# q-Hyperconvexity in Quasipseudometric Spaces and Fixed Point Theorems

Authors: Hans-Peter A. Künzi; Olivier Olela Otafudu
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

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 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 𝒫0(X) of nonempty subsets of X.)


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## Body

## 1. Introduction

In a previous work, we started investigating a concept of hyperconvexity in quasipseudometric spaces, which we calledq-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,y∈X. (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 ofq-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 [6–9].


## 2.  Preliminaries

In order to fix the terminology, we start with some basic concepts.Definition 2.1.
LetX be a set and let d:X×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 x∈X,(b)
d(x,z)≤d(x,y)+d(y,z) whenever x,y,z∈X.
We will say thatd is a T0-quasimetric provided that d also satisfies the following condition:  for each x,y∈X,
(2.1)d(x,y)=0=d(y,x)impliesthatx=y.Remark 2.2.
Letd be a quasipseudometric on a set X, then d-1:X×X→[0,∞) defined by d-1(x,y)=d(y,x) whenever x,y∈X is also a quasipseudometric, called the  conjugate quasipseudometric of d. As usual, a quasipseudometric d on X such that d=d-1 is called a  pseudometric. Note that for any (T0)-quasipseudometric d, the function ds=max{d,d-1}=d∨d-1 is a pseudometric (metric).
For anya,b∈[0,∞), we will set a-˙b=max{a-b,0}.
Let(X,d) be a quasipseudometric space. For each x∈X and ϵ>0, the set Bd(x,ϵ)={y∈X:d(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 byd on X. Similarly, for each x∈X and ϵ≥0, we define the ball Cd(x,ϵ)={y∈X:d(x,y)≤ϵ}. Note that this latter set is τ(d-1)-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)=⋃a∈A‍Bd(a,ϵ) and similarly Cd(A,ϵ)=⋃a∈A‍Cd(a,ϵ).
A pair(Cd(x,r);Cd-1(x,s)) with x∈X and nonnegative reals r,s will be called a double ball atx.
We shall also speak of a family[(Cd(xi,ri))i∈I;(Cd-1(xi,si))i∈I] of double balls, with xi∈X and ri,si≥0 whenever i∈I.
Let(X,d) be a quasipseudometric space and let 𝒫0(X) be the set of all nonempty subsets of X. Given C∈𝒫0(X), we will set dist(x,C)=inf{d(x,c):c∈C} and dist(C,x)=inf{d(c,x):c∈C} whenever x∈X.
For anyA,B∈𝒫0(X), we set
(2.2)dH(A,B)=max{supb∈Bdist(A,b),supa∈Adist(a,B)}
(compare [10]).
ThendH, 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)A∩clτ(d-1)A (compare [11, page 164]).
A mapf:(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,y∈X. 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,y∈X.
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)i∈I of points in X and families (ri)i∈I and (si)i∈I of nonnegative real numbers satisfying d(xi,xj)≤ri+sj whenever i,j∈I, the following condition holds:
(2.3)⋂i∈I(Cd(xi,ri)∩Cd-1(xi,si))≠∅.Definition 2.4 (see [1, Definition 5]).
Let(X,d) be a quasipseudometric space. A family of double balls [(Cd(xi,ri))i∈I;(Cd-1(xi,si))i∈I] with ri,si∈[0,∞) and xi∈X whenever i∈I is said to have  the mixed binary intersection property if for all indices i,j∈I,Cd(xi,ri)∩Cd-1(xj,sj)≠∅.Definition 2.5 (see [1, Definition 6]).
A quasipseudometric space(X,d) is called  q-hypercomplete (or Isbell-complete) if every family
(2.4)[(Cd(xi,ri))i∈I;(Cd-1(xi,si))i∈I]
of double balls, where ri,si≥0 and xi∈X whenever i∈I, having the mixed binary intersection property satisfies ⋂i∈I‍(Cd(xi,ri)∩Cd-1(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,y∈X and nonnegative real numbers r and s such that d(x,y)≤r+s, there exists z∈X 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 subsetA 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,y∈A. Note that a subset A of (X,d) is bounded if and only if there are x∈X and r,s≥0 such that A⊆Cd(x,r)∩Cd-1(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)i∈I be a nonempty family of points in X and let (ri)i∈I and (si)i∈I be two families of nonnegative reals such that d(xi,xj)≤ri+sj. Set D=⋂i∈I‍(Cd(xi,ri)∩Cd-1(xi,si)). Then D is nonempty and q-hyperconvex.Proof.
Note first thatD≠∅ by q-hyperconvexity of X. For each α∈S, let xα∈D and let rα,sα be nonnegative reals such that d(xα,xβ)≤rα+sβ whenever α,β∈S.
