Banach contraction principle for cyclical mappings on partial metric spaces
© Abdeljawad et al.; licensee Springer 2012
Received: 23 April 2012
Accepted: 30 August 2012
Published: 18 September 2012
We prove that the Banacah contraction principle proved by Matthews in 1994 on 0-complete partial metric spaces can be extended to cyclical mappings. However, the generalized contraction principle proved by (Ilić et al. in Appl. Math. Lett. 24:1326-1330, 2011) on complete partial metric spaces can not be extended for cyclical mappings. Some examples are given to illustrate our results. Moreover, our results generalize some of the results obtained by (Kirk et al. in Fixed Point Theory 4(1):79-89, 2003). An Edelstein type theorem is also extended when one of the sets in the cyclic decomposition is 0-compact.
Keywordspartial metric space fixed point cyclic mapping Banach contraction principle 0-compact set
1 Introduction and preliminaries
The Banach contraction mapping principle is considered to be the core 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 [7–14]). 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 a partial metric space. The contraction type condition used there should logically be called a partial contractive condition. In this work, we show that the contraction principle obtained in  can be generalized to cyclical mappings. In contrast, the principle proved in  cannot be extended for a cyclical case. An Edelstein type theorem is also extended when one of the sets in the cyclic decomposition is 0-compact. Some examples are also given to support our claims throughout the article.
(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)
Proof Let be given. Since T is continuous at z, find such that . Since , then , and hence find such that for all . That is for all . Thus and so for all . This shows our claim. □
A sequence is called 0-Cauchy  if . The partial metric space is called 0-complete 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 4 (see )
Let with the partial metric . Then is a 0-complete partial metric space which is not complete.
Theorem 5 ()
There exists a unique such that and .
Let and define .
Theorem 6 ()
the set is nonempty;
there is a unique such that ;
for each , the sequence converges, with respect to the metric , to u.
Definition 7 Let A and B be two nonempty closed subsets of a complete partial metric space such that . A mapping is called a cyclical contraction if it satisfies:
(C1): and .
(C2): There exists , and .
If (C2) in Definition 7 is replaced by the condition
(PC2): there exists and ,
then T is called a partial cyclical contraction. Note that partial cyclical contractions reflect the structure of a partial metric space better. The proof of the following lemma can be easily done by using the partial metric topology.
Lemma 8 A subset A of a partial metric space is closed if and only if whenever satisfies .
Definition 9 A subset A of a partial metric space is called 0-compact if, for any sequence in A, there exists a subsequence and such that .
Clearly, a closed subset of a 0-compact set is 0-compact.
Assume as in a PMS such that . Then for every .
2 Main results
We start this section with a theorem that will motivate us to obtain our main result for cyclic contraction mappings.
Then there exists such that and .
Observe that if the partial metric in Theorem 11 is replaced by a metric, then we conclude that z is a fixed point. The following theorem is an extension of Theorem 1.1 in .
Theorem 12 Let A and B be two nonempty closed subsets of a 0-complete partial metric space such that , and suppose is a cyclical contraction self-mapping of X. Then T has a unique fixed point in .
and this by (P4) implies that the sequence is 0-Cauchy for any . Consequently, converges to some point such that . However, in view of (C2), an infinite number of terms of the sequence lie in A and an infinite number of terms lie in B. Then by Lemma 8, we conclude that , so . Now (C1) and (C2) imply that the map T restricted to is a contraction. Then the result follows by Theorem 5. □
We next give an example showing that the generalization to a partial metric space in Theorem 12 is proper.
Example 13 Let , and . Then and . Provide X with the partial metric if both and otherwise. Then, clearly, is a complete partial metric space. Define by if and . Then it can be easily checked that T is a cyclical contraction with . Notice that the cyclical contractive condition of Theorem 12 is not satisfied when the partial metric p is replaced by the usual absolute value metric.
The following example shows that Theorem 6 cannot be extended for cyclical mappings when the cyclical contraction is replaced by a partial cyclical contraction.
for any , and any . However, .
Observe that the assumption that for all implies that T is well defined. □
Note that in the metric space case, the condition (4) implies that the map T is well defined.
Obviously Theorem 12 can be extended to the following version.
there exists such that , for .
Then T has a unique fixed point.
Proof One only needs to observe that given , infinitely many terms of the Cauchy sequence lie in each . Thus , and the restriction of T to this intersection is a contraction mapping. □
Remark 17 It is our belief that Theorem 12 can be extended to more general cyclical contraction mappings. However, it would be of more interest if the contractive type conditions are considered with control functions.
The following theorem is an extension of an Edelstein type theorem to partial metric spaces.
, for .
Then T has a unique fixed point.
and since and , we have a contradiction. Therefore, we conclude that and . Thus, by the assumption (1), .
This, in turn, implies that . Continuing step-by-step, we conclude that .
Uniqueness follows from the fact that any fixed point of T necessarily lies in by the assumption (1). □
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