Coupled fixed point theorems for partially contractive mappings
© Abdeljawad; licensee Springer 2012
Received: 14 May 2012
Accepted: 24 August 2012
Published: 12 September 2012
Recently, some authors have started to generalize fixed point theorems for contractive mappings in a class of generalized metric spaces in which the self-distance need not be zero. These spaces, partial metric spaces, were first introduced by Matthews in 1994. The proved fixed point theorems have been obtained for mappings satisfying contraction type conditions empty of the self-distance. In this article, we prove some coupled fixed point theorems for mappings satisfying contractive conditions allowing the appearance of self-distance terms. These partially contractive mappings do reflect the structure of the partial metric space, and hence their coupled fixed theorems generalize the previously obtained by (Aydi in Int. J. Math. Sci. 2011:Article ID 647091, 2011). Some examples are given to support our claims.
1 Introduction and preliminaries
The Banach contraction mapping principle is considered to be the soul of many extended fixed point theorems. It has widespread applications in many branches of mathematics, engineering and computer science. During the last decades many authors were able to generalize this principle [1–4]. After the appearance of partial metric spaces as a place for distinct research work into flow analysis, non-symmetric topology and domain theory [5, 6], many authors started to generalize this principle to these spaces (see [7–16]. However, the contraction type conditions used in those generalizations do not reflect the structure of a partial metric space apparently. Later, the authors in  proved a more reasonable contraction principle in the partial metric space in which they used self-distance terms. On the other hand, the theory of coupled fixed point theorems has recently attracted some authors (see [18–20]). Also, Meir-Keeler type common and tripled fixed point theorems have been recently considered over partial metric spaces [21, 22]. In this article, we prove a coupled partial contraction principle generalizing the recently published coupled fixed point theorems in . An example is presented to show that our coupled partial contraction principle is worthy of investigation.
(P2) If then (equality)
(P3) (small self-distances)
for all .
is a (usual) metric on X. Each partial metric p on X generates a topology on X with a base of the family of open p-balls , where for all and .
A sequence in a PMS converges to if and only if ;
A sequence in a PMS is called Cauchy if and only if exists (and is finite);
A PMS is said to be complete if every Cauchy sequence in X converges, with respect to , to a point such that ;
A mapping is said to be continuous at , if for every , there exists such that .
A sequence is Cauchy in a PMS if and only if is Cauchy in a metric space ;
- (B)A PMS is complete if and only if the metric space is complete. Moreover,(2)
A sequence is called 0-Cauchy  if . The partial metric space is called 0-complete [17, 23] if every 0-Cauchy sequence in x converges to a point with respect to p and . Clearly, every complete partial metric space is 0-complete. The converse need not be true.
Example 3 (see )
Let with the partial metric . Then is a 0-complete partial metric space which is not complete.
Let and define .
The following theorem was presented in .
the set is nonempty;
there is a unique such that ;
for each the sequence converges to u with respect to the metric .
Example 5 Let and provide X with the partial metric if both and otherwise. Then clearly is a complete partial metric space, and .
Aydi  proved the following coupled fixed point theorems in partial metric spaces:
where with . Then F has a unique coupled fixed point. That is, there exists unique such that and .
where with . Then F has a unique coupled fixed point.
2 Main results
- (b)there exists a unique such that(5)
- (c)for each the sequence defined by
We divide the proof of the claim (a) into the following steps:
where . On the other hand, (P3) implies and so (7) is obtained.
Substituting each of these cases in (12), letting and using (9) together with (11), we arrive at (10).
Step V: We prove that .
In particular and so .
From which it follows either and so implies that , or , and hence (P2) applied to implies . □
Remark 10 Although Theorem 9 does not imply the uniqueness of the fixed point, it is easy to see that, under the assumptions made, if and are both coupled fixed points for F satisfying , then . If the partially contractive condition (3) is replaced by the somewhat stronger condition below, the uniqueness of the coupled fixed point is guaranteed.
where and . From (23) we distinguish the following cases:
Case 1: If , then and so .
Case 2: If , then and so . □
As a corollary, we obtain the already mentioned result  stated in Theorem 7. As well, let us remark that the result of Aydi in Theorem 7 is valid also for 0-complete partial metric spaces.
where with . Then F has a unique coupled fixed point such that . Also for each the sequence converges to with respect to the metric .
Proof The condition (24) implies that , which, in turn, by (9) implies . But then by (10) we have so the -limit of the sequence is actually the unique coupled fixed point . □
- (a)There are no with such that , for all . That is, F does not verify the assumptions of Theorem 7. Indeed, if we assume that there exist with such that for all , we have
- (b)There are no with such that . That is, F does not verify the assumptions of Theorem 8. Indeed, if we assume that there exist with such that for all , we have(27)
F is partially contractive. That is F verifies the assumptions of Theorem 9. Hence, it has the coupled fixed point . The condition (3) is clearly satisfied since for all .
Note that and hence the coupled Picard sequence converges for and for may converge or not depending on the particular choice of .
If F is replaced by , then it is strong partially contractive, and hence by Theorem 11, we guarantee the uniqueness of the coupled fixed point .
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