We show that the family(3.1)[(Cd(xα,rα))α∈S,(Cd(xi,ri))i∈I;(Cd-1(xα,sα))α∈S,(Cd-1(xi,si))i∈I]
satisfies the hypothesis of q-hyperconvexity. Indeed, in particular, for each α∈S and i∈I, we have that d(xα,xi)≤si≤rα+si and d(xi,xα)≤ri≤ri+sα.
Hence, byq-hyperconvexity of X, we have that
(3.2)∅≠⋂i∈I(Cd(xi,ri)∩Cd-1(xi,si))∩⋂α∈S(Cd(xα,rα)∩Cd-1(xα,sα))=D∩⋂α∈S(Cd(xα,rα)∩Cd-1(xα,sα)).
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
(3.3)bicov(A)+=⋂{Cd(x,r):A⊆Cd(x,r),x∈X,r≥0},bicov(A)-=⋂{Cd-1(x,s):A⊆Cd-1(x,s),x∈X,s≥0}.
Furthermore, we define the bicover ofA by bicov(A):=bicov(A)+∩bicov(A)-.A nonempty bounded setA 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)∩Cd-1(x,ϵ2) where ϵ1,ϵ2≥0 and x∈X, that is, A=bicovA, will be called q-admissible in the following. By 𝒜q(X), we will denote the set  ofq-admissible subsets ofX. 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 covercovAofA as follows: covA=⋂{Cds(x):A⊆Cds(x),x∈X}. Obviously, we have A⊆bicov(A)⊆cov(A). The latter inclusion can be strict, as our first example shows.Example 3.2.
LetX=[0,1]×[1/4,3/4] be equipped with the T0-quasimetric d defined by d((α,β),(α',β'))=(α-˙α')∨(β-˙β') whenever (α,β),(α',β')∈X.
ConsiderA={(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]=Cd-1(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)∩Cd-1(b,s) for which {x,y}⊆Cd(a,r)∩Cd-1(b,s). Indeed, assume that the point z belongs to this segment. Then d(z,y)=0=d(x,z) and, therefore, z∈Cd(a,r)∩Cd-1(b,s) by the triangle inequality.
On the other hand,cov(A)=X, since {x,y}⊆Cds(z,ϵ) with z∈X implies that ϵ≥1/2. Indeed, assume that z=(a,b)∈X. Then, a≤ds((a,b),(0,1/2))≤ϵ and 1-a≤ds((a,b),(1,1/2))≤ϵ. Thus, ϵ≥max{a,1-a}≥1/2 with a∈[0,1]. In the light that the interval [1/4,3/4] has length 1/2, it follows that X⊆Cds(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 thatFix(T)≠∅. Note that T:(X,ds)→(X,ds) is nonexpansive, since for any x,y∈X, 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 thatFix(T) is indeed q-hyperconvex. Let
(3.4)[(Cd(xi,ri))i∈I;(Cd-1(xi,si))i∈I]
be a nonempty family of double balls, where xi∈Fix(T) and (ri)i∈I and (si)i∈I are two families of nonnegative reals such that d(xi,xj)≤ri+sj whenever i,j∈I. Since X is a q-hyperconvex T0-quasimetric space, the set
(3.5)X0=⋂i∈I(Cd(xi,ri)∩Cd-1(xi,si))≠∅.
Let x∈X0. Then, d(T(x),xi)=d(T(x),T(xi))≤d(x,xi)≤si and
(3.6)d(xi,T(x))=d(T(xi),T(x))≤d(xi,x)≤ri
whenever i∈I. Thus, T(x)∈X0 and we have T:X0→X0.
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)∩[⋂i∈I‍(Cd(xi,ri)∩Cd-1(xi,si)]≠∅. We have shown that Fix(T) is q-hyperconvex.


## 4. Chains ofq-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)i∈I be a descending family of nonempty q-hyperconvex subsets of X, where one assumes that I is totally ordered such that i1,i2∈I and i1≤i2 hold if and only if Hi2⊆Hi1. Then, ⋂i∈I‍Hi is nonempty and q-hyperconvex.Proof.
We begin by showing that⋂i∈I‍Hi≠∅. We first note that (X,ds) is a bounded metric space and (Hi)i∈I 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 ⋂i∈I‍Hi is nonempty and hyperconvex in (X,ds).
In order to complete the proof, we need to show thatH=⋂i∈I‍Hi is q-hyperconvex. Let a nonempty family (xα)α∈S of points in H and families of nonnegative real numbers (rα)α∈S and (sα)α∈S be given such that d(xα,xβ)≤rα+sβ whenever α,β∈S. Fix i∈I. Since Hi is a q-hyperconvex space and since xα∈Hi whenever α∈S, therefore, 𝒟i=⋂α∈S‍(Cd(xα,rα)∩Cd-1(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,(4.1)∅≠⋂i∈I𝒟i=⋂i∈I[⋂α∈S(Cd(xα,rα)∩Cd-1(xα,sα))∩Hi]=⋂α∈S(Cd(xα,rα)∩Cd-1(xα,sα))∩⋂i∈I‍Hi,
since (𝒟i)i∈I is descending. This proves that H=⋂i∈I‍Hi is q-hyperconvex.Definition 4.2.
