Geraghty–Pata–Suzuki-Type Proximal Contractions and Related Coincidence Best Proximity Point Results

: The objective of this research paper is to establish the existence and uniqueness of the best proximity and coincidence with best proximity point results, speciﬁcally focusing on Geraghty–Pata– Suzuki-type proximal mappings. To achieve this, we introduce three types of mappings, all within the context of a complete metric space: an α - θ -Geraghty–Pata–Suzuki-type proximal contraction; an α - θ -generalized Geraghty–Pata–Suzuki-type proximal contraction; and an α - θ -modiﬁed Geraghty– Pata–Suzuki-type proximal contraction. These new results generalize, extend, and unify various results from the existing literature. Symmetry plays a crucial role in solving nonlinear problems in operator theory, and the variables involved in the metric space are symmetric. Several illustrative examples are provided to showcase the superiority of our results over existing approaches.


Introduction and Preliminaries
The Banach-Caccioppoli fixed-point theorem is named after Stefan Banach (1892Banach ( -1945 and Renato Caccioppoli  and was first stated by Banach [1] in 1922. The Banach contraction principle, also known as the Banach fixed-point theorem, is one of the main pillars of metric fixed-point theory. This principle states that when a mapping T is a contraction on a complete metric space Ω and maps elements from Ω back to itself, there exists a unique fixed point µ in Ω. This fixed-point theorem has several applications in determining the existence of solutions of integral and differential equations. It is quite interesting to study contractive mapping cases that do not have a fixed point. It is also interesting if the contractive mapping T is a non-self-mapping, in which case it is impossible to find the fixed point such that Tµ = µ or M d (µ, Tµ) = 0. Then, it would be interesting to approximate the fixed point to minimize the error among µ and Tµ or to minimize min µ∈Ω M d (µ, Tµ).
The best proximity point result offers the necessary conditions to compute an approximate solution µ, which is considered optimal as it minimizes the error M d (µ, Tµ) in achieving the global minimum value M d (P, Q), where P and Q are nonempty subsets of a metric space (Ω, M d ) and T : P → Q is a non-self-mapping. Any point µ ∈ P is known as the best proximity point of the non-self-mapping T if M d (µ, Tµ) = M d (P, Q) = inf{M d (µ, y) : µ ∈ P , y ∈ Q}.
For more details, see the best proximity results in [2][3][4][5]. In 1969, Ky Fan [6] provided the first best approximation result. The coincidence of the best proximity point results is a generalization of the best proximity point results because it deals with two mappings, one of which is a non-self-mapping and the other is a self-mapping. Let P and Q be a nonempty subset of a metric space (Ω, M d ). If T : P → Q and g : P → P is a self-mapping and M d (gµ, Tµ) = M d (P, Q), then µ ∈ P is referred to as the coincidence of the best proximity point of the pair of mappings (g, T). The results concerning the coincidence of the best proximity points serve as a generalization of the best proximity point results and fixed-point results. This is evident when considering that if we set g as the identity mapping I P , each coincidence's best proximity point becomes the best proximity point of the mapping T. Moreover, if the mapping T is a self-mapping, then the concept of the best proximity point reduces to the notion of a fixed point. Researchers have explored various generalizations of the Banach fixed-point theorem in different directions, leading to numerous applications in various fields. Among them are two contractive conditions presented by V. Pata [7] and T. Suzuki [8], which shall be discussed here. Recently, Karapınar et al. in [9] modified these contractive conditions and proved some fixed-point results by introducing a new type of contraction called the α-Pata-Suzuki-type contraction. Geraghty [10] proposed another extension of the Banach contraction principle, known as the Geraghty contraction. Ayari [11] utilized this Geraghty contraction and proved the best proximity-point results for α-proximal Geraghty non-self-mappings. Recently, Saleem et al. in [12] introduced the Pata-type best proximal contraction and proved related results in the best proximity point. The Banach space is symmetric and is related to the fixed-point problems discussed in [13]. It has a certain importance, and several researchers are working on it around the globe. Recalling that symmetry is a mapping on an object Ω, preserving its underlying structure, Neugebaner [13] utilized this concept to derive various applications of a layered compression-expansion fixed-point theorem. These applications resulted in the derivation of solutions for a second-order difference equation with Dirichlet boundary conditions. In this paper, we introduce three different types of contractive conditions: (1) an α-θ-Geraghty-Pata-Suzuki-type proximal contraction, (2) an α-θ-generalized Geraghty-Pata-Suzuki-type proximal contraction, and (3) an α-θ-modified Geraghty-Pata-Suzuki-type proximal contraction. The purpose of this study is to prove the existence and uniqueness of the coincidence of best proximity and the best proximity-point results for the abovementioned proximal contractions in complete metric spaces. Several examples are given to highlight the superiority of our results. As an application, we shall derive certain recent fixed-point results as corollaries to our results.
