# Convergence Comparison of Several Iteration Algorithms for the Common Fixed Point Problems

- Yisheng Song
^{1}Email author and - Xiao Liu
^{1}

**2009**:824374

https://doi.org/10.1155/2009/824374

© Y. Song and X. Liu. 2009

**Received: **20 January 2009

**Accepted: **2 May 2009

**Published: **9 June 2009

## Abstract

We discuss the following viscosity approximations with the weak contraction for a non-expansive mapping sequence , , . We prove that Browder's and Halpern's type convergence theorems imply Moudafi's viscosity approximations with the weak contraction, and give the estimate of convergence rate between Halpern's type iteration and Mouda's viscosity approximations with the weak contraction.

## 1. Introduction

The following famous theorem is referred to as the Banach Contraction Principle.

Theorem 1.1 (Banach [1]).

Then has a unique fixed point.

In 2001, Rhoades [2] proved the following very interesting fixed point theorem which is one of generalizations of Theorem 1.1 because the weakly contractions contains contractions as the special cases .

Theorem 1.2 (Rhoades[2], Theorem 2).

for some is a continuous and nondecreasing function such that is positive on and . Then has a unique fixed point.

*nonexpansive*if

Actually, Xu [10, 11] and Wittmann [9] proved the following approximate fixed points theorem. Also see [12, 13].

Theorem 1.3.

Let be a nonempty closed convex subset of a Banach space . provided that is nonexpansive with , and is given by (1.5) and satisfies the condition , and (or ). Then is bounded and

and proved that converges to a fixed point of in a Hilbert space. They are very important because they are applied to convex optimization, linear programming, monotone inclusions, and elliptic differential equations. Xu [15] extended Moudafi's results to a uniformly smooth Banach space. Recently, Song and Chen [12, 13, 16–18] obtained a number of strong convergence results about viscosity approximations (1.8). Very recently, Petrusel and Yao [19], Wong, et al. [20] also studied the convergence of viscosity approximations, respectively.

We will prove that Browder's and Halpern's type convergence theorems imply Moudafi's viscosity approximations with the weak contraction, and give the estimate of convergence rate between Halpern's type iteration and Moudafi's viscosity approximations with the weak contraction.

## 2. Preliminaries and Basic Results

*normalized duality mapping*from into is defined by

Let denote the set of all fixed points for a mapping , that is, , and let denote the set of all positive integers. We write (resp. ) to indicate that the sequence weakly (resp. wea ) converges to ; as usual will symbolize strong convergence.

*a Gâteaux differentiable norm*(we also say that is

*smooth*), if the limit

exists for each
; (ii) *a uniformly Gâteaux differentiable norm*, if for each
in
, the limit (2.2) is uniformly attained for
; (iii) *a Fréchet differentiable norm*, if for each
, the limit (2.2) is attained uniformly for
; (iv) *a uniformly Fréchet differentiable norm* (we also say that
is *uniformly smooth*), if the limit (2.2) is attained uniformly for
. A Banach space
is said to be (v) *strictly convex* if
(vi) *uniformly convex* if for all
,
such that
For more details on geometry of Banach spaces, see [21, 22].

If
is a nonempty convex subset of a Banach space
and
is a nonempty subset of
, then a mapping
is called a *retraction* if
is continuous with
. A mapping
is called *sunny* if
whenever
and
. A subset
of
is said to be a *sunny nonexpansive retract* of
if there exists a sunny nonexpansive retraction of
onto
. We note that if
is closed and convex of a Hilbert space
, then the metric projection coincides with the sunny nonexpansive retraction from
onto
. The following lemma is well known which is given in [22, 23].

Lemma 2.1 (see [22, Lemma 5.1.6]).

Hence, there is at most one sunny nonexpansive retraction from onto .

In order to showing our main outcomes, we also need the following results. For completeness, we give a proof.

Proposition 2.2.

Let be a convex subset of a smooth Banach space . Let be a subset of and let be the unique sunny nonexpansive retraction from onto . Suppose is a weak contraction with a function on and is a nonexpansive mapping. Then

(i)the composite mapping is a weak contraction on ;

Proof.

Namely, is a weakly contractive mapping with a function . Thus, Theorem 1.2 guarantees that has a unique fixed point in , that is, satisfying (2.4) is uniquely defined for each . (i) and (ii) are proved.

Hence by the property of . This completes the proof.

*Browder's property*if for each , a sequence defined by

*Halpern's property*if for each , a sequence defined by

We know that if is a uniformly smooth Banach space or a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, is bounded, is a constant sequence , then has both Browder's and Halpern's property (see [7, 10, 11, 23], resp.).

Lemma 2.3 (see [24, Proposition 4]).

Let have Browder's property. For each , put , where is a sequence in defined by (2.10). Then is a nonexpansive mapping on .

Lemma 2.4 (see [24, Proposition 5]).

Let have Halpern's property. For each , put , where is a sequence in defined by (2.11). Then the following hold: (i) does not depend on the initial point . (ii) is a nonexpansive mapping on .

Proposition 2.5.

Let be a smooth Banach space, and have Browder's property. Then is a sunny nonexpansive retract of , and moreover, define a sunny nonexpansive retraction from to .

Proof.

By Lemma 2.1, is a sunny nonexpansive retraction from to .

We will use the following facts concerning numerical recursive inequalities (see [25–27]).

Lemma 2.6.

## 3. Main Results

We first discuss Browder's type convergence.

Theorem 3.1.

Then converges strongly to the unique point satisfying .

Proof.

Consequently, converges strongly to . This completes the proof.

We next discuss Halpern's type convergence.

Theorem 3.2.

Proof.

Consequently, we obtain the strong convergence of to , and the remainder estimates now follow from Lemma 2.6.

Theorem 3.3.

Let be a Banach space whose norm is uniformly Gâteaux differentiable, and satisfies the condition (C2). Assume that have Browder's property and for every , where is a bounded sequence in defined by (2.10). then have Halpern's property.

Proof.

(3.15) is proved. From (2.10) and , we have for all ,

where and The strong convergence of to follows from Lemma 2.6. Namely, have Halpern's property.

## 4. Deduced Theorems

Using Theorems 3.1, 3.2, and 3.3, we can obtain many convergence theorems. We state some of them.

We now discuss convergence theorems for families of nonexpansive mappings. Let be a nonempty closed convex subset of a Banach space . A (one parameter) nonexpansive semigroups is a family of selfmappings of such that

*uniformly asymptotically regular*(in short, u.a.r.) (see [28–31]) on if for all and any bounded subset of ,

Recently, Song and Xu [31] showed that have both Browder's and Halpern's property in a reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm whenever . As a direct consequence of Theorems 3.1, 3.2, and 3.3, we obtain the following.

Theorem 4.1.

Then as , both and strongly converge to , where is a sunny nonexpansive retraction from to .

Recently, Chen and Song [32] showed that have both Browder's and Halpern's property in a uniformly convex Banach space with a uniformly Gâeaux differentiable norm whenever . Then we also have the following.

Theorem 4.2.

where , and satisfies the condition , and satisfies the conditions and . Then as , both and strongly converge to , where is a sunny nonexpansive retraction from to .

## Declarations

### Acknowledgments

The authors would like to thank the editors and the anonymous referee for his or her valuable suggestions which helped to improve this manuscript.

## Authors’ Affiliations

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