Let(X,d) be a T0-quasimetric space and let a family of nonexpansive maps (Ti)i∈I, with Ti:(X,d)→(X,d), be given. We say that (Ti)i∈I is a commuting family if Ti∘Tj=Tj∘Ti whenever i,j∈I.
Our next lemma is motivated by [16, Corollary 8].Lemma 4.3.
If(Hα)α∈S is a family of bounded q-hyperconvex subsets of a T0-quasimetric space X such that ⋂α∈F‍Hα is nonempty and q-hyperconvex whenever F⊆S is finite, then the intersection ⋂α∈S‍Hα is nonempty and q-hyperconvex.Proof.
Considerℱ={I⊆S: for all J finite, J⊆S,   ⋂I∪J‍Hα is nonempty and q-hyperconvex}.
Obviously∅∈ℱ and ℱ satisfies the hypothesis of Zorn's lemma because of Theorem 4.1. Let I be maximal in ℱ. Then, I∪{α}∈ℱ whenever α∈S. Because of the maximality of I, we, therefore, have α∈I whenever α∈S.The next result is a consequence of Theorems3.3 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)i∈I, with Ti:(X,d)→(X,d), has a common fixed point. Moreover, the common fixed point set
(4.2)⋂i∈IFix(Ti),
is q-hyperconvex.Proof.
We observe that(X,ds) is a bounded hyperconvex metric space by [1, Proposition 2], and for each i∈I, 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 x∈X such that Ti(x)=x. We now show that, given any j∈I, we have that Tj(Fix(Ti))⊆Fix(Ti) Indeed, if for some x∈X, we have x=Ti(x), then Tj(x)=Tj(Ti(x))=Ti(Tj(x)). So Tj(x)∈Fix(Ti).
By Theorem3.3, we conclude that Tj:Fix(Ti)→Fix(Ti) has a fixed point y∈Fix(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)i∈F of nonexpansive self-maps on X the set of common fixed points is nonempty and q-hyperconvex.
Since⋂i∈F‍Fix(Ti) is nonempty subset and q-hyperconvex whenever F is a finite subset of I, by Lemma 4.3 we conclude that ⋂i∈I‍Fix(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,ϵ2≥0 the ϵ1,ϵ2-parallel set of A as
(5.1)Nϵ1,ϵ2(A)=⋃a∈A(Cd(a,ϵ2)∩Cd-1(a,ϵ1)).
(Note that for eachϵ>0 in particular Nϵ,ϵ(A)=⋃a∈A‍Cds(a,ϵ).)
Thus,x∈Nϵ1,ϵ2(A) if and only if there exists a∈A such that d(a,x)≤ϵ2 and d-1(a,x)≤ϵ1.
We next give a characterization ofNϵ1,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=⋂i∈I(Cd(xi,ri)∩Cd-1(xi,si)) with xi∈X and ri,si nonnegative reals whenever i∈I≠∅. Then, for each ϵ1,ϵ2≥0,
(5.2)Nϵ1,ϵ2(A)=⋂i∈I(Cd(xi,ri+ϵ2)∩Cd-1(xi,si+ϵ1)).Proof.
Suppose thaty∈Nϵ1,ϵ2(A). Then, d(a,y)≤ϵ2 and d(y,a)≤ϵ1 for some a∈A. But for each i∈I,
(5.3)d(xi,y)≤d(xi,a)+d(a,y)≤ri+ϵ2,d(y,xi)≤d(y,a)+d(a,xi)≤ϵ1+si.
Then, for each i∈I, we have y∈Cd(xi,ri+ϵ2) and y∈Cd-1(xi,si+ϵ1). This proves that  Nϵ1,ϵ2(A)⊆⋂i∈I(Cd(xi,ri+ϵ2)∩Cd-1(xi,si+ϵ1)).
Now suppose thaty∈⋂i∈I(Cd(xi,ri+ϵ2)∩Cd-1(xi,si+ϵ1)) and let i∈I.
Hence,(5.4)d(xi,y)≤ri+ϵ2,d(y,xi)≤ϵ1+si.
By definition of A and the triangle inequality, for any a∈A and any i,j∈I  we must have that
(5.5)d(xi,xj)≤d(xi,a)+d(a,xj)≤ri+sj.
Hence, [(Cd(xi,ri))i∈I,Cd(y,ϵ1);(Cd-1(xi,si))i∈I,Cd-1(y,ϵ2)] satisfies the hypothesis in the definition of q-hyperconvexity of (X,d).
