Strong Convergence Theorems of the General

(1)

Volume 2011, Article ID 643740,21pages doi:10.1155/2011/643740

Research Article

Strong Convergence Theorems of the General

Iterative Methods for Nonexpansive Semigroups in Banach Spaces

Rattanaporn Wangkeeree

^{1, 2}

1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Correspondence should be addressed to Rattanaporn Wangkeeree,rattanapornw@nu.ac.th Received 4 February 2011; Accepted 22 March 2011

Academic Editor: Yonghong Yao

Copyrightq2011 Rattanaporn Wangkeeree. 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.

LetEbe a real reflexive Banach space which admits a weakly sequentially continuous duality mapping fromEto E^∗. LetS {Ts : 0 ≤ s < ∞}be a nonexpansive semigroup onEsuch that FixS :

t≥0FixTt/∅, andf is a contraction onE with coeﬃcient 0 < α < 1. Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ λ > 1 and γ a positive real number such thatγ < 1/α1−

1−δ/λ. When the sequences of real numbers{αn}and {tn} satisfy some appropriate conditions, the three iterative processes given as follows:xn1 αnγfxn I −αnFTtnxn, n ≥ 0, yn1 αnγfTtnyn I −αnFTtnyn, n ≥ 0, and zn1Ttnαnγfzn I−αnFzn,n≥0 converge strongly tox, where xis the unique solution in FixSof the variational inequalityF−γfx, jx −x ≥ 0,x∈FixS. Our results extend and improve corresponding ones of Li et al.2009Chen and He2007, and many others.

1. Introduction

LetEbe a real Banach space. A mappingT ofEinto itself is said to be nonexpansive ifTx− Ty ≤ x−yfor eachx, y∈E. We denote by FixTthe set of fixed points ofT. A mapping f:E → Eis calledα-contraction if there exists a constant 0< α <1 such thatfx−fy ≤ αx−yfor allx, y∈E. A familyS{Tt: 0≤t <∞}of mappings ofEinto itself is called a nonexpansive semigroup onEif it satisfies the following conditions:

iT0xxfor allx∈E;

iiTst TsTtfor alls, t≥0;

(2)

iiiTtx−Tty ≤ x−yfor allx, y∈Eandt≥0;

ivfor allx∈E, the mappingt→Ttxis continuous.

We denote by FixSthe set of all common fixed points ofS, that is, FixS:{x∈E:Ttxx,0≤t <∞}

t≥0

FixTt. 1.1

In1, Shioji and Takahashi introduced the following implicit iteration in a Hilbert space

xnαnx 1−αn1 tn

_t_n

0

Tsxnds, ∀n∈, 1.2

where {αn} is a sequence in 0,1 and {tn} is a sequence of positive real numbers which diverges to ∞. Under certain restrictions on the sequence {αn}, Shioji and Takahashi 1 proved strong convergence of the sequence{xn}to a member ofFS. In2, Shimizu and Takahashi studied the strong convergence of the sequence{xn}defined by

xn1αnx 1−αn1 tn

_t_n

0

Tsxnds, ∀n∈ 1.3

in a real Hilbert space where{Tt : t ≥ 0} is a strongly continuous semigroup of nonexpansive mappings on a closed convex subsetC of a Banach spaceE and limn→ ∞tn ∞.

Using viscosity method, Chen and Song3studied the strong convergence of the following iterative method for a nonexpansive semigroup{Tt : t ≥ 0}with FixS/∅ in a Banach space:

xn1 αnfx 1−αn1 tn

_t_n

0

Tsxnds, ∀n∈, 1.4

wheref is a contraction. Note however that their iteratexnat stepnis constructed through the average of the semigroup over the interval0, t. Suzuki 4was the first to introduce again in a Hilbert space the following implicit iteration process:

xnαnu 1−αnTtnxn, ∀n∈, 1.5 for the nonexpansive semigroup case. In 2002, Benavides et al.5, in a uniformly smooth Banach space, showed that ifSsatisfies an asymptotic regularity condition and{αn}fulfills the control conditions lim_{n→ ∞}αn 0,_∞

n1αn ∞, and limn→ ∞αn/αn1 0, then both the implicit iteration process1.5and the explicit iteration process1.6,

xn1αnu 1−αnTtnxn, ∀n∈, 1.6 converge to a same point of FS. In 2005, Xu 6 studied the strong convergence of the implicit iteration process1.2and1.5in a uniformly convex Banach space which admits a

(3)

weakly sequentially continuous duality mapping. Recently, Chen and He7introduced the viscosity approximation process:

xn1αnfxn

1−βn Ttnxn, ∀n∈, 1.7

wherefis a contraction and{αn}is a sequence in0,1and a nonexpansive semigroup{Tt: t≥0}. The strong convergence theorem of{xn}is proved in a reflexive Banach space which admits a weakly sequentially continuous duality mapping. In8, Chen et al. introduced and studied modified Mann iteration for nonexpansive mapping in a uniformly convex Banach space.

