipseudometric space is -hyperconvex if and only if it is metrically convex and -hypercomplete.
As usual, a subset of a quasipseudometric space will be called bounded provided that there is a positive real constant such that whenever . Note that a subset of is bounded if and only if there are and such that .
3. Some First Results
Proposition 3.1 (compare [13, Proposition 4.5]). Let be a -hyperconvex quasipseudometric space. Let be a nonempty family of points in and let and be two families of nonnegative reals such that . Set . Then is nonempty and -hyperconvex.
Proof. Note first that by -hyperconvexity of . For each , let and let be nonnegative reals such that whenever .
We show that the family satisfies the hypothesis of -hyperconvexity. Indeed, in particular, for each and , we have that and .
Hence, by -hyperconvexity of , we have that Hence, the subspace of is -hyperconvex.
Let be a quasipseudometric space. For a nonempty bounded subset of , we set Furthermore, we define the bicover of by .
A nonempty bounded set in a quasipseudometric space that can be written as the intersection of a nonempty family of sets of the form where and , that is, , will be called -admissible in the following. By , we will denote the set of -admissible subsets of . Note that if is -hyperconvex, then any member of is -hyperconvex in the light of Proposition 3.1.
Let be a quasipseudometric space and let be a nonempty bounded subset in . Then, in accordance with [13, page 79], we can define the cover of as follows: Obviously, we have . The latter inclusion can be strict, as our first example shows.
Example 3.2. Let be equipped with the -quasimetric defined by whenever .
Consider . Then, is equal to the line segment in from to . This follows from the fact that, for each , we have and , and that the line segment is a subset of any set of the form for which . Indeed, assume that the point belongs to this segment. Then and, therefore, by the triangle inequality.
On the other hand, , since with implies that . Indeed, assume that . Then, and . Thus, with . In the light that the interval has length , it follows that . Therefore, .
By the results of [1, Example 1], is -hyperconvex, while the metric space is hyperconvex [1, Proposition 2], but not -hyperconvex (see [1, Example 2]).
The following result gives a quasipseudometric variant of a well-known result usually attibuted to Sine  (compare also ).
Theorem 3.3. If is a bounded -hyperconvex -quasimetric space and if is a nonexpansive map, then the fixed point set of in is nonempty and -hyperconvex.
Proof. We first show that . Note that is nonexpansive, since for any , we have and , and thus . By assumption is bounded. Furthermore, is a hyperconvex space according to [1, Proposition 2]. So, by [13, Theorem 4.8], we know that has a fixed point and is hyperconvex in .
We need to show that is indeed -hyperconvex. Let be a nonempty family of double balls, where and and are two families of nonnegative reals such that whenever . Since is a -hyperconvex -quasimetric space, the set Let . Then, and whenever . Thus, and we have .
Moreover, is a bounded -hyperconvex -quasimetric space by Proposition 3.1. So the first part of the proof implies that has a fixed point in , which implies that . We have shown that is -hyperconvex.
4. Chains of -Hyperconvex Subspaces
In this section, we will prove the analogue of a famous theorem due to Baillon .
Theorem 4.1. Let be a bounded -quasimetric space and let be a descending family of nonempty -hyperconvex subsets of , where one assumes that is totally ordered such that and hold if and only if . Then, is nonempty and -hyperconvex.
Proof. We begin by showing that . We first note that is a bounded metric space and is a descending chain of hyperconvex sets in by [1, Proposition 2]. By the well-known result of Baillon [16, Theorem 7], we conclude that is nonempty and hyperconvex in .
In order to complete the proof, we need to show that is -hyperconvex. Let a nonempty family of points in and families of nonnegative real numbers and be given such that whenever . Fix . Since is a -hyperconvex space and since whenever , therefore, is nonempty and -hyperconvex by the proof of Proposition 3.1 and thus a hyperconvex subset of by [1, Proposition 2].
Thus by the first part of our present proof, since is descending. This proves that is -hyperconvex.
Definition 4.2. Let be a -quasimetric space and let a family of nonexpansive maps , with , be given. We say that is a commuting family if whenever .
