Convergence of Iterative Sequences for Common Zero Points of a Family of m-Accretive Mappings in Banach Spaces

Yuan Qing,Sun Young Cho,Xiaolong Qin

doi:10.1155/2011/216173

Abstract

We introduce implicit and explicit viscosity iterative algorithms for a finite family of -accretive operators. Strong convergence theorems of the iterative algorithms are established in a reflexive Banach space which has a weakly continuous duality map.

Highlights

Let E be a real Banach space, and let J denote the normalized duality mapping from E into 2E∗ given byJ x f ∈ E∗ : x, f x 2 f 2, x ∈ E, 1.1 where E∗ denotes the dual space of E and ·, · denotes the generalized duality pairing
Recall that a mapping f : K → K is said to be a contraction if there exists a constant α ∈ 0, 1 such that f x − f y ≤ α x − y, ∀x, y ∈ K
Xu 10 studied the following iterative algorithm by viscosity approximation method x0 ∈ K, xn 1 αnf xn 1 − αn T xn, n ≥ 0, 1.15 where {αn} is a real sequence 0, 1, f : K → K is a contractive mapping, and T : K → K is a nonexpansive mapping with a fixed point

Summary

Introduction

Let E be a real Banach space, and let J denote the normalized duality mapping from E into 2E∗ given by. If H is a Hilbert space, T : K → K is a nonexpansive mapping and f : K → K is a contraction, he proved the following theorems. Xu 10 studied the following iterative algorithm by viscosity approximation method x0 ∈ K, xn 1 αnf xn 1 − αn T xn, n ≥ 0, 1.15 where {αn} is a real sequence 0, 1 , f : K → K is a contractive mapping, and T : K → K is a nonexpansive mapping with a fixed point. It is proved that the sequence {xn} generated in the iterative algorithms 1.17 and 1.18 converges strongly to a common zero point of a finite family of m-accretive mappings in reflexive Banach spaces, respectively

Preliminaries

Main Results

Adding them yields that