# Open image in new window -Functions on Quasimetric Spaces and Fixed Points for Multivalued Maps

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## Abstract

We discuss several properties of Open image in new window -functions in the sense of Al-Homidan et al.. In particular, we prove that the partial metric induced by any Open image in new window weighted quasipseudometric space is a Open image in new window -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 Open image in new window -function. In this context we also obtain some fixed point results for multivalued maps by using Bianchini-Grandolfi gauge functions.

### Keywords

Fixed Point Theorem Lower Semicontinuous Computational Biology Main Concept Nondecreasing Function## 1. Introduction and Preliminaries

Kada et al. introduced in [1] the concept of Open image in new window -distance on a metric space and extended the Caristi-Kirk fixed point theorem [2], the Ekeland variation principle [3] and the nonconvex minimization theorem [4], for Open image in new window -distances. Recently, Al-Homidan et al. introduced in [5] the notion of Open image in new window -function on a quasimetric space and then successfully obtained a Caristi-Kirk-type fixed point theorem,a Takahashi minimization theorem, an equilibrium version of Ekeland-type variational principle, and a version of Nadler's fixed point theorem for a Open image in new window *-* function on a complete quasimetric space, generalizing in this way, among others, the main results of [1] because every Open image in new window -distance is, in fact, a Open image in new window -function. This interesting approach has been continued by Hussain et al. [6], and by Latif and Al-Mezel [7], respectively. In particular, the authors of [7] have obtained a nice Rakotch-type theorem for Open image in new window -functions on complete quasimetric spaces.

In Section 2 of this paper, we generalize the basic theory of Open image in new window -functions to Open image in new window 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, Open image in new window quasipseudometric spaces (in particular, weightable Open image in new window quasipseudometric spaces and their equivalent partial metric spaces) rather than quasimetric spaces, play a crucial role (cf. [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], etc.). In particular, we prove that for every weighted Open image in new window quasipseudometric space the induced partial metric is a Open image in new window -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 Open image in new window -function. Finally, Section 3 is devoted to present a new fixed point theorem for Open image in new window -functions and multivalued maps on Open image in new window quasipseudometric spaces, by using Bianchini-Grandolfi gauge functions in the sense of [24]. Our result generalizes and improves, in several ways, well-known fixed point theorems.

Throughout this paper the letter Open image in new window and Open image in new window 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 Open image in new window quasipseudometric on a set Open image in new window , we mean a function Open image in new window such that for all Open image in new window ,

(i) Open image in new window ,

(ii) Open image in new window .

A Open image in new window quasipseudometric Open image in new window on Open image in new window that satisfies the stronger condition

is called a quasimetric on Open image in new window .

We remark that in the last years several authors used the term "quasimetric" to refer to a Open image in new window quasipseudometric and the term " Open image in new window quasimetric" to refer to a quasimetric in the above sense.

In the following we will simply write Open image in new window qpm instead of Open image in new window quasipseudometric if no confusion arises.

A Open image in new window qpm space is a pair Open image in new window such that Open image in new window is a set and Open image in new window is a Open image in new window qpm on Open image in new window . If Open image in new window is a quasimetric on Open image in new window , the pair Open image in new window is then called a quasimetric space.

Given a Open image in new window qpm Open image in new window on a set Open image in new window , the function Open image in new window defined by Open image in new window , is also a Open image in new window qpm on Open image in new window , called the conjugate of Open image in new window , and the function Open image in new window defined by Open image in new window is a metric on Open image in new window , called the supremum metric associated to Open image in new window .

Thus, every Open image in new window qpm Open image in new window on Open image in new window induces, in a natural way, three topologies denoted by Open image in new window , Open image in new window and Open image in new window , respectively, and defined as follows.

(i) Open image in new window is the Open image in new window topology on Open image in new window which has as a base the family of Open image in new window -open balls Open image in new window , where Open image in new window , for all Open image in new window and Open image in new window .

(ii) Open image in new window is the Open image in new window topology on Open image in new window which has as a base the family of Open image in new window -open balls Open image in new window , where Open image in new window , for all Open image in new window and Open image in new window .

(iii) Open image in new window is the topology on Open image in new window induced by the metric Open image in new window .

Note that if Open image in new window is a quasimetric on Open image in new window , then Open image in new window is also a quasimetric, and Open image in new window and Open image in new window are Open image in new window topologies on Open image in new window .

