Demiclosedness Principle for Total Asymptotically Nonexpansive Mappings inCAT(0)Spaces

Erdal Karapınar,Hero Salahifard,S Mansour Vaezpour

doi:10.1155/2014/738150

Abstract

We prove the demiclosedness principle for a class of mappings which is a generalization of all the forms of nonexpansive, asymptotically nonexpansive, and nearly asymptotically nonexpansive mappings. Moreover, we establish the existence theorem and convergence theorems for modified Ishikawa iterative process in the framework ofCAT(0)spaces. Our results generalize, extend, and unify the corresponding results on the topic in the literature.

Highlights

We prove the demiclosedness principle for a class of mappings which is a generalization of all the forms of nonexpansive, asymptotically nonexpansive, and nearly asymptotically nonexpansive mappings
A self-mapping T, on a metric space (X, d), is called asymptotic point-wise nonexpansive, introduced by Hussain and Khamsi [2], if there exists a sequence of mappings αn : K → [0, ∞) with lim supn → ∞ αn(x) ≤ 1 such that d (Tnx, Tny) ≤ αn (x) d (x, y), n ≥ 1, x, y ∈ X. (3)
A self-mapping T, on a metric space (X, d), is called asymptotic point-wise ψ-nonexpansive if there exists a sequence of mappings αn : K → [0, ∞) with lim supn → ∞ αn(x) ≤ 1 such that d (Tnx, Tny) ≤ αn (x) ψ (d (x, y)), n ≥ 1, x, y ∈ X. (5)

Summary

Introduction

A self-mapping T, on a metric space (X, d), is called nonexpansive, if d (Tx, Ty) ≤ d (x, y) , x, y ∈ X,. It is quite natural to extend (2) and (3) in the following way: a self-mapping T, on a metric space (X, d), is called asymptotically ψ-nonexpansive, if there exists a nonnegative sequence {kn}n≥1 with limn → ∞ kn = 1 such that d (Tnx, Tny) ≤ knψ (d (x, y)) , n ≥ 1, x, y ∈ X, (4). A self-mapping T, on a metric space (X, d), is said to be asymptotically nonexpansive in the intermediate sense, introduced by Bruck et al [4], if it is continuous and the following inequality holds: lim sup sup (d (Tnx, Tny) − d (x, y)) ≤ 0. Extend, and unify the corresponding results of [13, 20,21,22] and the references contained therein

Preliminary Remarks

Convergent Theorems

Approximation