Strong convergence for asymptotically nonexpansive mappings in the intermediate sense

Gang Eun Kim

doi:10.1186/1687-1812-2014-162

Abstract

In this paper, let C be a nonempty closed convex subset of a strictly convex Banach space. Then we prove strong convergence of the modified Ishikawa iteration process when T is an ANI self-mapping such that is contained in a compact subset of C, which generalizes the result due to Takahashi and Kim (Math. Jpn. 48:1-9, 1998). MSC:47H05, 47H10.

Highlights

1 Introduction Let C be a nonempty closed convex subset of a Banach space E, and let T be a mapping of C into itself
T is said to be asymptotically nonexpansive in the intermediate sense [ ] provided T is uniformly continuous and lim sup sup Tnx – Tny – x – y ≤
We denote by F(T) the set of all fixed points of T, i.e., F(T) = {x ∈ C : Tx = x}

Summary

Introduction

Let C be a nonempty closed convex subset of a Banach space E, and let T be a mapping of C into itself. T is said to be asymptotically nonexpansive in the intermediate sense (in brief, ANI) [ ] provided T is uniformly continuous and lim sup sup Tnx – Tny – x – y ≤. Takahashi and Kim [ ] proved the following result: Let E be a strictly convex Banach space and C be a nonempty closed convex subset of E and T : C → C be a nonexpansive mapping such that T(C) is contained in a compact subset of C. [ ] Let C be a nonempty compact convex subset of a strictly convex Banach space E with r = d(C) >. Let C be a nonempty compact convex subset of a Banach space E, and let T : C → C be an ANI mapping.

Suppose that the sequence

Suppose x

If we take y