CQ-Type Algorithm for Reckoning Best Proximity Points of EP-Operators

Abstract

We introduce a new class of non-self mappings by means of a condition which is called the (EP)-condition. This class includes proximal generalized nonexpansive mappings. It is shown that the existence of best proximity points for (EP)-mappings is equivalent to the existence of an approximate best proximity point sequence generated by a three-step iterative process. We also construct a CQ-type algorithm which generates a strongly convergent sequence to the best proximity point for a given (EP)-mapping.

A Modified Thakur Three-Step Iterative Algorithm to Garcia-Falset Mappings and Variational Inequalities

“…Following an argument similar to that in Case A and noticing (3.14), we derive After simplifying, we have The main results in this paper extend and generalize corresponding results in [7] [8] in the following senses:…”

“…1) The subset C of Banach space E does not have to be compact in our Theorem 3.1. However, this assumption is very necessary in Theorem 3.4 of Usurelu et al [7] and Corollary 2 of Houmani and Turcanu [8].…”

“…The study for Garcia-Falset mappings with iterative processes using a Banach space as underlying setting is only at the beginning (more precisely; there are just two research papers on uniformly convex Banach spaces that connect mappings endowed with property (E) (see [7] [8]). In order to establish strong convergence results for approximation of fixed points of Garcia-Falset mappings in Banach space, Gabriela et al [7] and Houmani and Turcanu [8] adopt different iteration schemes, respectively. However, the compactness assumption imposed on C is indispensable in both two papers.…”