On Best Proximity Results for a Generalized Modified Ishikawa’s Iterative Scheme Driven by Perturbed 2-Cyclic Like-Contractive Self-Maps in Uniformly Convex Banach Spaces

M De La Sen,Mujahid Abbas

doi:10.1155/2019/1356918

Abstract

This paper proposes a generalized modified iterative scheme where the composed self-mapping driving can have distinct step-dependent composition order in both the auxiliary iterative equation and the main one integrated in Ishikawa’s scheme. The self-mapping which drives the iterative scheme is a perturbed 2-cyclic one on the union of two sequences of nonempty closed subsets Ann=0∞ and Bnn=0∞ of a uniformly convex Banach space. As a consequence of the perturbation, such a driving self-mapping can lose its cyclic contractive nature along the transients of the iterative process. These sequences can be, in general, distinct of the initial subsets due to either computational or unmodeled perturbations associated with the self-mapping calculations through the iterative process. It is assumed that the set-theoretic limits below of the sequences of sets Ann=0∞ and Bnn=0∞ exist. The existence of fixed best proximity points in the set-theoretic limits of the sequences to which the iterated sequences converge is investigated in the case that the cyclic disposal exists under the asymptotic removal of the perturbations or under its convergence of the driving self-mapping to a limit contractive cyclic structure.

Highlights

The problem of existence of best proximity points in uniformly convex Banach spaces and in reflexive Banach spaces as well as the convergence of sequences built via cyclic contractions or cyclic φ-contractions to such points has been focused on and successfully solved in some classic pioneering works
A relevant attention has been recently devoted to the research of existence and uniqueness of fixed points of selfmappings as well as to the investigation of associated relevant properties like, for instance, stability of the iterations
The various related performed researches include the cases of strict contractive cyclic self-mappings and Meir-Keeler type cyclic contractions [3, 4, 6, 7]

Summary

Introduction

The problem of existence of best proximity points in uniformly convex Banach spaces and in reflexive Banach spaces as well as the convergence of sequences built via cyclic contractions or cyclic φ-contractions to such points has been focused on and successfully solved in some classic pioneering works. In general, that the sets involved in the cyclic disposal and their mutual distances can be subject to point-dependent perturbations so that the self-mapping is defined on the union of pairs of sequences of subsets of a normed space.

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