International Journal of Analysis

Volume 2013 (2013), Article ID 428561, 8 pages

http://dx.doi.org/10.1155/2013/428561

## Common Fixed Points in a Partially Ordered Partial Metric Space

Dipartimento di Matematica e Informatica, Università Degli Studi di Palermo, Via Archirafi, 34-90123 Palermo, Italy

Received 29 August 2012; Revised 12 September 2012; Accepted 17 September 2012

Academic Editor: Harumi Hattori

Copyright © 2013 Daniela Paesano and Pasquale Vetro. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the first part of this paper, we prove some generalized versions of the result of Matthews in (Matthews, 1994) using different types of conditions in partially ordered partial metric spaces for dominated self-mappings or in partial metric spaces for self-mappings. In the second part, using our results, we deduce a characterization of partial metric 0-completeness in terms of fixed point theory. This result extends the Subrahmanyam characterization of metric completeness.

#### 1. Introduction

In the mathematical field of domain theory, attempts were made in order to equip semantics domain with a notion of distance. In particular, Matthews [1] introduced the notion of a partial metric space as a part of the study of denotational semantics of data for networks, showing that the contraction mapping principle can be generalized to the partial metric context for applications in program verification. Moreover, the existence of several connections between partial metrics and topological aspects of domain theory has been lately pointed by other authors as O'Neill [2], Bukatin and Scott [3], Bukatin and Shorina [4], Romaguera and Schellekens [5], and others (see also [6–14] and the references therein).

After the result of Matthews [1], the interest for fixed point theory developments in partial metric spaces has been constantly growing, and many authors presented significant contributions in the directions of establishing partial metric versions of well-known fixed point theorems for the existence of fixed points, common fixed points, and coupled fixed points in classical metric spaces (see e.g., [15, 16]). Obviously, we cannot cite all these papers but we give only a partial list [17–49].

Recently, Romaguera [50] proved that a partial metric space is 0-complete if and only if every -Caristi mapping on has a fixed point. In particular, the result of Romaguera extended Kirk’s [51] characterization of metric completeness to a kind of complete partial metric spaces. Successively, Karapinar in [36] extended the result of Caristi and Kirk [52] to partial metric spaces.

In the first part of this paper, following this research direction, we prove some generalized versions of the result of Matthews by using different types of conditions in ordered partial metric spaces for dominated self-mappings or in partial metric spaces for self-mappings. The notion of dominated mapping of economics, finance, trade, and industry is also applied to approximate the unique solution of nonlinear functional equations. In the second part, using the results obtained in the first part, we deduce a characterization of partial metric 0-completeness in terms of fixed point theory. This result extends the Subrahmanyam [53] characterization of metric completeness. For other characterizations of metric completeness in terms of fixed point theory, the reader can see, for example, [54, 55] and for partial metric completeness, [41].

#### 2. Preliminaries

First, we recall some definitions and some properties of partial metric spaces that can be found in [1, 2, 40, 48, 50]. A partial metric on a nonempty set is a function such that for all , , , .

A partial metric space is a pair such that is a nonempty set and is a partial metric on . It is clear that if , then from and , it follows that . But if , may not be . A basic example of a partial metric space is the pair , where for all . Other examples of partial metric spaces which are interesting from a computational point of view can be found in [1].

Each partial metric on generates a topology on which has as a base the family of open -balls , where for all and .

If is a partial metric on , then the function given by is a metric on .

Let be a partial metric space. A sequence in converges to a point if and only if .

A sequence in is called a Cauchy sequence if there exists (and is finite) .

A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .

A sequence in is called 0-Cauchy if . We say that is 0-complete if every 0-Cauchy sequence in converges, with respect to , to a point such that .

On the other hand, the partial metric space , where denotes the set of rational numbers and the partial metric is given by , provides an example of a 0-complete partial metric space which is not complete.

It is easy to see that every closed subset of a complete partial metric space is complete.

