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

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

**2009**:824374

**DOI: **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

for , converges strongly.

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

(i) for

(ii) for and ;

(iii) for ;

*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

## References

- Banach S:
**Sur les opérations dans les ensembles abstraits et leur application aux equations intégrales.***Fundamenta Mathematicae*1922,**3:**133–181.MATHGoogle Scholar - Rhoades BE:
**Some theorems on weakly contractive maps.***Nonlinear Analysis: Theory, Methods & Applications*2001,**47**(4):2683–2693. 10.1016/S0362-546X(01)00388-1MathSciNetView ArticleMATHGoogle Scholar - Alber YaI, Guerre-Delabriere S:
**Principle of weakly contractive maps in Hilbert spaces.**In*New Results in Operator Theory and Its Applications, Operator Theory: Advances and Applications*.*Volume 98*. Edited by: Gohberg I, Lyubich Yu. Birkhäuser, Basel, Switzerland; 1997:7–22.View ArticleGoogle Scholar - Halpern B:
**Fixed points of nonexpanding maps.***Bulletin of the American Mathematical Society*1967,**73:**957–961. 10.1090/S0002-9904-1967-11864-0MathSciNetView ArticleMATHGoogle Scholar - Browder FE:
**Fixed-point theorems for noncompact mappings in Hilbert space.***Proceedings of the National Academy of Sciences of the United States of America*1965,**53:**1272–1276. 10.1073/pnas.53.6.1272MathSciNetView ArticleMATHGoogle Scholar - Reich S:
**Strong convergence theorems for resolvents of accretive operators in Banach spaces.***Journal of Mathematical Analysis and Applications*1980,**75**(1):287–292. 10.1016/0022-247X(80)90323-6MathSciNetView ArticleMATHGoogle Scholar - Takahashi W, Ueda Y:
**On Reich's strong convergence theorems for resolvents of accretive operators.***Journal of Mathematical Analysis and Applications*1984,**104**(2):546–553. 10.1016/0022-247X(84)90019-2MathSciNetView ArticleMATHGoogle Scholar - Xu H-K:
**Strong convergence of an iterative method for nonexpansive and accretive operators.***Journal of Mathematical Analysis and Applications*2006,**314**(2):631–643. 10.1016/j.jmaa.2005.04.082MathSciNetView ArticleMATHGoogle Scholar - Wittmann R:
**Approximation of fixed points of nonexpansive mappings.***Archiv der Mathematik*1992,**58**(5):486–491. 10.1007/BF01190119MathSciNetView ArticleMATHGoogle Scholar - Xu H-K:
**Another control condition in an iterative method for nonexpansive mappings.***Bulletin of the Australian Mathematical Society*2002,**65**(1):109–113. 10.1017/S0004972700020116MathSciNetView ArticleMATHGoogle Scholar - Xu H-K:
**Iterative algorithms for nonlinear operators.***Journal of the London Mathematical Society*2002,**66**(1):240–256. 10.1112/S0024610702003332MathSciNetView ArticleMATHGoogle Scholar - Song Y, Chen R:
**Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings.***Applied Mathematics and Computation*2006,**180**(1):275–287. 10.1016/j.amc.2005.12.013MathSciNetView ArticleMATHGoogle Scholar - Song Y, Chen R:
**Viscosity approximation methods for nonexpansive nonself-mappings.***Journal of Mathematical Analysis and Applications*2006,**321**(1):316–326. 10.1016/j.jmaa.2005.07.025MathSciNetView ArticleMATHGoogle Scholar - Moudafi A:
**Viscosity approximation methods for fixed-points problems.***Journal of Mathematical Analysis and Applications*2000,**241**(1):46–55. 10.1006/jmaa.1999.6615MathSciNetView ArticleMATHGoogle Scholar - Xu H-K:
**Viscosity approximation methods for nonexpansive mappings.***Journal of Mathematical Analysis and Applications*2004,**298**(1):279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleMATHGoogle Scholar - Song Y, Chen R:
**Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces.***Nonlinear Analysis: Theory, Methods & Applications*2007,**66**(3):591–603. 10.1016/j.na.2005.12.004MathSciNetView ArticleMATHGoogle Scholar - Song Y, Chen R:
**Convergence theorems of iterative algorithms for continuous pseudocontractive mappings.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(2):486–497. 10.1016/j.na.2006.06.009MathSciNetView ArticleMATHGoogle Scholar - Song Y, Chen R:
**Viscosity approximate methods to Cesàro means for non-expansive mappings.***Applied Mathematics and Computation*2007,**186**(2):1120–1128. 10.1016/j.amc.2006.08.054MathSciNetView ArticleMATHGoogle Scholar - Petruşel A, Yao J-C:
**Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(4):1100–1111. 10.1016/j.na.2007.06.016MathSciNetView ArticleMATHGoogle Scholar - Wong NC, Sahu DR, Yao JC:
**Solving variational inequalities involving nonexpansive type mappings.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(12):4732–4753. 10.1016/j.na.2007.11.025MathSciNetView ArticleMATHGoogle Scholar - Megginson RE:
*An Introduction to Banach Space Theory, Graduate Texts in Mathematics*.*Volume 183*. Springer, New York, NY, USA; 1998:xx+596.View ArticleMATHGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis: Fixed Point Theory and Its Application*. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Reich S:
**Asymptotic behavior of contractions in Banach spaces.***Journal of Mathematical Analysis and Applications*1973,**44:**57–70. 10.1016/0022-247X(73)90024-3MathSciNetView ArticleMATHGoogle Scholar - Suzuki T:
**Moudafi's viscosity approximations with Meir-Keeler contractions.***Journal of Mathematical Analysis and Applications*2007,**325**(1):342–352. 10.1016/j.jmaa.2006.01.080MathSciNetView ArticleMATHGoogle Scholar - Alber YaI, Iusem AN:
**Extension of subgradient techniques for nonsmooth optimization in Banach spaces.***Set-Valued Analysis*2001,**9**(4):315–335. 10.1023/A:1012665832688MathSciNetView ArticleMATHGoogle Scholar - Alber Y, Reich S, Yao J-C:
**Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces.***Abstract and Applied Analysis*2003,**2003**(4):193–216. 10.1155/S1085337503203018MathSciNetView ArticleMATHGoogle Scholar - Zeng L-C, Tanaka T, Yao J-C:
**Iterative construction of fixed points of nonself-mappings in Banach spaces.***Journal of Computational and Applied Mathematics*2007,**206**(2):814–825. 10.1016/j.cam.2006.08.028MathSciNetView ArticleMATHGoogle Scholar - Aleyner A, Censor Y:
**Best approximation to common fixed points of a semigroup of nonexpansive operators.***Journal of Nonlinear and Convex Analysis*2005,**6**(1):137–151.MathSciNetMATHGoogle Scholar - Benavides TD, Acedo GL, Xu H-K:
**Construction of sunny nonexpansive retractions in Banach spaces.***Bulletin of the Australian Mathematical Society*2002,**66**(1):9–16. 10.1017/S0004972700020621MathSciNetView ArticleMATHGoogle Scholar - Aleyner A, Reich S:
**An explicit construction of sunny nonexpansive retractions in Banach spaces.***Fixed Point Theory and Applications*2005,**2005**(3):295–305. 10.1155/FPTA.2005.295MathSciNetView ArticleMATHGoogle Scholar - Song Y, Xu S:
**Strong convergence theorems for nonexpansive semigroup in Banach spaces.***Journal of Mathematical Analysis and Applications*2008,**338**(1):152–161. 10.1016/j.jmaa.2007.05.021MathSciNetView ArticleMATHGoogle Scholar - Chen R, Song Y:
**Convergence to common fixed point of nonexpansive semigroups.***Journal of Computational and Applied Mathematics*2007,**200**(2):566–575. 10.1016/j.cam.2006.01.009MathSciNetView ArticleMATHGoogle Scholar

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