Strong and weak convergence of Ishikawa iterations for best proximity pairs

Moosa Gabeleh,S I Ezhil Manna,A Anthony Eldred,Olivier Olela Otafudu

doi:10.1515/math-2020-0002

Abstract

Abstract Let A and B be nonempty subsets of a normed linear space X. A mapping T : A ∪ B → A ∪ B is said to be a noncyclic relatively nonexpansive mapping if T(A) ⊆ A, T(B) ⊆ B and ∥Tx − Ty∥ ≤ ∥x − y∥ for all (x, y) ∈ A × B. A best proximity pair for such a mapping T is a point (p, q) ∈ A × B such that p = Tp, q = Tq and d(p, q) = dist(A, B). In this work, we introduce a geometric notion of proximal Opiaľs condition on a nonempty, closed and convex pair of subsets of strictly convex Banach spaces. By using this geometric notion, we study the strong and weak convergence of the Ishikawa iterative scheme for noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces. We also establish a best proximity pair theorem for noncyclic contraction type mappings in the setting of strictly convex Banach spaces.

Highlights

By using this geometric notion, we study the strong and weak convergence of the Ishikawa iterative scheme for noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces
Theorem 1.2. (Theorem 2.3 of [2]) Let (A, B) be a nonempty, bounded, closed and convex pair in a uniformly convex Banach space X and T : A ∪ B → A ∪ B be a noncyclic relatively nonexpansive mapping
Let (A, B) be a nonempty, closed and convex pair in a uniformly convex Banach space X such that either A or B is bounded and let T : A ∪ B → A ∪ B be a noncyclic relatively nonexpansive mapping

Summary

Introduction

It is well known that if A is a nonempty, compact and convex subset of a Banach space X, every nonexpansive mapping of A into itself has a xed point. In 1965, Kirk proved that if A is a nonempty, weakly compact and convex subset of a Banach space X with a geometric property, called normal structure, every nonexpansive self-mapping T : A → A has a xed point (Kirk’s xed point theorem [1]). A mapping T : A∪B → A ∪ B is said to be noncyclic relatively nonexpansive if T is noncyclic, that is, T(A) ⊆ A, T(B) ⊆ B, and Tx − Ty ≤ x − y for all (x, y) ∈ A × B Under this weaker assumption over T w.r.t. nonexpansiveness, the existence of the so-called best proximity pair, that is, a point (p, q) ∈ A × B such that p = Tp, q = Tq and d(p, q) = dist(A, B), was rst studied in [2] as below

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Convergence results

Therefore by

Noncyclic contraction type mappings