Hybrid Iteration Method for Fixed Points of Nonexpansive Mappings in Arbitrary Banach Spaces

M O Osilike,F O Isiogugu,P U Nwokoro

doi:10.1155/2007/64306

Abstract

AbstractWe prove that recent results of Wang (2007) concerning the iterative approximation of fixed points of nonexpansive mappings using a hybrid iteration method in Hilbert spaces can be extended to arbitrary Banach spaces without the strong monotonicity assumption imposed on the hybrid operator.

Highlights

Let H be a Hilbert space, T : H→H a nonexpansive mapping with F(T) = {x ∈ H : Tx = x}=∅, and F : H→H an L-Lipschitzian mapping which is η-strongly monotone, where T is η-strongly monotone if there exists η > 0 such that
(a) {xn}∞n=1 converges weakly to a fixed point of T, (b) {xn}∞n=1 converges strongly to a fixed point of T if and only if lim inf n→∞d(xn, F(T)) = 0, where d(x, F(T)) := inf { x − p : p ∈ F(T)}. It is our purpose in this paper to extend Lemma 1.1 and Theorem 1.2 from Hilbert spaces to arbitrary Banach spaces
If T satisfies condition (A), lim inf n→∞d(xn, F(T)) = 0; so under the conditions of Theorem 3.1, if T satisfies condition (A), {xn}∞n=1 converges strongly to a fixed point of T

Summary

Introduction

A mapping T : E→E is said to be L-Lipschitzian if there exists L > 0 such that. T is said to be nonexpansive if L = 1 in (1.1). Several authors have studied various methods for the iterative approximation of fixed points of nonexpansive mappings. Wang [1] studied the following iteration method in Hilbert spaces. Let H be a Hilbert space, T : H→H a nonexpansive mapping with F(T) = {x ∈ H : Tx = x}=∅, and F : H→H an L-Lipschitzian mapping which is η-strongly monotone, where T is η-strongly monotone if there exists η > 0 such that. Tx − T y, x − y ≥ η x − y 2, ∀x, y ∈ H

Fixed Point Theory and Applications

Preliminaries

Main results