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Volume 2009, Article ID 824374,13pages doi:10.1155/2009/824374

Research Article

Convergence Comparison of Several Iteration

Algorithms for the Common Fixed Point Problems

Yisheng Song and Xiao Liu

College of Mathematics and Information Science, Henan Normal University, 453007, China

Correspondence should be addressed to Yisheng Song,songyisheng123@yahoo.com.cn

Received 20 January 2009; Accepted 2 May 2009

Recommended by Naseer Shahzad

We discuss the following viscosity approximations with the weak contraction A for a non-expansive mapping sequence{Tn}, yn αnAyn 1−αnTnyn, xn1 αnAxn 1−αnTnxn. 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.

Copyrightq2009 Y. Song and X. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

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

Theorem 1.1Banach1. LetE, dbe a complete metric space and letAbe a contraction onX, that is, there existsβ∈0,1such that

dAx, Ayβdx, y, ∀x, y∈E. 1.1

ThenAhas 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 casesϕt 1−βt.

Theorem 1.2Rhoades2, Theorem 2. LetE, dbe a complete metric space, and letAbe a weak contraction onE, that is,

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for someϕ:0,∞ → 0,is a continuous and nondecreasing function such thatϕis positive on0,andϕ0 0. ThenAhas a unique fixed point.

The concept of the weak contraction is defined by Alber and Guerre-Delabriere3

in 1997. The natural generalization of the contraction as well as the weak contraction is

nonexpansive. Let K be a nonempty subset of Banach space E,T : KK is said to be

nonexpansiveif

TxTyxy, ∀x, yK. 1.3

One classical way to study nonexpansive mappings is to use a contraction to approximate a

nonexpansive mapping. More precisely, taket∈0,1and define a contractionTt:KK

byTtx tu 1−tTx, xK,whereuKis a fixed point. Banach Contraction Principle

guarantees thatTthas a unique fixed pointxtinK, that is,

xttu 1−tTxt. 1.4

Halpern4also firstly introduced the following explicit iteration scheme in Hilbert spaces:

foru, x0 ∈K,αn∈0,1,

xn1 αnu 1−αnTxn, n≥0. 1.5

In the case of T having a fixed point, Browder 5 resp. Halpern 4 proved that ifE is

a Hilbert space, then {xt} resp. {xn} converges strongly to the fixed point of T, that is,

nearest to u. Reich 6 extended Halpern’s and Browder’s result to the setting of Banach

spaces and proved that ifEis a uniformly smooth Banach space, then{xt}and{xn}converge

strongly to a same fixed point ofT, respectively, and the limit of {xt}defines theunique

sunny nonexpansive retraction fromKonto FixT. In 1984, Takahashi and Ueda7obtained

the same conclusion as Reich’s in uniformly convex Banach space with a uniformly Gˆateaux

differentiable norm. Recently, Xu8showed that the above result holds in a reflexive Banach

space which has a weakly continuous duality mapping. In 1992, Wittmann 9 studied

the iterative scheme 1.5 in Hilbert space, and obtained convergence of the iterations. In

particular, he proved a strong convergence result9, Theorem 2under the control conditions

C1lim

n→ ∞αn0, C2 ∞

n1

αn∞, C3 ∞

n1

nαn1| <∞. 1.6

In 2002, Xu10,11extended wittmann’s result to a uniformly smooth Banach space, and

gained the strong convergence of{xn}under the control conditionsC1,C2,and

C4 lim

n→ ∞ αn1

αn

1. 1.7

Actually, Xu 10, 11 and Wittmann 9 proved the following approximate fixed points

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Theorem 1.3. LetKbe a nonempty closed convex subset of a Banach spaceE. provided thatT :KKis nonexpansive withFixT/, and{xn}is given by1.5andαn ∈0,1satisfies the condition

C1,C2,andC3(orC4). Then{xn}is bounded andlimn→ ∞ xnTxn 0.

In 2000, for a nonexpansive selfmapping T with FixT/∅ and a fixed contractive

selfmappingf, Moudafi14introduced the following viscosity approximation method for

T:

xn1αnfxn 1−αnTxn, 1.8

and proved that {xn} converges to a fixed pointp of T 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 Chen12,13,16–18obtained 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.

