A Weak Convergence Theorem for Total Asymptotically Pseudocontractive Mappings in Hilbert Spaces

Xiaolong Qin,Sunyoung Cho,Shinmin Kang

doi:10.1155/2011/859795

Abstract

AbstractThe modified Ishikawa iterative process is investigated for the class of total asymptotically pseudocontractive mappings. A weak convergence theorem of fixed points is established in the framework of Hilbert spaces.

Highlights

Introduction and PreliminariesThroughout this paper, we always assume that H is a real Hilbert space, whose inner product and norm are denoted by ·, · and · . → and are denoted by strong convergence and weak convergence, respectively
T is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds: lim sup sup T nx − T ny − x − y ≤ 0
The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al 4 see 5. It is known 6 that if C is a nonempty closed convex bounded subset of a uniformly convex Banach space E and T is asymptotically nonexpansive in the intermediate sense, T has a fixed point

Summary

Introduction

Introduction and PreliminariesThroughout this paper, we always assume that H is a real Hilbert space, whose inner product and norm are denoted by ·, · and · . → and are denoted by strong convergence and weak convergence, respectively. They proved that if C is a nonempty closed convex bounded subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive mapping on C, T has a fixed point.

Objectives

Results

Conclusion