The sets mentioned below are important in the best proximity analysis.
then inequality (7) becomes which is a contradiction if we consider Then, from inequality (7), we have thus, {M d (µ n , µ n+1 )} is a decreasing sequence, and we have Now, we have to show that lim n→+∞ M d (µ n , µ n−1 ) = 0.
Then, inequality (8) can be written as which further implies that as γ ≤ µ. Suppose that s = Λc γ n [ 1+c 0 +c 1 c n + 1] γ and b = 3c 1 , then the above inequality can be written as for some constants s, b > 0. Thus, the sequence {c n } is bounded. Now, if there exists a divergent subsequence c n i , in that case, there is a subsequence s n i converging to Λ. If we and this leads to a contradiction. Moving forward, our next objective is to demonstrate that {µ n } is indeed a Cauchy sequence. To achieve this, we will show that Since m is a fixed natural number, define P n = n µ M d (µ n+m , µ n ), and If we take the limit → 0, then inequality (9) becomes where then inequality (10), can be written as then, from inequality (9), we have . . .
Since P 0 = 0, this gives Since n = 0, after division by n µ , we obtain that On taking the limit as n → +∞, then {µ n } is a Cauchy sequence. For each ∈ [0, 1], we have and assuming that then inequality (11) can be written as, and every Cauchy sequence in a complete metric space is convergent, there exists some µ ∈ P 0 such that µ n → µ in P 0 ⊆ P, Hence, P 0 is α-θ-proximal orbital complete.
Theorem 1. Let P and Q be nonempty subsets of a complete metric space (Ω, M d ). Furthermore, assume that the subset P is closed and we have a continuous mapping T : P → Q, which is an α-θ-proximal admissible and α-θ-Geraghty-Pata-Suzuki-type proximal contraction. Additionally, let T(P 0 ) ⊆ Q 0 , where P 0 = φ. If Q is approximately compact with respect to P, then µ is a unique best proximity point for T in P 0 .
Proof. Consider µ 0 in P 0 , from Lemma (4), the sequence {µ n } is a Cauchy α-θ-proximal admissible Picard sequence in P 0 . As a result, we can derive a sequence {µ n } in P 0 such that for all n ∈ N. Given that T is an α-θ-proximal admissible mapping, and it is also an α-θ-Geraghty-Pata-Suzuki-type proximal contraction, we can apply reasoning similar to that used in the proof of Lemma (4). Consequently, we can deduce that the sequence {µ n } ∈ P 0 is a Cauchy α-θ-proximal admissible Picard sequence. Since Ω is complete and P is closed, there exists some µ in P such that lim n→+∞ M d (µ n , µ) = 0. Since every Cauchy sequence in a complete metric space converges, there exists a µ in P 0 such that µ n → µ within P 0 . Furthermore, the mapping T is continuous, so we have Tµ n → Tµ. Now, based on the above inequality, we have: Thus, we can deduce that µ represents the α-θ-Geraghty-Pata-Suzuki-type best proximity point of the mapping T. Now, to show the uniqueness, we assume there exists y = µ (M d (µ, y) > 0) in P such that the mapping T is an α-θ-Geraghty-Pata-Suzuki-type proximal contraction, so Thus, the above inequality can be written as, Taking the limit → 0, then inequality (12) becomes ≤ m(µ, y), where Then, inequality (14) becomes which is a contradiction; therefore, µ = y, which shows that µ is the unique best proximity point of mapping Obviously, T(P 0 ) ⊆ Q 0 . Further, suppose that α(µ, y) = 2M d (µ, y) and θ(µ, y) = M d (µ, y). If we take µ = 1 3 , y = 3 3 ∈ P 0 , then α(µ, y) = 2( 2 3 ) > 2 3 = θ(µ, y), and we have .