Our next lemma is motivated by [16, Corollary 8].
Lemma 4.3. If is a family of bounded -hyperconvex subsets of a -quasimetric space such that is nonempty and -hyperconvex whenever is finite, then the intersection is nonempty and -hyperconvex.
Proof. Consider for all finite, , is nonempty and -hyperconvex.
Obviously and satisfies the hypothesis of Zorn's lemma because of Theorem 4.1. Let be maximal in . Then, whenever . Because of the maximality of , we, therefore, have whenever .
Theorem 4.4. Let be a bounded -hyperconvex -quasimetric space. Any commuting family of nonexpansive maps , with , has a common fixed point. Moreover, the common fixed point set is -hyperconvex.
Proof. We observe that is a bounded hyperconvex metric space by [1, Proposition 2], and for each , the map is nonexpansive, as we noted before (see proof of Theorem 3.3). By Theorem 3.3, each has a fixed point. Hence, there is such that . We now show that, given any , we have that Indeed, if for some , we have , then . So .
By Theorem 3.3, we conclude that has a fixed point , which then is a fixed point of and . Therefore, the set of common fixed points of and is -hyperconvex by Theorem 3.3. Hence, by induction for each finite family of nonexpansive self-maps on the set of common fixed points is nonempty and -hyperconvex.
Since is nonempty subset and -hyperconvex whenever is a finite subset of , by Lemma 4.3 we conclude that is nonempty and -hyperconvex.
5. Approximate Fixed Points
In this section, we investigate the approximation of fixed points by generalizing some results from  (compare ). We first define an -parallel set of a subset in a quasipseudometric space similarly to [13, page 89].
Definition 5.1. Let be a quasipseudometric space. Given a subset of , we define for the -parallel set of as
(Note that for each in particular
Thus, if and only if there exists such that and .
We next give a characterization of if is a -admissible set in a -hyperconvex quasipseudometric space (compare [13, Lemma 4.2]).
Lemma 5.2. Let be a -hyperconvex quasipseudometric space and let be a -admissible subset of , say with and nonnegative reals whenever . Then, for each ,
Proof. Suppose that . Then, and for some . But for each ,
Then, for each , we have and . This proves that
Now suppose that and let .
Hence, By definition of and the triangle inequality, for any and any we must have that Hence, satisfies the hypothesis in the definition of -hyperconvexity of .
So, by -hyperconvexity of ,
Therefore, there is such that and . Hence, 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 is a -hyperconvex -quasimetric space and let be a -admissible subset of . Then, for each there is a nonexpansive retraction of onto which has the property that and whenever .
Proof. Assume with . By Lemma 5.2, we know that is -admissible in and so is itself -hyperconvex by Proposition 3.1. Consider the family and is a nonexpansive retraction such that and whenever .
Let denote the identity map on . Note that . So . If one partially orders in the usual way if and only if and is an extension of , then each chain in is bounded above. So by Zorn's lemma has a maximal element which we denote by . We need to show that . Suppose that there exists such that , and consider the set First, we want to show that , and in order to do this by [1, Proposition 1], we need only to show that the family of double balls has the mixed binary intersection property.
First note that if , then Therefore, and intersect by metric convexity of .
Furthermore, by the definition of , for each , we see that and intersect.
Also for each , Hence, for any and and intersect, as well as for any and and intersect.
Since by Lemma 5.2 there is such that and, therefore, whenever .
Finally, if , then by assumption on , Thus, by metric convexity of , we have that and intersect as well as and intersect.
Of course, and intersect.
We have shown that the family of double balls has the mixed binary intersection property.
Hence, . Now, let and define by setting if and . Then, for each , we have so that is nonexpansive. Also, and . Therefore, we conclude that the pair contradicts the maximality of in . Consequently, and we are done.
Definition 5.5. Let be a -quasimetric space. For a map and for any , we use to denote the set of -approximate fixed points of ; that is, and .