On the other hand, iterative approximation methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example,9–11and the references therein. Let H be a real Hilbert space, whose inner product and norm are denoted by·,·and · , respectively. LetAbe a strongly positive bounded linear operator onH; that is, there is a constantγ >0 with property

Ax, x ≥γx² ∀x∈H. 1.8

A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH:

minx∈C

1

2Ax, x − x, b, 1.9

whereCis the fixed point set of a nonexpansive mappingTonHandbis a given point inH.

In 2003, Xu10proved that the sequence{xn}defined by the iterative method below, with the initial guessx0∈Hchosen arbitrarily,

x_n1 I−αnATxnαnu, n≥0, 1.10 converges strongly to the unique solution of the minimization problem 1.9 provided the sequence {αn} satisfies certain conditions. Using the viscosity approximation method, Moudafi12introduced the following iterative process for nonexpansive mappingssee13 for further developments in both Hilbert and Banach spaces. Letf be a contraction onH.

Starting with an arbitrary initialx0∈H, define a sequence{xn}recursively by

xn1 1−αnTxnαnfxn, n≥0, 1.11 where{αn}is a sequence in0,1. It is proved12,13that, under certain appropriate conditions imposed on{αn}, the sequence{xn}generated by1.11 strongly converges to the unique solutionx^∗inCof the variational inequality

I−f x^∗, x−x^∗

≥0, x∈H. 1.12

(4)

Recently, Marino and Xu14mixed the iterative method1.10and the viscosity approximation method1.11and considered the following general iterative method:

xn1 I−αnATxnαnγfxn, n≥0, 1.13 where A is a strongly positive bounded linear operator on H. They proved that if the sequence{αn}of parameters satisfies the certain conditions, then the sequence{xn}generated by1.13converges strongly to the unique solutionx^∗inHof the variational inequality

A−γf x^∗, x−x^∗

≥0, x∈H 1.14

which is the optimality condition for the minimization problem, min_x∈C1/2Ax, x −hx, wherehis a potential function forγf i.e., hx γfxforx∈H.

Very recently, Li et al. 15 introduced the following iterative procedures for the approximation of common fixed points of a one-parameter nonexpansive semigroup on a Hilbert spaceH:

x0x∈H, xn1 I−αnA1 tn

_t_n

0

Tsxndsαnγfxn, n≥0, 1.15

whereAis a strongly positive bounded linear operator onH.

Letδ andλbe two positive real numbers such thatδ, λ <1. Recall that a mappingF with domainDFand rangeRFinEis calledδ-strongly accretive if, for eachx, y∈DF, there existsjx−y∈Jx−ysuch that

Fx−Fy, j

x−y ≥δx−y², 1.16

whereJ is the normalized duality mapping fromEinto the dual spaceE^∗. Recall also that a mappingFis calledλ-strictly pseudocontractive if, for eachx, y∈DF, there existsjx−y∈ Jx−ysuch that

Fx−Fy, j

x−y ≤x−y²−λx−y −

Fx−Fy ². 1.17

It is easy to see that1.17can be rewritten as I−Fx−I−Fy, j

x−y ≥λI−Fx−I−Fy², 1.18 see16.

In this paper, motivated by the above results, we introduce and study the strong convergence theorems of the general iterative scheme{xn}defined by1.19in the framework of a reflexive Banach spaceEwhich admits a weakly sequentially continuous duality mapping:

x0 x∈E, xn1αnγfxn I−αnFTtnxn, n≥0, 1.19

(5)

whereF isδ-strongly accretive andλ-strictly pseudocontractive withδλ > 1,f is a contraction onEwith coeﬃcient 0< α <1,γ is a positive real number such thatγ <1/α1− 1−δ/λ, andS {Tt : 0 ≤ t < ∞}is a nonexpansive semigroup on E. The strong convergence theorems are proved under some appropriate control conditions on parameters {αn}and{tn}. Furthermore, by using these results, we obtain strong convergence theorems of the following new general iterative schemes{yn}and{zn}defined by

y0y∈E, yn1 αnγf

Ttnyn I−αnFTtnyn, n≥0, 1.20 z0 z∈E, zn1Ttn

αnγfzn I−αnFzn , n≥0. 1.21 The results presented in this paper extend and improve the main results in Li et al.15, Chen and He7, and many others.

2. Preliminaries

Throughout this paper, it is assumed thatEis a real Banach space with norm · and letJ denote the normalized duality mapping fromEintoE^∗given by

Jx

f ∈E^∗: x, f

x²f²

2.1 for eachx∈ E, whereE^∗denotes the dual space ofE, ·,·denotes the generalized duality pairing, and denotes the set of all positive integers. In the sequel, we will denote the single-valued duality mapping by j, and considerFT {x ∈ C : Tx x}. When{xn} is a sequence in E, then xn → x resp., xn x, xn ∗

x will denote strong resp., weak, weak^∗ convergence of the sequence {xn} to x. In a Banach space E, the following resultthe subdiﬀerential inequalityis well known17, Theorem 4.2.1: for allx, y∈E, for all jxy∈Jxy, for alljx∈Jx,

x²2 y, jx

≤xy²≤ x² y, j

xy . 2.2

A real Banach spaceE is said to be strictly convex if xy/2 < 1 for all x, y ∈ E with xy1 andx /y. It is said to be uniformly convex if, for all∈0,2, there exitsδ>0 such that

xy1 withx−y≥ implies xy

2 <1−δ. 2.3 The following results are well known and can be founded in17:

ia uniformly convex Banach spaceEis reflexive and strictly convex17, Theorems 4.2.1 and 4.1.6,

iiifEis a strictly convex Banach space andT :E → Eis a nonexpansive mapping, then fixed point setFTofT is a closed convex subset ofE17, Theorem 4.5.3.

