Abstract
We study -proximal Geraghty contractions in a -metric space endowed with graph We obtain some best proximity theorems for such contractions. An example and several consequences are given. As a consequence of our results, we also provide the best proximity point results in endowed with a binary relation.
1. Introduction and Preliminaries
Let and be nonempty subsets of a metric space and be a map. It is known that if is a nonself map, the equation does not always have a solution, and it clearly has no solution when and are disjoint. However, it is possible to determine an approximate solution such that the error is Such point is called the best proximity point of In the case that is a self-mapping, the best proximity point is simply a fixed point of . The best proximity point theorem was first studied in [1]. Then, there has been a wide range of research in this framework. Many researchers have studied and generalized the result in many aspects, for example, see [2–22].
In 2011, Raj [23] introduced the notion of -property and subsequently obtained a best proximity point result for a weakly contractive nonself map Best proximity point theorems for subsets of having the -property were also studied in great details in [24–26]. Zhang and Su [27] weakened the -property, called the weak -property, as well as improved the best proximity point theorem for Geraghty nonself contractions, see also [28].
Fixed-point theorems concerning a metric space endowed with graph , which generalizes the Banach contraction principle, were proposed by Jachymski [29]. Klanarong and Suantai [30] recently presented the notion of a -proximal generalized contraction. Several best proximity point results for these mappings were obtained. The concept of generalized metric spaces, also called -metric spaces, was introduced in [31]. It is a generalization of standard metric spaces covering many topological structures. Since then, many researchers have worked with these concepts and gave a large number of results, see [32–36], for example.
In this paper, we introduce a type of contractions called -proximal, Geraghty mappings. These maps are defined on subsets and of a -metric space which is endowed with graph Then, we establish a result on the existence and uniqueness of the best proximity point for these mappings. An example showing the validity of the main result is illustrated, and several corollaries are listed. Finally, by applying our main result, we obtain a best proximity point result in endowed with a symmetric binary relation.
2. Preliminaries and Definitions
Let be a nonempty set, and let be a function. For each , define
In 2015, Jleli and Samet [31] introduced a generalization of metric space as follows.
Definition 1 (see [31]). Let be a nonempty set. A function is said to be a generalized metric on if the following conditions hold:
(D1) For any , if , then
(D2) For any ,
(D3) There exists such that for any where
In this case, we say that is a generalized metric space, also known as a -metric space.
Later, Khemphet [37] modified the condition , which will be denoted by , as follows: “there exists such that for any , , where and .”
Clearly, is stronger than For convenience, when is replaced by , the -metric space will be called a -metric space.
Now, let be a -metric space if not otherwise specified. We are ready to discuss convergence and continuity in these spaces.
Definition 2 (see [31]). Let be a sequence in . The sequence is said to -converge to if Moreover, is called a -Cauchy sequence if Finally, is said to be -complete if each -Cauchy sequence in is a -convergent sequence in .
Any convergent sequence in a -metric space converges to a unique point.
Proposition 3 (see [31]). Let be a sequence in . For any , if , then .
Definition 4 (see [37]). A function is said to be -continuous at a point if for any , In addition, is said to be -continuous on if it is -continuous at each in .
Definition 5. A -metric space is said to be endowed with a graph ; if the set of vertices (denoted by ) is , the set of edges (denoted by ) contains the diagonal of but parallel edges.
We say that is transitive if for all ,
Definition 6. (see [38]). Let be endowed with a graph . A function is said to be -continuous at if for each with for all
Let and be nonempty subsets of We require the following notations:
Definition 7 (see [39]). Let be a mapping. An element is said to be a best proximity point of if . We denote the set of all best proximity points of by
Definition 8 (see [27]). Let be nonempty. Then, the pair is said to have the weak -property if where and .
Definition 9. Let . A mapping is said to be -proximal if and and for all .
3. Main Results
In this section, we assume that is a -metric space (or -metric space when specified) endowed with a transitive graph Let and be nonempty subsets of for which is nonempty.
The class of functions was used as an important tool in [24]. It is clearly a generalization of the well-known class of -valued functions introduced by Geraghty [40].
We now introduce a new type of Geraghty contractions.
Definition 10. A mapping is said to be a -proximal Geraghty mapping if the following hold: (i) is -proximal(ii)For all such that , there exists such that
Lemma 11. Let be a -proximal Geraghty mapping and the pair have the weak -property. Then, for any , (1)(2)If , then
Proof. (1)Let . Then, . Since , we have We can easily check that (2)Let such that . From (6), . By assumptions, and Similarly, and so .
Theorem 12. Let be a -metric space and be -complete. Let be a -proximal Geraghty map. Suppose that the following conditions hold: (i) and has the weak -property(ii)There exist such that , , and (iii) is -continuous and for such that , there exists such that
Then, there exists Moreover, has a unique best proximity point if for all
Proof. From (ii), there exist such that
Since Then, there exists such that and so
Since is -proximal, we finally have that
Continuing this process, we obtain a sequence such that
By using the weak -property with (11), we have that
If there exists such that , then
Now, suppose that for all . We shall prove that is a Cauchy sequence. We first show that .
