Best proximity point of modified Suzuki-Edelstein -Geraghty type proximal contractions

I n 1922, Banach [1] introduced a remarkable principle, namely Banach contraction principle which asserts that every contraction self-mapping on a complete metric space has a unique fixed point. This principle plays a leading role in the development of fixed point theory. Banach’s contraction principle has been generalized and extended in different directions. On his work, Edelstein [2] introduced the notion of contractive mapping and generalized Banach contraction principle. In 1973, Geraghty [3] generalized Banach’s contraction principle by replacing the contraction constant by a function having certain specified properties. In 2008, Suzuki [4] introduced a new type of mapping and presented a generalization of the Banach contraction principle in which the completeness can also be characterized by the existence of a fixed point of these mappings. All these generalizations are only applicable for self-mappings. In recent years, best proximity point theory attracted the attention of several authors. The purpose of best proximity point theory is to address a problem of finding the distance between two closed sets by using non self-mappings from one set to the other. The problem is known as the proximity point problem. Best proximity point theory analyzes the existence of an approximate solution that is optimal. Let A and B be two non-empty subsets of a metric space (X, d) and T : A → B is a mapping, then d(x, Tx) ≥ d(A, B) for all x ∈ A. In general, for non self-mapping T : A → B, the fixed point equation Tx = x may not have a solution. In this case, it is focused on the possibility of finding an element x ∈ A that is an approximate solution such that the error d(x, Tx) is minimum, possibly d(x, Tx) = d(A, B). A best proximity point becomes a fixed point if the underlying mapping is a self-mapping. Therefore, it can be concluded that best proximity point theorems generalize fixed point theorems in a natural way. In recent years, the existence and convergence of best proximity points is an interesting aspect of optimization theory which attracted the attention of many authors [5–9]. We recall the following notations and definitions: Let (X, d) be a metric space and let A and B be non-empty subsets of X.


Introduction and Preliminaries
I n 1922, Banach [1] introduced a remarkable principle, namely Banach contraction principle which asserts that every contraction self-mapping on a complete metric space has a unique fixed point. This principle plays a leading role in the development of fixed point theory. Banach's contraction principle has been generalized and extended in different directions. On his work, Edelstein [2] introduced the notion of contractive mapping and generalized Banach contraction principle. In 1973, Geraghty [3] generalized Banach's contraction principle by replacing the contraction constant by a function having certain specified properties. In 2008, Suzuki [4] introduced a new type of mapping and presented a generalization of the Banach contraction principle in which the completeness can also be characterized by the existence of a fixed point of these mappings. All these generalizations are only applicable for self-mappings.
In recent years, best proximity point theory attracted the attention of several authors. The purpose of best proximity point theory is to address a problem of finding the distance between two closed sets by using non self-mappings from one set to the other. The problem is known as the proximity point problem. Best proximity point theory analyzes the existence of an approximate solution that is optimal.
Let A and B be two non-empty subsets of a metric space (X, d) and T : A → B is a mapping, then d(x, Tx) ≥ d(A, B) for all x ∈ A. In general, for non self-mapping T : A → B, the fixed point equation Tx = x may not have a solution. In this case, it is focused on the possibility of finding an element x ∈ A that is an approximate solution such that the error d(x, Tx) is minimum, possibly d(x, Tx) = d(A, B).
A best proximity point becomes a fixed point if the underlying mapping is a self-mapping. Therefore, it can be concluded that best proximity point theorems generalize fixed point theorems in a natural way. In recent years, the existence and convergence of best proximity points is an interesting aspect of optimization theory which attracted the attention of many authors [5][6][7][8][9].
We recall the following notations and definitions: Let (X, d) be a metric space and let A and B be non-empty subsets of X.
We denote by F the class of all functions β : [0, ∞) → [0, 1) satisfying the following condition: We denote by Φ the class of all functions φ : [0, ∞) → [0, ∞) satisfying the following conditions: 1. φ is continuous, 2. φ is non-decreasing, and 3. φ(t) = 0 ⇐⇒ t = 0. Definition 1. [10] Let A and B be two non-empty subsets of a metric space (X, d) and α : A × A → [0, ∞) be a function. We say that a non self-mapping T : Definition 2. [11] Let A and B be two non-empty subsets of a metric space (X, d) and α : A × A → [0, ∞) be a function. We say that a non self-mapping T :

Definition 3. [12] Let
A and B be two non-empty subsets of a metric space (X, d) and A 0 = ∅, we say that the pair (A, B) has weak P-property if and only if for all x 1 , x 2 ∈ A and y 1 , y 2 ∈ B.

Definition 4.
[10] Let A and B be two non-empty subsets of a metric space (X, d) and α, η : A × A → [0, ∞) be functions. We say that a non self-mapping T : A → B is α−proximal admissible with respect to η if, for all x, y, u, v, z, w ∈ A,

Definition 5.
[13] Let A and B be two non-empty subsets of a metric space (X, d) and T : A → B be a mapping. We say that T has the RJ-property if for any sequence {x n } ⊆ A, Remark 1. [13] Any continuous mapping T : A → B has the RJ-property provided that A and B are non-empty closed subsets of a metric space (X, d). If A and B are not closed subsets of X, then T may not have RJ-property. In 2016, Hamzehnejadi and Lashkaripour [13] proved best proximity point results for non self-map satisfying the RJ-property. Definition 6. [13] Let A and B be two non-empty subsets of a metric space (X, d) and α : Theorem 1. [13] Let (X, d) be a complete metric space, A and B be non-empty subsets of X, α : X × X → [0, ∞) be a function and T : A → B be a mapping. If the following conditions are satisfied: denotes the set of best proximity points of T, then x * is a unique best proximity point of T.
In this paper, we denote by Φ ϕ the class of functions ϕ : [0, ∞) → [0, ∞) satisfying the following property: We denote by Ψ the set of non-decreasing functions ψ : Recently, Hussain et al., [14] proved the existence of best proximity point for modified Suzuki-Edelstein α-proximal contraction.
Definition 7. [14] Suppose A and B are two non-empty subsets of a metric space (X, d). A non self-mapping Motivated by the work of Suzuki, Edelstein and Geraghty, we introduce the notion of modified Suzuki-Edelstein-Geraghty proximal contraction and prove the existence and uniqueness of best proximity point for such mappings.

Main results
Definition 8. Let A and B be two non-empty subsets of metric space (X, d). Let T : A → B be non self-mapping and α : A × A → [0, ∞) be a function. T is said to be a modified Suzuki-Edelstein-Geraghty proximal contraction if there exist β ∈ F and φ ∈ Φ such that for all x, y ∈ A, where M(x, y) = max{d(x, y), d(x, Tx), d(y, Ty)}, m(x, y) = max{d(x, Tx), d(y, Ty)} and ϕ ∈ Φ ϕ .
Theorem 3. Let (X, d) be a complete metric space and A and B be non-empty closed subsets of X with A 0 is non-empty. If T : A → B be a modified Suzuki-Edelstein-Geraghty proximal contraction mapping such that the following conditions hold: 1. T(A 0 ) ⊆ B 0 and the pair (A, B) satisfies the weak P-property, 2. T is triangular α-proximal admissible with respect to η(x, y) = 2, 3. T is continuous, then T has a unique best proximity point in A 0 .
If φ(t) = t for all t ≥ 0, in Theorem 3, we have the following corollary.
We can prove the existence and uniqueness of best proximity point by replacing the continuity assumption with RJ-property in Theorem 3.