(6)

If a Banach spaceEadmits a sequentially continuous duality mappingJ from weak topology to weak star topology, then from Lemma 1 of 18, it follows that the duality mappingJis single-valued and alsoEis smooth. In this case, duality mappingJis also said to be weakly sequentially continuous, that is, for each{xn} ⊂Ewithxn x, thenJxn J^∗ x see18,19.

In the sequel, we will denote the single-valued duality mapping byj. A Banach space Eis said to satisfy Opial’s condition if, for any sequence{xn}inE,xn xasn → ∞implies

lim sup

n→ ∞ xn−x<lim sup

n→ ∞

xn−y ∀y∈Ewithx /y. 2.4

By Theorem 1 of18, we know that ifEadmits a weakly sequentially continuous duality mapping, thenEsatisfies Opial’s condition andEis smooth; for the details, see18.

Now, we present the concept of uniformly asymptotically regular semigroupalso see 20,21. Let Cbe a nonempty closed convex subset of a Banach spaceE,S {Tt : 0 ≤ t <∞}a continuous operator semigroup onC. Then,Sis said to be uniformly asymptotically regularin short, u.a.r.onCif, for allh≥0 and any bounded subsetDofC,

t→ ∞limsup

x∈D ThTtx−Ttx0. 2.5

The nonexpansive semigroup{σt :t > 0}defined by the following lemma is an example of u.a.r. operator semigroup. Other examples of u.a.r. operator semigroup can be found in20, Examples 17 and 18.

Lemma 2.1see3, Lemma 2.7. LetCbe a nonempty closed convex subset of a uniformly convex Banach spaceE,Da bounded closed convex subset ofC, andS{Ts: 0≤s <∞}a nonexpansive semigroup onCsuch thatFS/∅. For eachh >0, setσtx 1/t_t

0Tsxds, then

t→ ∞limsup

x∈Dσtx−Thσtx0. 2.6

Example 2.2. The set{σt:t > 0}defined byLemma 2.1is u.a.r. nonexpansive semigroup. In fact, it is obvious that{σt:t >0}is a nonexpansive semigroup. For eachh >0, we have

σtx−σhσtx

σtx− 1 h

_h

0

Tsσtxds

1 h

_h

0

σtx−Tsσtxds

≤ 1 h

_h

0

σtx−Tsσtxds.

2.7

(7)

ApplyingLemma 2.1, we have

t→ ∞lim sup

x∈Dσtx−σhσtx ≤ 1 h

_h

0

t→ ∞lim sup

x∈Dσtx−Tsσtxds0. 2.8 LetCbe a nonempty closed and convex subset of a Banach spaceEandDa nonempty subset ofC. A mappingQ:C → Dis said to be sunny if

QQxtx−Qx Qx, 2.9

wheneverQxtx−Qx∈Cforx∈Candt0. A mappingQ:C → Dis called a retraction ifQx xfor allx ∈ D. Furthermore,Qis a sunny nonexpansive retraction fromContoD ifQis a retraction fromContoDwhich is also sunny and nonexpansive. A subsetDofCis called a sunny nonexpansive retraction ofCif there exists a sunny nonexpansive retraction fromContoD. The following lemma concerns the sunny nonexpansive retraction.

Lemma 2.3see22,23. LetCbe a closed convex subset of a smooth Banach spaceE. LetDbe a nonempty subset ofCandQ : C → D be a retraction. Then,Qis sunny and nonexpansive if and only if

u−Qu, j

y−Qu ≤0 2.10

for allu∈Candy∈D.

Lemma 2.4see24, Lemma 2.3. Let{an}be a sequence of nonnegative real numbers satisfying the property

an1≤1−tnantncnbn, ∀n≥0, 2.11 where{tn},{bn}, and{cn}satisfy the restrictions

i_∞

n1tn∞;

ii_∞

n1bn<∞;

iiilim sup_n_{→ ∞}cn≤0.

Then, lim_n_{→ ∞}an0.

The following lemma will be frequently used throughout the paper and can be found in25.

Lemma 2.5see25, Lemma 2.7. LetEbe a real smooth Banach space andF :E → Ea mapping.

iIfFisδ-strongly accretive andλ-strictly pseudocontractive withδλ > 1, thenI−F is contractive with constant

1−δ/λ.

iIfFisδ-strongly accretive andλ-strictly pseudocontractive withδλ >1, then, for any fixed numberτ ∈0,1,I−τFis contractive with constant 1−τ1−

1−δ/λ.

(8)

3. Main Results

Now, we are in a position to state and prove our main results.

