Authors: S Sadiq Basha
Publish Date: 2010/08/12
Volume: 5, Issue: 4, Pages: 639-645
Abstract
Given nonvoid subsets A and B of a metric space and a nonself mapping TAlongrightarrow B the equation T x = x does not necessarily possess a solution Eventually it is speculated to find an optimal approximate solution In other words if T x = x has no solution one seeks an element x at which dx T x a gauge for the error involved for an approximate solution attains its minimum Indeed a best proximity point theorem is concerned with the determination of an element x called a best proximity point of the mapping T for which dx T x assumes the least possible value dA B By virtue of the fact that dx T x ≥ dA B for all x in A a best proximity point minimizes the real valued function xlongrightarrow dx Tx globally and absolutely and therefore a best proximity in essence serves as an ideal optimal approximate solution of the equation T x = x The aim of this article is to establish a best proximity point theorem for generalized contractions thereby producing optimal approximate solutions of certain fixed point equations In addition to exploring the existence of a best proximity point for generalized contractions an iterative algorithm is also presented to determine such an optimal approximate solution Further the best proximity point theorem obtained in this paper generalizes the wellknown Banach’s contraction principle
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