Best proximity results for proximal contractions in metric spaces endowed with a graph

Abstract: In this paper we define a generalized proximal G-contraction on a metric space having the additional structure of a directed graph. We obtain a best proximity point result for such contractions which is with a view to obtaining minimum distance between the domain and range sets. An example illustrating the main theorem is also discussed. The work is in the line of research on mathematical analysis as well as optimization in metric spaces with a graph.


Introduction and mathematical preliminaries
The purpose of this paper is to establish a best proximity point theorem for generalized rational proximal contractions. It is a study on metric spaces with the additional structure of a graph on it. We begin with the following technical details which are necessary for the discussion in the paper.
Throughout the paper (X, d) denotes a metric space and A, B ⊆ X. We use the following notations.

PUBLIC INTEREST STATEMENT
In this paper some results of mathematical analysis are established. It is a core area of mathematics on which stands a large part of the theoretical development of mathematics as well as many applications of mathematics. Particularly the results are in the domain of fixed point theory which is an extensive branch of mathematics having overlapping with various branches of pure and applied mathematics. The theory has also important implications in computer science. Although the present results are theoretical, there are potential applications of similar results in the literature. A noticeable aspect of the present work is the development of algorithm.
It is to be noted that if (A, B) is a nonempty, weakly compact and convex pair in a Banach space X, then A 0 and B 0 are nonempty (Basha & Veeramani, 2000;Gabeleh, 2015).
Definition 1.1 [P-property (Sankar Raj, 2011)] Let A and B be two nonempty subsets of a metric space (X, d) with A 0 ≠ ∅. Then the pair (A, B) is said to have the P-property if for any x 1 , x 2 ∈ A 0 and y 1 , y 2 ∈ B 0 , Abkar and Gabeleh (2012) have shown that every nonempty, bounded, closed and convex pair of subsets of a uniformly convex Banach space has the P-property. Some nontrivial examples of nonempty pairs of subsets which satisfy the P-property are given in Abkar and Gabeleh (2012).
Lemma 1.1 (Gabeleh, 2013) Let (A, B) be a pair of nonempty closed subsets of a complete metric space (X, d) such that A 0 is nonempty and (A, B) has the P-property. Then (A 0 , B 0 ) is a closed pair of subsets of X. Definition 1.2 An element x ∈ A is said to be a best proximity point the mapping S: coincides with X, that is, V(G) = X and the edge set E(G) contains all loops, that is, Δ ⊆ E(G). Assume that G has no parallel edges. By G −1 we denote the conversion of a graph G, that is, the graph obtained from G by reversing the directions of the edges. Thus we have Let G denote the undirected graph obtained from G by ignoring the directions of edges. Actually, it is convenient for us to treat G as a directed graph for which the set of its edges is symmetric. Under this convention, A graph G is connected if there is a path between any two vertices. G is weakly connected if G is connected.
Let G be such that E(G) is symmetric and x is a vertex in G, then the subgraph G x consisting of all edges and vertices which are contained in some path beginning at x is called the component of G We say a metric space (X, d) is endowed with a directed graph G, if G is a directed graph such that V(G) = X and Δ ⊆ E(G). We suppose that (X, d) is metric space endowed with a directed graph G. Definition 1.4 Let S:A ⟶ B be a mapping. Then Prox (S) and X S (G A 0 ) are defined as follows: (x, y, u, v), where k ∈ (0, 1) and M (x, y, u, v) is as in Definition 1.6.
As stated earlier, our purpose is to establish best proximity point results. Best proximity points are associated with non-self maps defined from one subset of a metric space to another. They are studied for the purpose of obtaining minimum distance between two sets. There are two aspects of this problem. Primarily, it is a global minimization problem where the quantity d(x, Sx) is minimized over x ∈ A subject to the condition that the minimum value is d (A, B). When this global minimum is attained at a point z, then we have a best proximity point for which d(z, Sz) = d(A, B). Another aspect is that it is an extension of the idea of fixed point to which it reduces in the cases where A ∩ B is nonempty. This is illustrated through the following. Let A = (−∞, 0] and B = [1, ∞) be two subsets of Then d(0, S(0)) = 1 = d (A, B). So that 0 is a best proximity point of the mapping S. This is not a fixed point of S. In fact fixed point of the non-self map S does not exist.
On the contrary if C = [0, ∞), then the mapping T:A ⟶ C given by Tx = − x 2 has a best proximity point which is also a fixed point.
In fact fixed points are best proximity points, but the converse is not true. The above is the reason for which fixed point methodologies are applicable to this category of problems. More elaborately, the problem can be treated as that of finding a global optimal approximate solution of the fixed point equation x = Sx even when the exact solution is nonexistent for A ∩ B = � which is the case of interest here. We adopt the later approach in this paper.
Metric spaces with the structure of graph have been considered in recent times especially in the context of fixed point theory of contractive type mappings. The line of research was originated in the work of Jachymski (2008) and was further pursued in Abbas, Nazir, Lampert, and Radenović (2016), Beg, Butt, and Radojević (2010), Bojor (2012), Eshi, Das, and Debnath (2016), Kumam, Salimi, and Vetro (2014), Tiammee and Suantai (2014), Shukla (2014). The essential feature of these works is that the metric inequality for the purpose of ensuring the fixed point need only be satisfied on certain pairs of points which are, in this case, connected by the edges of the graph. It is a further extension of metric spaces with a partial order structure on it.
In this paper, against the above background, we establish a best proximity point theorem in a metric space having a structure of graph defined on it by using generalized proximal G-contractions. In the last section we discuss an illustrative example.

