Abstract and Applied Analysis
Volume 2014 (2014), Article ID 979170, 7 pages
http://dx.doi.org/10.1155/2014/979170

Research Article

A Note on Best Approximation in 0-Complete Partial Metric Spaces

Marta Demma,¹ Mohamed Jleli,² Bessem Samet,² and Calogero Vetro¹

¹Università degli Studi di Palermo, Dipartimento di Matematica e Informatica, Via Archirafi 34, 90123 Palermo, Italy
²Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received 23 May 2014; Accepted 28 August 2014; Published 23 October 2014

Academic Editor: Erdal Karapınar

Copyright © 2014 Marta Demma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the existence and uniqueness of best proximity points in the setting of 0-complete partial metric spaces. We get our results by showing that the generalizations, which we have to consider, are obtained from the corresponding results in metric spaces. We introduce some new concepts and consider significant theorems to support this fact.

1. Introduction

Best approximation theory, in general, and best proximity point theory, in particular, have received a great attention in the last decades and have significant applications in convex optimization, differential inclusions, and optimal controls. We can say that these theories strongly relate nonlinear functional analysis with theory of functions and topologic studies.

Let be a metric space and let and be nonempty subsets of . A mapping is said to be cyclic if and ; see [1, 2]. Also, a point is called a best proximity point if , where . In light of these concepts, Eldred and Veeramani [3] proved the following existence theorem (see preliminaries for basic notions in this theorem); see also [4–7].

Theorem 1 (see [3]). Let and be nonempty closed convex subsets of a uniformly convex Banach space and let be a cyclic mapping on . Suppose that there exists such that

for all and . Then has a unique best proximity point in and converges to for each .

On the other hand, the concept of metric space has been generalized in many directions. In this paper, we consider the concept of partial metric introduced by Matthews [8] as a part of the study of denotational semantics. The reader is referred to Bukatin et al. [9] for more details and motivation in introducing the new context, which is also leading to interesting research in foundations of topology; see, for instance, [10–18]. In particular, Romaguera [19] introduced the notions of 0-Cauchy sequence and 0-complete partial metric spaces and proved an interesting characterization of partial metric spaces in terms of 0-completeness. For other results on this specific topic, we refer to [19–21].

Also, we point out that some recent results [22–24] showed that a lot of fixed point theorems in partial metric spaces can be directly reduced to their known metric counterparts.

In this paper, we study the existence and uniqueness of best proximity points in the setting of -complete partial metric spaces. We get our results by showing that the generalizations, which we have to consider, are obtained from the corresponding results in metric spaces. We introduce some new concepts and consider significant theorems to support this fact. The overall motivation of this work is in underlining the strong relation between standard metric spaces and their generalizations to better target the research on this topic. Using the approach described in this paper, the reader will succeed in obtaining the extensions, to partial metric setting, of many recent results in the literature on best approximation theory.

2. Preliminaries

We collect some notions and notations needed in the sequel. Let be the set of all nonnegative real numbers, the set of rational numbers, and the set of all positive integers.

2.1. Best Approximation

We recall the notion of the property UC and some basic properties of this notion.

Let be a Banach space. Then is said to be uniformly convex if for every there exists some such that the conditions , and imply that . On this basis, Suzuki et al. [25] introduced the notion of the property UC and extended Theorem 1 to metric spaces with the property UC.

Definition 2 (see [25]). Let and be nonempty subsets of a metric space . Then is said to satisfy the property UC if the following holds.

 If and are sequences in and is a sequence in such that

then

To clarify the value of the property UC, we give some examples.

Example 3 (see [26]). Let and be nonempty subsets of a uniformly convex Banach space . Assume that is convex. Then has the property UC.

Example 4 (see [25]). Let and be nonempty subsets of a metric space such that . Then satisfies the property UC.

Example 5 (see [25]). Let , , , and be nonempty subsets of a metric space such that , , and . If satisfies the property UC, then satisfies the property UC.

