Abstract
The purpose of this paper is to obtain best proximity point theorems for a weakly contractive mapping and a weakly Kannan mapping in partial metric spaces. In this paper, the P-operator technique, which changes a non-self mapping to a self mapping, provides a key method. Many recent results in this area have been improved.
MSC:47H05, 47H09, 47H10.
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1 Introduction and preliminaries
Let us recall some basic definitions of a partial metric space and its properties which can be found in [1].
Definition 1.1 A partial metric on a nonempty set X is a function such that for all :
(p1) ,
(p2) ,
(p3) ,
(p4) .
A partial metric space is a pair such that X is a nonempty set and p is a partial metric on X.
We can see from (p1) and (p2) that implies . However, the converse is not necessarily true. A typical example of this situation is provided by the partial metric space , where the function is defined by for all . Other examples of partial metric spaces which are interesting from a computational point of view may be found in [1] and [2].
Following [1], each partial metric p on X generates a topology on X, whose base is a family of open p-balls:
where for all and . Definitions of convergence, Cauchy sequence, completeness and continuity on partial metric spaces are as follows:
(d1) A sequence in a partial metric space converges to x if and only if .
(d2) A sequence in a partial metric space is called a Cauchy sequence if exists and is finite.
(d3) A partial metric space is called complete if every Cauchy sequence in X converges, with respect to , to a point such that .
(d4) A mapping is said to be continuous at if for every , there exists such that .
It can be easily verified that the function defined by
is a metric on X. The following useful remarks were introduced in [1]:
(r1) If a sequence converges in a partial metric space with respect to , then it converges with respect to . Of course, the converse is not true.
(r2) A sequence in a partial metric space is a Cauchy sequence if and only if it is a Cauchy sequence in the metric space .
(r3) A partial metric space is complete if and only if the metric space is complete.
(r4) Given a sequence in a partial metric space and , we have that
Let A and B be nonempty subsets of a metric space . An operator is said to be contractive if there exists such that for any . The well-known Banach contraction principle says: Let be a complete metric space, and let be a contraction of X into itself; then T has a unique fixed point in X.
In the last fifty years, the Banach contraction principle has been extensively studied and generalized on many settings. One of the generalizations is a weakly contractive mapping.
Definition 1.2 ([3])
Let be a metric space. A mapping is said to be weakly contractive provided that
for all , where the function , holds for every that
The fixed point theorem for a weakly contractive mapping was presented in [3].
Theorem 1.3 Let be a complete metric space. If is a weakly contractive mapping, then f has a unique fixed point and the Picard sequence of iterates converges, for every , to .
One type of contraction which is different from the Banach contraction is Kannan mappings. In [4], Kannan obtained the following fixed point theorem.
Theorem 1.4 ([4])
Let be a complete metric space, and let be a mapping such that
for all and some , then f has a unique fixed point . Moreover, the Picard sequence of iterates converges, for every , to .
In [5], the authors introduced a more general weakly Kannan mapping and obtained its fixed point theorem.
Definition 1.5 ([5])
Let be a metric space. A mapping is said to be weakly Kannan if there exists which satisfies, for every and for all , that
and
Theorem 1.6 ([5])
Let be a complete metric space. If is a weakly Kannan mapping, then f has a unique fixed point and the Picard sequence of iterates converges, for every , to .
Recently, Alghamdi et al. [6] generalized the weakly contractive and weakly Kannan mappings to partial metric spaces and obtained the following fixed point theorems.
Definition 1.7 ([6])
Let be a partial metric space. A mapping is said to be weakly contractive provided that there exists such that for every ,
and for every ,
Definition 1.8 ([6])
Let be a partial metric space. A mapping is said to be weakly Kannan if there exists which satisfies for every and for all that
and
Theorem 1.9 ([6])
Let be a complete partial metric space, and let be a weakly contractive mapping. Then f has a unique fixed point and the Picard sequence of iterates converges, with respect to , for every , to . Moreover, .
