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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 150970, 6 pages

http://dx.doi.org/10.1155/2013/150970

## On Best Proximity Points under the -Property on Partially Ordered Metric Spaces

^{1}Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia^{2}Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey

Received 11 May 2013; Accepted 16 June 2013

Academic Editor: Calogero Vetro

Copyright © 2013 Mohamed Jleli 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

Very recently, Abkar and Gabeleh (2013) observed that some best proximity point results under the -property can be obtained from the same results in fixed-point theory. In this paper, motivated by this mentioned work, we show that the most best proximity point results on a metric space endowed with a partial order (under the -property) can be deduced from existing fixed-point theorems in the literature. We present various model examples to illustrate this point of view.

#### 1. Introduction

Let and be two nonempty subsets of a metric space . Through this paper, we will use the following notations:

*Definition 1. *An element is said to be a best proximity point of the nonself-mapping if and only if it satisfies the condition that

The notion of -property was introduced in [1] as follows.

*Definition 2. *Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the -property if and only if
where and .

Various best proximity point results for different classes of nonself-mappings under the -property were established recently (see, e.g., [1–6] and references therein). Very recently, Abkar and Gabeleh [7] observed that some best proximity point results under the -property can be obtained from the same results in fixed-point theory. The main purpose of this paper is to show that in many cases best proximity point results (under the -property) on partially ordered metric spaces can also be deduced from the corresponding fixed-point theorems. We present various model examples to illustrate this point of view.

#### 2. Preliminaries

As model examples, we will consider some known fixed-point theorems in the framework of partially ordered metric spaces.

At first, we need to introduce two classes of real functions. Denote by the class of all functions satisfying the following conditions: is continuous and nondecreasing, .

We denote by the set of functions satisfying the following conditions: is nondecreasing, for each .

Through this paper, denotes the set of all natural numbers, and .

The following concepts will be useful later.

*Definition 3. *Let be a partially ordered set and be a giving mapping. We say that is nondecreasing (with respect to ) if and only if

*Definition 4. *Let be a partially ordered set and be a sequence in . We say that is nondecreasing (with respect to ) if and only if for all .

*Definition 5. *Let be a partially ordered set and be a metric on . We say that is regular if and only if the following condition holds:

*Definition 6. *Let be a partially ordered set. We say that is directed if and only if

In [8], Harjani and Sadarangani proved the following fixed point result.

Theorem 7. *Let be a partially ordered set and suppose that there exists a metric on such that is a complete metric space. Let be a giving mapping. Suppose that the following conditions hold: *(i)* is continuous or is regular,*(ii)* is nondecreasing,*(iii)*there exists such that ,*(iv)*for all such that , we have
where . **
Then has a fixed point. Moreover, the sequence converges to this fixed point. If, in addition, is directed, we have uniqueness of the fixed point. *

In [9], Agarwal et al. established the following fixed point result.

Theorem 8. *Let be a partially ordered set and suppose that there exists a metric in X such that is a complete metric space. Let be a giving mapping. Suppose that the following conditions hold: *(i)* is continuous or is regular,*(ii)* is nondecreasing,*(iii)*there exists such that ,*(iv)*for all such that , we have
where . **
Then has a fixed point. Moreover, the sequence converges to this fixed point. *

The above result was subsequently extended by Turinici in [10].

Through this paper, is a pair of nonempty subsets of a metric space , where is endowed with a partial order . We denote

*Definition 9. *An element is said to be a best proximity point of the nonself-mapping if and only if

The notion of -property was introduced in [1] as follows.

*Definition 10. *Let be a pair of nonempty subsets of a metric space with . Then the pair is said to have the -property if and only if
where and .

In [11], the authors introduced the concept of proximally increasing mappings.

*Definition 11. *A nonself-mapping is said to be proximally increasing if and only if
where .

*Definition 12. *Let be a pair of nonempty subsets of . A mapping is said to be an isometry if and only if

The following lemmas will be useful later.

Lemma 13. *Let be a pair of nonempty subsets of . Let be a bijective mapping. Then is an isometry if and only if is an isometry. *

*Proof. *Suppose that is an isometry. Let and . Since is an isometry, we have
that is,
Then is an isometry.

Lemma 14. *Consider two nonself-mappings and . Suppose that the following conditions hold: *(i)* is proximally increasing,*(ii)* is bijective,*(iii)* for all . **
Then the mapping is nondecreasing (with respect to ). *

*Proof. *Let such that . We have
Since is proximally increasing, we get that .

