# Generalized Contractive Set-Valued Maps on Complete Preordered Quasi-Metric Spaces

Authors: J. Marín; S. Romaguera; P. Tirado
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

By using a suitable modification of the notion of aw-distance we obtain some fixed point results for generalized contractive set-valued maps on complete preordered quasi-metric spaces. We also show that several distinguished examples of non-metrizable quasi-metric spaces and of cones of asymmetric normed spaces admit w-distances of this type. Our results extend and generalize some well-known fixed point theorems.


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## Body

## 1. Introduction and Preliminaries

Throughout this paper the lettersℝ, ℝ+, ℕ, and ω will denote the set of real numbers, the set of non-negative real numbers, the set of positive integer numbers and the set of non-negative integer numbers, respectively. Our basic references for quasi-metric spaces are [1, 2] and for asymmetric normed space it is [3].A quasi-pseudometric on a setX is a function d:X×X→ℝ+ such that for all x,y,z∈X: (i) d(x,x)=0; (ii) d(x,y)≤d(x,z)+d(z,y).Ifd satisfies conditions (i) and (ii) above but we allow d(x,y)=+∞, then d is said to be an extended quasi-pseudometric on X.Following the modern terminology, a quasi-pseudometricd on X satisfying (i′) d(x,y)=d(y,x)=0 if and only if x=y is called a quasi-metric on X.If the quasi-metricd satisfies the stronger condition (i′′) d(x,y)=0 if and only if x=y, we say that d is a T1 quasi-metric on X.A(T1) quasi-metric space is a pair (X,d) such that X is a nonempty set and d is a (T1) quasi-metric on X.Each extended quasi-pseudometricd on a set X induces a topology τd on X which has as a base the family of open balls {Bd(x,ε):x∈X,ε>0}, where Bd(x,ε)={y∈X:d(x,y)<ε} for all x∈X and ε>0.The closure with respect toτd of a subset A of X will be denoted by clτdA.Note that ifd is quasi-metric then τd is a T0 topology, and if d a T1 quasi-metric then τd is a T1 topology on X.Given a quasi-metricd on X, the function d-1 defined by d-1(x,y)=d(y,x) for all x,y∈X, is also a quasi-metric on X, and the function ds defined by ds(x,y)=max{d(x,y),d(y,x)} for all x,y∈X, is a metric on X.There exist several different notions of Cauchyness and quasi-metric completeness in the literature (see, e.g., [2]). In our context will be useful the following general notion.Definition 1.
A quasi-metricd on a set X will be called complete if every Cauchy sequence (xn)n∈ω in (X,d) converges with respect to the topology τd-1 (i.e., there exists z∈X such that limnd(xn,z)=0), where the sequence (xn)n∈ω is said to be Cauchy if for each ε>0 there exists n0∈ℕ such that d(xn,xm)<ε whenever n0≤n≤m. If d is complete we will say that the quasi-metric space (X,d) is complete.Kada et al. introduced in [4] the notion of w-distance for metric spaces and obtained, among other results, w-distance versions of the celebrated Ekeland variational principle [5] and the nonconvex minimization theorem [6]. In [7] Park extended this concept to quasi-metric spaces in order to generalize and unify different versions of Ekeland's variational principle. Park’s approach was continued by Al-Homidan et al. [8], and recently by Latif and Al-Mezel [9], and Marín et al. [10, 11], among others. Thus in [8] were obtained extensions and generalizations of Caristi-Kirk’s type fixed point theorem [12] as well as a Takahashi type minimization theorem and generalizations of Ekeland’s variational principle and of Nadler’s fixed point theorem [13], respectively, while in [9–11] were proved several fixed point theorems for single and set-valued maps on quasi-metric spaces by using Q-functions in the sense of [8] and w-distances.Definition 2 (see [7, 8]).