Theorem 3.1. Let E be a reflexive Banach space which admits a weakly sequentially continuous duality mappingJ. LetS {Tt : 0 ≤ t < ∞}be a u.a.r. nonexpansive semigroup on Esuch that FixS/∅. Suppose that the real sequences{αn} ⊂0,1,{tn} ⊂0,∞satisfy the conditions

n→ ∞limαn0,

∞ n0

αn∞, lim

n→ ∞tn∞. 3.1

LetF beδ-strongly accretive andλ-strictly pseudocontractive withδλ > 1,f : E → Ea con- traction mapping with coeﬃcientα ∈0,1, andγa positive real number such thatγ <1/α1− 1−δ/λ. Then, the sequence {xn} defined by 1.19 converges strongly tox, where x is the unique solution in FixSof the variational inequality

F−γf x, jx −x

≥0, x∈FixS 3.2

or equivalentlyx QFixSI−Fγfx, whereQFixS is the sunny nonexpansive retraction ofE onto FixS.

Proof. Note that FixSis a nonempty closed convex set. We first show that{xn}is bounded.

Letq∈FixS. Thus, byLemma 2.5, we have

xn1−qαnγfxn I−αnFTtnxn−I−αnFq−αnFq

≤αnγfxn−FqI−αnFTtnxn−q

≤αnγfxn−f

q αnγf

q −FqI−αnFxn−q

≤αnαγxn−qαnγf

q −Fq

⎛

⎝1−αn

⎛

⎝1−

1−δ λ

⎞

⎠

⎞

⎠xn−q

⎛

⎝1−αn

⎛

⎝1−

1−δ λ −αγ

⎞

⎠

⎞

⎠xn−q αn

⎛

⎝1−

1−δ λ −αγ

⎞

⎠ γf

q −Fq 1−

1−δ/λ−αγ

≤max

xn−q, 1 1−

1−δ/λ−αγγf

q −Fq

, ∀n≥0.

3.3

(9)

By induction, we get xn−q≤max

x0−q, 1 1−

1−δ/λ−αγ γf

q −Fq

, n≥0. 3.4

This implies that{xn}is bounded and, hence, so are{fxn}and{FTtnxn}. This implies that

nlim→ ∞xn1−Ttnxn lim

n→ ∞αnγfxn−FTtnxn0. 3.5 Since{Tt}is a u.a.r. nonexpansive semigroup and limn→ ∞tn∞, we have, for allh >0,

nlim→ ∞ThTtnxn−Ttnxn ≤ lim

n→ ∞ sup

x∈{xn}ThTtnx−Ttnx0. 3.6 Hence, for allh >0,

xn1−Thxn1 ≤ xn1−TtnxnTtnxn−ThTtnxnThTtnxn−Thxn1

≤2x_n1−TtnxnTtnxn−ThTtnxn −→0.

3.7

That is, for allh >0,

nlim→ ∞xn−Thxn0. 3.8

LetΦ QFixS. Then,ΦI−F−γfis a contraction onE. In fact, fromLemma 2.5i, we have Φ

I−F−γf x−Φ

I−F−γf y≤I−F−γf x−

I−F−γf y

≤I−Fx−I−Fyγfx−f y

≤

1−δ

λ x−yαγx−y

⎛

⎝

1−δ λ αγ

⎞

⎠x−y, ∀x, y∈E.

3.9

Therefore,ΦI−F−γfis a contraction onEdue to

1−δ/λαγ∈0,1. Thus, by Banach contraction principle,QFixSI−F−γfhas a unique fixed pointx. Then, using Lemma 2.3,

xis the unique solution in FixSof the variational inequality3.2. Next, we show that lim sup

n→ ∞

γfx−Fx, j xn−x

≤0. 3.10

(10)

Indeed, we can take a subsequence{xnk}of{xn}such that

lim sup

n→ ∞

γfx−Fx, j xn−x lim

k→ ∞

γfx−Fx, jx nk −x

. 3.11

We may assume thatxnk p∈Eask → ∞, since a Banach spaceEhas a weakly sequentially continuous duality mapping J satisfying Opial’s condition 13. We will prove that p ∈ FixS. Suppose the contrary,p /∈ FixS, that is,Th0p /pfor someh0 >0. It follows from 3.8and Opial’s condition that

lim inf

k→ ∞ xnk−p<lim inf

k→ ∞ xnk−Th0p

≤lim inf

k→ ∞

xnk−Th0xnkTh0xnk −Th0p

≤lim inf

k→ ∞

xnk −Th0xnkxnk−p lim inf

k→ ∞ xnk−p.

3.12

This is a contradiction, which shows thatp ∈ FTh for allh > 0, that is,p ∈ FixS. In view of the variational inequality3.2and the assumption that duality mappingJis weakly sequentially continuous, we conclude

lim sup

n→ ∞

γfx −Fx, jx n−x lim

k→ ∞

γfx−Fx, j xnk −x

≤

γfx −Fx, j

p−x ≤0.

3.13

Finally, we will show thatxn → x. For each n≥0, we have

xn1−x ²αnγfxn I−αnFTtnxn−I−αnFx−αnFx²

≤αnγfxn−αnFx I−αnFTtnxn−I−αnFx² I−αnFTtnxn−I−αnFx²2αn

γfxn−Fx, jx n1−x

≤

⎛

⎝1−αn

⎛

⎝1−

1−δ λ

⎞

⎠

⎞

⎠

2

xn−x ²2αn

γfxn−γfx, j xn1−x

2αn

γfx −Fx, jx n1−x .

