Best Proximity Coincidence Point Results for ðα,DÞ-Proximal Generalized Geraghty Mappings in JS-Metric Spaces

We introduce a type of Geraghty contractions in a JS-metric space X, called ðα,DÞ-proximal generalized Geraghty mappings. By using the triangular-ðα,DÞ-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 X endowed with a binary relation.


Introduction and Preliminaries
Let T : A → B be a map where A and B are two nonempty subsets of a metric space X: It is known that if T is a nonself-map, the equation Tx = x does not always have a solution, and it clearly has no solution when A and B are disjoint. However, it is possible to determine an approximate solution x * such that the error is dðx * , Tx * Þ = dðA, BÞ: Such point x * is called a best proximity point of T: 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][3][4][5][6][7][8][9][10][11][12][13][14][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 θ : ½0,∞Þ → ½0, 1Þ such that t n → 0 when θðt n Þ → 1: By including 1 in the ranges of those mappings θ, Ayari [20] provided a new result on the existence and uniqueness of the best proximity point for α-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 a is called a best proximity coincidence point of the pair ðg, TÞ, where g is a self-map on A, if dðga, TaÞ = dðA, BÞ: Clearly, if g is the identity map, then each best proximity coincidence point of the pair ðg, TÞ is a best proximity point for T: 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][24][25], including the recent article [26] where best proximity point results for Suzuki-Edelstein proximal contractions were studied. (See also, [27][28][29][30][31] for his work. ) Zhang and Su [32] weakened the P-property, called the weak P-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 α-Geraghty contractions ðg, TÞ in metric spaces by using the weak P-property where g is an isometry.
The concept of generalized metric spaces (or JS-metric spaces) was introduced in [34] in 2015. It is a generalization of standard metric spaces covering many topological structures.
Let X be a nonempty set, and let D : X × X → ½0,∞ be a function. For each x ∈ X, we set Definition 1 (see [34]). A function D : X × X → ½0,∞ is called a generalized metric on X if it satisfies the following conditions. (D 1 )For any x, y ∈ X, Dðx, yÞ = 0 implies x = y. (D 2 )For any x, y ∈ X, Dðx, yÞ = Dðy, xÞ.
whenever x, y ∈ X and fx n g ∈ CðD, X, xÞ.
In this case, we say that ðX, DÞ is a generalized metric space. It is, however, usually called a JS-metric space.
Remark 2. We note that, in general, results of best proximity points using the weak P -property in usual metric spaces might not be attained in the setting of JS -metric spaces. For example, Dðx, xÞ is not necessarily equal to 0, and Dðx n , y n Þ might not converge to Dðx, yÞ when x n → x and y n → y: Let X ≔ ðX, DÞ be a JS-metric space. We now discuss the convergence and the continuity in these spaces.
Definition 3 (see [34]). Let fx n g be a sequence in X. The sequence fx n g is said to D -converge to x ∈ X if fx n g ∈ CðD, X, xÞ: Moreover, fx n g is called a D -Cauchy sequence if lim m,n→∞ Dðx n , x m Þ = 0: Finally, ðX, DÞ is said to be D -complete if each D -Cauchy sequence in X is a D -convergent sequence in X.
Definition 5 (see [34]). A function f : X → X is said to be D -continuous at a point x 0 ∈ X if for any fx n g ∈ CðD, X, x 0 Þ, f f x n g ∈ CðD, X, f x 0 Þ: In addition, f is said to be D -continuous on X if it is D -continuous at each point in X: The concept of α-admissible mapping was introduced by Samet et al. [35] in 2012. The notion of triangular α-admissible mappings was then given by Karapinar [36]. Recently, Khemphet [37] extended the concept as follows.
Definition 6 (see [37]). Let ðX, DÞ be a generalized metric space, and let f and g be self-mappings on X . Given that α : X × X → ½0,∞Þ is a function, f is said to be triangular-ðα, DÞ -admissible w.r.t. g if, for all x, y, z ∈ X , the following conditions hold.
In this article, we introduce a type of Geraghty contractions which will be called ðα, DÞ-proximal generalized Geraghty mappings. These maps are motivated by the work of Khemphet [37]. Using the weak P-property in the setting of JS-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 X endowed with a binary relation. Note that some other results of best proximity points in X endowed with binary relations can be deduced from our result.

Main Results
Throughout this article, let X ≔ ðX, DÞ be a JS-metric space, and let A and B be nonempty disjoint subsets of X: Also, we require the following notations: Clearly, if one of A 0 and B 0 is nonempty, then so is the other.

