We establish common fixed point results for two pairs of weakly compatible mappings on a partial metric space, satisfying a weak contractive condition involving generalized control functions. The presented theorems extend and unify various known fixed point results. Examples are given to show that our results are proper extensions of the known ones.
MSC: 47H10, 54H25, 54H10.
Keywords:partial metric space; generalized altering distance; coincidence point; common fixed point; weakly compatible mappings
In , Matthews introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. He showed that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification.
Subsequently, several authors (see, e.g., Altun and Erduran , Oltra et al., Romaguera and Schellekens , Romaguera and Valero , Rus , Djukić et al., Nashine et al., Di Bari and Vetro , Paesano and Vetro , Shatanawi et al., Shatanawi and Nashine , Aydi et al.) derived fixed point theorems in partial metric spaces.
Altering distance functions (also called control functions) were introduced by Khan et al.. Subsequently, they were used by many authors to obtain fixed point results, including those in partial metric spaces (e.g., Abdeljawad , Abdeljawad et al.[16,17], Altun et al., Ćirić et al., Karapinar and Yüksel ). Generalized altering distance functions with several variables were used on metric spaces by Berinde , Choudhury  and Rao et al..
In this paper, an attempt has been made to derive some common fixed point theorems for two pairs of weakly compatible mappings on partial metric spaces, satisfying a weak contractive condition involving generalized control functions. The presented theorems extend and unify various known fixed point results. Examples are given to show that our results are proper extensions of the known ones.
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 . A sequence in converges to a point , with respect to , if . This will be denoted as , or . If is a partial metric space, and is a mapping, continuous at (in ) then, for each sequence in X, we have
It is easy to see that every closed subset of a complete partial metric space is complete.
1 ψ is continuous;
2 ψ is increasing in each of its variables;
Simple examples of generalized altering distance functions with, say, four variables are:
Recall also the following notions. Let X be a nonempty set and be given self-maps on X. If for some , then x is called a coincidence point of and , and w is called a point of coincidence of and . The pair is said to be weakly compatible if , whenever for some t in X.
3.1 Some auxiliary results
Assertions similar to the following lemma (see, e.g., ) were used (and proved) in the course of proofs of several fixed point results in various papers.
3.2 Main results
(a) IandShave a coincidence point,
(b) JandThave a coincidence point.
Without loss of the generality, we may assume that
which is a contradiction. It follows that and hence . Following similar arguments, we obtain . Thus becomes an eventually constant sequence and is a point of coincidence of I and S, while is a point of coincidence of J and T.
Assume further that (3.5) holds. We claim that
This implies that
By a similar reasoning, we obtain that
Combining (3.8) and (3.9), we obtain
By (3.6), we have
Next, we claim that is a Cauchy sequence in the space (and also in the metric space by Lemma 1). For this it is sufficient to show that is a Cauchy sequence. Suppose that this is not the case. Then, using Lemma 3 we get that there exist and two sequences and of positive integers such that and sequences (3.2) tend to ε when . Applying condition (3.3) to elements and , we get that
Finally, we prove the existence of a common fixed point of the four mappings I, J, S and T.
This implies that
Thus we have
Now we can suppose, without loss of generality, that IX is a closed subset of the partial metric space . From (3.15), there exists such that . We claim that . Suppose, to the contrary, that . By (p4) we get
It follows by (3.15) that
Now, from (3.3)
Therefore, from (3.16) we have
which is a contradiction. Thus we deduce that
which is a contradiction. Then, we deduce that
Again from (3.15) we get that
Now, from (3.3)
Therefore, from (3.20) we have
a contradiction. This implies that
Hence, we have
Now, combining (3.21) and (3.22), we deduce
so z is a common fixed point of the four mappings I, J, S and T.
It is easy to state the corollary of Theorem 1 involving a contraction of integral type.
Corollary 1LetT, S, IandJas well as, satisfy the conditions of Theorem 1, except that condition (3.3) is replaced by the following: there exists a positive Lebesgue integrable functionuonsuch thatfor eachand that
Putting and for , , Theorem 9] is obtained.
It is clear from the proof of Theorem 1 that condition (3.3), resp. (3.23), can be replaced by
where . Hence, Theorem 1 can be considered an extension of , Theorem 2.1] to the frame of partial metric spaces (since semi-compatible mappings are weakly compatible).
Putting and , with , in Theorem 2 (with condition (3.25)), we obtain , Theorem 8]. The same substitution in Theorem 1 (with (3.24)) gives an improvement of , Theorem 12] (since only weak compatibility and not commutativity of the respective mappings is assumed).
Putting and for in Theorem 2 (with condition (3.25)), , Theorem 5] is obtained.
Of course, several known results from the frame of standard metric spaces (see, e.g.,  and ) are also special cases of these theorems. For example, the following corollary can be obtained as a consequence of Theorem 2, which is a generalization and extension of , Corollary 3.2].
Remark 2 However, it is not possible to use in Theorems 1 and 2, as the following example, adapted from , Example 2.3], shows.
holds for all . However, these mappings have no common fixed points; hence, condition (3.23) (or (3.25)) of Theorem 2 cannot be replaced by the respective condition with 5 variables. At the same time, condition (3.25) is not satisfied since, for , , and
This example also shows (as in , Remark 2.4]) the importance of the second generalized altering distance function in Theorems 1 and 2.
The next example shows that Theorems 1 and 2 are proper extensions of the respective results in standard metric spaces.
On the other hand,
Thus, condition (3.25) for does not hold and the existence of a common fixed point of these mappings cannot be derived from , Theorem 2.1].
The authors declare that they have no competing interests.
All authors contributed equally and significantly to writing this paper. All authors read and approved the final manuscript.
The first and second author would like to acknowledge the financial support received from University Kebangsaan Malaysia under the research grant OUP-UKM-FST-2012. The fourth and fifth author are thankful to the Ministry of Science and Technological Development of Serbia.
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