In this paper, we naturally introduce viscosity approximations1.9and1.10with

the weak contractionAfor a nonexpansive mapping sequence{Tn},

ynαnAyn 1−αnTnyn, 1.9

xn1αnAxn 1−αnTnxn. 1.10

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

Throughout this paper, a Banach spaceEwill always be over the real scalar field. We denote

its norm by · and its dual space byE∗. The value ofx∗∈E∗atyEis denoted byy, x∗

and thenormalized duality mappingJfromEinto 2Eis defined by

Jx fE∗:x, f x f, x f , ∀x∈E. 2.1

Let FixTdenote the set of all fixed points for a mappingT, that is, FixT {x∈E:Txx},

and letNdenote the set of all positive integers. We writexn xresp.xn x∗ to indicate

that the sequencexn weaklyresp. weak∗converges tox; as usualxnxwill symbolize

strong convergence.

In the proof of our main results, we need the following definitions and results. Let

SE :{x ∈E; x 1}denote the unit sphere of a Banach spaceE.Eis said to haveia

Gˆateaux differentiable normwe also say thatEissmooth, if the limit

lim

t→0

xty− x

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exists for eachx, ySE;iia uniformly Gˆateaux differentiable norm, if for eachy inSE,

the limit 2.2 is uniformly attained for xSE; iii a Fr´echet differentiable norm, if for

each xSE, the limit 2.2 is attained uniformly forySE;iv a uniformly Fr´echet

differentiable norm we also say that E is uniformly smooth, if the limit 2.2 is attained

uniformly for x, ySE×SE. A Banach space E is said to be v strictly convex if

x y 1, x /yimplies xy/2 <1;viuniformly convexif for allε∈0,2,∃δε>0

such that x y 1 with x−y ≥ εimplies xy /2 < 1−δε.For more details on

geometry of Banach spaces, see21,22.

IfCis a nonempty convex subset of a Banach spaceE,andD is a nonempty subset

ofC, then a mappingP : CD is called aretractionif P is continuous with FixP D.

A mappingP : CD is calledsunnyifPP xtxP x P x,for allxCwhenever

P xtxP xC,andt >0. A subsetDofCis said to be asunny nonexpansive retractofC

if there exists a sunny nonexpansive retraction ofContoD. We note that ifKis closed and

convex of a Hilbert spaceE, then the metric projection coincides with the sunny nonexpansive

retraction fromContoD. The following lemma is well known which is given in22,23.

Lemma 2.1see22, Lemma 5.1.6. LetCbe nonempty convex subset of a smooth Banach space

E,/DC,J :EEthe normalized duality mapping ofE, andP :CDa retraction. Then

Pis both sunny and nonexpansive if and only if there holds the inequality:

xP x, JyP x≤0, ∀x∈C, yD. 2.3

Hence, there is at most one sunny nonexpansive retraction fromContoD.

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

Proposition 2.2. LetKbe a convex subset of a smooth Banach spaceE. LetCbe a subset ofKand letP be the unique sunny nonexpansive retraction fromKontoC. SupposeAis a weak contraction with a functionϕonK,andTis a nonexpansive mapping. Then

ithe composite mappingTAis a weak contraction onK;

iiFor eacht∈0,1, a mappingTt 1−tTtAis a weak contraction onK. Moreover,

{xt}defined by2.4is well definition:

xttAxt 1−tTxt; 2.4

iiiz PAz if and only if z is a unique solution of the following variational

inequality:

Azz, Jyz≤0, ∀y∈C. 2.5

Proof. For anyx, yK, we have

TAxT

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So,TA is a weakly contractive mapping with a functionϕ. For each fixed t ∈ 0,1,and

ψs tϕs, we have

TtxTtytAx 1−tTxtAy 1−tTy ≤1−tTxTytAxAy

≤1−txytxytϕxy

xyψxy.

2.7

Namely,Ttis a weakly contractive mapping with a functionψ. Thus,Theorem 1.2guarantees

thatTthas a unique fixed pointxtinK, that is,{xt}satisfying2.4is uniquely defined for

eacht∈0,1.iandiiare proved.

Subsequently, we showiii. Indeed, by Theorem 1.2, there exists a unique element

zK such thatz PAz. Such azCfulfils2.5byLemma 2.1. Next we show that the

variational inequality2.5has a unique solutionz. In fact, supposepCis another solution

of2.5. That is,

App, Jzp≤0, Azz, Jpz≤0. 2.8

Adding up gets

ϕpzpzpz2−ApAzpzpzApAz, Jpz≤0.