Aw-distance for a quasi-metric space (X,d) is a function q:X×X→ℝ+ satisfying the following three conditions:(W1)
q(x,y)≤q(x,z)+q(z,y) for all x,y,z∈X;(W2)
q(x,·):X→ℝ+ is lower semicontinuous on (X,τd-1) for all x∈X;(W3)
for eachε>0 there exists δ>0 such that q(x,y)≤δ and q(x,z)≤δ, imply d(y,z)≤ε.Note that every quasi-metricd on X satisfies conditions (W1) and (W2) above.Ifd is a metric on X, then Definition 2 provides the notion of a w-distance for the metric space (X,d) as defined in [4]. In particular, every metric d on X is a w-distance for (X,d).Unfortunately, the situation is quite different whend is a quasi-metric. In fact, it was shown in [10] that if a quasi-metric d on X is also a w-distance for (X,d), then the topology τd induced by d is metrizable. Hence, many distinguished examples of nonmetrizable quasi-metrizable topological spaces do not admit any compatible quasi-metric which is also a w-distance.Motivated by this fact, in Section2 we will show that the use of (pre)ordered quasi-metric spaces, with a suitable adaptation of the notion of w-distance to this setting, allows us to generate several interesting examples of preordered quasi-metric spaces (X,d) for which the quasi-metric d is a w-distance in this new sense. In Section 3 we will prove a fixed point theorem for set-valued maps on complete preordered quasi-metric spaces by means of the modified notion of w-distance, that generalizes and extends several well-known fixed point theorems and allows us to deduce fixed point results involving the lower Hausdorff distance of a complete preordered quasi-metric space. We illustrate these results with some examples.


## 2. Preordered Quasi-Metric Spaces,w⪯-Distances, and Examples

We start this section by recalling some pertinent concepts.A preorder on a (nonempty) setX is a reflexive and transitive (binary) relation ⪯ on X. If, in addition, ⪯ is antisymmetric (i.e., condition x⪯y and y⪯x, implies x=y), ⪯ is called a partial order or, simply, an order on X. The usual order on ℝ is denoted by ≤.Let⪯ be a preorder on X. Given x∈X the set {y∈X:x⪯y}will be denoted by ↑{x}. A sequence (xn)n∈ω in X is said to be nondecreasing if xn⪯xn+1 for all n∈ω.Remark 3.
Given a (nonempty) setX, the (trivial) relation ⪯t given by x⪯ty if and only if x,y∈X is obviously a preorder on X.According to [14], a (pre)ordered quasi-metric space is a triple (X,⪯,d) such that ⪯ is a (pre)order on X and d is a quasi-metric on X.Observe that if(X,d) is a quasi-metric space, then the relation ≤d on X defined by x≤dy if and only if d(x,y)=0 is a partial order on X called the specialization order of (X,d). So (X,≤d,d) is an ordered quasi-metric space.Definition 4.
Aw⪯-distance for a preordered quasi-metric space (X,⪯,d) is a function q:X×X→ℝ+ satisfying conditions (W1) and (W2) of Definition 2, and: (W⪯3) for each ε>0 there exists δ>0 such that q(x,y)≤δ, q(x,z)≤δ, and y⪯z, imply d(y,z)≤ε.Example 5.
Letq be a w-distance for a quasi-metric space (X,d). Then q is obviously a w⪯-distance for the preordered quasi-metric space (X,⪯t,d).Example 6.
Let(X,d) be a quasi-metric space. Consider the ordered quasi-metric space (X,≤d,d). Of course, d satisfies conditions (W1) and (W2). Moreover, it trivially satisfies condition (W⪯3) of Definition 4. Hence d is a w⪯-distance for (X,≤d,d).Example 7.
LetX=ℝ and let dS be the quasi-metric on X given by dS(x,y)=y-x if x≤y, and dS(x,y)=1 if x>y. Then dS induces the Sorgenfrey topology on X. We show that dS is is a w⪯-distance for the ordered T1 quasi-metric (X,≤,dS). Indeed, since dS is a quasi-metric, we only need to show condition (W⪯3) of Definition 4. To this end, choose ε>0. Put δ=min{1/2,ε}, and let dS(x,y)≤δ and dS(x,z)≤δ with y≤z. Therefore dS(x,y)=y-x≤ε and dS(x,z)=z-x≤ε. Since y≤z, we have dS(y,z)=z-y≤z-x≤ε. We conclude that dS is a w⪯-distance for (X,≤,dS).Our next example should be compared with Example 3.1 of [8]. Recall [3, 15] that an asymmetric norm on a real vector space X is a function p:X→ℝ+ such that for each x,y∈X and r∈ℝ+: (i) p(x)=p(-x)=0 if and only if x=0; (ii) p(rx)=rp(x); (iii) p(x+y)≤p(x)+p(y).Then, the pair(X,p) is called an aysmmetric normed space. Asymmetric norms are called quasi-norms in [16, 17], and so forth.Example 8.
