Results on the Existence and Convergence of Best Proximity Points

Ali Abkar,Moosa Gabeleh

doi:10.1155/2010/386037

Abstract

AbstractWe first consider a cyclic "Equation missing"-contraction map on a reflexive Banach space "Equation missing" and provide a positive answer to a question raised by Al-Thagafi and Shahzad on the existence of best proximity points for cyclic "Equation missing"-contraction maps in reflexive Banach spaces in one of their works (2009). In the second part of the paper, we will discuss the existence of best proximity points in the framework of more general metric spaces. We obtain some new results on the existence of best proximity points in hyperconvex metric spaces as well as in ultrametric spaces.

Highlights

Let X X, d be a metric space, and let A, B be two subsets of X
Later on, Eldred and Veeramani 2 considered the class of cyclic contractions
In the last section we study the existence of best proximity points in spherically complete ultrametric spaces, as well as in hyperconvex metric spaces

Summary

Introduction

Let X X, d be a metric space, and let A, B be two subsets of X. Let A and B be two nonempty closed convex subsets of a uniformly convex Banach space X, and let T : A∪B → A∪B be a cyclic contraction map. They proved the existence of a best proximity point for a cyclic contraction map in a reflexive Banach space X see 3, Theorems 10, 11 .

Results

Conclusion