A Best Proximity Point Theorem for G-Proximal (delta, 1-delta) Weak Contraction in Complete Metric Space Endowed with a Graph

Clement Boateng Ampadu

doi:10.34198/ejms.1119.9196

Abstract

The notion of (delta, 1-delta) weak contraction appeared in [1]. In this paper, we consider that the map satisfying the (delta, 1-delta) weak contraction is a non-self map, and obtain a best proximity point theorem in complete metric space endowed with a graph.

Highlights

Introduction and PreliminariesAt first we recall the followingDefinition 1.1. [1] Let (X, d ) be a metric space
We consider that the map satisfying the (δ, 1 − δ) weak contraction is a non-self map, and obtain a best proximity point theorem in complete metric space endowed with a graph
The notion of G-proximal Kannan mapping appeared in [3], we introduce the following

Summary

Introduction and Preliminaries

A map T : X ֏ X is called a (δ, 1 − δ) weak contraction if there exists δ ∈ (0, 1) such that the following holds d(Tx, Ty) ≤ δd(x, y) + (1 − δ) d( y, Tx). Let W and V be two nonempty subsets of a metric space (X , d ). A non-self mapping S : W ֏ V is called a G-proximal (δ, 1 − δ) weak contraction, if there exists δ ∈ (0, 1) such that (x, y) ∈ E(G), d (u, Sx) = d(W , V ) and d (v, Sy) = d (W , V ) implies d(u, v) ≤ δd(x, y) + (1 − δ) d( y, u), where x, y, u, v ∈ W. A Best Proximity Point Theorem for G-Proximal (δ, 1 − δ) Weak Contraction

Main Result

Open Problem