Lemma 1 (see [1, 40]). *Let be a partial metric space. Then*(a)* is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space . *(b)* A partial metric space is complete if and only if the metric space is complete. Furthermore, if and only if
*

The following lemma is obvious.

Lemma 2. *Let be a partial metric space and . If and , then for all .*

Define . Then , where denotes the closure of (for details see [22, Lemma 1]). From for every , we deduce that .

Let be a nonempty set and . The mappings are said to be weakly compatible if they commute at their coincidence points (i.e., whenever ). A point is called a point of coincidence of and if there exists a point such that .

Lemma 3 (see [56]). *Let be a nonempty set and the mappings have a unique pointof coincidence inIf and are weakly compatible, then and have a unique common fixed point.*

Let be a nonempty set. If is a partial metric space and is a partially ordered set, then is called a partially ordered partial metric space. are called comparable if or holds. Let be a partially ordered set and two mappings. is called an -dominated mapping if for every .

#### 3. Main Results

Let be a partial metric space and be such that . For every we consider the sequence defined by for all and we say that is a --sequence of the initial point (see [57]).

Denote with the family of non-decreasing functions such that and for each , where is the th iterate of .

Lemma 4. *For every function , the following holds, if for each , then .*

The following theorem is one of our main results, and it ensures the existence of a common fixed point for two self-mappings in the setting of partially ordered partial metric spaces.

Theorem 5. *Let be a partially ordered partial metric space and two mappings such that . Assume that there exists such that
**
for all with and comparable. If the following conditions hold: *(i)* is a -dominated mapping, *(ii)*either or is a 0-complete subspace of , *(iii)*for a non-increasing sequence converging to , we have for all and ,**then and have a point of coincidence. Moreover, if and are weakly compatible, then and have a common fixed point.*

*Proof. *Let be fixed and be a --sequence of the initial point . As for all , then the sequence is non-increasing.

If for some , then is a point of coincidence of and . Suppose that for all . Since and are comparable for all , we have

If from
we obtain a contradiction and so
Then, we have
and hence
Fix and we choose such that
for all . Let and we show that

Clearly, (12) is true for . Suppose that (12) holds for some , as and are comparable for all , then

This implies that (12) holds for and by induction, it holds for all . From (12), we deduce that there exists
and hence is a 0-Cauchy sequence.

Suppose that is a 0-complete subspace of , then there exists such that
This holds also if is 0-complete with .

Let be such that . We show that is a point of coincidence of and . If not, we have . This implies that there exists such that
for every . By condition (iii), and are comparable for every and hence, by condition (5) with and , we deduce that
for every . Letting in the previous inequality and using Lemma 2, we obtain
which implies , that is, . Thus, we have shown that is a point of coincidence of and . If and are weakly compatible, then . By condition (iii), , that is, and are comparable. Using the contractive condition (5), we get
which implies and hence is a common fixed point of and .

We shall give a sufficient condition for the uniqueness of the common fixed point in Theorem 5.

Theorem 6. *Let all the conditions of Theorem 5 be satisfied. If the following condition holds:*(iv)* for all there exists such that and , where is the --sequence of the initial point , ** then and have a unique common fixed point.*

*Proof. *Let be two common fixed points of and with . If and are comparable, then using the contractive condition (5), we deduce that . If and are not comparable, then there exists such that , . As is a -dominated mapping, we get that
To continue, we obtain
and hence and are comparable.

Using the contractive condition (5) with and , we get
for all . Since, the contractive condition (5) ensures that , we have
Now, by condition (iv), and hence for sufficiently large, we have
Without loss of generality, assuming that (24) holds for all , it follows that
Now, letting in (25), we obtain

With similar arguments, we deduce that . Hence
as , which is a contradiction. Thus and have a unique common fixed point.

As a consequence of Theorem 5, we state the following result.

