Abstract

We introduce a type of Geraghty contractions in a -metric space called -proximal generalized Geraghty mappings. By using the triangular--proximal admissible property, we obtain the existence and uniqueness theorem of best proximity coincidence points for these mappings together with some corollaries and illustrative examples. As an application, we give a best proximity coincidence point result in endowed with a binary relation.

1. Introduction and Preliminaries

Let be a map where and are two nonempty subsets of a metric space 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 a best proximity 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 [2–15]). For some recent articles regarding these points, see [16, 17] where Geraghty type mappings were studied and [18] where cyclic and noncyclic nonexpansive mappings were considered.

One of the well-known generalizations of the Banach contraction principle is the result given by Geraghty [19] which enriches the principle by considering the class of mappings such that when By including in the ranges of those mappings , Ayari [20] provided a new result on the existence and uniqueness of the best proximity point for -proximal Geraghty mappings.

The concept of the best proximity coincidence point, which is an extension of a best proximity point problem, was mentioned in [21] (see also [22]) where some results of mappings in generalized metric spaces were presented. A point is called a best proximity coincidence point of the pair , where is a self-map on , if Clearly, if is the identity map, then each best proximity coincidence point of the pair is a best proximity point for

A large number of results concerning these point problems in various metric spaces have been investigated since then. Hussain and the coauthors contributed several interesting results and generalizations in [23–25], including the recent article [26] where best proximity point results for Suzuki-Edelstein proximal contractions were studied. (See also, [27–31] for his work.)

Zhang and Su [32] weakened the -property, called the weak -property, and improved a best proximity point theorem for Geraghty nonself-contractions. In 2018, Komal et al. [33] obtained best proximity coincidence point theorems for -Geraghty contractions in metric spaces by using the weak -property where is an isometry.

The concept of generalized metric spaces (or -metric spaces) was introduced in [34] in 2015. It is a generalization of standard metric spaces covering many topological structures.

Let be a nonempty set, and let be a function. For each , we set

Definition 1 (see [34]). A functionis called a generalized metric onif it satisfies the following conditions.
(D₁)For any , implies .
(D₂)For any , .
(D₃)There exists a constant such that whenever and .

In this case, we say that is a generalized metric space. It is, however, usually called a -metric space.

Remark 2. We note that, in general, results of best proximity points using the weak-property in usual metric spaces might not be attained in the setting of-metric spaces. For example,is not necessarily equal toandmight not converge towhen

Let be a -metric space. We now discuss the convergence and the continuity in these spaces.

Definition 3 (see [34]). Letbe a sequence in. The sequenceis said to-converge toifMoreover,is called a-Cauchy sequence ifFinally,is said to be-complete if each-Cauchy sequence inis a-convergent sequence in.

Proposition 4 (see [34]). For any, if, then.

Definition 5 (see [34]). A functionis said to be-continuous at a pointif for any In addition,is said to be-continuous onif it is-continuous at each point in

The concept of -admissible mapping was introduced by Samet et al. [35] in 2012. The notion of triangular -admissible mappings was then given by Karapinar [36]. Recently, Khemphet [37] extended the concept as follows.

Definition 6 (see [37]). Letbe a generalized metric space, and letandbe self-mappings on. Given thatis a function,is said to be triangular--admissible w.r.t.if, for all, the following conditions hold.(i)If , then and .(ii)If and , then .

In this article, we introduce a type of Geraghty contractions which will be called -proximal generalized Geraghty mappings. These maps are motivated by the work of Khemphet [37]. Using the weak -property in the setting of -metric space, we establish a result on the existence and uniqueness of the best proximity coincidence point for these mappings. Examples showing the validity of the main result and some corollaries are listed. Finally, by applying our main result, we obtain a best proximity coincidence point result in endowed with a binary relation. Note that some other results of best proximity points in endowed with binary relations can be deduced from our result.

2. Main Results

Throughout this article, let be a -metric space, and let and be nonempty disjoint subsets of Also, we require the following notations:

Clearly, if one of and is nonempty, then so is the other.

Definition 7 (see [21]). Letandbe mappings. An elementis said to be a best proximity coincidence point of the pairif. The set of all best proximity coincidence points of the pairis denoted by

Definition 8 (see [32]). Suppose thatis nonempty. The pairis said to have the weak-property if and only ifimplies, whereand

Definition 9. Letandbe mappings. The pairis said to be triangular--proximal admissible if the following conditions hold.(i)If and then .(ii)If and , then , for all .

We consider the class of mappings which is a slight generalization of the well-known class of -valued functions introduced by Geraghty [19]:

Now, we introduce a class of our contractions as follows.

Definition 10. Letandbe mappings. Given thatis a function, the pairis said to be an-proximal generalized Geraghty mapping if the following conditions hold.(i) is triangular--proximal admissible.(ii)There is such that for all , if and , thenwhere .

We first give a useful lemma.

Lemma 11. Letbe a function. Letandbe two mappings such thatis an-proximal generalized Geraghty mapping, and lethave the weak-property. Iffor allthen.

Proof. Let , we have that From the assumption, , and . Since , is triangular--proximal admissible and (6), we have that Also, since and is triangular--proximal admissible, then Similarly, we can show that
Note that Since is an -proximal generalized Geraghty mapping, and has the weak -property, for some . From the property of , we can conclude that . Similarly, we also have that . Then, Since , we have that for some . Thus, which implies that .

Theorem 12. Letand letbe-complete. Given thatis a function, and letandbe mappings such thatis an-proximal generalized Geraghty mapping. Suppose that the following conditions hold.(i) and the pair has the weak -property.(ii)There exist such that and .(iii)For such that for all , there is a subsequence with for all .

Then, there exists such that . Moreover, if for all and is injective, then has a unique best proximity coincidence point.

Proof. From (ii), there exist such that Since , , and is triangular--proximal admissible, there exists such that Continuing in this way, we obtain a sequence such that for all Using the weak -property to (13), for and , we have that If there exists such that , then from (13), Now suppose that for all By the definition of , We will first show that Let . Since is an -proximal generalized Geraghty mapping together with (13) and (14), we obtain that where If , then by (16), Since for all , and thus, By the definition of ,
If , we again have that Since is arbitrary, is nonnegative and nonincreasing. Therefore, converges to . Suppose on the contrary that . From (21), It follows that . Since , we have that which is a contradiction. Thus, must be and that Next, we shall show that is a -Cauchy sequence. Suppose that this is not the case. Then, there exists such that for any , there are subsequences and of satisfying for .
Since is triangular--proximal admissible, it is easy to see that It follows from (13) and (24) that for any Since is an -proximal generalized Geraghty mapping and has the weak -property, we obtain that where If is either or , then, by (23), . This contradicts the assumption that is not -Cauchy. Thus,
As a consequence, By repeating the same steps, it follows that where . Therefore, Let such that Define If , which is impossible. Thus, . Without loss of generality, we may assume that By the definition of , Then, there exists such that Now, which is a contradiction. Therefore, is a -Cauchy sequence.
Since is -complete, there exists such that Equivalently, Since and , it follows that there exists such that By (13) and (iii), there is a subsequence of such that for all . From (13), we have that By the weak -property, (37) and (38), we obtain that
Since and is an -proximal generalized Geraghty mapping, where By (23) and (35), we immediately have that If , by letting in (39), We subsequently have that By the property of , which is a contradiction. It follows that must be equal to , and thus . Therefore, from (37), there exists such that Suppose further that and By Lemma 11, Since is injective, The proof is now completed.