Research Article | Open Access

# A Note on Best Proximity Point Theorems under Weak -Property

**Academic Editor:**Mohamed Boussairi Jleli

#### Abstract

In the very recent paper of Akbar and Gabeleh (2013), by using the notion of -property, it was proved that some late results about the existence and uniqueness of best proximity points can be obtained from the versions of associated existing results in the fixed point theory. Along the same line, in this paper, we prove that these results can be obtained under a weaker condition, namely, weak -property.

#### 1. Introduction and Preliminaries

Let and be two nonempty subsets of a metric space . By and , we denote the following sets: where .

*Definition 1. *Let and be a pair of nonempty subsets of a metric space and let be a mapping. One will say that is a best proximity of if

In [1], the authors introduced the notion of -property as follows.

*Definition 2 (see [1]). *Let be a pair of nonempty subsets of a metric space with . Then, the pair is said to have the -property if
where , and , .

Very recently, Zhang et al. [2] introduced the weak -property.

*Definition 3 (see [2]). *Let be a pair of nonempty subsets of a metric space with . Then, the pair is said to have the -property if
where , and , .

It is evident that if a pair has the -property, then it has the weak -property. In [2], the authors established some examples to prove that the converse of the statement above is false.

In the literature, a great number of fixed point theorems have appeared to generalize, extend, and improve the celebrated Banach's contraction principle. Among them, we present two of these results as examples and we prove that their versions in the context of the best proximity point theory can be deduced as consequences of results in the setting of fixed point theory.

Let denote the set of all functions which satisfyis continuous, and nondecreasing; is positive on and .

Theorem 4 (see [3]). *Let be a complete metric space and an operator satisfying
**
where which satisfy . Then, has a unique fixed point.*

*Remark 5. *Notice that Theorem 4 remains valid if we remove the assumption that .

*Definition 6. *Let denote the family of all functions which satisfy for each , where is the th iterate of . is nondecreasing.

*Remark 7. *A function, , is known as a comparison function or Bianchini-Grandolfi gauge function in the literature [4]. Moreover, it is easily seen that such functions satisfy the conditions

In 2012, Romaguera [5] proved the analog of the following fixed point theorem in the context of partial metric spaces. It is evident that its metric version remains true.

Theorem 8 (see [5]). *Let be a complete metric space and an operator satisfying
**
where and
**
Then, has a unique fixed point.*

#### 2. Main Results

We start this section with the following lemma which appeared implicitly in [2]. We prove this lemma for that the paper is self-contained.

Lemma 9. *Let be a pair of nonempty closed subsets of a complete metric space . Suppose that the following conditions hold: *(i)*;*(ii)*the pair has the weak -property.** Then, the set is closed.*

*Proof. *We have due to the assumption that . Let be a sequence such that . We shall prove that . In fact, since , we find a sequence such that
By using the weak -property, we infer that
As is a Cauchy sequence in ; from the inequality (10), we derive that the sequence is a Cauchy sequence in . Therefore, since is closed, we deduce that for certain .

Finally, by the continuity of the metric together with (9), it gives us . Hence, .

Lemma 10. *Let be a pair of nonempty closed subsets of a complete metric space . Suppose that the following conditions hold: *(i)*;*(ii)*the pair has the weak -property;*(iii)* is a continuous mapping with .** Then, we have .*

*Proof. *It is sufficient to prove that if , then . In fact, since , we can find a sequence such that . Due to (iii), we have . Since the mapping is continuous and is closed by Lemma 9, we conclude that .

Before stating the main result of this paper, we need to recall the main result of Sankar Raj [1].

Theorem 11. *Let be a pair of nonempty closed subsets of a complete metric space with . Let be a mapping satisfying
**
where . Suppose also that the pair has the -property and . Then, has a unique best proximity point.*

Now, we present the main results of the paper. If in Theorem 11 we replace -property by the weak -property, we derive the following result.

Theorem 12. *Theorem 11 under the the assumption of the weak -property instead of the -property is a consequence of Theorem 4.*

*Proof. *Suppose that all the conditions of Theorem 11 are fulfilled under the the assumption of the weak -property instead of the -property. We first note that is continuous, since
for any . Hence, by Lemma 10, we have .

Next, we define an operator by such that . As , we can find such that . Moreover, is a well-defined mapping. Suppose, on the contrary, that there exists another such that . By using the fact that the pair has the weak -property, we derive that
and, consequently, .

Now, we consider the operator . In the sequel, we shall prove that the operator satisfies all assumptions of Theorem 4. Notice that is a complete metric space since is closed.

On the other hand, we have
for . As the pair has the weak -property, we infer from (14) that
Regarding (11), we derive that
for any , .

Therefore, by Theorem 4, there exists a unique fixed point of , that is, . By definition, we have
Therefore, is a best proximity point of the mapping and this completes the existence part of the proof.

We shall show that is the unique best proximity of the mapping . Suppose, on the contrary, that is another best proximity of the mapping . Consequently, we have
and also . Moreover, taking into account the definition of the operator , this means that
Hence, is a fixed point of . By the uniqueness of the fixed point of , we deduce that , which completes the proof.

Theorem 4 imposes Theorem 11 because we have used a weaker condition, namely, the notion of the weak -property.

Theorem 13. *Let be a pair of nonempty closed subsets of a complete metric space with . Let be a continuous mapping satisfying
**
where and
**
Suppose also that the pair has the weak -property and . Then, has a unique best proximity point.*

By using the same techniques used in Theorem 12, we derive Theorem 13.

Theorem 14. *Theorem 13 is a consequence of Theorem 8.*

To avoid the repetition, we omit the proof Theorem 14.

*Remark 15. *Notice that in Theorem 14, we assume that is continuous since the contractive condition (20) appeared in Theorem 13 does not imply the continuity of .

Corollary 16. *Let be a pair of nonempty closed subsets of a complete metric space with . Let be a continuous mapping satisfying
**
where and
**
Suppose also that the pair has the weak -property and . Then, has a unique best proximity point.*

We skip the proof since the mapping is nondecreasing and hence we get Therefore, Corollary 16 is a consequence of Theorem 13.

*Remark 17. *Corollary 16 can be considered as a partial version of Theorem 20 of [6].

#### Conflict of Interests

The authors declare that they have no conflict of interests.

#### Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

#### Acknowledgment

The authors are grateful to the reviewers for their careful reviews and useful comments.

#### References

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#### Copyright

Copyright © 2014 Angel Almeida 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.