Convergence of Picard’s iteration using projection algorithm for noncyclic contractions

Abstract

Given A andB two nonempty subsets of a metric space, a mapping T:A∪B→A∪B is noncyclic provided that T(A)⊆A and T(B)⊆B. A point (p,q)∈A×B is called a best proximity pair for the noncyclic mapping T if p=Tp,q=Tq and d(p,q)=dist(A,B). In this article, we survey the convergence of Picard’s iteration to a best proximity pair for noncyclic contractions using a projection algorithm in uniformly convex Banach spaces, where the initial point is in the proximal set of A. We also provide some sufficient conditions to ensure the existence of a common best proximity pairs for a pair of noncyclic mappings.