Lemma 15. *Consider two nonself-mappings and . Suppose that the following conditions hold: *(i)* is bijective,*(ii)* for all ,*(iii)*there exist such that and ,*(iv)*the pair satisfies the -property. **
Then . *

*Proof. *Using (ii) with , we have . From condition (iii), we have . It follows from the property (condition (iv)) that , which implies that . From (i) and (iii), we obtain that .

Lemma 16. *Let be a pair of nonempty closed subsets of a complete metric space such that is nonempty and has the property. Then is a closed pair of subsets of . *

The proof of Lemma 16 can be found in [5].

Lemma 17. *Let be a pair of nonempty closed subsets of a metric space such that is nonempty. Assume that the pair has the property. Then there exists a bijective isometry such that
*

The proof of Lemma 17 can be found in [7].

#### 3. Discussions and Results

In [11], the authors established the following best proximity point result.

Theorem 18. *Let be a nonempty set such that is a partially ordered set and is a complete metric space. Let be a pair of nonempty closed subsets of the metric space such that . Let be a nonself-mapping. Suppose that the following conditions hold: *(i)* and satisfy the property,*(ii)* is a continuous on ,*(iii)* is proximally increasing,*(iv)*there exist such that
*(v)*for all such that , we have
where . **
Then there exists an element such that
**
Further, the sequence , defined by
**
converges to the element . *

We shall prove that the best proximity point result given by Theorem 18 is a consequence of the fixed point result given by Theorem 7.

##### 3.1. Proof of Theorem 18

Denote by the restriction of the mapping to the subset of . Since , we have . From Lemma 17, there exists a bijective isometry such that Consider the self-mapping . We shall prove that the mapping satisfies all the conditions of Theorem 7.

*Claim 1. * is complete.

From Lemma 16, is a closed subset of the complete metric space . Then is complete.

*Claim 2. * is continuous and nondecreasing mapping.

From Lemma 13, is an isometry, so is continuous on . Since is continuous on is continuous on . Thus, the self-mapping is continuous on . On the other hand, since is proximally increasing, then is also a proximally increasing mapping. Now, our claim follows immediately from Lemma 14.

*Claim 3. *There exists such that .

We have
Using the property, we obtain that , which implies that , that is, . Since , our claim holds with .

*Claim 4. *The mapping satisfies condition (iv) of Theorem 7.

Let such that . Since is an isometry, it follows from condition (v) that
This proves our claim.

Now, the mapping satisfies all the conditions of Theorem 7. We deduce that has a fixed point , and the sequence , defined by converges to .

We claim that is a best proximity point of . Indeed, we have which implies that , that is, . From , we obtain that . Then is a best proximity point of .

Let be the sequence defined by Since from the property, we get that which implies that It follows from (25) and (30) that as . This makes end to the proof of Theorem 18.

*Remark 19. *In Theorem 18, the authors considered only the continuous case. However, from Theorem 7, we can also remove the continuity assumption of and replace it by the regularity assumption of . Moreover, by assuming that is directed, we obtain uniqueness of the best proximity point.

##### 3.2. Additional Result

Using the same techniques, we can obtain a best proximity version of Theorem 8.

Theorem 20. *Let be a nonempty set such that is a partially ordered set and is a complete metric space. Let be a pair of nonempty closed subsets of the metric space such that . Let be a nonself-mapping. Suppose that the following conditions hold: *(i)* and satisfy the property, *(ii)* is continuous on or is regular,*(iii)* is proximally increasing,*(iv)*there exist such that
*(v)*for all such that , we have
where and
**
Then there exists an element such that
**
Further, the sequence , defined by
**
converges to the element . *

* Proof. * We continue to use the same notations as in the proof of Theorem 18. Following the proof of Theorem 18, we have only to check that the self-mapping satisfies condition (iv) of Theorem 8. Indeed, let such that , we have
On the other hand, one has the evaluation
where we denoted
*Estimation of *.

We have
*Estimation of *.

We have
*Estimation of *.

We have

Now, we deduce that
Since is nondecreasing, we get that
Finally, it follows from (36) and (43) that
Thus, we proved that the mapping satisfies condition (iv) of Theorem 8. Applying Theorem 8, we obtain the existence of a fixed point of . The rest of the proof is exactly like that of Theorem 18, so we omit it.

#### Acknowledgment

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project no RGP-VPP-237.

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