Let(X,⪯,∥·∥) be a normed lattice. Denote by X+ the positive cone of X, that is, X+:={x∈X:0⪯x}, and define ∥·∥+:X→ℝ+ as ∥x∥+=∥x∨0∥ for all x∈X. Then ∥·∥+ is an aysmmetric norm on X (see, e.g., [17, Theorem 3.1]), and thus the function d defined by d(x,y)=∥y-x∥+ for all x,y∈X is a quasi-metric on X, so (X,⪯,d) is an ordered quasi-metric space. Hence (X+,⪯,d+) is also an ordered quasi-metric space, where d+ denotes the restriction of d to X+.
We will show that the functionq defined by q(x,y)=∥y∥ for all x,y∈X+, is a w⪯-distance for (X+,⪯,d+). Indeed, first note that condition (W1) is trivially satisfied. Now fix x∈X+ and let (yn)n∈ω be a sequence in X+ such that limd+(yn,y)=0 for some y∈X+. Since
(1)q(x,y)=∥y∥=∥y∥+=∥y-yn+yn∥+≤∥y-yn∥++∥yn∥+=d+(yn,y)+q(x,yn)
for all n∈ω, we deduce that q(x,·) is lower semicontinuous for (X+,τ(d+)-1), and thus condition (W2) is satisfied. Finally, choose ε>0 and put δ=ε/2. Suppose q(x,y)≤δ and q(x,z)≤δ with y⪯z. Therefore
(2)d+(y,z)=∥z-y∥+=∥(z-y)∨0∥=∥z-y∥≤∥z∥+∥y∥=q(x,z)+q(x,y)≤2δ=ε.
Consequently condition (W⪯3) is also satisfied, so q is a w⪯-distance for (X+,⪯,d+).Definition 9.
A preordered quasi-metric space(X,⪯,d) is called complete if for each nondecreasing Cauchy sequence (xn)n∈ω the following two conditions hold:(i1)
there existsz∈X satisfying limnd(xn,z)=0;(i2)
eachz∈X satisfying limnd(xn,z)=0 verifies that xn⪯z for all n∈ω.Next we give some examples of complete preordered quasi-metric spaces.Example 10.
Let(X,d) be any complete quasi-metric space. Then (X,⪯t,d) is obviously a complete preordered quasi-metric space.Example 11.
Let⪯ be a partial order on a set X. Then, for every complete quasi-metric d on X such that d(x,y)=0 if and only if x⪯y, we have that (X,⪯,d) is a complete ordered quasi-metric space (note that in this case the partial order ⪯ coincides with the specialization order ≤d). Indeed, let (xn)n∈ω be a nondecreasing Cauchy sequence and let z∈X be such that limnd(xn,z)=0. Choose any n∈ω. Then, for each arbitrary ε>0 there is m>n such that d(xm,z)<ε. Since xn⪯xm, we have d(xn,xm)=0, so by the triangle inequality, d(xn,z)<ε. Since ε is arbitrary we deduce that d(xn,z)=0. Hence xn⪯z.Example 12.
Let(X,≤,dS) be the ordered T1 quasi-metric space of Example 7. If (xn)n∈ω is a Cauchy sequence in (X,dS) that is also nondecreasing, then it is clear that limndS(xn,z)=0 only for z=sup{xn:n∈ω}. Therefore (X,≤,dS) is a complete ordered T1 quasi-metric space.Example 13.
LetX=ℝ+ and let d be the complete quasi-metric on X given by d(x,y)=max{y-x,0} for all x,y∈X. Then (X,≤,d) is not a complete preordered quasi-metric space in our sense because any (nondecreasing Cauchy) sequence (xn)n∈ω in X satisfies limnd(xn,0)=0, so condition (i2) of Definition 9 does not hold. However, since d(x,y)=0 if and only if x≥y, it follows from Example 11 that (X,≥,d) is a complete ordered quasi-metric space.


## 3. Fixed Point Results

Answering a question posed by Reich [18], Mizoguchi and Takahashi [19] (see also [20, 21]) obtained a set-valued generalization-improvement of the Rakotch fixed point theorem [22, Corollary of Theorem 2]. Recently, Latif and Al-Mezel [9, Theorem 2.3] extended Mizoguchi-Takahashi’s theorem to the framework of complete T1 quasi-metric spaces by using w-distances (actually they states their result in a slightly more general form by using Q-functions in the sense of [8], instead of w-distances). Here we obtain a fixed point theorem for complete preordered quasi-metric spaces from which [9, Theorem 2.3] can be deduced as a special case. Several other consequences are deduced and some illustrative examples are given.We first introduce the notions of contractiveness that we will use in the rest of the paper.If(X,d) is a quasi-metric space, we denote by 2X the set of all nonempty subsets of X and by Cd(X) the set of all nonempty τd-closed subsets of X.Definition 14.