So, byq-hyperconvexity of X,
(5.6)∅≠(⋂i∈ICd(xi,ri))∩Cd(y,ϵ1)∩(⋂i∈ICd-1(xi,si))∩Cd-1(y,ϵ2)=⋂i∈I(Cd(xi,ri)∩Cd-1(xi,si))∩(Cd(y,ϵ1)∩Cd-1(y,ϵ2))=A∩(Cd(y,ϵ1)∩Cd-1(y,ϵ2)).
Therefore, there isa∈A such that d(y,a)≤ϵ1 and d(a,y)≤ϵ2. Hence, y∈Nϵ1,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,ϵ2≥0 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 x∈Nϵ1,ϵ2(A).Proof.
Assume∅≠A=⋂i∈I(Cd(xi,ri)∩Cd-1(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 ℱ={(D,RD):A⊆D⊆Nϵ1,ϵ2(A) and RD:D→A is a nonexpansive retraction such that d(x,R(x))≤ϵ1 and d(R(x),x)≤ϵ2 whenever x∈D}.
LetId denote the identity map on A. Note that (A,Id)∈ℱ. So ℱ≠∅. If one partially orders ℱ in the usual way ((D,RD)≼(H,RH) if and only if D⊆H and RH is an extension of RD), then each chain in (ℱ,≼) is bounded above. So by Zorn's lemma ℱ has a maximal element which we denote by (D,RD). We need to show that D=Nϵ1,ϵ2(A). Suppose that there exists x∈Nϵ1,ϵ2(A) such that x∉D, and consider the set
(5.7)C=[⋂w∈D(Cd(RD(w),d(w,x))∩Cd-1(RD(w),d(x,w)))]∩[⋂i∈I(Cd(xi,ri)∩Cd-1(xi,si))]∩[Cd(x,ϵ1)∩Cd-1(x,ϵ2)].
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
(5.8)[(Cd(RD(w),d(w,x)))w∈D,(Cd(xi,ri))i∈I,Cd(x,ϵ1);(Cd-1(RD(w),d(x,w)))w∈D,(Cd-1(xi,si))i∈I,Cd-1(x,ϵ2)]
of double balls has the mixed binary intersection property.
First note that ifw1,w2∈D, then
(5.9)d(RD(w1),RD(w2))≤d(w1,w2)≤d(w1,x)+d(x,w2).
Therefore, Cd(RD(w1),d(w1,x)) and Cd-1(RD(w2),d(x,w2)) intersect by metric convexity of (X,d).
Furthermore, by the definition ofA, for each i,j∈I,  we see that Cd(xi,ri) and Cd-1(xj,sj) intersect.
Also for eachw∈D, RD(w)∈A=⋂i∈I(Cd(xi,ri)∩Cd-1(xi,si)). Hence, for any w∈D and i∈I,Cd(RD(w),d(w,x)) and Cd-1(xi,si) intersect, as well as for any w∈D and i∈I,Cd-1(RD(w),d(x,w)) and Cd(xi,ri) intersect.
Since(5.10)x∈Nϵ1,ϵ2(A)=⋂i∈I(Cd(xi,ri+ϵ2)∩Cd-1(xi,si+ϵ1)),
by Lemma 5.2 there is a∈A such that x∈Cd(a,ϵ2)∩Cd-1(a,ϵ1) and, therefore, (Cd(x,ϵ1)∩Cd-1(x,ϵ2))∩(Cd(xi,ri)∩Cd-1(xi,si))≠∅ whenever i∈I.
Finally, ifw∈D, then by assumption on RD,
(5.11)d(RD(w),x)≤d(RD(w),w)+d(w,x)≤ϵ2+d(w,x),d(x,RD(w))≤d(x,w)+d(w,RD(w))≤d(x,w)+ϵ1.
Thus, by metric convexity of (X,d), we have that Cd(RD(w),d(w,x)) and Cd-1(x,ϵ2) intersect as well as Cd-1(RD(w),d(x,w)) and Cd(x,ϵ1) intersect.
Of course,Cd(x,ϵ1) and Cd-1(x,ϵ2) intersect.
We have shown that the family(5.12)[Cd(RD(w),d(w,x))w∈D,(Cd(xi,ri))i∈I,Cd(x,ϵ1);Cd-1(RD(w),d(x,w))w∈D,(Cd-1(xi,si))i∈I,Cd-1(x,ϵ2)]
of double balls has the mixed binary intersection property.
Hence,∅≠C⊆A. Now,  let u∈C and define R′:D∪{x}→A by setting R′(w)=RD(w) if w∈D and R′(x)=u. Then, for each w∈D, we have
(5.13)d(R′(x),R′(w))=d(u,RD(w))≤d(x,w),d(R′(w),R′(x))=d(RD(w),u)≤d(w,x),
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 (ℱ,≼). 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 infx∈Xds(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,ϵ2≥0, we use Fϵ1,ϵ2(T) to denote the set of ϵ1,ϵ2-approximate fixed points of T; that is, Fϵ1,ϵ2(T)={x∈X:d(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,ϵ2≥0 one has that Fϵ1,ϵ2(T) is nonempty. Then, the set Fϵ1,ϵ2(T) is q-hyperconvex.Proof.