Since and is a -proximal, Geraghty mapping, then there exists such that
Clearly, is nonincreasing. Thus, . Suppose that and let in (14). Then,
It follows that By the definition of , which is a contradiction. Thus, must be
Now, suppose that is not a Cauchy sequence. Then, there exists such that for all , there are subsequences and of such that for
Since is transitive, for all . Since is a -proximal Geraghty mapping and (12),
Consequently,
For , we have that
Set If , then which is a contradiction. If , without loss of generality, we may assume that Then, we have that
This implies that there exists such that
Thus,
which is a contradiction. Therefore, is a -Cauchy sequence in .
Since is a -complete, there exists such that and so Since is -continuous,
By and (iii), there exist and such that
It follows that Suppose that such that By Lemma 11, The proof is now completed.
Theorem 13. Let be -complete and be a -proximal Geraghty map. Suppose that the following conditions hold: (i) and has the weak -property(ii)There exist such that , , and (iii)For , if for all , then there exists a subsequence with for all
Then, there exists Moreover, has a unique best proximity point if for all
Proof. From the proof of Theorem 12, by using the assumptions (i)-(ii), we obtain a sequence such that
Equivalently, Since and , . It follows that there exists such that
By (11) and (iii), there exists a subsequence of such that for all . Then, from (11),
By the weak -property of , (24) and (25), we have that
Since and is a -proximal Geraghty mapping, we obtain that
Taking in (26),
Therefore, It follows from Proposition 3 that . From (24), there exists such that Finally, if such that By Lemma 11, we have that The proof is now completed.
Example 1. Let be equipped with a -metric given by
Let and We can see that the pair have the weak -property. Also, we have that
Let be a mapping defined by Then,
Define a graph by and Clearly, is transitive and there is such that
We first check that is -proximal. Let such that and
Thus,
Now, suppose that Then, and . Since , we consider the following two cases.
If , then, and It follows that Thus, which is a contradiction.
If , then and so This implies that which is a contradiction.
Therefore, and so is -proximal.
We consider a constant map
Let Then, or .
If , then
If , then
Thus, is a -proximal Geraghty map.
Finally, we will show that the condition (iii) in Theorem 13 holds. Let for some such that for all Then,
If for all , then for all . Assume that there exists an such that . By (35), for all . Suppose that . Then,
This contradicts to the fact that . Thus, and so . Therefore, by Theorem 13.
We next present some consequences from our main results.
Definition 14. A mapping is said to be a -proximal contraction if the following hold: (i) is -proximal(ii)For all such that , there exists such that We immediately have the following corollaries.
Corollary 15. Let be a -metric space and be -complete. Let be a -proximal contraction. Suppose that the following conditions hold: (i) and has the weak -property(ii)There exist such that , , and (iii) is -continuous and for such that , for some
Then, there exists . Moreover, if for all , has a unique best proximity point.
Corollary 16. Let be -complete and be a -proximal contraction. Suppose that the following conditions hold: (i) and has the weak -property(ii)There exist such that , , and (iii)For any sequence in , if for all , then there exists a subsequence with for all
Then, there exists . Moreover, if for all , has a unique best proximity point.
Definition 17. A mapping is said to be a -proximal, R-type mapping if the following hold: (i) is -proximal(ii)For all such that , Applying in Theorems 12 and 13, we obtain two corollaries as follows.
Corollary 18. Let be a -metric space and be -complete. Let be a -proximal, R-type mapping. Suppose that the following conditions hold: (i) and has the weak -property(ii)There exist such that , and (iii) is -continuous and for such that , for some
Then, there exists . Moreover, if for all , has a unique best proximity point.
Corollary 19. Let be -complete and be a -proximal, R-type mapping. Suppose that the following conditions hold: (i) and has the weak -property(ii)There exist such that , , and (iii)If for any sequence in , if for all , then there is a subsequence with for all
Then, there exists . Moreover, if for all , has a unique best proximity point.
4. Application
In this section, we apply our result on best proximity points on a metric space endowed with binary relation. Let and be nonempty subset of a -metric space with a binary relation , and let be a nonself mapping. The mapping is said to be a -proximally comparative if and and for all .
Definition 20. The mapping is said to be -proximally comparative, Geraghty mapping if the following hold: (1) is a -proximally comparative(2)There exists such that for all , if , then
Corollary 21. Let be symmetric and transitive, and let be -complete. Let be a -proximally comparative, Geraghty map. Suppose that the following conditions hold: (i) and has the weak -property(ii)There exist such that , , and (iii)For any sequence in , if for all , then there exists a subsequence with for all
Then, there exists . Moreover, if for all , has a unique best proximity point.
Proof. Define a graph by and Let such that and .
By the definition of , we have that . Since is a -proximally comparative, It follows that Therefore, is a -proximal Geraghty mapping. The condition (ii) implies that there exist such that and . Also, the condition (iii) of Theorem 13 follows from the property of and the condition (iii). By applying Theorem 13, we have that Moreover, if , then which implies that Again, by Theorem 13, .
Note that when is a -metric space, and the condition (iii) in the above corollary is replaced by the condition (iii) in Theorem 12, the result also follows.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors have no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was partially supported by Chiang Mai University and by the Centre of Excellence in Mathematics, CHE, Thailand.