Theorem 6 (see [25]). Let be a metric space and let and be nonempty subsets of such that satisfies the property UC. Assume that is complete and let be a cyclic mapping on . Assume that there exists such that

for all and . Then the following hold:

(i) has a unique best proximity point in ;(ii) is a unique fixed point of   in ;(iii) converges to for every ;(iv) has at least one best proximity point in ;(v)if satisfies the property UC, then is a unique best proximity point in and converges to for every .

2.2. Partial Metric Spaces

We give the definitions and some characterizations of partial metric and partial metric space.

Definition 7 (see [8]). A partial metric on a nonempty set is a mapping such that, for all , the following conditions are satisfied:

(p1);(p2);(p3);(p4).

A nonempty set equipped with a partial metric is called partial metric space; we denote this space by .

Notice that if , then (p1) and (p2) imply , but the converse does not hold true in general.

Also, each partial metric on generates a topology on which has as a base the family of the open balls (-balls) , where

for all and .

Definition 8 (see [8, 27]). Let be a partial metric space. Then a sequence is called

(i)convergent, with respect to , if there exists some in such that ;(ii)Cauchy sequence if there exists (and is finite) .

A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to , to a point such that .

As shown in [19], one can introduce a weaker form of completeness in partial metric spaces. Precisely, we recall the following statements:

(i)a sequence in is called -Cauchy if ;(ii) is -complete if every -Cauchy sequence in converges, with respect to the partial metric , to a point such that ;(iii)if is complete, then is -complete, but the converse does not hold.

Example 9 (see [19]). The partial metric space , where for all , is a -complete partial metric space which is not complete.

3. Fixed Point Results

Hitzler and Seda in [28] proved a useful tool: Proposition 10. Later on, Haghi et al. [22] used the same argument to show that many fixed point generalizations to partial metric spaces can be obtained from the corresponding results in metric spaces. Also, they considered some significant cases to support their work. Here we recall, without giving the proof, the key result of the above authors; the interested reader is referred to [22, 28] for more details.

Proposition 10 (see [22, 28]). Let be a partial metric space. Then the function given by

is a metric on such that is complete if and only if is -complete.

Inspired by this fact, we introduce two new notions in partial metric spaces and use these notions to extend the approach in [22] to best proximity point results.

Definition 11. Let be a partial metric space and let be a nonempty subset of . Then, is said to be a -closed subset of if, for each sequence in converging to a point with , we have .

Example 12. In the partial metric space , where for all , any real line interval containing zero is -closed. Also the interval , for every , is -closed. We note that is -closed but it is not closed.

Proposition 13. Let be a partial metric space and let be a nonempty -closed subset of . Then is a closed subset of the metric space , where is the metric given in Proposition 10.

Proof. Let be any nonempty -closed subset of . Assume that a sequence in converges, with respect to the metric , to a point ; that is whenever . Also, we assume that for all . This implies that for all . The triangular inequality gives us

and hence, for , we deduce that . It follows that the sequence converges to in . Now, since is -closed, we have and so is a closed subset of .

In view of the results in [22], we recall some known results in partial metric spaces and relate these results to their metric counterparts. Obviously we start with the partial metric version of the Banach-Caccioppoli theorem, due to Matthews [8]. However, for convenience, we formulate this result in the setting of -complete partial metric spaces.

Theorem 14 (see [19]). Let be a -complete partial metric space and let be a mapping. If there exists a real number such that

for all , then there exists a unique point such that . Moreover, .

As known, the above statement affirms that the fixed point of any contraction (i.e., self-mapping satisfying condition (8)) has zero self-distance.

Then, we give the following short proof of the above theorem.

Proof. Let be the metric defined in Proposition 10. Since satisfies condition (8), then satisfies the following condition:

Indeed, if with , then ; on the contrary, if the contraction condition is trivially satisfied. Thus, is a contraction in the metric space . Now, the metric space is complete in view of Proposition 10, since is a -complete partial metric space. Consequently, the existence and uniqueness of a point such that follows by application of the Banach-Caccioppoli fixed point theorem in metric space. Finally, from (8), by choosing , we deduce that .

