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Functions on Quasimetric Spaces and Fixed Points for Multivalued Maps
Fixed Point Theory and Applications volume 2011, Article number: 603861 (2011)
Abstract
We discuss several properties of functions in the sense of AlHomidan et al.. In particular, we prove that the partial metric induced by any weighted quasipseudometric space is a function and show that both the Sorgenfrey line and the Kofner plane provide significant examples of quasimetric spaces for which the associated supremum metric is a function. In this context we also obtain some fixed point results for multivalued maps by using BianchiniGrandolfi gauge functions.
1. Introduction and Preliminaries
Kada et al. introduced in [1] the concept of distance on a metric space and extended the CaristiKirk fixed point theorem [2], the Ekeland variation principle [3] and the nonconvex minimization theorem [4], for distances. Recently, AlHomidan et al. introduced in [5] the notion of function on a quasimetric space and then successfully obtained a CaristiKirktype fixed point theorem,a Takahashi minimization theorem, an equilibrium version of Ekelandtype variational principle, and a version of Nadler's fixed point theorem for a  function on a complete quasimetric space, generalizing in this way, among others, the main results of [1] because every distance is, in fact, a function. This interesting approach has been continued by Hussain et al. [6], and by Latif and AlMezel [7], respectively. In particular, the authors of [7] have obtained a nice Rakotchtype theorem for functions on complete quasimetric spaces.
In Section 2 of this paper, we generalize the basic theory of functions to quasipseudometric spaces. Our approach is motivated, in part, by the fact that in many applications to Domain Theory, Complexity Analysis, Computer Science and Asymmetric Functional Analysis, quasipseudometric spaces (in particular, weightable quasipseudometric spaces and their equivalent partial metric spaces) rather than quasimetric spaces, play a crucial role (cf. [8–23], etc.). In particular, we prove that for every weighted quasipseudometric space the induced partial metric is a function. We also show that the Sorgenfrey line and the Kofner plane provide interesting examples of quasimetric spaces for which the associated supremum metric is a function. Finally, Section 3 is devoted to present a new fixed point theorem for functions and multivalued maps on quasipseudometric spaces, by using BianchiniGrandolfi gauge functions in the sense of [24]. Our result generalizes and improves, in several ways, wellknown fixed point theorems.
Throughout this paper the letter and will denote the set of positive integer numbers and the set of nonnegative integer numbers, respectively.
Our basic references for quasimetric spaces are [25, 26].
Next we recall several pertinent concepts.
By a quasipseudometric on a set , we mean a function such that for all ,
(i),
(ii).
A quasipseudometric on that satisfies the stronger condition
(i′)
is called a quasimetric on .
We remark that in the last years several authors used the term "quasimetric" to refer to a quasipseudometric and the term " quasimetric" to refer to a quasimetric in the above sense.
In the following we will simply write qpm instead of quasipseudometric if no confusion arises.
A qpm space is a pair such that is a set and is a qpm on . If is a quasimetric on , the pair is then called a quasimetric space.
Given a qpm on a set , the function defined by , is also a qpm on , called the conjugate of , and the function defined by is a metric on , called the supremum metric associated to .
Thus, every qpm on induces, in a natural way, three topologies denoted by , and , respectively, and defined as follows.
(i) is the topology on which has as a base the family of open balls , where , for all and .
(ii) is the topology on which has as a base the family of open balls , where , for all and .
(iii) is the topology on induced by the metric .
Note that if is a quasimetric on , then is also a quasimetric, and and are topologies on .
Note also that a sequence in a qpm space is convergent (resp., convergent) to if and only if (resp., .
It is well known (see, for instance, [26, 27]) that there exists many different notions of completeness for quasimetric spaces. In our context we will use the following notion.
A qpm space is said to be complete if every Cauchy sequence is convergent, where a sequence is called Cauchy if for each there exists such that whenever .
In this case, we say that is a complete qpm on .
2. Functions on qpmSpaces
We start this section by giving the main concept of this paper, which was introduced in [5] for quasimetric spaces.
Definition 2.1.
A function on a qpm space is a function satisfying the following conditions:
(Q1), for all ,
(Q2) if , and is a sequence in that converges to a point and satisfies , for all , then ,
(Q3) for each there exists such that and imply .
If is a metric space and satisfies conditions (Q1) and (Q3) above and the following condition:
(Q2′) is lower semicontinuous for all , then is called a wdistance on (cf. [1]).
Clearly is a distance on whenever is a metric on .