Let(X,⪯,d) be a preordered quasi-metric space and let T:X→2X be a set-valued map such that Tx∩↑{x}≠∅ for all x∈X. We say that T is w⪯-contractive if there exist a w⪯-distance q for (X,⪯,d) and a constant r∈(0,1), such that for each x,y∈X, with x⪯y, and u∈Tx∩↑{x} there is v∈Ty∩↑{y} satisfying q(u,v)≤rq(x,y).Definition 15.
Let(X,⪯,d) be a preordered quasi-metric space and let T:X→2X be a set-valued map such that Tx∩↑{x}≠∅ for all x∈X. We say that T is generalized w⪯-contractive if there exist a w⪯-distance q for (X,⪯,d) and a function α:ℝ+→[0,1) with limsupr→t+α(r)<1 for all t∈ℝ+, and such that for each x,y∈X, with x⪯y, and u∈Tx∩↑{x} there is v∈Ty∩↑{y} satisfying q(u,v)≤α(q(x,y))q(x,y).Theorem 16.
Let(X,⪯,d) be a complete preordered quasi-metric space and T:X→Cd(X) be a generalized w⪯-contractive set-valued map. Then T has a fixed point.Proof.
SinceT is generalized w⪯-contractive, there is a w⪯-distance q for (X,⪯,d) and a function α:ℝ+→[0,1) with limsupr→t+α(r)<1 for all t∈ℝ+, and such that for each x,y∈X, with x⪯y, and u∈Tx∩↑{x} there is v∈Ty∩↑{y} satisfying
(3)q(u,v)≤α(q(x,y))q(x,y).
Fixx0∈X. Since Tx0∩↑{x0}≠∅ there exists x1∈Tx0 such that x0⪯x1. Taking x=x0 and y=u=x1, we deduce the existence of an x2∈Tx1 such that x1⪯x2 and
(4)q(x1,x2)≤α(q(x0,x1))q(x0,x1).
Repeating the above argument, there isx3∈Tx2 such that x2⪯x3 and
(5)q(x2,x3)≤α(q(x1,x2))q(x1,x2).
Hence, following this process we construct a sequence(xn)n∈ω in X such that for every n∈ℕ,(a)
xn+1∈Txn,(b)
xn⪯xn+1, and(c)
q(xn,xn+1)≤α(q(xn-1,xn))q(xn-1,xn).
Next we show that(xn)n∈ω is a Cauchy sequence in the quasi-metric space (X,d).
To this end, first suppose that there isk∈ω such that q(xk,xk+1)=0. Thus q(xn,xm)=0 whenever k<n<m, by conditions (c) and (W1). Then, from conditions (b) and (W⪯3) we deduce that (xn)n∈ω is a Cauchy sequence in (X,d).
Now suppose thatq(xn,xn+1)>0 for all n∈ω. Put rn=q(xn,xn+1), n∈ω. Then (rn)n∈ω is a strictly decreasing sequence of non-negative real numbers. Let c∈ℝ+ be such that limnrn=c. Then
(6)limsuprn→cα(rn)<1.
Hence there existb∈(0,1) and n0∈ℕ such that α(rn)<b for all n≥n0. By condition (c) we deduce that
(7)q(xn,xn+1)<bq(xn-1,xn)<b2q(xn-2,xn-1)<⋯<bn-n0q(xn0,xn0+1),
for all n>n0. Now choose ε>0. Then, there is δ>0 for which condition (W⪯3) follows. Since by (7) and (W1) there is n1∈ℕ such that q(xn,xm)≤δ whenever n1≤n<m, we deduce from (W⪯3) that d(xn,xm)≤ε whenever n1<n<m. Therefore (xn)n∈ω is a nondecreasing Cauchy sequence in (X,⪯,d).
Since(X,⪯,d) is a complete preordered quasi-metric space, there exists z∈X such that limnd(xn,z)=0 and xn⪯z for all n∈ω.
Next we show thatlimnq(xn,z)=0.
Indeed, chooseε>0. Then, there is n0∈ℕ such that q(xn,xm)<ε/2 whenever n0≤n<m. Given n≥n0 there is, by condition (W2), an n1>n such that
(8)q(xn,z)-q(xn,xn1)<ε2.
