Some new common fixed point results through generalized altering distances on partial metric spaces
© Ahmad et al.; licensee Springer 2012
Received: 21 March 2012
Accepted: 4 July 2012
Published: 23 July 2012
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.
Keywordspartial 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.
Definition 1 A partial metric on a nonempty set X is a function such that for all :
The pair is called a partial metric space.
It is clear that, if , then from (p1) and (p2), it follows that . But may not be 0.
Clearly, a limit of a sequence in a partial metric space need not be unique. Moreover, the function need not be continuous in the sense that and implies .
A sequence in is called a Cauchy sequence if exists (and is finite).
The space is said to be complete if every Cauchy sequence in X converges, with respect to , to a point such that .
It is easy to see that every closed subset of a complete partial metric space is complete.
is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space .
The space is complete if and only if the metric space is complete.
ψ is continuous;
ψ is increasing in each of its variables;
if and only if .
The set of generalized altering distance functions with n variables will be denoted by . If , we will write (obviously, this function belongs to ).
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.
As a corollary (putting for a partial metric p), we obtain
3.2 Main results
one of the ranges IX, JX, TX and SX is a closed subset of .
I and S have a coincidence point,
J and T have a coincidence point.
Moreover, if the pairs and are weakly compatible, then I, J, T and S have a unique common fixed point.
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.
which implies that , and thus . Hence, (3.7) is proved.
which implies that , that is a contradiction since . We deduce that is a Cauchy sequence.
Finally, we prove the existence of a common fixed point of the four mappings I, J, S and T.
We get that , so u is a coincidence point of I and S.
We get that , so v is a coincidence point of J and S.
so z is a common fixed point of the four mappings I, J, S and T.
a contradiction. Hence , that is, . We conclude that S, T, I and J have only one common fixed point in X. The proof is complete. □
It is easy to state the corollary of Theorem 1 involving a contraction of integral type.
for all . Then, S, T, I and J have a unique common fixed point.
If in Theorem 1 is the identity mapping on X, then we have the following consequence:
where and are altering distance functions, and . Then T and S have a unique common fixed point.
Remark 1 Several corollaries of Theorems 1 and 2 could be derived for particular choices of and . We state some of them.
Putting and for , , Theorem 9] is obtained.
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].
where m and n are positive integers, and are altering distance functions, and . Then T and S have a unique common fixed point.
Remark 2 However, it is not possible to use in Theorems 1 and 2, as the following example, adapted from , Example 2.3], shows.
whatever is chosen.
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.
Hence, condition (3.25) is satisfied, as well as other conditions of Theorem 2. Mappings have a common fixed point .
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 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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