3.14

(11)

On the other hand,

γfxn−γfx, j xn1−x

≤γαxn−xx n1−x

≤γαxn−x

⎡

⎢⎢

⎣

⎛

⎝1−αn

⎛

⎝1−

1−δ λ

⎞

⎠

⎞

⎠

2

xn−x ²2αn γfxn−Fx, j x_n1−x

⎤

⎥⎥

⎦

≤γα

⎛

⎝1−αn

⎛

⎝1−

1−δ λ

⎞

⎠

⎞

⎠xn−x ²

γαxn−x $

2 γfxn−Fx, j xn1−x √ αn

≤γα

⎛

⎝1−αn

⎛

⎝1−

1−δ λ

⎞

⎠

⎞

⎠xn−x ²√ αnM0,

3.15 whereM0is a constant satisfyingM0≥γαxn−x $

2|γfxn−Fx, jx n1−x|. Substitut- ing3.15in3.14, we obtain

xn1−x ²≤

⎛

⎝1−αn

⎛

⎝1−

1−δ λ

⎞

⎠

⎞

⎠

2

xn−x ²2αnγα

⎛

⎝1−αn

⎛

⎝1−

1−δ λ

⎞

⎠

⎞

⎠

× xn−x ²2αn√

αnM02αn

γfx−Fx, j xn1−x

⎛

⎜⎝1−2αn

⎛

⎝1−

1−δ λ

⎞

⎠α²_n

⎛

⎝1−

1−δ λ

⎞

⎠

2⎞

⎟⎠xn−x ²

2αnγα

⎛

⎝1−αn

⎛

⎝1−

1−δ λ

⎞

⎠

⎞

⎠xn−x ²

2αn√

αnM02αn

γfx −Fx, jx n1−x

⎛

⎝1−2αn

⎡

⎣

⎛

⎝1−

1−δ λ

⎞

⎠−αγαnγα

⎛

⎝1−

1−δ λ

⎞

⎠

⎤

⎦

⎞

⎠xn−x ²

αn

⎡

⎢⎣αn

⎛

⎝1−

1−δ λ

⎞

⎠

2

xn−x ²2M0

√αn2

⎤

⎥⎦

1−αnγn xn−x ²αnγn

βn

γn,

3.16

(12)

where

γn2

⎡

⎣

⎛

⎝1−

1−δ λ

⎞

⎠−αγαnγα

⎛

⎝1−

1−δ λ

⎞

⎠

⎤

⎦,

βn

⎡

⎢⎣αn

⎛

⎝1−

1−δ λ

⎞

⎠

2

xn−x ²2M0√ αn2

⎤

⎥⎦.

3.17

It is easily seen that_∞

n1αnγn∞. Since{xn}is bounded and limn→ ∞αn0, by3.46, we obtain lim sup_{n→ ∞}βn/γn≤0, applyingLemma 2.4to3.16to concludexn → xasn → ∞.

This completes the proof.

UsingTheorem 3.1, we obtain the following two strong convergence theorems of new iterative approximation methods for a nonexpansive semigroup{Tt: 0≤t <∞}.

Corollary 3.2. Let E be a reflexive Banach space which admits a weakly sequentially continuous duality mappingJ. LetS {Tt : 0≤ t <∞}be a u.a.r. nonexpansive semigroup onEsuch that FixS/∅. Suppose that the real sequences{αn} ⊂0,1,{tn} ⊂0,∞satisfy the conditions

n→ ∞limαn0,

∞ n0

αn∞, lim

n→ ∞tn∞. 3.18

LetF beδ-strongly accretive andλ-strictly pseudocontractive withδλ > 1,f : E → Ea con- traction mapping with coeﬃcientα ∈0,1, andγa positive real number such thatγ <1/α1− 1−δ/λ. Then, the sequence {yn} defined by 1.20 converges strongly tox, where x is the unique solution in FixSof the variational inequality

≥0, x∈FixS 3.19

Proof. Let{xn}be the sequence given byx0y0and

xn1 αnγfxn I−αnFTtnxn, ∀n≥0. 3.20 FormTheorem 3.1,xn → x. We claim that yn → x. Indeed, we estimate

xn1−yn1

≤αnγf

Ttnyn −fxnI−αnFTtnxn−Ttnyn

≤αnγαTtnyn−xn

⎛

⎝1−αn

⎛

⎝1−

1−δ λ

⎞

⎠

⎞

⎠xn−yn

(13)

≤αnγαTtnyn−TtnxαnγαTtnx−xn

⎛

⎝1−αn

⎛

⎝1−

1−δ λ

⎞

⎠

⎞

⎠xn−yn

≤αnγαyn−xαnγαx−xn

⎛

⎝1−αn

⎛

⎝1−

1−δ λ

⎞

⎠

⎞

⎠xn−yn

≤αnγαyn−xnαnγαxn−x αnγαx−xn

⎛

⎝1−αn

⎛

⎝1−

1−δ λ

⎞

⎠

⎞

⎠xn−yn

⎛

⎝1−αn

⎛

⎝1−

1−δ λ −γα

⎞

⎠

⎞

⎠xn−yn αn

⎛

⎝1−

1−δ λ −γα

⎞

⎠ 2αγ '

1−

1−δ/λ−γα(x−xn.