For our scope, we need to introduce a new notion of -continuity.

Definition 15. Let be a partial metric space and let be a mapping. We say that is -continuous in with if and for each sequence converging to we have

Also, is -continuous mapping in if is -continuous in each with .

Remark 16. Any contraction in Banach-Caccioppoli sense, in a partial metric space, is a -continuous mapping.

Now, we give an example of a -continuous mapping that is not continuous.

Example 17. Let be a partial metric space, where for all . Let be the mapping defined by

We note that is a contraction with and hence is a -continuous mapping. On the other hand, is not continuous, for instance, in . In fact, for every , let ; then we have as , but

Now, we recall that is said to be a Banach operator if there exists a real number such that , for all ([29], Definition 4). It is well known that every continuous Banach operator on a complete metric space has a fixed point (see, for instance, [29], Corollary 2). Then, we give the following existence result in partial metric spaces.

Theorem 18. Let be a -complete partial metric space and let be a -continuous mapping such that

for all , where . Then has a fixed point in .

Proof. Let be the metric given in Proposition 10. Since satisfies condition (13), then is a Banach operator; that is

Indeed, if , then . Again, if , the above contractive condition is trivially satisfied. This implies that, for each starting point , the sequence is Cauchy in and hence -Cauchy in . Thus, there exists such that

Also, since is -continuous, we deduce that

It follows that is a fixed point of ; that is .

Next, we show that this theorem can be viewed as a direct consequence of its metric counterpart. First, we need the following auxiliary result.

Proposition 19. Let be a partial metric space and let be a -continuous mapping. Then is continuous with respect to the metric given in Proposition 10; that is is continuous.

Proof. We assume that is -continuous mapping and show that is continuous with respect to the metric . Fix and consider a sequence converging to with respect to . If for all , then and so for all . On the contrary, assume that for all (the same holds if for infinite many values of ). Therefore, from we deduce that

Since is -continuous, then

and so we conclude that is continuous.

We give the following short proof of Theorem 18.

Proof. Since in Theorem 18 is -continuous, in virtue of Proposition 19, we have that is continuous with respect to the metric given in Proposition 10. Thus, we can apply Corollary 2 of [29] to conclude.

Briefly, by using the same technique, we show that also the fixed point results for cyclic mappings in metric spaces can be translated in partial metric spaces. Firstly, we recall the following definition.

Definition 20. Let be a partial metric space and let and be two nonempty subsets of . A mapping is called cyclic mapping if and . Also is a cyclic contraction if there exists a real number such that

Now, we state and prove the following.

Theorem 21. Let be a -complete partial metric space and let and be two nonempty -closed subsets of . If is a cyclic contraction, then has a unique fixed point in .

Proof. Let be the metric given in Proposition 10. In virtue of Proposition 13, and are closed subsets of the metric space , since and are -closed sets. Obviously, since satisfies condition (19), then satisfies also the following condition:

Indeed, if and , then . On the other hand, for , the above condition is trivially true. We obtain that is a cyclic contraction with respect to the metric . Therefore, from Theorem 1.1 of [2] (in view of the comments in the introduction section of [2]) readily follows the existence and uniqueness of a point such that . Also, from (19) for , we get .

4. Best Proximity Point Results

In the light of extending the technique in the previous section to obtain best proximity point results in partial metric space, we recall some notions and notations.

Definition 22. Let be a partial metric space and let and be two nonempty subsets of . For all , put

If is the metric related to the partial metric in Proposition 10, then if and if .

Then we adapt the property UC in partial metric spaces.

Definition 23. Let and be nonempty subsets of a partial metric space . Then is said to satisfy the property if the following holds.

 If and are sequences in and is a sequence in such that

then

Following the approach in [22], we present an auxiliary lemma; see Lemma 2.2 of [22].

