Abstract
In this paper, we prove some results of a common fixed point for two self-mappings on partial metric spaces. Our results generalize some interesting results of Ilić et al. (Appl. Math. Lett. 24:1326-1330, 2011). We conclude with a result of the existence of a fixed point for set-valued mappings in the context of 0-complete partial metric spaces.
MSC:54H25, 47H10.
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1 Introduction
In the mathematical field of domain theory, attempts were made 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 have been lately pointed out by other authors such as O’Neill [2], Bukatin and Scott [3], Bukatin and Shorina [4], Romaguera and Schellekens [5] and others.
After the definition of the concept of a partial metric space, Matthews [1] obtained a Banach type fixed point theorem on complete partial metric spaces. This result was recently generalized by Ilić et al. [6]. In this paper, we prove some results of a common fixed point for two self-mappings on partial metric spaces. Our results generalize some interesting results of Ilić et al. We conclude this paper with a new existence result of a fixed point for set-valued mappings in a partial metric space.
2 Preliminaries
First, we recall some definitions and some properties of partial metric spaces that can be found in [2, 5–27]. A partial metric on a nonempty set X is a function such that for all ,
() ,
() ,
() ,
() .
It is clear that () implies the triangular inequality. A partial metric space is a pair such that X is a nonempty set and p is a partial metric on X. It is clear that if , then from and , it follows that . But if , may not be 0. 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 p on X generates a topology on X which has as a base the family of open p-balls , where
for all and .
If p is a partial metric on X, then the function given by
is a metric on X.
Definition 1 Let be a partial metric space.
-
(i)
A sequence in converges to a point if and only if .
-
(ii)
A sequence in is called a Cauchy sequence if there exists (and is finite) .
-
(iii)
A partial metric space is said to be complete if every Cauchy sequence in X converges, with respect to , to a point such that .
-
(iv)
A sequence in is called 0-Cauchy if .
We say that is 0-complete if every 0-Cauchy sequence in X 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 p 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 ([9])
Let be a partial metric space and . If and , then for all .
Define . Then , where denotes the closure of A (for details see [10], Lemma 1).
Let X be a non-empty set and . The mappings T, f are said to be weakly compatible if they commute at their coincidence point (i.e., whenever ). A point is called a point of coincidence of T and f if there exists a point such that .
Lemma 2 (Proposition 1.4 of [28])
Let X be a non-empty set and the mappings T, have a unique point of coincidence v in X. If T and f are weakly compatible, then v is a unique common fixed point of T and f.
3 Main results
Let be a partial metric space and T, be such that . For every , we consider a sequence defined by for all and we say that is a T-f-sequence of the initial point (see [29]). Set
It is not always . This is true if and only if . If , , , and , then and .
Let be a partial metric space. We denote with the family of pairs such that:
-
(i)
T and f are self-mappings on X with ;
-
(ii)
for each the following condition holds:
(3.1)
where .
Remark 1 If , then for each , we have
Indeed, because , we have that from which follows , that is, .
Lemma 3 Let be a partial metric space with and . If are coincidence points for T and f, then .
Proof From
it follows that either or . Now, from (), it follows , that is, . □
Lemma 4 Let be a partial metric space and . If fX is a complete subspace of X, then for each , there is such that
Proof We fix and prove that each T-f-sequence of the initial point is a Cauchy sequence in TX.
From Remark 1, we deduce that . Hence, is a nonincreasing sequence. Set
and
We prove that
Obviously, (3.2) is true for . Assume that (3.2) is true for some , then
We fix and choose such that for all and . For each , we have
Now, from
for each , we obtain that
Since fX is a complete subspace of X, there is such that
Let such that . We show that .
Now, from (3.1), we deduce that there exist such that
-
(i)
for all ;
-
(ii)
for all ;
-
(iii)
for all .
Clearly, at least one of the sets , , is infinite.
Then, from
if () is infinite, taking the limit as and , it follows that and so .
Now, if we choose such that , we deduce that . If , also . □
Lemma 5 Let be a partial metric space and . If fX is a complete subspace of X, then there exists such that .