Theorem 1.10 ([6])
Let be a complete partial metric space, and let be a weakly Kannan mapping. Then f has a unique fixed point and the Picard sequence of iterates converges, with respect to , for every , to . Moreover, .
In this paper, we first obtain best proximity point theorems for a weakly contractive mapping and a weakly Kannan mapping in partial metric spaces. The P-operator technique, which changes a non-self mapping to a self mapping, provides a key method. Many recent results in this area have been improved.
Before giving the main results, we need the following notations and basic facts.
Let A, B be two nonempty subsets of a complete partial metric space and consider a mapping . The best proximity point problem is whether we can find an element such that , where . Since for any , in fact, the optimal solution to this problem is the one for which the value is attained. Some works on the best proximity point problem can be found in [7–11].
Let A and B be two nonempty subsets of a partial metric space . We denote by and the following sets:
2 Best proximity point theorems in partial metric spaces
Definition 2.1 Let be a pair of nonempty subsets of a partial metric space . A mapping is said to be weakly contractive provided that
for all , where the function , holds for every that
Definition 2.2 Let be a pair of nonempty subsets of a partial metric space . A mapping is said to be weakly Kannan provided that
for all , where the function , holds for every that
We rewrite the P-property in the setting of partial metric spaces as follows.
Definition 2.3 Let be a pair of nonempty subsets of a partial metric space with . Then the pair is said to have the P-property if and only if, for any and ,
Lemma 2.4 Let be a partial metric space, then p is a continuous function, that is, for any , if , , then as .
Proof Since
From the above inequality, we can get that
On the other hand, we have
Then we can obtain
Above all, we can get that
This completes the proof. □
Remark For (r4) we know that, for any , if , , then as .
Theorem 2.5 Let be a pair of nonempty closed subsets of a complete partial metric space such that . Let be a continuous weakly contractive mapping. Suppose that and the pair has the P-property. Then T has a unique best proximity point and the iteration sequence defined by
converges, with respect to , for every , to .
Proof We first prove that is closed with respect to . Let be a sequence such that . It follows from the P-property that
Hence
as , where and , . Then is a Cauchy sequence in , so that converges to a point . By the continuity of a partial metric p, we have , that is, , and hence is closed with respect to .
Let be the closure of in a metric space , we claim that . In fact, if , then there exists a sequence such that . By the continuity of T and the closedness of , we have ; that is, .
Define an operator by . Since the pair has the P-property, we have
for any . This shows that is a weak contraction from a complete partial metric subspace into itself. Using Theorem 1.9, we can get that has a unique fixed point ; that is, , which implies that
Therefore, is the unique one in such that . And the Picard iteration sequence converges, with respect to , for every , to . Since the iteration sequence defined by
is exactly the subsequence of , so that it converges, for every , to . This completes the proof. □
Theorem 2.6 Let be a pair of nonempty closed subsets of a complete partial metric space such that . Let be a continuous weakly Kannan mapping. Suppose that and the pair has the P-property. Then T has a unique best proximity point and the iteration sequence defined by
converges, with respect to , for every , to .
Proof We can prove that is closed and in the same way as in Theorem 2.5. Now define an operator by . Since the pair has the P-property, we have
for any . This shows that is a weakly Kannan mapping from a complete partial metric subspace into itself. Using Theorem 1.10, we can get that has a unique fixed point ; that is, , which implies that
Therefore, is the unique one in such that . And the Picard iteration sequence converges, with respect to , for every , to . Since the iteration sequence defined by
is exactly the subsequence of , so that it converges, for every , to . This completes the proof. □
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This project is supported by the National Natural Science Foundation of China under grant (11071279).
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Zhang, J., Su, Y. Best proximity point theorems for weakly contractive mapping and weakly Kannan mapping in partial metric spaces. Fixed Point Theory Appl 2014, 50 (2014). https://doi.org/10.1186/1687-1812-2014-50
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DOI: https://doi.org/10.1186/1687-1812-2014-50