Abstract
Motivated by experience from computer science, Matthews (1994) introduced a nonzero self-distance called a partial metric. He also extended the Banach contraction principle to the setting of partial metric spaces. In this paper, we show that fixed point theorems on partial metric spaces (including the Matthews fixed point theorem) can be deduced from fixed point theorems on metric spaces. New fixed point theorems on metric spaces are established and analogous results on partial metric spaces are deduced.
MSC:47H10, 54H25.
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1 Introduction and preliminaries
Over the last decades, fixed point theory has been revealed as a very powerful tool in the study of nonlinear phenomena. In particular, fixed point techniques and results have been developed in pure and applied analysis, topology and geometry. A fundamental result of this theory is the well-known Banach contraction principle [1]. In the last fifty years, the Banach contraction principle has been extensively studied and generalized on many settings (see, for example, [2–12]). In the same decades, several authors worked on domain theory in order to equip semantics domain with the notion of distance. In 1994, Matthews [8] introduced the notion of partial metric space as a part of the study of denotational semantics of dataflow networks and showed that the Banach contraction principle can be generalized to the partial metric context for applications in program verification. Later on, many researchers studied fixed point theorems in partial metric spaces. For more details, the reader can refer to [13–22].
First, we start by recalling some basic definitions and properties of partial metric spaces.
Definition 1.1 (see [8])
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.
It is clear that if , then from (p1) and (p2), ; but if , may not be 0. A basic example of a partial metric space is the pair , where for all . Other examples of partial metric spaces which are interesting from a computational point of view may be found in [8]. We observe that 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 . If p is a partial metric on X, then the function given by
is a metric on X.
Definition 1.2 (see [8])
Let be a partial metric space and be a sequence in X. Then
-
(i)
converges to a point if and only if . We may write this as ;
-
(ii)
is called a Cauchy sequence if exists and is finite;
-
(iii)
is said to be complete if every Cauchy sequence in X converges, with respect to , to a point such that .
Lemma 1.1 (see [8])
Let be a partial metric space. Then
-
(a)
is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space ;
-
(b)
A partial metric space is complete if and only if the metric space is complete. Furthermore, if and only if
In 2008, Rus [20] investigated the basic problems of metric fixed point theory in the case of partial metric spaces: existence and uniqueness, the convergence of successive approximations and well posedness of the fixed point problem. Consequently, he presented some interesting open questions: If is a generalized contraction, which condition satisfies T with respect to ? How to use fixed point theorems in a metric space to give analogous fixed point results in a partial metric space ?
In this paper, motivated and inspired by Rus [20], we present new fixed point theorems on complete metric spaces. As an application of our results, we show that the Matthews fixed point theorem on partial metric spaces is a particular case of our first theorem on complete metric spaces. Moreover, we obtain some useful corollaries in the setting of partial metric spaces. In so doing, we provide useful answers to the above open questions.
2 New fixed point theorems on metric spaces
In this section, we present new fixed point theorems on the setting of complete metric spaces. Our first result is the following.
Theorem 2.1 Let be a complete metric space, be a lower semi-continuous function and be a given mapping. Suppose that for any , there exists such that for all ,
Then T has a unique fixed point . Moreover, we have .
Proof Let be an arbitrary point in X. Consider the sequence defined by for all .
Step I. We prove that
If for some n we have , then will be a fixed point of T. So, we can suppose that
Suppose that for some ,
From (4) and (5), we have
From (2), there exists such that
From (4) and (5), we get
which implies that , a contradiction. Thus, our assumption (5) is wrong. We deduce that
This implies that is a non-increasing sequence of positive real numbers. Hence, there is such that
Suppose that . Then for all . From (2), there exists such that
Letting in the above inequality and using (6), we get , which implies that , a contradiction. Thus, and (3) holds.
Step II. We prove that is a Cauchy sequence in .
Suppose that is not a Cauchy sequence. Then there exists for which we can find two sequences of positive integers and such that for all positive integer k,
From (7), we have
Thus, for all k, we have
Letting in the above inequality and using (3), we obtain
On the other hand, we have
and
Thus, we have
Letting in the above inequality, using (3) and (8), we obtain that
On the other hand, for all k, we have
where Φ is the upper bound of (note that from (3), as ). Applying (2), we obtain that there exists such that
for all k. Letting in the above inequality, using (3), (8) and (9), we obtain that , which implies that , a contradiction with . Hence, we deduce that is a Cauchy sequence in . Since is complete, there exists such that as .
Step III. We prove that
From (3), we have as . Since φ is lower semi-continuous, it follows that
which implies (10).
Step IV. We prove that is a fixed point of T. Consider the sets defined by
As , at least one of these subsets is infinite. So, we consider two cases.
Case 1. I is an infinite subset. In this case, we can find a subsequence of such that for all p. Since and as , we have necessarily , that is, is a fixed point of T.
Case 2. J is an infinite subset. In this case, we can find a subsequence of such that for all p. Then, for every p, we can find such that
From (2), we have
Since , for all p we deduce that
Since as (see (3) and (10)), letting in the above inequality, we obtain that
and so as . By the uniqueness of the limit, we have necessarily . Then is a fixed point of T.
Step V. Uniqueness of the fixed point. Suppose that is another fixed point of T, that is, and . Since , we can find such that . From (2), we have
that is,
which implies that , a contradiction with . Then is the unique fixed point of T. This makes end to the proof. □
An immediate consequence of Theorem 2.1 is the following.
