Strong Convergence Theorems of the General Iterative Methods for Nonexpansive Semigroups in Banach Spaces

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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

1_{Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand} 2_{Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand}

Correspondence should be addressed to Rattanaporn Wangkeeree,[email protected]

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, wherexis the unique solution

in FixSof the variational inequalityF−γfx, j x−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;

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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−αn

1 tn

_t_n

0

Tsxnds, ∀n∈ 1.3

in a real Hilbert space where{Tt : t ≥ 0} is a strongly continuous semigroup of

nonex-pansive 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−αn

1 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 limn→ ∞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

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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:

min x∈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,

xn1 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

con-ditions imposed on{αn}, the sequence{xn}generated by1.11 strongly converges to the

unique solutionx∗inCof the variational inequality

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Recently, Marino and Xu14mixed the iterative method1.10and the viscosity

approxima-tion 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, minx∈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−αnA

1 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, jx−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, jx−y ≤x−y2−λx−y −Fx−Fy 2. 1.17

It is easy to see that1.17can be rewritten as

I−Fx−I−Fy, jx−y ≥λI−Fx−I−Fy2, 1.18

see16.

In this paper, motivated by the above results, we introduce and study the strong

con-vergence theorems of the general iterative scheme{xn}defined by1.19in the framework of

a reflexive Banach spaceEwhich admits a weakly sequentially continuous duality mapping:

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whereF isδ-strongly accretive andλ-strictly pseudocontractive withδλ > 1,f is a

con-traction 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, fx2f2 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≤ x2y, jxy . 2.2

A real Banach spaceE is said to be strictly convex if xy/2 < 1 for all x, y ∈ E with

xy1 and_{x /}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,

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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_∈_Ewith_{x /}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,

lim

t→ ∞sup_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/t₀tTsxds, then

lim

t→ ∞sup_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−σhσtxσtx− 1

h

_h

0

Tsσtxds

1

h

_h

0

σtx−Tsσtxds

≤ 1

h

_h

0

σtx−Tsσtxds.

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ApplyingLemma 2.1, we have

lim

t→ ∞ sup_x∈Dσtx−σhσtx ≤

1 h

h

0

lim

t→ ∞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, jy−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, limn→ ∞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 constant1−δ/λ.

iIfFisδ-strongly accretive andλ-strictly pseudocontractive withδλ >1, then, for any

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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

lim

n→ ∞αn0,

∞

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₋_q_{αnγfxn I}₋_αnFT_tnxn₋_I₋_αnFq₋_αnFq

≤αnγfxn−FqI−αnFTtnxn−q

≤αnγfxn−fq αnγfq −FqI−αnFxn−q

≤αnαγxn−qαnγfq −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.

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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

lim

n→ ∞xn1−Ttnxnnlim→ ∞αnγfxn−FTtnxn0. 3.5

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

lim

n→ ∞ThTtnxn−Ttnxn ≤nlim→ ∞_x∈{xsup

n}

ThTtnx−Ttnx0. _3.6

Hence, for allh >0,

xn1−Thxn1 ≤ xn1−TtnxnTtnxn−ThTtnxnThTtnxn−Thxn1

≤2xn1−TtnxnTtnxn−ThTtnxn −→0.

3.7

That is, for allh >0,

lim

n→ ∞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−ΦI−F−γf y≤I−F−γf x−I−F−γf y

≤I−Fx−I−Fyγfx−fy

≤

1−δ

λ x−yαγx−y

⎛ ⎝

1−δ

λ αγ

⎞

⎠_x₋_y_, _{∀x, y}_∈_E.

3.9

Therefore,ΦI−F−γfis a contraction onEdue to1−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→ ∞

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

lim sup

n→ ∞

γfx−Fx, j xn−x lim

k→ ∞

γfx−Fx, jxn k −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, jxn −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−αnFx2

2αnγfxn−Fx, jxn 1−x

≤ ⎛

⎝₁₋_αn

⎛

⎝₁₋

1−δ

λ ⎞ ⎠

⎞ ⎠

2

xn−x 2_2αn_γfxn₋_γf_{x, j} _xn

1−x

2αnγfx −Fx, jxn 1−x

.