However, the situation is very different in the quasimetric case. Indeed, it is obvious that if is a qpm space, then satisfies conditions (Q1) and (Q2), whereas Example 3.2 of [5] shows that there exists a qpm space such that does not satisfy condition (Q3), and hence it is not a Qfunction on . In this direction, we next present some positive results.
Lemma 2.2.
Let q be a Qfunction on a qpm space . Then, for each , there exists such that and imply .
Proof.
By condition (Q3), . Interchanging and , it follows that , so .
Proposition 2.3.
Let be a qpm space. If is a Qfunction on , then , and hence, is a metrizable topology on .
Proof.
Let be a sequence in which is convergent to some . Then, by Lemma 2.2, . We conclude that .
Remark 2.4.
It follows from Proposition 2.3 that many paradigmatic quasimetrizable topological spaces , as the Sorgenfrey line, the Michael line, the Niemytzki plane and the Kofner plane (see [25]), do not admit any compatible quasimetric which is a function on .
In the sequel, we show that, nevertheless, it is possible to construct an easy but, in several cases, useful function on any quasimetric space, as well as a suitable functions on any weightable qpm space.
Recall that the discrete metric on a set is the metric on defined as , for all , and , for all with .
Proposition 2.5.
Let be a quasimetric space. Then, the discrete metric on is a function on .
Proof.
Since is a metric it obviously satisfies condition (Q1) of Definition 2.1.
Now suppose that is a sequence in that converges to some , and let and such that , for all . If , then . If , we deduce that , for all . Since , it follows that , so , and thus . Hence, condition (Q2) is also satisfied.
Finally, satisfies condition (Q3) taking for every
Example 2.6.
On the set of real numbers define as if , and if . Then, is a quasimetric on and the topological space is the celebrated Sorgenfrey line. Since is the discrete metric on , it follows from Proposition 2.5 that is a function on .
Example 2.7.
The quasimetric on the plane , constructed in Example 7.7 of [25], verifies that is the socalled Kofner plane and that is the discrete metric on , so, by Proposition 2.5, is a function on .
Matthews introduced in [14] the notion of a weightable qpm space (under the name of a "weightable quasimetric space"), and its equivalent partial metric space, as a part of the study of denotational semantics of dataflow networks.
A qpm space is called weightable if there exists a function such that for all . In this case, we say that is a weightable qpm on . The function is said to be a weighting function for and the triple is called a weighted qpm space.
A partial metric on a set is a function such that, for all :
(i),
(ii),
(iii),
(iv).
A partial metric space is a pair such that is a set and is a partial metric on .
Each partial metric on induces a topology on which has as a base the family of open balls , where , for all and .
The precise relationship between partial metric spaces and weightable qpm spaces is provided in the next result.
Theorem 2.8 (Matthews [14]).

(a)
Let be a weightable qpm space with weighting function. Then, the function defined by , for all , is a partial metric on . Furthermore .

(b)
Conversely, let be a partial metric space. Then, the function defined by , for all is a weightable qpm on with weighting function given by for all . Furthermore .
Remark 2.9.
The domain of words, the interval domain, and the complexity quasimetric space provide distinguished examples of theoretical computer science that admit a structure of a weightable qpm space and, thus, of a partial metric space (see, e.g., [14, 20, 21]).
Proposition 2.10.
Let be a weighted qpm space. Then, the induced partial metric is a Qfunction on .
Proof.
We will show that satisfies conditions (Q1), (Q2), and Q(3) of Definition 2.1.
(Q1) Let , then
(Q2)Let be a sequence in which is convergent to some . Let and such that , for all .
Choose . Then, there exists such that , for all . Therefore,
Since is arbitrary, we conclude that .
(Q3) Given , put . If and , it follows
3. Fixed Point Results
Given a qpm space , we denote by the collection of all nonempty subsets of , by the collection of all nonempty closed subsets of , and by the collection of all nonempty closed subsets of .
Following AlHomidan et al. [5, Definition 6.1] if is a quasimetric space, we say that a multivalued map is contractive if there exists a function on and such that for each and there is satisfying .
Latif and AlMezel (see [7]) generalized this notion as follows.
If is a quasimetric space, we say that a multivalued map is generalized contractive if there exists a function on such that for each and there is satisfying
where is a function such that for all .
Then, they proved the following improvement of the celebrated Rakotch fixed point theorem (see [28]).
Theorem 3.1 (Lafit and AlMezel [7, Theorem 2.3]).
Let be a complete quasimetric space. Then, for each generalized qcontractive multivalued map there exists such that .
On the other hand, Bianchini and Grandolfi proved in [29] the following fixed point theorem.