Theorem 5.6 (compare [13, Theorem 4.11]). Suppose that is a -hyperconvex -quasimetric space and that the map is nonexpansive. Furthermore suppose that for some one has that is nonempty. Then, the set is -hyperconvex.
Proof. For each in some nonempty index set , let , and let and satisfy
We need to show that
We know that is -hyperconvex according to Proposition 3.1, since is -hyperconvex. Furthermore, is obviously bounded in .
Also, if , then for each ,
This proves that by Lemma 5.2. Now, by Lemma 5.3, there is a nonexpansive retraction of onto for which and whenever . Also since is a nonexpansive map of into , it must have a fixed point by Theorem 3.3.
Suppose that for some . Then, Thus, the proof is complete, since .
6. External -Hyperconvexity
We next define an externally -hyperconvex subset of a quasipseudometric space in analogy to [17, Definition 3.5]. Note that this definition strengthens the concept of a -hyperconvex subset of (compare also [21, Definition 3]).
Definition 6.1. Let be a quasipseudometric space. A subspace of is said to be externally -hyperconvex (relative to ) if given any family of points in and families of nonnegative real numbers and the following condition holds: if whenever , and whenever , then
In the following, will denote the set of nonempty externally -hyperconvex subsets of .
Example 6.2 (compare [21, Theorem 7]). Let be a nonempty externally -hyperconvex subset in a quasipseudometric space and let be any point of . Set and . Then, by applying external -hyperconvexity of to the double ball , we conclude that there is . Thus, and .
Lemma 6.3 (compare [17, Lemma 3.8]). Let be a -hyperconvex space and let . Furthermore, let where is a nonempty family of points in and and are families of nonnegative reals. Then, there is such that and .
Proof. Evidently, satisfies the mixed binary intersection property. Thus, there is by -hyperconvexity of . Obviously, then satisfies the stated condition.
Lemma 6.4 (compare [18, Lemma 2]). Let be a -hyperconvex quasipseudometric space. Suppose that is externally -hyperconvex relative to and suppose that is a -admissible subset of such that . Then is externally -hyperconvex relative to .
Proof. Assume that a given nonempty family of points in and families of nonnegative real numbers and satisfy , , and whenever .
Since is -admissible, with and whenever . Because and whenever , it follows that for each , and for chosen according to Lemma 6.3 we have Also, since for each , , and since , it follows that and that whenever . Trivially, we have and whenever .
Therefore, by external -hyperconvexity of , we conclude that Thus, the proof is complete.
We next show that the intersection of a descending family of externally -hyperconvex nonempty subspaces of a bounded -hyperconvex -quasimetric space behaves as expected.
Theorem 6.5 (compare [18, Theorem ]). Let be a bounded -hyperconvex -quasimetric space. Moreover, let be a descending family of nonempty externally -hyperconvex subsets of , where is assumed to be totally ordered such that and if and only if . Then is nonempty and externally -hyperconvex relative to .
Proof. Theorem 4.1 implies that . In order to show that is externally -hyperconvex, let a nonempty family of points in and families of nonnegative real numbers and be given such that , and and whenever .
Since is -hyperconvex, we know that . Also, since for each , and , we have and whenever , so that, by external -hyperconvexity of , we conclude that whenever .
By Lemma 6.4, is a descending chain of nonempty (externally) -hyperconvex subsets of , so that again by Theorem 4.1 .
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 be a -hyperconvex -quasimetric space, let be any set, and let a map be given. Then, there exists a map for which whenever and for which whenever .
Proof. Let denote the collection of all pairs , where whenever , and whenever . Note that , since for any choice of and . Define a partial order relation on by setting if and only if , and, .
Let be an increasing chain in . Then it follows that where . By Zorn's lemma, has a maximal element, say . Assume that and select . Set and consider the set Since , by definition of external -hyperconvexity, if for each , we have and and for any , also We are going to check that these conditions hold.
Let . For each , we have and by definition of the Hausdorff quasipseudometric.
Since , for each , there is such that , and there is such that Therefore, and .
We finally also note that by assumption on , for each we have that Thus, we have shown that . Choose and define
if and if .
Since for each