Common Best Proximity Coincidence Point Theorem for Dominating Proximal Generalized Geraghty in Complete Metric Spaces

The best proximity point problems have been attracted to many researchers as there are various applications in realworld problems. The optimization problem is one of the applications that benefit from the best proximity point theory. In other words, it helps finding an approximate solution to the fixed point problems even the mapping itself does not have a fixed point (see [1–23]). In literature, most works focus on suggesting suitable conditions to promise the existence of approximate optimal solutions. These results give the best proximity point theorem in a variety of approaches. For instance, the work of Geraghty [24] is one of several important results inspired by the Banach contraction principle for the existence of fixed points for self mappings in metric spaces. In fact, this result generalizes previous concepts by introducing the class Θ of all mappings θ : 1⁄20,∞Þ⟶ 1⁄20, 1Þ such that


Introduction
The best proximity point problems have been attracted to many researchers as there are various applications in realworld problems. The optimization problem is one of the applications that benefit from the best proximity point theory. In other words, it helps finding an approximate solution to the fixed point problems even the mapping itself does not have a fixed point (see ). In literature, most works focus on suggesting suitable conditions to promise the existence of approximate optimal solutions. These results give the best proximity point theorem in a variety of approaches.
For instance, the work of Geraghty [24] is one of several important results inspired by the Banach contraction principle for the existence of fixed points for self mappings in metric spaces. In fact, this result generalizes previous concepts by introducing the class Θ of all mappings θ : ½0,∞Þ ⟶ ½0, 1Þ such that lim n→∞ θ t n ð Þ = 1 ⟹ lim n→∞ t n = 0: In 2012, Basha [25] proposed a result on common best proximity points with a property called proximal commutativity of mappings. Later, Kumam and Mongkolekeha [26] considered common best proximity point theorems for proximity commuting mappings. In addition, this study has been done according to Geraghty's work in complete metric spaces. After that, Chen [27] established the definition of a mapping T generally dominates a mapping S and accomplished theorems of existence and uniqueness of common best proximity points for a pair of nonself mappings. Lately, Ayari [28] improved the class Θ of Geraghty and defined a new class B of the mappings β : ½0,∞Þ ⟶ ½0, 1 such that Accordingly, the existence and uniqueness of best proximity points is guaranteed for α-proximal Geraghty nonself mappings on a closed subset of a complete metric space.
To generalize previous results, we are interested to extend our study to common best proximity coincidence points for two mappings under certain conditions. Specifically, we investigate the existence and uniqueness of common best proximity coincidence points for any pairs of two mappings that are dominating proximal generalized Geraghty on a complete metric space. In particular, this work is organized into three sections. First, the motivation of the present study is given as described above. Next, we recall some essential definitions needed in our work. In Section 3, a new concept of dominating proximal generalized Geraghty for two mappings is introduced. Then, we show that a common best proximity coincidence point of these mappings uniquely exists under some additional assumptions. Moreover, an example is provided to support the main result. Lastly, we consider some further results following from our main theorem.

Preliminaries
In this section, we review some notations and important definitions to be used in the next section. Let ðA, BÞ be a pair of nonempty subsets of a metric space ðX, dÞ. We adopt the following notations: Definition 1 (see [1,26,29]). Let S, T : A ⟶ B and g : A ⟶ A be mappings.
An element x * ∈ A is said to be (ii) A best proximity coincidence point of the pair ðg, (iii) A common best proximity coincidence point of the pair ðS, TÞ if Definition 2 (see [29]). Let S, T : A ⟶ B be mappings. A pair ðS, TÞ is said to commute proximally if for each x, u, v ∈ A,