Thus(9)q(xn,z)<ε2+q(xn,xn1)<ε.
Thereforelimnq(xn,z)=0.
Finally, sincexn⪯z for all n∈ω, we can find a sequence (vn)n∈ℕ in Tz such that z⪯vn and
(10)q(xn,vn)≤α(q(xn-1,z))q(xn-1,z)
for all n∈ℕ. Hence limnq(xn,vn)=0. We deduce from (W⪯3) that limnd(z,vn)=0. So z∈Tz because Tz∈Cd(X). This concludes the proof.Corollary 17.
Let(X,⪯,d) be a complete preordered quasi-metric space and T:X→Cd(X) be a w⪯-contractive set-valued map. Then T has a fixed point.Corollary 18.
Let(X,⪯,d) be a complete preordered T1 quasi-metric space for which d is a w⪯-distance and let T:X→X be a self-map. If there is a function α:ℝ+→[0,1) with limsupr→t+α(r)<1 for all t∈ℝ+, and such that for each x,y∈X, with x⪯y, one has
(11)d(Tx,Ty)≤α(d(x,y))d(x,y),
then T has a fixed point.Proof.
Sinceτd is a T1 topology, then Tx∈Cd(X) for all x∈X. The result is now an immediate consequence of Theorem 16.Remark 19.
Putting⪯=⪯t and taking into account Example 5, we deduce that [9, Theorem 2.3] and [8, Theorem 6.1] are, for w-distances, special cases of Theorem 16 and Corollary 17 respectively, whereas Corollary 18 provides a quasi-metric generalization of Rakotch’s fixed point theorem.Next we give an easy example where Corollary17, and hence Theorem 16, can be applied to the involved complete ordered T1 quasi-metric space (X,⪯,d), but not to the complete ordered metric space (X,⪯,ds).Example 20.
Let(X,≤dS) be the complete ordered T1 quasi-metric space of Example 12 and let T:X→X defined by Tx=x/2 for all x∈X. Since d is a w⪯-distance for (X,≤,dS) (see Example 7), and for each x,y∈X with x≤y, we have
(12)dS(Tx,Ty)=y-x2=12dS(x,y),
then all conditions of Corollary 17, and thus of Theorem 16, are satisfied. However, for x,y∈X with 0≤x<y≤1, we have
(13)(dS)s(Tx,Ty)=1=(dS)s(x,y),
so Theorem 16 cannot be applied to the complete ordered metric space (X,≤,(dS)s) and the self-map T.In the sequel we will apply Corollary17 to deduce a fixed point result for set-valued maps on complete preordered T1 quasi-metric spaces involving the (lower) Hausdorff distance.Let(X,d) be a quasi-metric space. For each A,B∈Cd(X) let
(14)Hd-(A,B)=supa∈Ad(a,B),Hd+(A,B)=supb∈Bd(A,b),Hd(A,B)=max{Hd-(A,B),Hd+(A,B)}.ThenHd-, Hd+ and Hd will be called the lower Hausdorff distance of (X,d), the upper Hausdorff distance of (X,d) and the Hausdorff distance of (X,d), respectively (compare e.g., [23–26]).It is interesting to note thatHd-, Hd+ and Hd are extended quasi-pseudometrics on Cd(X), but not quasi-metrics, in general.Corollary 21.
Let(X,⪯,d) be a complete preordered T1 quasi-metric space for which d is a w⪯-distance and let T:X→Cd(X) be a set-valued map such that Tx∩↑{x}≠∅ for all x∈X. If there is r∈(0,1) such that for each x,y∈X, with x⪯y,
(15)Hd-(Tx∩↑{x},Ty∩↑{y})≤rd(x,y),
then T has a fixed point.Proof.
Takes∈(r,1). Then T is a w⪯-contractive set-valued map for the w⪯-distance d and the constant s. By Corollary 17, T has a fixed point.Remark 22.
Observe that for the ordered quasi-metric space(X,≤d,d), any set-valued map T:X→Cd(X) such that Tx∩↑{x}≠∅ for all x∈X,satisfies that every x∈X is a fixed point of T. Indeed, condition Tx∩↑{x}≠∅ implies Tx∩clτd-1{x}≠∅, so x∈clτdTx, that is, x∈Tx. Note also that the contraction condition (15) is, in this case, equivalent to the following:
(16)d(x,y)=0⇒Hd-(Tx∩clτd-1{x},Ty∩clτd-1{y})=0.