3.21 It follows from _∞

n1αn ∞, limn→ ∞xn −x 0, and Lemma 2.4 thatxn −yn → 0.

Consequently,yn → xas required.

Corollary 3.3. Let E be a reflexive Banach space which admits a weakly sequentially continuous duality mappingJ. LetS {Tt : 0≤ t <∞}be a u.a.r. nonexpansive semigroup onEsuch that FixS/∅. Suppose that the real sequences{αn} ⊂0,1,{tn} ⊂0,∞satisfy the conditions

n→ ∞limαn0,

∞ n0

αn∞, lim

n→ ∞tn∞. 3.22

LetF beδ-strongly accretive andλ-strictly pseudocontractive withδ λ > 1,f : E → Eacon- traction mapping with coeﬃcientα ∈0,1, andγa positive real number such thatγ <1/α1− 1−δ/λ. Then, the sequence{zn}defined by1.21converges strongly tox, where xis the unique solution in FixSof the variational inequality

≥0, x∈FixS 3.23

Proof. Define the sequences{yn}and{βn}by

yn αnγfzn I−αnFzn, βnαn1 ∀n∈. 3.24

(14)

Takingp∈FixS, we have

zn1−pTtnyn−Ttnp≤yn−p

αnγfzn I−αnFzn−I−αnFp−αnFp

≤

⎛

⎝1−αn

⎛

⎝1−

1−δ λ

⎞

⎠

⎞

⎠zn−pαnγfzn−F p

⎛

⎝1−αn

⎛

⎝1−

1−δ λ

⎞

⎠

⎞

⎠zn−pαn

⎛

⎝1−

1−δ λ

⎞

⎠γfzn−F ' p

1−

1−δ/λ(. 3.25

It follows from induction that zn1−p≤max

z0−p,γfz0−F p 1−

1−δ/λ

, n≥0. 3.26

Thus, both{zn}and{yn}are bounded. We observe that yn1αn1γfzn1 I−αn1Fzn1βnγf

Ttnyn

I−βnF Ttnyn. 3.27 Thus,Corollary 3.2implies that{yn}converges strongly to some pointx. In this case, we also have

zn−x ≤ zn−ynyn−xαnγfzn−Fznyn−x−→0. 3.28 Hence, the sequence{zn}converges strongly to some pointx. This complete the proof.

UsingTheorem 3.1,Lemma 2.1, andExample 2.2, we have the following result.

Corollary 3.4. Let E be a uniformly convex Banach space which admits a weakly sequentially continuous duality mappingJ. LetS {Tt: 0≤t <∞}be a nonexpansive semigroup onEsuch that FixS/∅. Suppose that the real sequences{αn} ⊂0,1,{tn} ⊂0,∞satisfy the conditions

n→ ∞limαn0,

∞

n0αn∞, lim

n→ ∞tn∞. 3.29

LetF beδ-strongly accretive andλ-strictly pseudocontractive withδλ > 1,f : E → Ea con- traction mapping with coeﬃcientα ∈0,1, andγa positive real number such thatγ <1/α1− 1−δ/λ. Then, the sequence{xn}defined by

x0x∈E, xn1 αnγfxn I−αnF1

tn

_t_n

0

Ttxnds, n≥0

3.30

(15)

converges strongly tox, where xis the unique solution in FixSof the variational inequality F−γf x, jx −x

≥0, x∈FixS 3.31

or equivalentlyxQFixSI−Fγfx, whereQFixSis the sunny nonexpansive retraction ofE onto FixS.

Corollary 3.5. LetH be a real Hilbert space. Let S {Tt : 0 ≤ t < ∞} be a nonexpansive semigroup onHsuch that FixS/∅. Suppose that the real sequences{αn} ⊂ 0,1,{tn} ⊂0,∞ satisfy the conditions

nlim→ ∞αn 0, ∞ n0

αn∞, lim

n→ ∞tn∞. 3.32

Letf :E → Ebe a contraction mapping with coeﬃcientα∈0,1andAa strongly positive bounded linear operator with coeﬃcient γ > 1/2 and 0 < γ < 1−$

2−2γ/α. Then, the sequence {xn} defined by

x0x∈E,

xn1 αnγfxn I−αnA1 tn

_t_n

0

Ttxnds, n≥0

3.33

converges strongly tox, wherexis the unique solution in FixSof the variational inequality A−γf x, jx −x

≥0, x∈FixS 3.34

or equivalentlyxQFixSI−Aγfx, whereQFixSis the sunny nonexpansive retraction ofE onto FixS.

Proof. SinceAis a strongly positive bounded linear operator with coeﬃcientγ, we have Ax−Ay, x−y

≥γx−y². 3.35 Therefore,Aisγ-strongly accretive. On the other hand,

I−Ax−I−Ay²

x−y −

Ax−Ay ,

x−y −

Ax−Ay

x−y, x−y

−2

Ax−Ay, x−y

Ax−Ay, Ax−Ay x−y²−2

Ax−Ay, x−y

Ax−Ay²

≤x−y²−2

Ax−Ay, x−y

A²x−y².