Theorem 2.1 Let (X, d) be a complete metric space endowed with a directed graph G. Let (A, B) be a pair of nonempty and closed subsets of X such that A 0 is nonempty and closed. Let S:A ⟶ B be a mapping with the properties that S(A 0 ) ⊆ B 0 and S is generalized proximal G-contraction on A 0 . Suppose that (a) S is continuous or (b) the triple (X, d, G) is regular. Then the the following statements hold:
(1) For any x ∈ X S (G A 0 ), S has a best proximity point in (2) If X S (G A 0 ) ≠ � and G A 0 is weakly connected, then S has a best proximity point in A 0 .
Proof (1) It follows from the definition of A 0 and B 0 that for every x ∈ A 0 there exists y ∈ B 0 such that d(x, y) = d(A, B) and conversely, for every y � ∈ B 0 there exists (A, B). (A, B). Now x 1 ∈ A 0 and S(A 0 ) ⊆ B 0 imply the existence of a point x 2 ∈ A 0 such that d(x 2 , Sx 1 ) = d (A, B). As S is generalized proximal G-contraction on A 0 , we get (x 1 , x 2 ) ∈ E(G). In this way we obtain a sequence {x n } in A 0 such that for all n ≥ 0, and Now, for all n ≥ 0 we have x n ∈ A 0 , (x n , x n+1 ) ∈ E(G), d(x n+1 , Sx n ) = d(A, B) and d(x n+2 , Sx n+1 ) = d (A, B). Since S is generalized proximal G-contraction on A 0 , we have x n+2 ), for some positive integer n. Then d(x n+1 , x n+2 ) > 0. Then it follows from (2.3) that which is a contradiction. Therefore,

Hence we have from (2.3) and (2.4) that
By repeated application of (2.5), we have For arbitrary m, n ∈ ℕ with m > n, Therefore, {x n } is a Cauchy sequence in A 0 . Since A 0 is a closed subset of complete metric space (X, d), there exists z ∈ A 0 such that • Suppose that S is continuous.
Taking n ⟶ ∞ in (2.2) and using the continuity of S, we have d(z, Sz) = d (A, B); that is, z is a best proximity point of S.
• Next we suppose that the triple (X, d, G) is regular.
By (2.1) and (2.7), we have Now z ∈ A 0 and S(A 0 ) ⊆ B 0 imply the existence of a point p ∈ A 0 for which By (2.2), (2.8) and (2.9), we have for all n ≥ 0 (2.7) x n ⟶ z as n ⟶ ∞.
Since S is generalized proximal G-contraction on A 0 , we have where Using (2.7), we have Taking the limit as n ⟶ ∞ in (2.10), using (2.7) and (2.11), we have d(z, p) ≤ k d(z, p), which is a contradiction unless d(z, p) = 0; that is, p = z. Then by (2.9) we have that d(z, Sz) = d(A, B); that is, z is a best proximity point of S. By (2.8), it is obvious that ( (1) and (2), S has a best proximity point in X * .
(4) Let Prox (S) ≠ �. Then there exists atleast one element x ∈ Prox (S). Now x ∈ Prox (S) means  (x, y, u, v), where k ∈ (0, 1) and Also, suppose that (a) S is continuous or (b) the triple (X, d, G) is regular. Then the following statements hold : (1) For any x ∈ X S (G A 0 ), S has a best proximity point in (2) If X S (G A 0 ) ≠ � and G A 0 is weakly connected, then S has a best proximity point in A 0 .
M (x, y, u, v) (x, y, u, v), where k ∈ (0, 1) and Hence S is generalized proximal G-contraction on A 0 . Therefore, we have the required proof from that of Theorem 2.1. ✷

Example
Example 3.1 Let X = R 2 (R denotes the set of real numbers) and d be a metric on X de- Let k ∈ (0, 1) be such that 1 − 1 b 2 ≤ k < 1. The function S satisfies all the postulates of Theorem 2.1. The set of best proximity points of the mapping S, that is, Prox (S) is nonempty. Here Prox(S) = (0, 1), (b, 1) ⊆ A 0 (Figures 1-3).