Note also that a sequence Open image in new window in a Open image in new window qpm space Open image in new window is Open image in new window -convergent (resp., Open image in new window -convergent) to Open image in new window if and only if Open image in new window (resp., Open image in new window .

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 Open image in new window qpm space Open image in new window is said to be complete if every Cauchy sequence is Open image in new window -convergent, where a sequence Open image in new window is called Cauchy if for each Open image in new window there exists Open image in new window such that Open image in new window whenever Open image in new window .

In this case, we say that Open image in new window is a complete Open image in new window qpm on Open image in new window .

## 2. Open image in new window-Functions on Open image in new window qpm-Spaces

We start this section by giving the main concept of this paper, which was introduced in [5] for quasimetric spaces.

Definition 2.1.

A Open image in new window -function on a Open image in new window qpm space Open image in new window is a function Open image in new window satisfying the following conditions:

(Q1) Open image in new window , for all Open image in new window ,

(Q2) if Open image in new window , and Open image in new window is a sequence in Open image in new window that Open image in new window -converges to a point Open image in new window and satisfies Open image in new window , for all Open image in new window , then Open image in new window ,

(Q3) for each Open image in new window there exists Open image in new window such that Open image in new window and Open image in new window imply Open image in new window .

If Open image in new window is a metric space and Open image in new window satisfies conditions (Q1) and (Q3) above and the following condition:

(Q2′) Open image in new window is lower semicontinuous for all Open image in new window , then Open image in new window is called a *w*-distance on Open image in new window (cf. [1]).

Clearly Open image in new window is a Open image in new window -distance on Open image in new window whenever Open image in new window is a metric on Open image in new window .

However, the situation is very different in the quasimetric case. Indeed, it is obvious that if Open image in new window is a Open image in new window qpm space, then Open image in new window satisfies conditions (Q1) and (Q2), whereas Example 3.2 of [5] shows that there exists a Open image in new window qpm space Open image in new window such that Open image in new window does not satisfy condition (Q3), and hence it is not a *Q*-function on Open image in new window . In this direction, we next present some positive results.

Lemma 2.2.

Let q be a Q-function on a Open image in new window qpm space Open image in new window . Then, for each Open image in new window , there exists Open image in new window such that Open image in new window and Open image in new window imply Open image in new window .

Proof.

By condition (Q3), Open image in new window . Interchanging Open image in new window and Open image in new window , it follows that Open image in new window , so Open image in new window .

Proposition 2.3.

Let Open image in new window be a Open image in new window qpm space. If Open image in new window is a Q-function on Open image in new window , then Open image in new window , and hence, Open image in new window is a metrizable topology on Open image in new window .

Proof.

Let Open image in new window be a sequence in Open image in new window which is Open image in new window -convergent to some Open image in new window . Then, by Lemma 2.2, Open image in new window . We conclude that Open image in new window .

Remark 2.4.

It follows from Proposition 2.3 that many paradigmatic quasimetrizable topological spaces Open image in new window , as the Sorgenfrey line, the Michael line, the Niemytzki plane and the Kofner plane (see [25]), do not admit any compatible quasimetric Open image in new window which is a Open image in new window -function on Open image in new window .

In the sequel, we show that, nevertheless, it is possible to construct an easy but, in several cases, useful Open image in new window -function on any quasimetric space, as well as a suitable Open image in new window -functions on any weightable Open image in new window qpm space.

Recall that the discrete metric on a set Open image in new window is the metric Open image in new window on Open image in new window defined as Open image in new window , for all Open image in new window , and Open image in new window , for all Open image in new window with Open image in new window .

Proposition 2.5.

Let Open image in new window be a quasimetric space. Then, the discrete metric on Open image in new window is a Open image in new window -function on Open image in new window .

Proof.

Since Open image in new window is a metric it obviously satisfies condition (Q1) of Definition 2.1.

Now suppose that Open image in new window is a sequence in Open image in new window that Open image in new window -converges to some Open image in new window , and let Open image in new window and Open image in new window such that Open image in new window , for all Open image in new window . If Open image in new window , then Open image in new window . If Open image in new window , we deduce that Open image in new window , for all Open image in new window . Since Open image in new window , it follows that Open image in new window , so Open image in new window , Open image in new window and thus Open image in new window . Hence, condition (Q2) is also satisfied.

Finally, Open image in new window satisfies condition (Q3) taking Open image in new window for every Open image in new window

Example 2.6.