Define ψ(s) = se s for all s ≥ 0. If µ = 2 and γ ∈ [0, 2], assuming that λ = γ, taking = 1 10 and verifying that T satisfies the conditions of an α-θ-Geraghty-Pata-Suzuki-type proximal contraction, it is evident the mapping T satisfies the following condition: (15) for all u, v, µ and y ∈ P 0 . For this setting and the following simple calculation, we have With simple calculation steps, one can verify that inequality (15) holds for µ = 1 3 and y = 3 3 , and the mapping T satisfies an α-θ-Geraghty-Pata-Suzuki-type proximal contraction for 76.25768383 ≤ Λ. Since the mapping T satisfies an α-θ-Geraghty-Pata-Suzuki-type proximal contraction for every pair of µ, y ∈ P 0 , there exists some u, v ∈ P 0 for Λ = 164, and mapping T satisfies the conditions of Theorem (1), and µ = 1 3 is the unique best proximity point of the mapping T.
Lemma 5. Let P and Q be two nonempty subsets of a complete metric space (Ω, M d ), and T : P → Q be a continuous α-Geraghty-Pata-Suzuki-type proximal contraction with T(P 0 ) ⊆ Q 0 and P 0 = φ. Then, P 0 is α-proximal orbital complete.
Proof. By taking θ(µ, y) = 1 for all µ, y ∈ P in an α-θ-proximal admissible, we have an α-proximal admissible. Continuing on the same line of proof as Lemma (4), then P 0 is α-proximal orbital complete.

Corollary 1.
Consider a continuous mapping T : P → Q, which is both an α-proximal admissible mapping and an α-Geraghty-Pata-Suzuki-type proximal contraction, with A 0 = φ and T(A 0 ) ⊆ Q 0 . If the set Q is approximately compact with respect to the set P, then the mapping T has a unique best proximity point µ in A 0 .
Proof. By taking θ(µ, y) = 1 in the proof of Theorem (1), we have the desired result.
Theorem 2. Let P and Q be nonempty subsets of a complete metric space (Ω, M d ), and suppose that T : P → Q is a modified Geraghty-Pata-Suzuki proximal contraction satisfying T(P 0 ) ⊆ Q 0 . If Q is approximately compact with respect to P, then T has a unique best proximity point µ in P 0 .
Definition 21. Let P and Q be two nonempty subsets of a metric space (Ω, M d ), T : P → Q and g : P → P be a non-self-mapping and α, θ : P × P → [0, +∞), then T is said to be α-θ-generalized proximal admissible if for all µ, y, u, v ∈ P. Remark 5. If the mapping g = I P , then an α-θ-generalized proximal admissible mapping becomes an α-θ-proximal admissible mapping.
Consider that lim n→+∞ M d (µ n , µ n−1 ) = r = 0. Then, inequality (20) can be written as Taking the limit as n → +∞, we have and taking the limit as → 0, then the above inequality can be written as which can aldo be written as M d (gµ n , gµ n−1 ) ≤ M d (µ n−1 , µ n−2 ).
Proof. By choosing q(µ, y) = M d (µ, y) in Theorem (4), and following on the same line of proof, we have the desired result.