For eachi in some nonempty index set I, let xi∈Fϵ1,ϵ2(T), and let ri≥0 and si≥0 satisfy
(5.14)d(xi,xj)≤ri+sj.
We need to show that
(5.15)[⋂i∈I(Cd(xi,ri)∩Cd-1(xi,si))]∩Fϵ1,ϵ2(T)≠∅.
We know that ∅≠J=⋂i∈I(Cd(xi,ri)∩Cd-1(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, ifx∈J, then for each i∈I,
(5.16)d(xi,T(x))≤d(xi,T(xi))+d(T(xi),T(x))≤ϵ2+d(xi,x)≤ϵ2+ri,d(T(x),xi)≤d(T(x),T(xi))+d(T(xi),xi)≤d(x,xi)+ϵ1≤si+ϵ1.
This proves thatT(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 x∈Nϵ1,ϵ2(J). Also since R∘T is a nonexpansive map of J into J, it must have a fixed point by Theorem 3.3.
Suppose that(R∘T)(x0)=x0 for some x0∈J. Then,
(5.17)d(x0,T(x0))=d((R∘T)(x0),T(x0))≤ϵ2,d(T(x0),x0)=d(T(x0),(R∘T)(x0))≤ϵ1.
Thus, the proof is complete, since x0∈J∩Fϵ1,ϵ2(T).


## 6. Externalq-Hyperconvexity

We next define an externallyq-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)i∈I of points in X and families of nonnegative real numbers (ri)i∈I and (si)i∈I the following condition holds:ifd(xi,xj)≤ri+sj whenever i,j∈I, dist(xi,E)≤ri and dist(E,xi)≤si whenever i∈I, then ⋂i∈I(Cd(xi,ri)∩Cd-1(xi,si))∩E≠∅.
In the following,ℰq(X) will denote the set of nonempty externally q-hyperconvex subsets of (X,d).Example 6.2 (compare [21, Theorem 7]).
LetE 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);Cd-1(x,s)), we conclude that there is p∈Cd(x,r)∩Cd-1(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 x∈X. Furthermore, let ∅≠A=⋂i∈I‍(Cd(xi,ri)∩Cd-1(xi,si)) where (xi)i∈I is a nonempty family of points in X and (ri)i∈I and (si)i∈I are families of nonnegative reals. Then, there is p∈A such that dist(x,A)=d(x,p) and dist(A,x)=d(p,x).Proof.
Evidently,(6.1)[(Cd(xi,ri))i∈I,(Cd(x,dist(x,A)+ϵ))ϵ>0;(Cd-1(xi,si))i∈I,(Cd-1(x,dist(A,x)+ϵ))ϵ>0]
satisfies the mixed binary intersection property. Thus, there is
(6.2)p∈A∩Cd(x,dist(x,A))∩Cd-1(x,dist(A,x))
by q-hyperconvexity of (X,d). Obviously, p then satisfies the stated condition.The following lemma, which makes use of Lemma6.3, 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 E⊆X is externally q-hyperconvex relative to X and suppose that A is a q-admissible subset of (X,d) such that E∩A≠∅. Then E∩A is externally q-hyperconvex relative to X.Proof.
Assume that a given nonempty family(xα)α∈S of points in X and families of nonnegative real numbers (rα)α∈S and (sα)α∈S satisfy d(xα,xβ)≤rα+sβ, dist(xα,A∩E)≤rα, and dist(A∩E,xα)≤sα whenever α,β∈S.
SinceA is q-admissible, ∅≠A=⋂i∈I(Cd(xi,ri)∩Cd-1(xi,si)) with xi∈X and ri,si≥0 whenever i∈I. Because dist(xα,A)≤dist(xα,A∩E)≤rα and dist(A,xα)≤dist(A∩E,xα)≤sα whenever α∈S, it follows that for each  α∈S, i∈I and for p∈A chosen according to Lemma 6.3 we have
(6.3)d(xα,xi)≤d(xα,p)+d(p,xi)≤rα+si,d(xi,xα)≤d(xi,p)+d(p,xα)≤ri+sα.
Also, since for each i∈I, A⊆Cd(xi,ri)∩Cd-1(xi,si), and since A∩E≠∅, it follows that
(6.4)dist(xi,E)≤ri,dist(E,xi)≤si,
and that d(xi,xj)≤ri+sj whenever i,j∈I. Trivially, we have dist(xα,E)≤rα and dist(E,xα)≤sα whenever α∈S.