2.9

Hencezpby the property ofϕ. This completes the proof.

Let {Tn} be a sequence of nonexpansive mappings with F

n0FixTn/∅ on a

closed convex subsetKof a Banach spaceEand let{αn}be a sequence in0,1withC1.

E, K,{Tn},{αn}is said to haveBrowder’s propertyif for eachuK, a sequence{yn}defined

by

yn 1−αnTnynαnu, 2.10

for n ∈ N, converges strongly. Let {αn}be a sequence in 0,1 with C1and C2. Then

E, K,{Tn},{αn}is said to haveHalpern’s propertyif for eachuK, a sequence{yn}defined

by

yn1 1−αnTnynαnu, 2.11

forn∈N, converges strongly.

We know that if E is a uniformly smooth Banach space or a uniformly convex

Banach space with a uniformly Gˆateaux differentiable norm, K is bounded, {Tn} is a

constant sequence T, thenE, K,{Tn},{1/n} has both Browder’s and Halpern’s property

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Lemma 2.3see24, Proposition 4. LetE, K,{Tn},{αn}have Browder’s property. For eachK, putP ulimn→ ∞ yn, where{yn}is a sequence inKdefined by2.10. ThenPis a nonexpansive

mapping onK.

Lemma 2.4 see24, Proposition 5. Let E, K,{Tn},{αn} have Halpern’s property. For eachK, putP u limn→ ∞ yn, where{yn}is a sequence inKdefined by 2.11. Then the following

hold: (i)P udoes not depend on the initial pointy1. (ii)Pis a nonexpansive mapping onK.

Proposition 2.5. LetEbe a smooth Banach space, andE, K,{Tn},{αn}have Browder’s property.

Then F is a sunny nonexpansive retract of K, and moreover, P u limn→ ∞ yn define a sunny

nonexpansive retraction fromKtoF.

Proof. For eachpF, it is easy to see from2.10that

uyn, J

pyn

1−αn αn

ynpTnpTnyn, J

pyn

≤ 1−αn αn

TnpTnynpynpyn2

≤0,

2.12

uyn, J

pyn

uP u, Jpyn

P uyn, J

pyn

. 2.13

This implies for anypFand someL≥ ynp ,

uP u, Jpyn

ynP u, J

pyn

LynP u → 0. 2.14

The smoothness ofEimplies the norm weak∗continuity ofJ22, Theorems 4.3.1, 4.3.2, so

lim

n→ ∞

uP u, Jpyn

uP u, JpP u. 2.15

Thus

uP u, JpP u≤0, ∀p∈F. 2.16

ByLemma 2.1,P ulimn→ ∞ynis a sunny nonexpansive retraction fromKtoF.

We will use the following facts concerning numerical recursive inequalitiessee25–

27.

Lemma 2.6. Letn},andn}be two sequences of nonnegative real numbers, andn}a sequence

of positive numbers satisfying the conditionsn0γn∞,andlimn→ ∞ βn/αn0. Let the recursive

inequality

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be given whereψλis a continuous and strict increasing function for allλ≥0withψ0 0. Then ( 1)n}converges to zero, asn → ∞; ( 2) there exists a subsequencenk} ⊂ {λn}, k 1,2, . . . , such that

λnkψ

−1

1

nk

m0αm

βnk

αnk

,

λnk1≤ψ

−1

1

nk

m0αm

βnk

αnk

βnk,

λnλnk1−

n−1

mnk1

αm θm, nk

1< n < nk1, θm m

i0

αi,

λn1≤λ0− n

m0

αm

θmλ0, 1≤nnk−1,

1≤nksmaxmax

s;

s

m0

αm θmλ0

.

2.18

3. Main Results

We first discuss Browder’s type convergence.

Theorem 3.1. Let E, K,{Tn},{αn} have Browder’s property. For each uK, put P u

limn→ ∞yn, where{yn}is a sequence inKdefined by2.10. LetA:KKbe a weak contraction

with a functionϕ. Define a sequence{xn}inKby

xnαnAxn 1−αnTnxn, n∈N. 3.1

Then{xn}converges strongly to the unique pointzKsatisfyingPAz z.