Theorem 3.2 (Bianchini and Grandolfi [29]).
Let be a complete metric space and let be a map such that for each
where is a nondecreasing function satisfying , for all ( denotes the nth iterate of ). Then, has a unique fixed point.
A function satisfying the conditions of the preceding theorem is called a BianchiniGrandolfi gauge function (cf [24, 30]).
It is easy to check (see [30, Page 8]) that if is a BianchiniGrandolfi gauge function, then , for all , and hence .
Our next result generalizes BianchiniGrandolfi's theorem for Qfunctions on complete qpm spaces.
Theorem 3.3.
Let be a complete qpm space, q a Qfunction on , and a multivalued map such that for each and , there is satisfying
where is a BianchiniGrandolfi gauge function. Then, there exists such that and .
Proof.
Fix and let . By hypothesis, there exists such that . Following this process, we obtain a sequence with and , for all . Therefore
for all .
Now, choose . Let for which condition (Q3) is satisfied. We will show that there is such that whenever .
Indeed, if , then and thus , for all , so, by condition (Q1), whenever .
If , , so there is such that
Then, for , we have
In particular, and whenever , so, by Lemma 2.2, whenever .
We have proved that is a Cauchy sequence in (in fact, it is a Cauchy sequence in the metric space . Since is complete there exists such that .
Next, we show that .
To this end, we first prove that . Indeed, choose . Fix . Since whenever , it follows from condition (Q2) that whenever .
Now for each take such that
If , it follows that . Otherwise we obtain .
Hence, , and by Lemma 2.2,
Therefore, .
It remains to prove that .
Since , we can construct a sequence in such that , and
Since , it follows that , and thus . So, by Lemma 2.2, is a Cauchy sequence in (in fact, it is a Cauchy sequence in . Let such that . Given , there is such that , for all . By applying condition (Q2), we deduce that , so . Since , it follows from condition (Q1) that . Therefore, , for all , by condition (Q3). We conclude that , and thus .
The next example illustrates Theorem 3.3.
Example 3.4.
Let and let be the qpm on given by . It is well known that is weightable with weighting function given by , for all . Let be partial metric induced by . Then, is a function on by Proposition 2.10. Note also that, by Theorem 2.8 (a),
for all . Moreover is clearly complete because is the Euclidean metric on and thus is a compact metric space.
Now define by
for all . Note that because the nonempty closed subsets of are the intervals of the form , .
Let be such that , for all , and , for all . We wish to show that is a BianchiniGrandolfi gauge function.
It is clear that is nondecreasing.
Moreover, , for all . Indeed, if we have whenever , while for , we have so,
and following this process we deduce the known fact that , for all . We have shown that is a BianchiniGrandolfi gauge function.
Finally, for each and , there exists such that . Choose . Then and
If , then , and thus .
We have checked that conditions of Theorem 3.3 are fulfilled, and hence, there is with . In fact is the only point of satisfying and (actually . The following consequence of Theorem 3.3, which is also illustrated by Example 3.4, improves and generalizes in several directions the Banach Contraction Principle for partial metric spaces obtained in Theorem 5.3 of [14].
Corollary 3.5.
Let be a partial metric space such that the induced weightable qpm is complete and let be a multivalued map such that for each and , there is satisfying
where is a BianchiniGrandolfi gauge function. Then, there exists such that and .
Proof.
Since (see Theorem 2.8), we deduce from Proposition 2.10 that is a function for the complete (weightable) qpm space . The conclusion follows from Theorem 3.3.
Observe that if is a nondecreasing function such that , for all , then the function given by , is a BianchiniGrandolfi gauge function (compare [31, Proposition 8]). Therefore, the following variant of Theorem 3.1, which improves Corollary 2.4 of [7], is now a consequence of Theorem 3.3.
Corollary 3.6.
Let be a complete qpm space. Then, for each generalized qcontractive multivalued map with q nondecreasing, there exists such that and .
Remark 3.7.
The proof of Theorem 3.3 shows that the condition that is complete can be replaced by the more general condition that every Cauchy sequence in the metric space is convergent.
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Acknowledgments
The authors thank one of the reviewers for suggesting the inclusion of a concrete example to which Theorem 3.3 applies. They acknowledge the support of the Spanish Ministry of Science and Innovation, Grant no. MTM200912872C0201.
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Marín, J., Romaguera, S. & Tirado, P. Functions on Quasimetric Spaces and Fixed Points for Multivalued Maps. Fixed Point Theory Appl 2011, 603861 (2011). https://doi.org/10.1155/2011/603861
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DOI: https://doi.org/10.1155/2011/603861