Therefore, by externalq-hyperconvexity of E, we conclude that
(6.5)[⋂i∈I(Cd(xi,ri)∩Cd-1(xi,si))]∩[⋂α∈S(Cd(xα,rα)∩Cd-1(xα,sα))∩E]=⋂α∈S(Cd(xα,rα)∩Cd-1(xα,sα))∩(E∩A)≠∅.
Thus, the proof is complete.We next show that the intersection of a descending family of externallyq-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)i∈I be a descending family of nonempty externally q-hyperconvex subsets of X, where I is assumed to be totally ordered such that i1,i2∈I and i1≤i2 if and only if Xi2⊆Xi1. Then⋂i∈IXi is nonempty and externally q-hyperconvex relative to X.Proof.
Theorem4.1 implies that D=⋂i∈IXi≠∅. In order to show that D is externally q-hyperconvex, let a nonempty family (xα)α∈S of points in X and families of nonnegative real numbers (rα)α∈S and (sα)α∈S be given such that d(xα,xβ)≤rα+sβ, and dist(xα,D)≤rα and dist(D,xα)≤sα whenever α,β∈S.
SinceX is q-hyperconvex, we know that A:=⋂α∈S(Cd(xα,rα)∩Cd-1(xα,sα))≠∅. Also, since for each α∈S, dist(xα,D)≤rα and dist(D,xα)≤sα, we have dist(xα,Xi)≤rα and dist(Xi,xα)≤sα whenever i∈I, so that, by external q-hyperconvexity of Xi, we conclude that A∩Xi≠∅ whenever i∈I.
By Lemma6.4,  (A∩Xi)i∈I is a descending chain of nonempty (externally) q-hyperconvex subsets of (X,d), so that again by Theorem 4.1  ⋂i∈I(A∩Xi)=A∩D≠∅.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 T*:X→ℰq(H) be given. Then, there exists a map T:X→H for which T(x)∈T*(x) whenever x∈X and for which d(T(x),T(y))≤dH(T*(x),T*(y)) whenever x,y∈X.Proof.
Letℱ denote the collection of all pairs (D,T), where D⊆X,T:D→H,T(d)∈T*(d) whenever d∈D, and d(T(x),T(y))≤dH(T*(x),T*(y)) whenever x,y∈D. Note that ℱ≠∅, since ({x0},T)∈ℱ for any choice of x0∈X and T(x0)∈T*(x0). Define a partial order relation on ℱ by setting (D1,T1)≼(D2,T2) if and only if D1⊆D2, and, T2|D1=T1.
Let((Dα,Tα))α∈S be an increasing chain in (ℱ,≼). Then it follows that (⋃α∈S‍Dα,T)∈ℱ where T|Dα=Tα. By Zorn's lemma, (ℱ,≼) has a maximal element, say (D,T). Assume that D≠X and select x0∈X∖D. Set D~=D∪{x0} and consider the set
(6.6)J=⋂x∈D[Cd(T(x),dH(T*(x),T*(x0)))∩Cd-1(T(x),dH(T*(x0),T*(x)))]∩T*(x0).
Since T*(x0)∈ℰq(H), by definition of external q-hyperconvexity, J≠∅ if for each x∈D, we have dist(T(x),T*(x0))≤dH(T*(x),T*(x0)) and
(6.7)dist(T*(x0),T(x))≤dH(T*(x0),T*(x)),
and for any x,y∈D, also
(6.8)d(T(x),T(y))≤dH(T*(x),T*(x0))+dH(T*(x0),T*(y)).
We are going to check that these conditions hold.
Letx∈D. For each ϵ>0, we have T*(x)⊆Bd-1(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.
SinceT(x)∈T*(x), for each ϵ>0, there is a∈T*(x0) such that d(T(x),a)≤dH(T*(x),T*(x0))+ϵ, and there is b∈T*(x0) such that
(6.9)d(b,T(x))≤dH(T*(x0),T*(x))+ϵ.
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 onT, for each x,y∈D we have that
(6.10)d(T(x),T(y))≤dH(T*(x),T*(y))≤dH(T*(x),T*(x0))+dH(T*(x0),T*(y)).
Thus, we have shown that J≠∅. Choose y0∈J and define
T
~
(
x
)
=
y
0 if x=x0 and T~(x)=T(x) if x∈D.
Since for eachx∈D,   d(T~(x0),T~(x))=d(y0,T(x))≤dH(T*(x0),T*(x)) and d(T~(x),T~(x0))=d(T(x),y0)≤dH(T*(x),T*(x0)), we conclude that (D∪{x0},T~)∈ℱ, contradicting the maximality of (D,T). Therefore, D=X.Corollary 6.7 (compare [18, Corollary 1]).
Let(H,d) be a q-hyperconvex T0-quasimetric space. Moreover, let (X,ρ) be a T0-quasimetric space, and suppose that T*:X→ℰq(H) is nonexpansive, that is, dH(T*(x),T*(y))≤ρ(x,y) whenever x,y∈X. Then, there is a nonexpansive map T:(X,ρ)→(H,d) for which T(x)∈T*(x) whenever x∈X.Proof.