On the set Open image in new window of real numbers define Open image in new window as Open image in new window if Open image in new window , and Open image in new window if Open image in new window . Then, Open image in new window is a quasimetric on Open image in new window and the topological space Open image in new window is the celebrated Sorgenfrey line. Since Open image in new window is the discrete metric on Open image in new window , it follows from Proposition 2.5 that Open image in new window is a Open image in new window -function on Open image in new window .

Example 2.7.

The quasimetric Open image in new window on the plane Open image in new window , constructed in Example 7.7 of [25], verifies that Open image in new window is the so-called Kofner plane and that Open image in new window is the discrete metric on Open image in new window , so, by Proposition 2.5, Open image in new window is a Open image in new window -function on Open image in new window .

Matthews introduced in [14] the notion of a weightable Open image in new window 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 Open image in new window qpm space Open image in new window is called weightable if there exists a function Open image in new window such that for all Open image in new window . In this case, we say that Open image in new window is a weightable Open image in new window qpm on Open image in new window . The function Open image in new window is said to be a weighting function for Open image in new window and the triple Open image in new window is called a weighted Open image in new window qpm space.

A partial metric on a set Open image in new window is a function Open image in new window such that, for all Open image in new window :

(i) Open image in new window ,

(ii) Open image in new window ,

(iii) Open image in new window ,

(iv) Open image in new window .

A partial metric space is a pair Open image in new window such that Open image in new window is a set and Open image in new window is a partial metric on Open image in new window .

Each partial metric Open image in new window on Open image in new window induces a Open image in new window topology Open image in new window on Open image in new window which has as a base the family of open Open image in new window -balls Open image in new window , where Open image in new window , for all Open image in new window and Open image in new window .

The precise relationship between partial metric spaces and weightable Open image in new window qpm spaces is provided in the next result.

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

- (b)
Conversely, let Open image in new window be a partial metric space. Then, the function Open image in new window defined by Open image in new window , for all Open image in new window is a weightable Open image in new window qpm on Open image in new window with weighting function Open image in new window given by Open image in new window for all Open image in new window . Furthermore Open image in new window .

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 Open image in new window qpm space and, thus, of a partial metric space (see, e.g., [14, 20, 21]).

Proposition 2.10.

Let Open image in new window be a weighted Open image in new window qpm space. Then, the induced partial metric Open image in new window is a Q-function on Open image in new window .

Proof.

We will show that Open image in new window satisfies conditions (Q1), (Q2), and Q(3) of Definition 2.1.

(Q2)Let Open image in new window be a sequence in Open image in new window which is Open image in new window -convergent to some Open image in new window . Let Open image in new window and Open image in new window such that Open image in new window , for all Open image in new window .

Since Open image in new window is arbitrary, we conclude that Open image in new window .

(Q3) Given Open image in new window , put Open image in new window . If Open image in new window and Open image in new window , it follows

## 3. Fixed Point Results

Given a Open image in new window qpm space Open image in new window , we denote by Open image in new window the collection of all nonempty subsets of Open image in new window , by Open image in new window the collection of all nonempty Open image in new window -closed subsets of Open image in new window , and by Open image in new window the collection of all nonempty Open image in new window -closed subsets of Open image in new window .

Following Al-Homidan et al. [5, Definition 6.1] if Open image in new window is a quasimetric space, we say that a multivalued map Open image in new window is Open image in new window -contractive if there exists a Open image in new window -function Open image in new window on Open image in new window and Open image in new window such that for each Open image in new window and Open image in new window there is Open image in new window satisfying Open image in new window .

Latif and Al-Mezel (see [7]) generalized this notion as follows.

where Open image in new window is a function such that Open image in new window for all Open image in new window .

Then, they proved the following improvement of the celebrated Rakotch fixed point theorem (see [28]).

Theorem 3.1 (Lafit and Al-Mezel [7, Theorem 2.3]).

Let Open image in new window be a complete quasimetric space. Then, for each generalized q-contractive multivalued map Open image in new window there exists Open image in new window such that Open image in new window .

On the other hand, Bianchini and Grandolfi proved in [29] the following fixed point theorem.

Theorem 3.2 (Bianchini and Grandolfi [29]).

where Open image in new window is a nondecreasing function satisfying Open image in new window , for all Open image in new window ( Open image in new window denotes the nth iterate of Open image in new window ). Then, Open image in new window has a unique fixed point.

A function Open image in new window satisfying the conditions of the preceding theorem is called a Bianchini-Grandolfi gauge function (cf [24, 30]).

