Best Proximity Point Results in 𝐺-Metric Spaces

Abstract

-metric spaces proved to be a rich source for fixed point theory; however, the best proximity point problem has not been considered in such spaces. The aim of this paper is to introduce certain new classes of proximal contraction mappings and establish the best proximity point theorems for such kind of mappings in -metric spaces. As a consequence of these results, we deduce certain new best proximity and fixed point results. Moreover, we present an example to illustrate the usability of the obtained results.

1. Introduction and Preliminaries

The best approximation results provide an approximate solution to the fixed point equation , when the non-self-mapping has no fixed point. In particular, a well-known best approximation theorem, due to Fan [1], asserts the fact that if is a nonempty compact convex subset of a Hausdorff locally convex topological vector space and is a continuous mapping, then there exists an element satisfying the condition , where is a metric on .

The best proximity point evolves as a generalization of the concept of the best approximation. The best approximation theorem guarantees the existence of an approximate solution; the best proximity point theorem is contemplated for solving the problem to find an approximate solution which is optimal. Given nonempty closed subsets and of , when a non-self-mapping has not a fixed point, it is quite natural to find an element such that is minimum. The best proximity point theorems guarantee the existence of an element such that ; this element is called the best proximity point of . Moreover, if the mapping under consideration is a self-mapping, the best proximity point theorem reduces to a fixed point result. For some results in this direction, we refer to [2–7] and references therein.

On the other hand, Mustafa and Sims introduced the notion of -metric and investigated the topology of such spaces. The authors also characterized some celebrated fixed point results in the context of -metric space. Following this initial paper, a number of authors have published so many fixed point results on the setting of -metric space (see [8–14] and references therein). Samet et al. [15] and Jleli and Samet [16] reported that some published results can be considered a straight consequence of the existence theorem in the setting of usual metric space. More recently, Asadi et al. [17] proved some fixed point theorems in the framework of -metric space that cannot be obtained from the existence results in the context of associated metric space. -metric spaces proved to be rich for fixed point theory but the best proximity problem remains open. In this paper we prove certain best proximity point results and as consequence we deduce some recent fixed point results as corollaries.

First we recollect some necessary definitions and results in this direction. The notion of -metric spaces is defined as follows.

Definition 1 (see [18]). Let be a nonempty set and let be a function satisfying the following properties:(G1), if ,(G2) for all with ,(G3) for all with ,(G4) (symmetry in all three variables),(G5) (rectangle inequality) for all .
Then the function is called a generalized metric or, more specifically, a -metric on , and the pair is called a -metric space.

Note that every -metric on induces a metric on defined by

For a better understanding of the subject we give the following examples of -metrics.

Example 2. Let be a metric space. The function , defined by for all , is a -metric on .

Example 3 (see, e.g., [18]). Let . The function , defined by for all , is a -metric on .

In their initial paper, Mustafa and Sims [18] also defined the basic topological concepts in -metric spaces as follows.

Definition 4 (see [18]). Let be a -metric space and let be a sequence of points of . We say that is -convergent to if that is, for any , there exists such that , for all . We call the limit of the sequence and write or .

Proposition 5 (see [18]). Let be a -metric space. The following are equivalent:(1) is -convergent to ,(2) as ,(3) as ,(4) as .

Definition 6 (see [18]). Let be a -metric space. A sequence is called a -Cauchy sequence if, for any , there exists such that for all ; that is, as .

Proposition 7 (see [18]). Let be a -metric space. Then the following are equivalent:(1)the sequence is -Cauchy,(2)for any , there exists such that , for all .

Definition 8 (see [18]). A -metric space is called -complete if every -Cauchy sequence is -convergent in .

Definition 9. Let be a -metric space. A mapping is said to be continuous if, for any three -convergent sequences , , and converging to , , and , respectively, is -convergent to .

Mustafa [19] extended the well-known Banach contraction principle mapping in the framework of -metric spaces as follows.

Theorem 10 (see [19]). Let be a complete -metric space and let be a mapping satisfying the following condition for all : where . Then has a unique fixed point.

Theorem 11 (see [19]). Let be a complete -metric space and let be a mapping satisfying the following condition for all : where . Then has a unique fixed point.

Remark 12. We notice that condition (5) implies condition (6). The converse is true only if . For details see [19].

Lemma 13 (see [19]). By the rectangle inequality (G5) together with the symmetry (G4), we have

2. Main Results

At first we assume that where if and only if .

Recall that every -metric on induces a metric on defined by

Let be a -metric space. Suppose that and are nonempty subsets of a -metric space . We define the following sets: where .

Definition 14. Let be a -metric space and let and be two nonempty subsets of . Then is said to be approximatively compact with respect to if every sequence in , satisfying the condition for some in , has a convergent subsequence.

Definition 15. Let and be two nonempty subsets of a -metric space . Let be a non-self-mapping. We say is a ---proximal contractive mapping if, for , holds where and .

Theorem 16. Let be two nonempty subsets of a -metric space such that is a complete -metric space, is nonempty, and is approximatively compact with respect to . Assume that is a ---proximal contractive mapping such that . Then has the unique best proximity point; that is, there exists unique such that .