We finish the paper with two examples that illustrate Corollary21 and Remark 22, respectively.Example 23.
LetX be the set of all continuous functions from [0,1] into itself and let d be the T1 quasi-metric on X defined as (compare [27, Example 4]):
(17)d(f,g)=sup{g(x)-f(x):x∈[0,1]}iff(x)≤g(x)∀x∈[0,1],d(f,g)=1,otherwise.
Let⪯ be the usual pointwise partial order on X, that is, f⪯g if and only if f(x)≤g(x) for all x∈[0,1]. By standard arguments we deduce that (X,⪯,d) is a complete ordered T1 quasi-metric space: Indeed, given a nondecreasing Cauchy sequence (fn)n∈ω in (X,⪯,d), then limnd(fn,f)=0 only for the function f∈X defined by f(x)=sup{fn(x):n∈ω} for all x∈[0,1].
Moreoverd is a w⪯-distance for (X,⪯,d) because given ε>0 we take δ=min{1/2,ε}, and then for d(f,g)≤δ, d(f,h)≤δ and g⪯h, we obtain
(18)d(g,h)=sup{h(x)-g(x):x∈[0,1]}≤sup{h(x)-f(x):x∈[0,1]}=d(f,h)≤δ≤ε.
Now construct the set-valued mapT:X→Cd(X) given by
(19)Tf={{f(x)+2n-12n}fn∈X:fn(x)=f(x)+2n-12n∀x∈[0,1],n∈ℕ}.
Note thatTf∈Cd(X). Indeed, suppose that there is h∈cldTf∖Tf. Then, there is a subsequence (fnk)k∈ω of (fn)n∈ω such that d(h,fnk)<2-k for all k∈ω. Since fn⪯fn+1 for all n, we can assume, without loss of generality, that fnk⪯fnk+1 for all k. Consequently, we have for each x∈[0,1] and each k,
(20)fn0(x)≤fnk(x)<h(x)+2-k.
Sinceh(x)≤fn0(x), we deduce that fn0(x)=h(x) for all x∈[0,1], which contradicts that h∉Tf. We conclude that Tf∈Cd(X).
MoreoverTf∩↑{f}≠∅ for all f∈X because f⪯fn for all n∈ℕ and thus Tf∩↑{f}=Tf.
Finally, letf,g∈X, with f⪯g, and u∈Tf∩↑{f}. Then, there is n∈ℕ such that u(x)=(f(x)+2n-1)/2n for all x∈[0,1]. Taking v(x)=(g(x)+2n-1)/2n for all x∈[0,1], we have v∈Tg∩↑{g}, u⪯v, and
(21)d(u,v)=sup{g(x)-f(x)2n:x∈[0,1]}≤12d(f,g).
Hence(22)Hd-(Tf∩↑{f},Tg∩↑{g})≤12d(f,g).
By Corollary21, T has a fixed point. In fact, the function h defined by h(x)=1 for all x∈[0,1], satisfies h∈Th.Example 24.
Consider the Banach lattice(l1,⪯,∥·∥), where l1 denotes the vector space of all infinite sequences x:=(xn)n∈ω of real numbers such that ∑n=0∞‍|xn|<∞, ⪯ denotes the usual order on l1 and ∥x∥:=∑n=0∞‍|xn| for all x:=(xn)n∈ω∈l1.
Now denote byl1+ the positive cone of l1 and by d+ the quasi-metric on l1+ defined by d+(x,y)=∥(y-x)∨0∥ for all x,y∈l1+ (compare Example 8). Then (l1+,d+) is a complete quasi-metric space by [28, Theorem 2].
Letψ:l1+→l1+ be nondecreasing and such that ψ(x)⪯x for all x∈l1+. Define T:l1+→Cd+(l1+) as
(23)Tx={y∈l1+:ψ(x)⪯y},
for all x∈l1+. Then Tx∩clτd-1{x}={y∈l1+:ψ(x)⪯y⪯x} (compare Remark 22). In fact x∈Tx for all x∈l1+.
Finally note that givenx,y∈l1+ with d+(x,y)=0 and u∈Tx∩clτd-1{x}, we have that y⪯x and ψ(x)⪯u⪯x, so ψ(y)⪯ψ(x)⪯u, and hence
(24)d+(u,ψ(y))=0.
Sinceψ(y)∈Ty∩clτd-1{y}, we deduce that condition (16) of Remark 22 is also satisfied.



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*Source: 269246-2013-01-29.xml*