Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings

Yongfu Su,Dongxing Wang,Meijuan Shang

doi:10.1155/2008/284613

Yongfu Su, Dongxing Wang + Show 1 more

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https://doi.org/10.1155/2008/284613

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Abstract

AbstractThe purpose of this article is to prove strong convergence theorems for fixed points of closed hemi-relatively nonexpansive mappings. In order to get these convergence theorems, the monotone hybrid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S. Matsushita and W. Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping. The results of this paper modify and improve the results of S. Matsushita and W. Takahashi (2005), and some others.

Highlights

In 2005, Shin-ya Matsushita and Wataru Takahashi 1 proposed the following hybrid iteration method it is called the CQ method with generalized projection for relatively nonexpansive mapping T in a Banach space E: x0 ∈ C chosen arbitrarily, yn J−1 αnJxn 1 − αn JT xn, Cn z ∈ C : φ z, yn ≤ φ z, xn, Qn z ∈ C : xn − z, Jx0 − Jxn ≥ 0, xn 1 Π Cn ∩Qn x0They proved the following convergence theorem.Fixed Point Theory and ApplicationsTheorem 1.1 MT
Note that the hybrid iteration method presented by S.Matsushita and W
Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping

Summary

Introduction

In 2005, Shin-ya Matsushita and Wataru Takahashi 1 proposed the following hybrid iteration method it is called the CQ method with generalized projection for relatively nonexpansive mapping T in a Banach space E: x0 ∈ C chosen arbitrarily, yn J−1 αnJxn 1 − αn JT xn , Cn z ∈ C : φ z, yn ≤ φ z, xn , Qn z ∈ C : xn − z, Jx0 − Jxn ≥ 0 , xn 1.

Preliminaries

Strong convergence for hemi-relatively nonexpansive mappings