Theorem 2.2 Let be a complete metric space, be a lower semi-continuous function and be a given mapping. Suppose that there exists a constant such that for all ,
Then T has a unique fixed point . Moreover, we have .
Remark 2.1 Taking in Theorem 2.2, we obtain the Banach contraction principle.
The following result is a generalization of Kannan’s fixed point theorem.
Theorem 2.3 Let be a complete metric space, be a lower semi-continuous function and be a given mapping. Suppose that there exists a constant such that for all ,
Then T has a unique fixed point . Moreover, we have .
Proof Let be an arbitrary point. Define the sequence in X by for all . Applying (11) with and , we get immediately
Continuing this process, by induction, we obtain that
for all . Note that since , we have , so from inequality (12), the sequence is Cauchy in the metric space . Since is a complete metric space, there exists such that as . On the other hand, from (12), we have
for all . Letting in the above inequality, we get as . Since φ is lower semi-continuous, it follows that
that is, . Now, we prove that is a fixed point of T. Indeed, applying condition (11), we obtain that
for all . Letting , we obtain
which implies (since ) that
that is, .
Suppose now that is another fixed point of T. Applying (11) with , we obtain
which implies (since ) that . Now, applying (11) with and , taking into consideration that , we obtain , which implies that . □
Remark 2.2 Taking in Theorem 2.3, we obtain Kannan’s fixed point theorem [6].
Now, we state and prove a generalization of Reich’s fixed point theorem.
Theorem 2.4 Let be a complete metric space, be a lower semi-continuous function and be a given mapping. Suppose that there exist , with , such that
for all . Then T has a unique fixed point . Moreover, we have .
Proof Let be an arbitrary point. Define the sequence in X by for all . Applying (13) with and , using the triangular inequality, we have
This implies that
where . Continuing this process, by induction, we obtain that
for all . The rest of the proof is similar to the proof of Theorem 2.3. □
Remark 2.3 Taking in Theorem 2.4, we obtain Reich’s fixed point theorem [10].
Now, we give a generalization of Chatterjea’s fixed point theorem.
Theorem 2.5 Let be a complete metric space, be a lower semi-continuous function and be a given mapping. Suppose that there exists a constant such that
for all . Then T has a unique fixed point . Moreover, we have .
Proof Let be an arbitrary point. Define the sequence in X by for all . Applying (14) with and , using the triangular inequality, we have
This implies that
Continuing this process, by induction, we obtain that
for all . The rest of the proof is similar to the proof of Theorem 2.3. □
Remark 2.4 Taking in Theorem 2.5, we obtain Chatterjea’s fixed point theorem [3].
3 From metric to partial metric
As we have said in Section 1, partial metric spaces arose from the need to develop a version of the Banach contraction principle which would work for partially computed sequences as well as totally computed ones [23].
In this section, from our previous obtained results on metric spaces, we show that we can deduce easily various fixed point theorems on partial metric spaces including the Matthews fixed point theorem.
Corollary 3.1 Let be a complete partial metric space and be a given mapping. Suppose that for any , there exists such that for all ,
Then T has a unique fixed point . Moreover, we have .
Proof From (1), for all , we have
Note that since is complete, from Lemma 1.1, is a complete metric space. Consider the function defined by
Then from (15), for any , there exists such that for all ,
Now, T satisfies condition (2) of Theorem 2.1 with . Finally, to apply Theorem 2.1, we have to check the lower semi-continuity of the function φ (with respect to the topology of ). Let be a sequence in X such that , where . Then by Lemma 1.1, we get , that is, . Thus, φ is continuous and therefore the desired result follows immediately from Theorem 2.1. □
Now, the Matthews fixed point theorem follows immediately from Corollary 3.1.
Corollary 3.2 Let be a complete partial metric space and be a given mapping. Suppose that there exists such that for all ,
Then T has a unique fixed point . Moreover, we have .
Now, from our Theorem 2.3, we deduce the following Kannan’s fixed point theorem on partial metric spaces.
Corollary 3.3 Let be a complete partial metric space and be a given mapping. Suppose that there exists a constant such that
for all . Then T has a unique fixed point . Moreover, we have .
Proof Using the same notations as in the proof of Corollary 3.1, one can show easily from (16) that for all , we have
So, the result follows from Theorem 2.3 with and . □
Analogously, from Theorem 2.4, we derive the following Reich’s fixed point theorem on partial metric spaces.
Corollary 3.4 Let be a complete partial metric space and be a given mapping. Suppose that there exist , with , such that
for all . Then T has a unique fixed point . Moreover, we have .
Proof From (17), one can show easily that for all , we have
So, the result follows from Theorem 2.4 with and . □
From Theorem 2.5, we get the following Chatterjea’s fixed point theorem on partial metric spaces.
Corollary 3.5 Let be a complete partial metric space and be a given mapping. Suppose that there exists a constant such that
for all . Then T has a unique fixed point . Moreover, we have .
Proof From (18), one can show easily that for all , we have
So, the result follows from Theorem 2.5 with and . □
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This work is supported by the Research Center, College of Science, King Saud University.
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Samet, B., Vetro, C. & Vetro, F. From metric spaces to partial metric spaces. Fixed Point Theory Appl 2013, 5 (2013). https://doi.org/10.1186/1687-1812-2013-5
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DOI: https://doi.org/10.1186/1687-1812-2013-5