It is easy to check (see [30, Page 8]) that if Open image in new window is a Bianchini-Grandolfi gauge function, then Open image in new window , for all Open image in new window , and hence Open image in new window .

Our next result generalizes Bianchini-Grandolfi's theorem for *Q*-functions on complete Open image in new window qpm spaces.

Theorem 3.3.

where Open image in new window is a Bianchini-Grandolfi gauge function. Then, there exists Open image in new window such that Open image in new window and Open image in new window .

Proof.

for all Open image in new window .

Now, choose Open image in new window . Let Open image in new window for which condition (Q3) is satisfied. We will show that there is Open image in new window such that Open image in new window whenever Open image in new window .

Indeed, if Open image in new window , then Open image in new window and thus Open image in new window , for all Open image in new window , so, by condition (Q1), Open image in new window whenever Open image in new window .

In particular, Open image in new window and Open image in new window whenever Open image in new window , so, by Lemma 2.2, Open image in new window whenever Open image in new window .

We have proved that Open image in new window is a Cauchy sequence in Open image in new window (in fact, it is a Cauchy sequence in the metric space Open image in new window . Since Open image in new window is complete there exists Open image in new window such that Open image in new window .

Next, we show that Open image in new window .

To this end, we first prove that Open image in new window . Indeed, choose Open image in new window . Fix Open image in new window . Since Open image in new window whenever Open image in new window , it follows from condition (Q2) that Open image in new window whenever Open image in new window .

If Open image in new window , it follows that Open image in new window . Otherwise we obtain Open image in new window .

Therefore, Open image in new window .

It remains to prove that Open image in new window .

Since Open image in new window , it follows that Open image in new window , and thus Open image in new window . So, by Lemma 2.2, Open image in new window is a Cauchy sequence in Open image in new window (in fact, it is a Cauchy sequence in Open image in new window . Let Open image in new window such that Open image in new window . Given Open image in new window , there is Open image in new window such that Open image in new window , for all Open image in new window . By applying condition (Q2), we deduce that Open image in new window , so Open image in new window . Since Open image in new window , it follows from condition (Q1) that Open image in new window . Therefore, Open image in new window , for all Open image in new window , by condition (Q3). We conclude that Open image in new window , and thus Open image in new window .

The next example illustrates Theorem 3.3.

Example 3.4.

for all Open image in new window . Moreover Open image in new window is clearly complete because Open image in new window is the Euclidean metric on Open image in new window and thus Open image in new window is a compact metric space.

for all Open image in new window . Note that Open image in new window because the nonempty Open image in new window -closed subsets of Open image in new window are the intervals of the form Open image in new window , Open image in new window .

Let Open image in new window be such that Open image in new window , for all Open image in new window , and Open image in new window , for all Open image in new window . We wish to show that Open image in new window is a Bianchini-Grandolfi gauge function.

It is clear that Open image in new window is nondecreasing.

and following this process we deduce the known fact that Open image in new window , for all Open image in new window . We have shown that Open image in new window is a Bianchini-Grandolfi gauge function.

If Open image in new window , then Open image in new window , and thus Open image in new window .

We have checked that conditions of Theorem 3.3 are fulfilled, and hence, there is Open image in new window with Open image in new window . In fact Open image in new window is the only point of Open image in new window satisfying Open image in new window and Open image in new window (actually Open image in new window . 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.

where Open image in new window is a Bianchini-Grandolfi gauge function. Then, there exists Open image in new window such that Open image in new window and Open image in new window .

Proof.

Since Open image in new window (see Theorem 2.8), we deduce from Proposition 2.10 that Open image in new window is a Open image in new window -function for the complete (weightable) Open image in new window qpm space Open image in new window . The conclusion follows from Theorem 3.3.

Observe that if Open image in new window is a nondecreasing function such that Open image in new window , for all Open image in new window , then the function Open image in new window given by Open image in new window **,** is a Bianchini-Grandolfi 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 Open image in new window be a complete Open image in new window qpm space. Then, for each generalized q-contractive multivalued map Open image in new window with q nondecreasing, there exists Open image in new window such that Open image in new window and Open image in new window .

Remark 3.7.

The proof of Theorem 3.3 shows that the condition that Open image in new window is complete can be replaced by the more general condition that every Cauchy sequence in the metric space Open image in new window is Open image in new window -convergent.

## Notes

### 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. MTM2009-12872-C02-01.

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