Proof. We note that Proposition 2.2ii assures the existence and uniqueness of {xn}. It

follows fromProposition 2.2iandLemma 2.3thatP Ais a weak contraction onK, then by

Theorem 1.2, there exists the unique elementzKsuch thatPAz z. Define a sequence

{yn}inKby

ynαnAz 1−αnTnyn, for any n∈N. 3.2

Then by the assumption,{yn}converges strongly toPAz. For everyn, we have

xnyn1αnTnxnTnynαn AxnAz

≤1−αnxnynαnAxnAynαnAynAzxnynαnϕxnynαnynzϕ xnz

,

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then

ϕxnynynz. 3.4

Therefore,

lim

n→ ∞ϕxnyn≤0, i.e., nlim→ ∞xnyn0. 3.5

Hence,

lim

n→ ∞ xnz ≤nlim→ ∞xnynynz0. 3.6

Consequently,{xn}converges strongly toz. This completes the proof.

We next discuss Halpern’s type convergence.

Theorem 3.2. LetE, K,{Tn},{αn}have Halpern’s property. For eachuK, putP ulimn→ ∞yn,

where{yn}is a sequence inKdefined by2.11. LetA:KKbe a weak contraction with a function ϕ. Define a sequence{xn}inKbyx1∈Kand

xn1αnAxn 1−αnTnxn, n∈N. 3.7

Then{xn}converges strongly to the unique pointzKsatisfyingPAz z. Moreover, there exist

a subsequence{xnk} ⊂ {xn}, k1,2, . . ., and∃{εn} ⊂0,withlimn→ ∞εn0such that

yn

kxnkϕ− 1

1

nk

m0αm

εnk

,

xn

k1−ynk1≤ϕ− 1

1

nk

m0αm

εnk

αnkεnk,

xnynxn

k1−ynk1−

n−1

mnk1

αm θm, nk

1< n < nk1, θm m

i0

αi,

yn1xn1x0y0n m0

αm θm

y0x0, 1≤nnk−1,

1≤nksmaxmax

s;

s

m0

αm θm

y0x0.

3.8

Proof. It follows fromProposition 2.2iandLemma 2.4thatP Ais a weak contraction onK,

then byTheorem 1.2, there exists a unique elementzKsuch thatzPAz. Thus we may

define a sequence{yn}inKby

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Then by the assumption,ynPAzasn → ∞. For everyn, we have

xn1yn1αn AxnAz 1αnTnxnTnyn

αnAxnAynAynAz 1−αnxnynxnynαnϕxnynαnynzϕynz.

3.10

Thus, we get forλn xnyn the following recursive inequality:

λn1≤λnαnϕλn βn, 3.11

whereβnαnεn, andεn ynz . Thus byLemma 2.6,

lim

n→ ∞xnyn0. 3.12

Hence,

lim

n→ ∞ xnz ≤nlim→ ∞xnynynz0. 3.13

Consequently, we obtain the strong convergence of {xn} toz PAz, and the remainder

estimates now follow fromLemma 2.6.

Theorem 3.3. Let E be a Banach space E whose norm is uniformly Gˆateaux differentiable, andn} satisfies the condition (C2). Assume that E, K,{Tn},{αn} have Browder’s property and

limn→ ∞ ynTmyn 0 for every m ∈ N, where{yn} is a bounded sequence inK defined by

2.10. thenE, K,{Tn},{αn}have Halpern’s property.

Proof. Define a sequence{zm}inKbyuKand

zmαmu 1−αmTmzm, m∈N. 3.14

It follows fromProposition 2.5and the assumption thatP ulimm→ ∞zmis the unique sunny

nonexpansive retraction fromKtoF.Subsequently, we approved that

∀ε >0, lim sup

n→ ∞

uP u, JynP u

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In fact, sinceP uF, then we have

zmyn2 1

αm

Tmzmyn, J

zmyn

αm

uyn, J

zmyn

1−αm

TmzmTmyn, J

zmyn

Tmynyn, J

zmyn

αm

uP u, Jzmyn

αm

P uzm, J

zmyn

αm

zmyn, J

zmyn

ynzm2TmynynMαm

uP u, Jzmyn

αm zmP u M,

3.16

then

uP u, Jynzm

ynTmyn

αm MM zmP u ,

3.17

where Mis a constant such that M ≥ ynzm by the boundedness of{yn}, and {zm}.