Main Results
In this section, we introduce a class of pairs of some proximal generalized Geraghty contractions with dominating property and prove common best proximity theorem for this class.
Definition 3. Let S, T : A ⟶ B be mappings. A pair ðS, TÞ is said to be dominating proximal generalized Geraghty if there exists β ∈ B such that for each x 1 , where Proof. Let x 0 be a fixed element in A 0 . From the assumption that SðA 0 Þ ⊆ TðA 0 Þ, we get that for each element x ∈ A 0 , there is an element y ∈ A 0 such that Sx = Ty. Then, we obtain a sequence fx n g in A 0 satisfying for each n ≥ 0. Since SðA 0 Þ ⊆ B 0 , there exists an element u n ∈ A 0 such that for all n ≥ 0: Further, we obtain that for all n ≥ 0.
Our first goal is to show that Su = Tu for some u ∈ A 0 . In the case that u n 0 = u n 0 +1 for some n 0 ≥ 0, by (11) and (12), we get that Since S and T commute proximally, Sðu n 0 Þ = Tðu n 0 +1 Þ = Tðu n 0 Þ, and so we are done. Now, for the harder part, assume that u n ≠ u n+1 for all n ≥ 0. From (12), note that for all n ≥ 1. Since ðS, TÞ is dominating proximal generalized Geraghty, we have that 2 Journal of Function Spaces Consider that This implies that Mðu n−1 , u n , u n , u n+1 Þ = max fdðu n−1 , u n Þ, dðu n , u n+1 Þg for all n ≥ 1.
Next, we prove that the sequence fdðu n , u n+1 Þg converges to 0.
Consider the following two cases.
Case 2. Mðu n−1 , u n , u n , u n+1 Þ = dðu n , u n+1 Þ. Similarly, by (15), we have that Since dðu n , u n+1 Þ > 0 for all n ≥ 0, we get that 1 ≤ βðdðu n , u n+1 ÞÞ ≤ 1, and hence, lim n→∞ βðdðu n , u n+1 ÞÞ = 1: By the definition of β, we also have that lim n→∞ dðu n , u n+1 Þ = 0: Due to both cases, we obtain the desired limit Now, we claim that fu n g is a Cauchy sequence. Suppose contradiction, that is, fu n g is not a Cauchy sequence. Then, there exists ε > 0 such that there are subsequences fu m k g and fu n k g of fu n g so that for all k ∈ ℕ with m k > n k > k, we obtain In addition, we can choose the smallest n k satisfying (22) for all k ∈ ℕ so that By using (22) and (23), we have that Since lim n→∞ dðu n , u n+1 Þ = 0, taking the limit as k ⟶ ∞ in (24) implies Consider, by the triangular inequality, that Consequently, ε = lim k→∞ dðu m k , u n k Þ ≤ lim k→∞ dðu m k +1 , u n k +1 Þ.
In the same way, we get that and so lim k→∞ dðu m k +1 , u n k +1 Þ ≤ lim k→∞ dðu m k , u n k Þ = ε. Thus, Since fu m k g and fu n k g satisfy equations (11) and (12), we obtain that for each k ≥ 1. Since ðS, TÞ is dominating proximal generalized Geraghty,

Journal of Function Spaces
where By (21), we observe that and, as a consequence, Hence, (25) implies that lim k→∞ Mðu n k , u m k , u n k +1 , u m k +1 Þ = lim k→∞ dðu n k , u m k Þ = ε > 0. Then, by (28) and (30), we obtain that By the property of β, we obtain that a contradiction. Therefore, we can conclude that fu n g is a Cauchy sequence.
The essential observation is that fu n g is a Cauchy sequence in the closed subset A 0 of the complete metric space X. Then, there exists u ∈ A 0 such that lim n→∞ u n = u. Consider, by (11) and (12), that dðu n , Sx n Þ = dðu n−1 , Tx n Þ = dðA, BÞ. Since S and T commute proximally, We are now in a position to show that a best proximity coincidence point of ðS, TÞ exists. Since SðA 0 Þ ⊆ B 0 , there exists x * ∈ A 0 such that By the assumption that S and T commute proximally, S x * = Tx * . According to the assumption that SðA 0 Þ ⊆ B 0 , there exists z * ∈ A 0 such that Next, we claim that x * = z * . Suppose that x * ≠ z * , i.e., d ðx * , z * Þ > 0. We observe that Since dðx * , z * Þ > 0, we have 1 ≤ βðdðx * , z * ÞÞ ≤ 1. By the property of β, dðx * , z * Þ = 0. This contradicts the assumption that x * ≠ z * . Thus, x * = z * , and hence That is, the element x * ∈ A is a common best proximity coincidence point of ðS, TÞ.
Next, consider, by the definition of A 0 and B 0 , that A 0 = A and B 0 = B. Additionally, Thus, S and T commute proximally. Finally, by Theorem 4, we can conclude that there is a unique common best proximity coincidence point of the pair ðS, TÞ. In fact, the point ð0, 1Þ is the unique common best proximity coincidence point of ðS, TÞ.
As a consequence of our result, the following corollaries are given. Precisely, these are the special cases of Theorem 4 when βðtÞ = k for k ∈ ½0, 1Þ, and βðtÞ = e −kt for k > 0, respectively.