Proof By Lemma 4, there exists a sequence such that
for all . First, we prove that
For fixed, we choose . From Remark 1, it follows
which implies
Now, for all , we have
If , then
and so . This implies that .
If
then
This implies that
In both cases, (3.4) holds. Since fX is a complete subspace of X, there is such that
Let such that . From it follows that . We prove that . As and for all , we get that
As , we obtain and so . □
The following theorem of a common fixed point in a partial metric space is one of our main results.
Theorem 1 Let be a partial metric space with . If fX is a complete subspace of X, then T and f have a unique point of coincidence with . Moreover, if T and f are weakly compatible and , then T and f have a unique common fixed point .
Proof By Lemma 5 and Lemma 3, there exists such that is a unique point of coincidence for T and f with . If T and f are weakly compatible and , by Lemma 3, from , that is , it follows that . By Lemma 2, z is a unique common fixed point for T and f belonging to . □
Theorem 2 Let be a partial metric space and . Suppose that the following condition holds:
for each , where . If fX is a complete subspace of X, and T and f are weakly compatible, then T and f have a unique common fixed point .
Proof Clearly, , by Lemma 5, there exists such that . If is such that and , then by Lemma 3. If , from (3.5) we obtain either or . In each of these cases, we deduce that , and so T and f have a unique point of coincidence. By Lemma 2, since T and f are weakly compatible, T and f have a unique common fixed point . □
If in Theorems 1 and 2 we choose , we obtain Theorems 3.1 and 3.3 of Ilić et al.
Example 1 Let and be defined by . Then is a complete partial metric space. Let be defined by
and
respectively.
In order to show that T and f satisfy the contractive condition (3.5) in Theorem 2 with , we consider the following cases.
Case 1. . We have
Case 2. and . We have
Case 3. , and . We have
Case 4. and . We have
Case 5. . We have
Since fX is a complete subspace of X, and T and f are weakly compatible, by Theorem 2, T and f have a unique common fixed point .
4 Fixed points for set-valued mappings
Investigations of the existence of fixed points of set-valued contractions in metric spaces were initiated by Nadler [30]. The following theorem is motivated by Nadler’s results and also generalizes the well-known Banach contraction theorem in several ways.
Denote with Ψ the family of nondecreasing functions such that for each , where is the n th iterate of ψ.
Lemma 6 For every function , the following holds: for each .
Definition 2 Let be a partial metric space and let , where is the family of nonempty closed subsets of X. T is ψ-contractive if there exists such that, for any and , there is with
Theorem 3 Let be a 0-complete partial metric space and let be a ψ-contractive mapping. Then there exists such that , i.e., z is a fixed point of T, and .
Proof Fix and let . If , then and is a fixed point of T. Assume, hence, ; then there exists such that . Proceeding in this way, we have a sequence in X such that and for every . Consequently,
Since, the series converges, we get that converges too. It follows, for ,
Now, from (4.1), we deduce that and hence is a 0-Cauchy sequence in X. Since X is a 0-complete space, there exists such that and . For there is such that . From
as , we obtain that ; since Tz is closed, we have . □
Example 2 Let and be defined by . Then is a 0-complete partial metric space. Let be defined by
Note that Tx is closed and bounded for all under the given partial metric p.
We show that T is a ψ-contractive mapping with respect to defined by for all . In fact, for all and , if , , then we choose . It implies that
By Theorem 3, T has a fixed point .
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Acknowledgements
The authors thank the referees for their valuable comments that helped us to improve the text. The first and fourth authors are supported by Università degli Studi di Palermo (Local University Project ex 60%). The second and third authors are thankful to the Ministry of Science and Technological Development of Serbia (project III44006).
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Di Bari, C., Milojević, M., Radenović, S. et al. Common fixed points for self-mappings on partial metric spaces. Fixed Point Theory Appl 2012, 140 (2012). https://doi.org/10.1186/1687-1812-2012-140
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DOI: https://doi.org/10.1186/1687-1812-2012-140