BecauseT* is nonexpansive, the selection obtained from Theorem 6.6 is also nonexpansive.Corollary 6.8 (compare [18, Corollary 2]).
LetH be a bounded and q-hyperconvex T0-quasimetric space and suppose that T*:H→ℰq(H) is nonexpansive. Then T* has a fixed point, that is, there exists x∈H such that x∈T*(x).Proof.
The existence of a fixed point for the nonexpansive selectionT of T*, which exists by Corollary 6.7, follows from Theorem 3.3.In the following theorem, we setFix(T*)={x∈H:x∈T*(x)}. According to Corollary 6.8,  Fix(T*)≠∅ if H is bounded and q-hyperconvex, and T* is nonexpansive.Theorem 6.9 (compare [18, Theorem 2]).
Let(H,d) be a q-hyperconvex T0-quasimetric space, let T*:H→ℰq(H) be a nonexpansive map and suppose that Fix(T*)≠∅. Then, there exists a nonexpansive map T:H→H with T(x)∈T*(x) whenever x∈H and for which Fix(T)=Fix(T*).Proof.
Letℱ denote the collection of all pairs (D,T), where Fix(T*)⊆D⊆H,T:D→H,T(d)∈T*(d) whenever d∈D,T(x)=x whenever x∈Fix(T*), and d(T(x),T(y))≤d(x,y) whenever x,y∈D. By assumption (Fix(T*),Id)∈ℱ, so ℱ≠∅. The argument now follows from a modification of the proof of Theorem 6.6. One defines a partial order on ℱ by setting (D1,T1)≼(D2,T2) if and only if D1⊆D2 and T2|D1=T1.
Let((Dα,Tα))α∈S be an increasing chain in (ℱ,≼). Then, it follows that (⋃α∈S‍Dα,T)∈ℱ where T|Dα=Tα whenever α∈S. By Zorn's lemma, (ℱ,≼) has a maximal element, say (D,T). Assume D≠H and find x0∈H∖D. Set D~=D∪{x0} and consider the set:
(6.11)J=⋂x∈D[Cd(T(x),d(x,x0))∩Cd-1(T(x),d(x0,x))]∩T*(x0).
Since T*(x0)∈ℰq(H), by definition of external q-hyperconvexity, J≠∅ if for each x∈D, we have dist(T(x),T*(x0))≤d(x,x0) and dist(T*(x0),T(x))≤d(x0,x), and for any x,y∈D we have d(T(x),T(y))≤d(x,x0)+d(x0,y).
We are going to check these conditions next. Letx∈D. For each ϵ>0, we have T*(x)⊆Bd-1(T*(x0),dH(T*(x),T*(x0))+ϵ)⊆Bd-1(T*(x0),d(x,x0)+ϵ) and T*(x)⊆Bd(T*(x0),dH(T*(x0),T*(x))+ϵ)⊆Bd(T*(x0),d(x0,x)+ϵ) by definition of the Hausdorff quasipseudometric.
SinceT(x)∈T*(x), for each ϵ>0, there is a∈T*(x0) such that d(T(x),a)≤d(x,x0)+ϵ, and there is b∈T*(x0) such that d(b,T(x))≤d(x0,x)+ϵ. Therefore, dist(T(x),T*(x0))≤d(x,x0) and dist(T*(x0),T(x))≤d(x0,x).
We finally also note that by assumption onT for each x,y∈D we have that d(T(x),T(y))≤d(x,y)≤d(x,x0)+d(x0,y).
Thus, we have shown thatJ≠∅. Choose y0∈J and define T~(x)=y0 if x=x0 and T~(x)=T(x) if x∈D.
Since for eachx∈D,d(T~(x0),T~(x))=d(y0,T(x))≤d(x0,x) and
(6.12)d(T~(x),T~(x0))=d(T(x),y0)≤d(x,x0),
we conclude that (D∪{x0},T~)∈ℱ, contradicting the maximality of (D,T). Therefore, D=H.We will next establish theq-hyperconvexity of the space of all bounded λ-Lipschitzian self-maps on a q-hyperconvex T0-quasimetric space.Theorem 6.10 (compare [18, Theorem 3]).
Let(X,d) be a q-hyperconvex T0-quasimetric space and for λ>0 let ℱλ denote the family of all bounded λ-Lipschitzian self-maps on (X,d) equipped with the T0-quasimetric d^(f,g)=supx∈Xd(f(x),g(x)) whenever f,g∈ℱλ. Then (ℱλ,d^) is itself a q-hyperconvex T0-quasimetric space.Proof.
