Construction of Fixed Points by Some Iterative Schemes

A El-Sayed Ahmed,A Kamal

doi:10.1155/2009/612491

A El-Sayed Ahmed, A Kamal

Open Access

https://doi.org/10.1155/2009/612491

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Abstract

We obtain strong convergence theorems of two modifications of Mann iteration processes with errors in the doubly sequence setting. Furthermore, we establish some weakly convergence theorems for doubly sequence Mann's iteration scheme with errors in a uniformly convex Banach space by a Frechet differentiable norm.

Highlights

Let X be a real Banach space and let C be a nonempty closed convex subset of X
It is assumed throughout this paper that T is a nonexpansive mapping such that Fix T / ∅
Let X be a uniformly smooth Banach space and let T : C → C be a nonexpansive mapping with a fixed point

Summary

Introduction

Let X be a real Banach space and let C be a nonempty closed convex subset of X. The following modification of Mann’s iteration method 1.1 in a Hilbert space H is given by Nakajo and Takahashi 6 : x0 x ∈ C, yn αnxn 1 − αn T xn, Cn z ∈ C : yn − z ≤ xn − z , Qn z ∈ C : xn − z, x0 − xn ≥ 0 , xn 1 P cn Qn x0 , where PK denotes the metric projection from H onto a closed convex subset K of H They proved that if the sequence {αn} is bounded from one, {xn} defined by 1.2 converges strongly to PFix T x0. In 7 , it is proved, in a uniformly smooth Banach space and under certain appropriate assumptions on the sequences {αn} and {rn}, that {xn} defined by 1.4 converges strongly to a zero of A

Preliminaries

A Fixed Point of Nonexpansive Mappings

Convergence to a Zero of Accretive Operator

Weakly Convergence Theorems