Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces
© Aydi et al.; licensee Springer 2012
Received: 23 February 2012
Accepted: 10 July 2012
Published: 25 July 2012
We prove some coincidence and common fixed point results for three mappings satisfying a generalized weak contractive condition in ordered partial metric spaces. As application of the presented results, we give a unique fixed point result for a mapping satisfying a weak cyclical contractive condition. We also provide some illustrative examples.
1 Introduction and preliminaries
In the last decades, several authors have worked on domain theory in order to equip semantics domain with a notion of distance. In 1994, Matthews  introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks and showed that the Banach contraction principle  can be generalized to the partial metric context for applications in program verification. Later on, many researchers studied fixed point theorems in partial metric spaces as well as ordered partial metric spaces. For more details, see [5, 6, 9–15, 19, 20, 33, 34, 36].
Recently, there have been so many exciting developments in the field of existence of fixed points in partially ordered sets. For instance, Ran and Reurings  extended the Banach contraction principle in partially ordered sets with some applications to matrix equations. For more details on fixed point theory in partially ordered sets, we refer the reader to [1–4, 7, 8, 17, 18, 24, 28, 30–32, 39, 41] and the references cited therein.
In this paper, we establish some coincidence and common fixed point results for three self-mappings on an ordered partial metric space satisfying a generalized weak contractive condition. The presented theorems extend some recent results in the literature. Moreover, as application, we give a unique fixed point theorem for a mapping satisfying a weak cyclical contractive condition.
Throughout this paper, will denote the set of all non-negative real numbers. First, we start by recalling some known definitions and properties of partial metric spaces.
Definition 1.1 ()
A partial metric on a nonempty set X is a function such that for all :
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 (p1) and (p2), ; but if , may not be 0. A basic example of a partial metric space is the pair , where for all .
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 .
is a metric on X.
Definition 1.2 ()
converges to a point if and only if . We may write this as .
is called a Cauchy sequence if exists and is finite.
is said to be complete if every Cauchy sequence in X converges, with respect to , to a point , such that .
Lemma 1.3 ()
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
Definition 1.4 ()
Let be a partial metric space and be a given mapping. We say that T is continuous at , if for every , there exists such that .
Lemma 1.5 (Sequential characterization of continuity)
is a partial metric space,
is a partially ordered set.
Definition 1.7 Let be a partially ordered set. Then are called comparable if or holds.
Definition 1.8 ()
Remark 1.9 If is the identity mapping ( for all , shortly ), then the fact that S and T are weakly increasing with respect to R implies that S and T are weakly increasing mappings, that is, and for all . Finally, a mapping is weakly increasing if and only if for all .
Now, and , then S and T are weakly increasing with respect to R.
Definition 1.11 Let be an ordered partial metric space. We say that X is regular if and only if the following hypothesis holds: is a non-decreasing sequence in X with respect to ⪯ such that as , then for all .
Finally, we recall the following definition of partial-compatibility introduced by Samet et al. .
Definition 1.12 Let be a partial metric space and be given mappings. We say that the pair is partial-compatible if the following conditions hold:
(b1) implies that .
(b2) , whenever is a sequence in X such that and for some .
Note that Definition 1.12 extends and generalizes the notion of compatibility introduced by Jungck .
2 Main results
We start this section with some auxiliary results (see also ).
As a corollary, applying Lemma 2.1 to the associated metric of a partial metric p, and using Lemma 1.3, we obtain the following lemma (see also ).
In the sequel, let Ψ be the set of functions such that ψ is continuous, strictly increasing and if and only if . Also, let Φ be the set of functions such that φ is lower semi-continuous and if and only if . Such ψ and φ are called control functions.
Our first main result is the following.
T, S and R are continuous,
the pairs and are partial-compatible,
T and S are weakly increasing with respect to R.
where and . Then T, S and R have a coincidence point , that is, .
We claim that is a Cauchy sequence in the partial metric space . To this aim, we distinguish the following two cases.
This implies that . Continuing this process, we obtain for all . This implies that , therefore is Cauchy in . The same conclusion holds if for some .
Also, since , it follows . Thus, from (2.18), and so .
that is, u is a coincidence point of T, S and R. □
Remark 2.4 We point out that the order in which the mappings in condition (b) of Theorem 2.3 are considered is crucial. Trivially, Theorem 2.3 remains true if we assume that the partial-compatible pairs are and .