Journal of Function Spaces
We consider the class of mappings Θ which is a slight generalization of the well-known class of ½0, 1Þ-valued functions introduced by Geraghty [19]: Now, we introduce a class of our contractions as follows.
Definition 10. Let T : A → B and S : A → A be mappings. Given that α : X × X → ½0,∞Þ is a function, the pair ðS, TÞ is said to be an ðα, DÞ -proximal generalized Geraghty mapping if the following conditions hold.
Thus, s must be 0 and that Next, we shall show that fSx n g is a D-Cauchy sequence. Suppose that this is not the case. Then, there exists ε > 0 such that for any k ∈ ℕ, there are subsequences fSx n k g and fSx m k g of fSx n g satisfying DðSx n k , Sx m k Þ ≥ ε for m k ≥ n k ≥ k.
Since ðS, TÞ is triangular-ðα, DÞ-proximal admissible, it is easy to see that It follows from (13) and (24) that for any k ∈ ℕ, Since ðS, TÞ is an ðα, DÞ-proximal generalized Geraghty mapping and ðA, BÞ has the weak P-property, we obtain that where If Mðx n k −1 , x m k −1 , x n k , x m k Þ is either DðSx n k −1 , Sx n k Þ or D ðSx m k −1 , Sx m k Þ, then, by (23), lim k→∞ DðSx n k , Sx m k Þ = 0. This contradicts the assumption that fSx n g is not D-Cauchy. Thus, Mðx n k −1 , x m k −1 , x n k , x m k Þ = DðSx n k −1 , Sx m k −1 Þ: As a consequence, By repeating the same steps, it follows that where i = 0, 1, 2, ⋯, n k − 1. Therefore,

Journal of Function Spaces
Let i k ∈ f1, 2, ⋯, n k g such that Define If η < 1, lim k→∞ DðSx n k , Sx m k Þ = 0 which is impossible. Thus, η = 1. Without loss of generality, we may assume that lim k→∞ θðDðSx n k −i k , Sx n k +m k −i k ÞÞ = 1: By the definition of θ, lim k→∞ DðSx n k −i k , Sx n k +m k −i k Þ = 0: Then, there exists k 0 ∈ ℕ such that Now, which is a contradiction. Therefore, fSx n g is a D-Cauchy sequence.
Since ðSðA 0 Þ, DÞ is D-complete, there exists x * ∈ A 0 such that Equivalently, Since A 0 ⊆ SðA 0 Þ and TðA 0 Þ ⊆ B 0 , it follows that there exists a ∈ A 0 such that By (13) and (iii), there is a subsequence fSx n k g of fSx n g such that αðSx n k , x * Þ ≥ 1 for all k ∈ ℕ. From (13), we have that By the weak P-property, (37) and (38), we obtain that D ðSx n k +1 , SaÞ ≤ DðTx n k , Tx * Þ: Since αðSx n k , x * Þ ≥ 1 and ðS, TÞ is an ðα, DÞ-proximal generalized Geraghty mapping, where By (23) and (35), we immediately have that If DðSx * , SaÞ > 0, by letting k → ∞ in (39), We subsequently have that By the property of θ, which is a contradiction. It follows that DðSx * , SaÞ must be equal to 0, and thus Sx * = Sa. Therefore, from (37), there exists x * ∈ A such that Suppose further that x * , y * ∈ BCðS, TÞ and αðx * , y * Þ ≥ 1: By Lemma 11, Sx * = Sy * : Since S is injective, x * = y * : The proof is now completed. x j j + y j j, x ≠ 0 and y ≠ 0, Choose A = ½−2, 0 and B = ½0, 1: Let T : A → B be a mapping defined by and let a mapping S : A → A be defined by It is not difficult to see that DðA, BÞ = 0 and ðA, BÞ has the weak P-property. Next, define the map α : X × X → ½0,∞Þ by
We note that there is a map θ ∈ Θ defined by θðtÞ = 2/3. Now, for x, y satisfying αðSx, SyÞ ≥ 1, we have that Sx ≠ 0 or Sy = 0. We consider the following two cases.
Example 14. Let X = ℝ 2 be equipped with the JS -metric D given by We consider the disjoint subsets A and B of X given by A = fð−1, yÞ ; 0 ≤ y ≤ 1g and B = fð1, yÞ ; 0 ≤ y ≤ 1g: We can check that DðA, BÞ = 2 and the pair ðA, BÞ has the weak P -property.
Next, we present a corollary of our result. The following definition is required.
Definition 15. Let T : A → B and S : A → A be mappings. Let α : X × X → ½0,∞Þ be a function. Then, the pair ðS, TÞ is said to be an ðα, DÞ -proximal mapping if the following conditions hold.

Corollary 16.
Let A 0 ⊆ SðA 0 Þ and ðSðA 0 Þ, DÞ be D -complete. Given that α : X × X → ½0,∞Þ is a function, and let T : A → B and S : A → A be mappings such that ðS, TÞ is an ðα, DÞ -proximal mapping. Suppose that the following conditions hold.

Consequence
We will apply our result on the best proximity coincidence point on a JS-metric space endowed with a binary relation R: Let T : A → B and S : A → A be mappings. The pair ðS, TÞ is said to be ðR, DÞ-proximally comparative if SxRSy and DðSu 1 , TxÞ = DðSu 2 , TyÞ = DðA, BÞ ⇒ Su 1 RSu 2 and DðSu 1 , Su 2 Þ < ∞ for all x, y, u 1 , u 2 ∈ A.
Definition 17. Let T : A → B and S : A → A be mappings. The pair ðS, TÞ is said to be an ðR, DÞ -proximally comparative generalized Geraghty mapping if the following hold.

Conclusion and Open Questions
We have introduced new classes of Geraghty's type mappings called ðα, DÞ-proximal generalized Geraghty mappings. Then, we investigated some conditions for this type of mappings to have a best proximity coincidence point in JS-metric spaces using the weak P-property. The question is whether one can extend Theorem 12 to the framework of common best proximity point in a JS-metric space X: Can we also extend the result when X is other generalized metric spaces?

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.