We leave it to the reader to verify thatd^ is an extended T0-quasimetric on the set ℱλ. We next note that d^ is a T0-quasimetric, since d^ does not attain ∞. Indeed, let x0,x∈X and f,g∈ℱλ. Then,
(6.13)d(f(x),g(x))≤|d(f(x),g(x))-d(f(x0),g(x0))|+d(f(x0),g(x0))≤ds(f(x0),f(x))+ds(g(x),g(x0))+d(f(x0),g(x0))≤Mf+Mg+d(f(x0),g(x0))
for some positive real constants Mf and Mg, since f and g are bounded. We conclude that d^(f,g)≠∞.
Suppose that(fα)α∈S is a nonempty family of functions in ℱλ and let
(6.14)(rα)α∈S,(sα)α∈S,
be families of nonnegative reals such thatd^(fα,fβ)≤rα+sβ whenever α,β∈S. Then, for each x∈X, we have d(fα(x),fβ(x))≤rα+sβ whenever α,β∈S. So because of the q-hyperconvexity of (X,d), we have that
(6.15)J(x)=⋂α∈S(Cd(fα(x),rα)∩Cd-1(fα(x),sα))≠∅.
Note that, by Lemma 6.4 applied to E=X, we see that J(x)∈ℰq(X) whenever x∈X.
We next show thatdH(J(x),J(y))≤λd(x,y) whenever x,y∈X. To this end, it suffices to show that for each x,y∈X we have  J(y)⊆Cd(J(x),λd(x,y)) (i.e., J(x)⊆Cd(J(y),λd(y,x)), and that J(x)⊆Cd-1(J(y),λd(x,y)).
Fixx,y∈X. If z∈J(x), then for each α∈S, by the λ-Lipschitzian condition satisfied by fα,
(6.16)d(z,fα(y))≤d(z,fα(x))+d(fα(x),fα(y))≤d(z,fα(x))+λd(x,y)≤sα+λd(x,y),d(fα(y),z)≤d(fα(y),fα(x))+d(fα(x),z)≤λd(y,x)+rα.
By Lemma 5.2 applied to Nλd(x,y),λd(y,x)(J(y)), we then have
(6.17)z∈⋂α∈S[Cd(fα(y),rα+λd(y,x))∩Cd-1(fα(y),sα+λd(x,y))]=Nλd(x,y),λd(y,x)(J(y))=⋃a∈J(y)[Cd(a,λd(y,x))∩Cd-1(a,λd(x,y))].
Therefore,  J(x)⊆Cd-1(J(y),λd(x,y)), and J(x)⊆Cd(J(y),λd(y,x)) and thus J(y)⊆Cd(J(x),λd(x,y)) whenever x,y∈X. Hence, our claim is verified.
In the light of Theorem6.6 for each x∈X, it is possible to find f(x)∈J(x) so that we get f∈ℱλ, since d(f(x),f(y))≤dH(J(x),J(y))≤λd(x,y) whenever x,y∈X. In particular, we also note that f is bounded. Indeed, fix α∈S. Then, for any x,y∈X and some positive real constant Mfα, we have d(f(x),f(y))≤d(f(x),fα(x))+d(fα(x),fα(y))+d(fα(y),f(y))≤sα+Mfα+rα by the choice of f. Thus, f is indeed bounded.
Sincef∈⋂α∈S‍(Cd^(fα,rα)∩Cd^-1(fα,sα)), we have shown that (ℱλ,d^) is q-hyperconvex.We conclude this article with a curious observation in the spirit of [18, Proposition 2].Proposition 6.11.
Suppose that(X,d) is a bounded q-hyperconvex T0-quasimetric space and let U=⋂i∈I‍(Cd(xi,ri)∩Cd-1(xi,si)) and V=⋂i∈I‍(Cd(yi,ri)∩Cd-1(yi,si)) with two nonempty families (xi)i∈I,(yi)i∈I of points in X and two families (ri)i∈I,(si)i∈I of nonnegative reals. Then, dH(V,U)≤sup{d(yi,xi):i∈I}.Proof.
LetρUV=sup{d(xi,yi):i∈I} and similarly, let ρVU=sup{d(yi,xi):i∈I}, and let x∈U. Then, for each i∈I,  d(x,yi)≤d(x,xi)+d(xi,yi)≤si+ρUV and d(yi,x)≤d(yi,xi)+d(xi,x)≤ρVU+ri. Consequently, x∈⋂i∈I‍(Cd(yi,ri+ρVU)∩Cd-1(yi,si+ρUV))=⋃a∈V‍(Cd(a,ρVU)∩Cd-1(a,ρUV)) by Lemma 5.2.
Therefore,U⊆Cd(V,ρVU) and U⊆Cd-1(V,ρUV), and similarly, by interchanging U and V, hence, V⊆Cd-1(U,ρVU). We have shown that dH(V,U)≤ρVU.



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*Source: 765903-2012-09-13.xml*