3.36

(16)

SinceA is strongly positive if and only if1/AAis strongly positive, we may assume, without loss of generality, thatA1, so that

Ax−Ay, x−y

≤x−y²−1

2I−Ax−I−Ay² x−y²−1

2x−y −

Ax−Ay ².

3.37

Hence,Ais 12-strongly pseudocontractive. ApplyingCorollary 3.4, we conclude the result.

Theorem 3.6. Let E be a reflexive Banach space which admits a weakly sequentially continuous duality mappingJ. LetS {Tt : 0 < t < ∞}be a u.a.r. nonexpansive semigroup on Esuch that FixS/∅. Let{αn}and{tn}be sequences of real number satisfying

0< αn<1, ∞ n0

αn ∞, tn>0, lim

n→ ∞αn lim

n→ ∞

αn

tn 0. 3.38

x0x∈E,

x_n1αnγfxn I−αnFTtnxn, n≥0

3.39

≥0, x∈FixS 3.40

Proof. By the same argument as in the proof ofTheorem 3.1, we can obtain that{xn},{fxn}, and{FTtnxn}are bounded andQFixSI−F−γfis a contraction onE. Thus, by Banach contraction principle,QFixSI−F−γfhas a unique fixed pointx. Then, using Lemma 2.3,

xis the unique solution in FixSof the variational inequality3.40. Next, we show that lim sup

n→ ∞

γfx−Fx, j xn−x

≤0. 3.41

Indeed, we can take a subsequence{xnk}of{xn}such that lim sup

n→ ∞

γfx−Fx, j xn−x lim

k→ ∞

γfx−Fx, jx nk −x

. 3.42

(17)

We may assume thatxnk p∈Eask → ∞. Now, we show thatp∈FixS. Put

xkxnk, αkαnk sktnk ∀k∈. 3.43 Fixt >0, then we have

xk−Ttp^t/sⁱ⁻¹

i0

Ti1skxk−Tiskxk

T

)* t sk

+ sk

, xk−T

)* t sk

+ sk

, p

T

)* t sk

+ sk

,

p−Ttp

≤

* t sk

+

Tskxk−xk1xk1−p T

) t−

*t sk

+ sk

, p−p

≤

* t sk

+

αkFTskxk−fxkxk1−p T

) t−

* t sk

+ sk

, p−p

≤ )tαk

sk

,FTskxk−fxkxk1−pmaxTsp−p: 0≤s≤sk

. 3.44

Thus, for allk∈, we obtain lim sup

k→ ∞

xk−Ttp≤lim sup

k→ ∞

xk1−plim sup

k→ ∞

xk−p. 3.45 Since Banach space E has a weakly sequentially continuous duality mapping satisfying Opial’s condition 13, we can conclude thatTtp pfor allt > 0, that is,p ∈ FixS. In view of the variational inequality3.2and the assumption that duality mappingJis weakly sequentially continuous, we conclude

lim sup

n→ ∞

γfx −Fx, jx n−x lim

k→ ∞

γfx−Fx, j xnk −x

≤

γfx −Fx, J

p−x ≤0.

3.46

By the same argument as in the proof ofTheorem 3.1, we conclude thatxn → xasn → ∞.

This completes the proof.

Using Theorem 3.6 and the method as in the proof of Corollary 3.7, we have the following result.

Corollary 3.7. Let E be a reflexive Banach space which admits a weakly sequentially continuous duality mappingJ. LetS {Tt : 0< t < ∞}be a u.a.r. nonexpansive semigroup onEsuch that FixS/∅. Let{αn}and{tn}be sequences of real number satisfying

0< αn <1, ∞ n0

αn∞, tn>0, lim

n→ ∞αn lim

n→ ∞

αn

tn 0. 3.47

(18)

Let F be aδ-strongly accretive andλ-strictly pseudocontractive withδ λ > 1, f : E → E a contraction mapping with coeﬃcientα∈0,1, andγis a positive real number such thatγ <1/α1− 1−δ/λ. Then, the sequence{yn}defined by

y0y∈E, yn1αnγf

Ttnyn I−αnFTtnyn, n≥0

3.48

≥0, x∈FixS 3.49

Using Theorem 3.6 and the method as in the proof of Corollary 3.8, we have the following result.

Corollary 3.8. Let E be a reflexive Banach space which admits a weakly sequentially continuous duality mappingJ. LetS {Tt : 0< t < ∞}be a u.a.r. nonexpansive semigroup onEsuch that FixS/∅. Let{αn}and{tn}be sequences of real number satisfying

0< αn <1, ∞

n0αn∞, tn>0, lim

n→ ∞αn lim

n→ ∞

αn

tn 0. 3.50

LetF be aδ-strongly accretive andλ-strictly pseudocontractive withδλ > 1,f :E → Ea con- traction mapping with coeﬃcientα∈0,1, andγis a positive real number such thatγ <1/α1− 1−δ/λ. Then, the sequence{zn}defined by

z0z∈E, zn1Ttn

αnγfzn I−αnFzn , n≥0

3.51

≥0, x∈FixS 3.52

UsingTheorem 3.6,Lemma 2.1, andExample 2.2, we have the following result.