Therefore, using limn→ ∞ ynTmyn 0, andzmP u, we get

lim sup

m→ ∞

lim sup

n→ ∞

uP u, Jynzm

≤0. 3.18

On the other hand, since the duality mapJ is norm topology to weak∗ topology uniformly

continuous in a Banach spaceEwith uniformly Gˆateaux differentiable norm, we get that as

m → ∞,

uP u, J

ynP u

Jynzm → 0, ∀n . 3.19

Therefore for anyε >0,∃N >0 such that for allm > Nand alln≥0, we have that

uP u, JynP u

<uP u, Jynzm

ε. 3.20

Hence noting3.18, we get that

lim sup

n→ ∞

uP u, JynP u

≤lim sup

m→ ∞ lim supn→ ∞

uP u, Jynzm

εε. 3.21

3.15is proved. From2.10andP uF, we have for alln≥0,

yn1P u2

αn

uP u, Jyn1−P u

1−αn

TnynP u, J

yn1−P u

≤1−αn

TnynP u2

Jyn1−P u2

2 αn

uP u, Jyn1−P u

≤1−αn

ynP u2

2

yn1P u2

2 αn

uP u, Jyn1−P u

.

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Thus,

yn1P u2

ynP u2−αnynP u22αn

uP u, Jyn1−P u

. 3.23

Consequently, we get forλn ynP u 2the following recursive inequality:

λn1 ≤λnαnψλn βn, 3.24

whereψt t,andβn2αnε.The strong convergence of{yn}toP ufollows fromLemma 2.6.

Namely,E, K,{Tn},{αn}have Halpern’s property.

4. Deduced Theorems

Using Theorems3.1,3.2, and3.3, we can obtain many convergence theorems. We state some

of them.

We now discuss convergence theorems for families of nonexpansive mappings. LetK

be a nonempty closed convex subset of a Banach spaceE. Aone parameternonexpansive

semigroups is a familyF{Tt:t >0}of selfmappings ofKsuch that

iT0xxforxK;

iiTtsxTtTsxfort, s >0,andxK;

iiilimt→0TtxxforxK;

ivfor eacht >0, Ttis nonexpansive, that is,

TtxTtyxy, ∀x, yK. 4.1

We will denote byFthe common fixed point set ofF, that is,

F:FixF {x∈K:Ttxx, t >0}

t>0

FixTt. 4.2

A continuous operator semigroupFis said to beuniformly asymptotically regularin

short, u.a.r. see28–31onKif for allh≥0 and any bounded subsetCofK,

lim

t→ ∞supxC ThTtxTtx 0. 4.3

Recently, Song and Xu31showed thatE, K,{Ttn},{αn}have both Browder’s

and Halpern’s property in a reflexive strictly convex Banach space with a uniformly

Gˆateaux differentiable norm whenevertn → ∞n → ∞. As a direct consequence

of Theorems3.1,3.2, and3.3, we obtain the following.

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that limn→ ∞ tn ∞, and βn ∈ 0,1satisfies the condition C1, and αn ∈ 0,1 satisfies the

conditionsC1andC2. If{yn}and{xn}defined by

ynβnAyn

1−βn

Ttnyn, n∈N,

xn1αnAxn 1−αnTtnxn, n≥1.

4.4

Then as n → ∞, both {yn}, and{xn} strongly converge to z PAz, where P is a sunny

nonexpansive retraction fromKtoF.

Let{tn}a sequence of positive real numbers divergent to∞, and for each t > 0 and

xK,σtxis the average given by

σtx

1

t t

0

Tsxds. 4.5

Recently, Chen and Song 32 showed that E, K,tn},{αn} have both Browder’s and

Halpern’s property in a uniformly convex Banach space with a uniformly Gˆaeaux

differentiable norm whenevertn → ∞n → ∞. Then we also have the following.

Theorem 4.2. LetEbe a uniformly convex Banach space with uniformly Gˆateaux differentiable norm,

and letK, Abe as inTheorem 4.1. Suppose that{Tt}a nonexpansive semigroups fromKinto itself such thatF:FixF t>0 FixTt/,{yn},and{xn}defined by

ynβnAyn

1−βn

σtn

yn

, n∈N,

xn1αnAxn 1−αnσtnxn, n∈N,

4.6

wheretn → ∞, andβn ∈0,1satisfies the conditionC1, andαn ∈0,1satisfies the conditions

C1andC2. Then asn → ∞, both{yn},and{xn}strongly converge tozPAz, wherePis a

sunny nonexpansive retraction fromKtoF.

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.

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References

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