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On the other hand,

γfxn−γfx, j xn1−x

≤γαxn−xxn 1−x

≤γαxn−x

⎡ ⎢ ⎢ ⎣ ⎛

⎝₁₋_αn

⎛

⎝₁₋

1−δ

λ ⎞ ⎠ ⎞ ⎠ 2

xn−x 2

2αn γfxn−Fx, j xn1−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, jxn 1−x|. Substitut-

ing3.15in3.14, we obtain

xn1−x 2≤

⎛

⎝₁₋_αn

⎛

⎝₁₋

1−δ

λ ⎞ ⎠ ⎞ ⎠ 2

xn−x 2

2αnγα ⎛

⎝₁₋_αn

⎛

⎝₁₋

1−δ

λ ⎞ ⎠

⎞ ⎠

× xn−x 2_2αn√_αnM

02αn

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, jxn 1−x

⎛

⎝₁₋_2αn

⎡ ⎣

⎛

⎝₁₋

1−δ

λ ⎞

⎠₋_αγ_αnγα

⎛

⎝₁₋

1−δ

λ ⎞ ⎠ ⎤ ⎦ ⎞

⎠_xn₋_x 2

αn

⎡ ⎢ ⎣αn

⎛

⎝₁₋

1−δ

λ ⎞ ⎠

2

xn−x 2

2M0

√

αn2γfx −Fx, jxn 1−x

⎤ ⎥ ⎦

1−αnγn xn−x 2αnγnβn

γn,

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where

γn2

⎡ ⎣

⎛

⎝1−

1−δ

λ ⎞

⎠₋_αγ_αnγα

⎛

⎝1−

1−δ

λ ⎞ ⎠ ⎤

⎦_,

βn

⎡ ⎢ ⎣αn

⎛

⎝₁₋

1−δ

λ ⎞ ⎠

2

xn−x 2

2M0√αn2

γfx −Fx, jxn 1−x

⎤ ⎥ ⎦.

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

lim

n→ ∞αn0,

∞

n0

αn∞, lim

n→ ∞tn∞. 3.18

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

F−γf x, jx −x ≥0, x∈FixS 3.19

onto FixS.

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

≤αnγfTtnyn −fxnI−αnFTtnxn−Ttnyn

≤αnγαTtnyn−xn

⎛

⎝₁₋_αn

⎛

⎝₁₋

1−δ

λ ⎞ ⎠

⎞

(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.

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

lim

n→ ∞αn0,

∞

n0

αn∞, lim

n→ ∞tn∞. 3.22

LetF beδ-strongly accretive andλ-strictly pseudocontractive withδ λ > 1,f : E → E

acon-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

F−γf x, jx −x ≥0, x∈FixS 3.23

onto FixS.

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

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Takingp∈FixS, we have

_zn₁₋_p_Ttn_yn₋_Ttnp_≤_yn₋_p

αnγfzn I−αnFzn−I−αnFp−αnFp

≤ ⎛

⎝₁₋_αn

⎛

⎝₁₋

1−δ

λ ⎞ ⎠

⎞

⎠_zn₋_p_αn_γfzn₋_F_p

⎛

⎝₁₋_αn

⎛

⎝₁₋

1−δ

λ ⎞ ⎠

⎞

⎠_zn₋_p_αn

⎛

⎝₁₋

1−δ

λ ⎞

⎠_'γfzn−Fp

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

lim

n→ ∞αn0,

∞

n0

αn∞, lim

n→ ∞tn∞. 3.29

1−δ/λ. Then, the sequence{xn}defined by

x0x∈E,

xn1 αnγfxn I−αnF

1 tn

_t_n

0

Ttxnds, n≥0

(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

lim

n→ ∞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−αnA

1 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−2Ax−Ay, x−yAx−Ay, Ax−Ay

x−y2−2Ax−Ay, x−yAx−Ay2

≤x−y2−2Ax−Ay, x−yA2x−y2.

(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−Ay

2

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→ ∞αnnlim→ ∞

αn

tn 0. 3.38

x0x∈E,

xn1αnγfxn I−αnFTtnxn, n≥0

3.39

F−γf x, jx −x ≥0, x∈FixS 3.40

onto FixS.

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, jxn k −x

(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₋_T_tpt/si−1

i0

Ti1skxk−Tiskxk

T )* t sk + sk ,

xk−T

)* t sk + sk ,

pT

)* 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₋_T_tp_≤_{lim sup}

k→ ∞

_xk₁₋_p_{lim 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, jxn −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.

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 nlim→ ∞

αn

(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

F−γf x, jx −x ≥0, x∈FixS 3.49

onto FixS.

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

following result.

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 nlim→ ∞

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

F−γf x, jx −x ≥0, x∈FixS 3.52

onto FixS.

(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

αn

tn 0. 3.53

x0x∈E,

xn1αnγfxn I−αnF

1 tn

_t_n

0

Ttxnds, n≥0

3.54

F−γf x, jx −x ≥0, x∈FixS 3.55

onto FixS.

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

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

0

Ttxnds, n≥0

3.57

A−γf x, jx −x ≥0, x∈FixS 3.58

or equivalentlyxQFixSI−Aγfx, whereQFixSis the sunny nonexpansive retraction ofE

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Acknowledgment

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

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