(19)

Corollary 3.9. Let E be a uniformly convex Banach space which admits a weakly sequentially continuous duality mappingJ. LetS {Tt: 0< t <∞}be a nonexpansive semigroup onEsuch that FixS/∅. Let{αn}and{tn}be sequences of real numbers satisfying

0< αn<1, ∞ n0

αn∞, tn>0, lim

n→ ∞αn lim

n→ ∞

αn

tn 0. 3.53

x0x∈E,

xn1αnγfxn I−αnF1 tn

_t_n

0

Ttxnds, n≥0

3.54

≥0, x∈FixS 3.55

Corollary 3.10. LetH be a real Hilbert space. Let S {Tt : 0 ≤ t < ∞}be a nonexpansive semigroup onHsuch that FixS/∅. Suppose that the real sequences{αn} ⊂ 0,1,{tn} ⊂0,∞ satisfy the conditions

0< αn<1, ∞ n0

αn∞, tn>0, lim

n→ ∞αn lim

n→ ∞

αn

tn 0. 3.56

Letf :E → Ebe a contraction mapping with coeﬃcientα∈0,1andAa strongly positive bounded linear operator with coeﬃcient γ > 1/2 and 0 < γ < 1−$

2−2γ/α. Then, the sequence {xn} defined by

x0x∈E,

xn1αnγfxn I−αnA1 tn

_t_n

0

Ttxnds, n≥0

3.57

converges strongly tox, where xis the unique solution in FixSof the variational inequality A−γf x, jx −x

≥0, x∈FixS 3.58

or equivalentlyxQFixSI−Aγfx, whereQFixSis the sunny nonexpansive retraction ofE onto FixS.

(20)

Acknowledgment

The project was supported by the “Centre of Excellence in Mathematics” under the Commis- sion on Higher Education, Ministry of Education, Thailand.

References

1 N. Shioji and W. Takahashi, “Strong convergence theorems for asymptotically nonexpansive semi- groups in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 1, pp. 87–99, 1998.

2 T. Shimizu and W. Takahashi, “Strong convergence to common fixed points of families of nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 71–83, 1997.

3 R. Chen and Y. Song, “Convergence to common fixed point of nonexpansive semigroups,” Journal of Computational and Applied Mathematics, vol. 200, no. 2, pp. 566–575, 2007.

4 T. Suzuki, “On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2133–2136, 2003.

5 T. D. Benavides, G. L ´opez Acedo, and H.-K. Xu, “Construction of sunny nonexpansive retractions in Banach spaces,” Bulletin of the Australian Mathematical Society, vol. 66, no. 1, pp. 9–16, 2002.

6 H.-K. Xu, “A strong convergence theorem for contraction semigroups in Banach spaces,” Bulletin of the Australian Mathematical Society, vol. 72, no. 3, pp. 371–379, 2005.

7 R. Chen and H. He, “Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space,” Applied Mathematics Letters, vol. 20, no. 7, pp. 751–757, 2007.

8 R. D. Chen, H. M. He, and M. A. Noor, “Modified Mann iterations for nonexpansive semigroups in Banach space,” Acta Mathematica Sinica, vol. 26, no. 1, pp. 193–202, 2010.

9 F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization. An International Journal, vol. 19, no. 1-2, pp. 33–56, 1998.

10 H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003.

11 H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society.

Second Series, vol. 66, no. 1, pp. 240–256, 2002.

12 A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.

13 H.-K. Xu, “Approximations to fixed points of contraction semigroups in Hilbert spaces,” Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 157–163, 1998.

14 G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,”

Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.

15 S. Li, L. Li, and Y. Su, “General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 9, pp. 3065–3071, 2009.

16 E. Zeidler, Nonlinear Functional Analysis and Its Applications. III: Variational Methods and Optimization, Springer, New York, NY, USA, 1985.

17 W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000.

18 J.-P. Gossez and E. Lami Dozo, “Some geometric properties related to the fixed point theory for nonexpansive mappings,” Pacific Journal of Mathematics, vol. 40, pp. 565–573, 1972.

19 J. S. Jung, “Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 509–520, 2005.

20 A. Aleyner and Y. Censor, “Best approximation to common fixed points of a semigroup of nonexpansive operators,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 137–151, 2005.

21 A. Aleyner and S. Reich, “An explicit construction of sunny nonexpansive retractions in Banach spaces,” Fixed Point Theory and Applications, no. 3, pp. 295–305, 2005.

22 R. E. Bruck Jr., “Nonexpansive retracts of Banach spaces,” Bulletin of the American Mathematical Society, vol. 76, pp. 384–386, 1970.

23 S. Reich, “Asymptotic behavior of contractions in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 44, pp. 57–70, 1973.

(21)

24 K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods

& Applications, vol. 67, no. 8, pp. 2350–2360, 2007.

25 H. Piri and H. Vaezi, “Strong convergence of a generalized iterative method for semigroups of nonexpansive mappings in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 907275, 16 pages, 2010.