Lemma 24. Let be a partial metric space and let and be two nonempty subsets of . Assume that is a cyclic mapping and is the metric given in Proposition 10. Then, for all and with , we have , where

Proof. Let and ; if , and , then

Also, if , and , in view of the fact that , we deduce

Analogous reasoning shows that in the case , and , and in the case , and . This concludes the proof. However, notice that for all we have

This implies that

for all .

A direct consequence of Lemma 24 is the following.

If and are nonempty and disjoint subsets of a partial metric space , then

for all ,  , where .

We are ready to state and prove the following best approximation result in partial metric spaces.

Theorem 25. Let be a partial metric space and let and be nonempty and disjoint subsets of such that satisfies the property . Assume that is -complete and is a cyclic mapping on . Also suppose that there exists a real number such that

for all and . Then the following hold:

(i) has a unique best proximity point in ;(ii) is a unique fixed point of in ;(iii) converges to for every ;(iv) has at least one best proximity point in ;(v)if satisfies the property , then is a unique best proximity point in and converges to for every .

Proof. Let be the metric given in Proposition 10. Since has the property in and , then has the property UC in . Also, since is -complete, then is complete in . It follows that all the hypotheses of Theorem 6 are satisfied with respect to the metric and consequently also the assertions (i)–(v) in Theorem 25 hold true.

Recently, Caballero et al. [30] proved the following theorem for noncyclic mappings.

Theorem 26 (see [30]). Let be a complete metric space and let and be nonempty closed subsets of such that and

where and . Let be a mapping such that and

for all and some function satisfying the condition: implies . Then has a unique best proximity point .

From the above theorem we are ready to derive the partial metric counterpart.

Theorem 27. Let be a -complete partial metric space and let and be nonempty and disjoint -closed subsets of such that and

where and . Let be a mapping such that and

for all and some function satisfying the condition: implies . Then has a unique best proximity point .

Proof. Let be the metric given in Proposition 10 and hence, since is -complete, is complete. In virtue of Proposition 13, and are closed subsets of the metric space , since and are -closed sets. Also, since , then (31) and (32) hold true. It follows that all the hypotheses of Theorem 26 are satisfied with respect to the metric and consequently also Theorem 27 holds true.

Very recently, Gabeleh and Shahzad [31] proved the following theorem for a pair of mappings.

Theorem 28 (see [31]). Let be a complete metric space, let and be nonempty closed subsets of with , as in Theorem 26 and let be defined by . Assume that and are two mappings satisfying the following conditions:

(i)there exists such that implies for all ;(ii) and ;(iii) is an isometry; that is for all .

If the pair satisfies the condition

where and , then there exists a unique such that . Moreover, for any , the iterative sequence , defined by the algorithm , converges to .

Also from this theorem we are ready to derive the partial metric counterpart.

Theorem 29. Let be a -complete partial metric space, let and be nonempty and disjoint -closed subsets of with , as in Theorem 27, and let be defined by . Assume that and are two mappings satisfying the following conditions:

(i)there exists such that implies for all ;(ii) and ;(iii) is an injective function such that for all .

If the pair satisfies the condition

where and , then there exists a unique such that . Moreover, for any , the iterative sequence , defined by the algorithm , converges to .

Proof. Let be the metric given in Proposition 10; since is -complete, then is complete. In virtue of Proposition 13, and are closed subsets of the metric space , since and are -closed sets. Also, since , then conditions (i) and (iii) of Theorem 28 and (35) hold true. It follows that all the hypotheses of Theorem 28 are satisfied with respect to the metric and consequently also Theorem 29 holds true.

5. Conclusion

The development of best approximation theory is an actual and relevant topic in solving various minimization problems. In particular, the constructive proofs of the existence and uniqueness of best proximity points consent to obtain algorithms for approaching these problems. On the other hand, over the years, an interesting matter was to introduce generalized notions of distance to suit better specific problems (i.e., here we consider partial metrics used in Computer Science). Then, the approach described in this paper suggests how to obtain theoretical results in generalized metric spaces from the corresponding results in standard metric spaces. As shown above, the approach has several advantages, such as the ability to define and use new notions for simplifying proofs and calculations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This project was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.

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