Strong Convergence of Iterative Schemes for Zeros of Accretive Operators in Reflexive Banach Spaces

Jongsoo Jung

doi:10.1155/2010/103465

Abstract

We introduce composite iterative schemes by the viscosity iteration method for finding a zero of an accretive operator in reflexive Banach spaces. Then, under certain differen control conditions, we establish strong convergence theorems on the composite iterative schemes. The main theorems improve and develop the recent corresponding results of Aoyama et al. (2007), Chen and Zhu (2006, 2008), Jung (2010), Kim and Xu (2005), Qin and Su (2007) and Xu (2006) as well as Benavides et al. (2003), Kamimura and Takahashi (2000), Maingé (2006), and Nakajo (2006).

Highlights

Let E be a real Banach space and C a nonempty closed convex subset of E
Recall that a mapping f : C → C is a contraction on C if there exists a constant k ∈ 0, 1 such that f x −f y ≤ k x−y, x, y ∈ C
A mapping T : C → C is said to be nonexpansive if T x − T y ≤ x − y, x, y ∈ C, and F T denote the set of fixed points of T ; that is, F T {x ∈ C : x T x}

Summary

Introduction

Let E be a real Banach space and C a nonempty closed convex subset of E. In 2007, Qin and Su 19 considered the following iterative scheme in either a uniformly smooth Banach space or a reflexive Banach space having a weakly sequentially continuous duality mapping, which is a simpler modification of the iterative scheme 1.2 : for resolvent Jrn of an m-accretive operator A, u ∈ D A and {αn}, {βn} ⊂ 0, 1 , x0 x ∈ E, yn βnxn 1 − βn Jrn xn, 1.4 xn 1 αnu 1 − αn yn, n ≥ 0 They proved that the sequence {xn} generated by 1.4 converges strongly to a zero of an m-accretive operator A under the conditions C1 , C2 , and C3 on {αn} and the condition. Our results complement the corresponding results of Benavides et al and Kamimura and Takahashi

Preliminaries and Lemmas

Main Results

14 Putting aj t