Proof. Since the subset is not empty, we take in . Taking into account, we can find such that . Further, since , it follows that there is an element in such that . Recursively, we obtain a sequence in satisfying This shows that where , , , , and . Therefore from (11) we have which implies . So the sequence is decreasing sequence in and thus it is convergent to . We claim that . Suppose, on the contrary, that . Taking limit as in (14) we get which implies . That is, which is a contrary. Hence, . That is, We will show that is a -Cauchy sequence.
Suppose, on the contrary, that there exists and a sequence of such that with . Further, corresponding to , we can choose in such a way that it is the smallest integer with and satisfying (17). Hence, By Proposition 5(iii) and (G5) we have Letting in (19) we derive that Also, by Proposition 5(iii) and (G5) we obtain the following inequalities: Letting in (21) and applying (20) we find that Again by Proposition 5(iii) and (G5) we have
Taking limit as in (23) and applying (22) we have By (11) with , , , , and we have Taking limit as in the above inequality we have which implies which is a contradiction. Thus, That is, is a Cauchy sequence. Since is a complete -metric space, so there exists such that as . On the other hand, for all , we can write Taking the limit as in the above inequality, we get Since is approximatively compact with respect to , so the sequence has a subsequence that converges to some . Hence, and so . Now, since , there exists such that .
From (11) with , , , , and we have Taking limit as we get Then . That is, . Thus . Therefore has the best proximity point. To prove uniqueness, suppose that , such that and . Now by (65) with and we get which implies ; that is, .

Example 17. Let and be a -metric on . Then . Let and . Define by Also define by and . Clearly, , , , and . Let and . Then . Also, if , then . Therefore, if then Now since so, . Hence, That is, Thus is a ---proximal contractive mapping. All conditions of Theorem 16 hold true and has the unique best proximity point. Here, is the unique best proximity point of .

If in Theorem 16 we take and , where , then we deduce the following corollary.

Corollary 18. Let be two nonempty subsets of a -metric space such that is a complete -metric space, is nonempty, and is approximatively compact with respect to . Assume that is a non-self-mapping such that and, for , holds where . Then has the unique best proximity point. That is, there exists unique such that .

Theorem 19. Let be two nonempty subsets of a -metric space such that is a complete -metric space, is nonempty, and is approximatively compact with respect to . Assume that is a non-self-mapping such that and, for , holds where and . Then has the best proximity point. Moreover, if , then the best proximity point of is unique.

Proof. Following the same lines in the proof of Theorem 16, we can construct a sequence in satisfying From (40) with , , , , , , and we have which implies where . Thus, for all . From we know that with and by Proposition 5(iii) we know that Thus using (44) we obtain Moreover, for all , , we have by rectangle inequality and so . Thus is a Cauchy sequence. Due to the completeness of , there exists such that converges to . As in proof of Theorem 16 we have for some . Again, since , so there exists such that .
From (40) with , , , , , , and we have By taking limit as in the above inequality we get which implies Assume to contrary, . Therefor from we have . Then from (51) we deduce which is a contradiction. Hence, ; that is, . That is, has the best proximity point. To prove uniqueness, suppose that ,  , and . Now by (40) with and we have which implies which is a contradiction. Hence, . That is, has the unique best proximity point.

Theorem 20. Let be two nonempty subsets of a -metric space such that is a complete -metric space, is nonempty, and is approximatively compact with respect to . Assume that is a non-self-mapping such that and, for , holds where and . Then has the best proximity point. Moreover, if , then has the unique best proximity point.

Proof. Following the same lines in the proof of Theorem 16, we can construct a sequence in satisfying From (55) with , , , and we have for all . This implies where . Now, for all , , we have by rectangle inequality which implies . Thus is a Cauchy sequence. Due to the completeness of , there exists such that converges to . Now as in proof of Theorem 16 there exists such that . Now from (55) with , , , and we deduce By taking limit as in the above inequality we get ; that is, . Hence, ; that is, has the best proximity point. To prove uniqueness, assume that , such that and . Now by (55) with and we have which implies which is a contradiction. Hence, .

By taking , in Theorem 20, we obtain the following Corollary.

Corollary 21. Let be two nonempty subsets of a -metric space such that is a complete -metric space, is nonempty, and is approximatively compact with respect to . Assume that is a non-self-mapping such that and, for , holds where . Then has the best proximity point. Moreover, if , then has the unique best proximity point.

3. Application to Fixed Point Theory

In this section, as an application of our best proximity results, we will derive certain new fixed point results.

Note that if and , then , , and . That is, . Therefore, if in Theorem 16 we take , we deduce the following recent result.

Theorem 22 (Theorem 2.3 of [17]). Let be a complete -metric space and let be a mapping satisfying the following condition, for all , where and : Then has a unique fixed point.

Corollary 23. Let be a complete -metric space and let be a mapping satisfying the following condition for all , where : Then has a unique fixed point.

Similarly we can deduce the following fixed point result from Theorem 19.

Theorem 24 (Theorem 2.2 of [17]). Let be a complete -metric space and let be a mapping satisfying the following condition for all , where with : Then has a fixed point. Moreover, if , then has a unique fixed point.

Finally, we can deduce the following fixed point result from Theorem 20.

Theorem 25. Let be a complete -metric space and be a mapping satisfying the following condition for all , where with : Then has a fixed point. Moreover, if , then has a unique fixed point.

By taking in the above theorem we have the following corollary.

Corollary 26 (Theorem 2.1 of [17]). Let be a complete -metric space and let be a mapping satisfying the following condition for all , where : Then has a fixed point. Moreover, if , then has a unique fixed point.

Conflict of Interests

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

Acknowledgment

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first and second authors acknowledge with thanks DSR, KAU, for the financial support.