All the other hypotheses of Theorem 2.3 are satisfied and T, S and R have a coincidence point . (Moreover, is the unique common fixed point of T, S and R.)
Note that Theorem 2.3 is not applicable in respect of the usual order of real numbers because T is not weakly increasing. It follows that the partial order may be fundamental.
Under different hypotheses, the conclusion of Theorem 2.3 remains true without assuming the continuity of T, S and R, and the partial-compatibility of the pairs and . This is the purpose of the next theorem.
RX is a closed subspace of ,
T and S are weakly increasing with respect to R,
X is regular.
where and . Then, T, S and R have a coincidence point , that is, .
which is true if . This means that .
We conclude that u is a coincidence point of T, S and R. □
If is the identity mapping , by Theorem 2.6, we obtain the following common fixed point result involving two mappings.
where and . Then, T and S have a common fixed point , that is, .
The following example shows that the hypothesis ‘T and S are weakly increasing (with respect to R)’ has a key role for the validity of our results.
Consider the mappings defined by and , for all . Also, define the functions by and , for all . It is easy to show that and , for all , that is, T and S are weakly increasing. Now, take x and y comparable and, without loss of generality, assume , so that . It is easy to show that (2.27) holds and all the other hypotheses of Corollary 2.7 are satisfied. Then, T and S have a unique common fixed point .
Note that Corollary 2.7 is not applicable in respect of the usual order of real numbers because T and S are not weakly increasing.
Now, we shall prove the existence and uniqueness of a common fixed point for three mappings.
Theorem 2.9 In addition to the hypotheses of Theorem 2.3, suppose that for any , there exists such that and . Then, T, S and R have a unique common fixed point, that is, there exists a unique such that .
Now, and from (2.35), (2.36), we obtain , and so (2.28) holds.
that is, and u is a coincidence point of T and R.
This proves that u is a common fixed point of the mappings T, S and R.
which yields the uniqueness of the common fixed point of T, S and R. This completes the proof. □
for all such that Rx and Ry are comparable.
3 Application to cyclical contractions
In this section we use the previous results to prove a fixed point theorem for a mapping satisfying a weak cyclical contractive condition. In 2003, Kirk et al.  studied existence and uniqueness of a fixed point for mappings satisfying cyclical contractive conditions in complete metric spaces.
Definition 3.1 Let be a metric space, m a positive integer and nonempty subsets of X. A mapping T on is called a m-cyclic mapping if , , where .
Definition 3.2 Let Y be a nonempty set, m a positive integer and an operator. By definition, is a cyclic representation of Y with respect to T if T is a m-cyclic mapping and are nonempty sets.
Example 3.3 Let . Assume and , so that . Define such that , for all . It is clear that is a cyclic representation of Y.
is a cyclic representation of Y with respect to T,
- (ii)there exist and such that(3.1)
for every comparable , ().
Now, we state and prove the following result.
T is a cyclic weak -contraction,
T is weakly increasing and continuous,
the pair is partial-compatible,
for any , there exists such that and .
Then, T has a unique fixed point , that is, .
For any , there is such that and . Then, following the lines of the proof of Theorem 2.3, it is easy to show that is a Cauchy sequence in the partial metric space , which is complete, so converges to some . On the other hand, by condition (i) of Definition 3.4, it follows that the iterative sequence has an infinite number of terms in for each . Since is complete, from each , , one can extract a subsequence of that converges to y. In virtue of the fact that each , , is closed, we conclude that and thus . Obviously, is closed and complete. Now, consider the restriction of T on , that is which satisfies the assumptions of Theorem 2.3 and thus, has a unique fixed point in , say u, which is obtained by iteration from the starting point . To conclude, we have to show that, for any initial value , we get the same limit point . Due to condition (c) and using the analogous ideas of the proof of Theorem 2.9, it can be obtained that, for any initial value , as . This completes the proof. □
The authors are really thankful to the anonymous referee for his/her precious suggestions useful to improve the quality of the paper. The third author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the forth author would like to thank the Commission on Higher Education and the Thailand Research Fund under Grant MRG no. 5380044 for financial support during the preparation of this manuscript.
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