Theorem 7. *Let be a partial metric space and two mappings such that . Assume that there exists such that
**
for all . If or is a 0-complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique common fixed point.*

*Proof. *Proceeding as in the proof of Theorem 5, we get that and have a unique point of coincidence and, by Lemma 3, and have a unique common fixed point.

From Theorem 7, we can deduce the following corollaries.

Corollary 8 (Banach type). *Let be a partial metric space and two mappings such that . Assume that
**
for all , where . If or is a 0-complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique common fixed point.*

Corollary 9 (Bianchini type). *Let be a partial metric space and let be two mappings such that . Assume that
**
for all , where . If or is a 0-complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique common fixed point.*

Corollary 10 (Reich type [58]). *Let be a partial metric space and let be two mappings such that . Assume that
**
for all , where and . If or is a 0-complete subspace of , then and have a unique point of coincidence. Moreover, if and are weakly compatible, then and have a unique common fixed point.*

The following example shows that there exist mappings that satisfy the contractive condition (28), but are not quasi-contractions [59].

*Example 11. *Consider the set and the function given by , and . Obviously, is a partial metric on , but it is not a metric (since for and ). Clearly, is a 0-complete partial metric space. Let be defined by and for every . Take for every .

First, we will check that and satisfy the contractive condition (28). If , then and (28) trivially holds. Let, for example, , then we have the following three cases:

Thus, all the conditions of Theorem 7 are satisfied and the existence of a common fixed point of and (which is ) follows. The same conclusion cannot be obtained by the main results from [59]. Indeed, using , and then taking instead of , in (5), we obtain
Since , the conclusion follows.

The following example shows that there exist mappings that satisfy the contractive condition (5), but do not satisfy the contractive condition (28).

*Example 12. *Let be endowed with the partial metric
Clearly, is a 0-complete partial metric space. Let be defined by
and for each . As and have many common fixed points (each is a common fixed point), then it is immediate to show that and do not satisfy the contractive condition (28).

If is ordered by
then and satisfy the contractive condition (5) where is defined by
Using Theorem 5, we deduce that and have a common fixed point.

#### 4. Completeness in Partial Metric Spaces and Fixed Points

In this section, we characterize those partial metric spaces for which every Bianchini mapping has a fixed point in the style of Subrahmanyam characterization of metric completeness. This will be done by means of the notion of 0-completeness which was introduced by Romaguera in [50].

Let be a partial metric space and be a mapping. We recall that is a Bianchini [60] mapping if for all , where .

Theorem 13. *Let be a partial metric space. If every mapping satisfying the following conditions: *(i)* for all , for a fixed , *(ii)* TX is countable ** has a fixed point, then is 0-complete.*

*Proof. *Suppose that there is a 0-Cauchy sequence of distinct points in which is not convergent in . We put and we note that for every , we have .

Now, implies that . Since is a 0-Cauchy sequence in , there exists a least positive integer such that

In particular

For fixed , since and so , there is such that

Now, let be defined by
From the definition of , we deduce that satisfies the condition (ii). On the other hand, satisfies also the condition (i). In fact, (i) is verified by assuming and , and noting that
It is clear that has not fixed points since , . Thus, the assumption that there is a 0-Cauchy sequence which is not convergent in leads to a contradiction to Theorem 13 and thereby establishes the same.

If in Theorem 13 we choose , by Corollary 9, we obtain the following characterization of 0-completeness for partial metric spaces.

Theorem 14. *A partial metric space is 0-complete if and only if every mapping satisfying the following conditions: *(i)* for all , for a fixed , *(ii)* TX is countable ** has a fixed point.*

In Theorem 14, the class of mappings satisfying (i) and (ii) can be replaced by the class of mappings satisfying (ii) and the following condition:(i).

#### Acknowledgment

The authors are grateful to the editor and referees for their valuable suggestions and critical remarks for improving this paper. P. Vetro is supported by Università degli Studi di Palermo (Local University Project R.S. ex 60%).

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