A New Composite General Iterative Scheme for Nonexpansive Semigroups in Banach Spaces

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Volume 2011, Article ID 560671,18pages doi:10.1155/2011/560671

Research Article

A New Composite General Iterative Scheme for

Nonexpansive Semigroups in Banach Spaces

Pongsakorn Sunthrayuth and Poom Kumam

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Correspondence should be addressed to Poom Kumam,[email protected]

Received 1 February 2011; Accepted 19 March 2011

Academic Editor: Yonghong Yao

Copyrightq2011 P. Sunthrayuth and P. Kumam. 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.

We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping. A strong convergence theorem of the purposed iterative approximation method is established under some certain control conditions. Our results improve and extend announced by many others.

1. Introduction

Throughout this paper we denoted byÆandÊ

_{the set of all positive integers and all positive}

real numbers, respectively. LetX be a real Banach space, and letC be a nonempty closed convex subset ofX. A mappingTofCinto itself is said to be nonexpansive ifTx−Ty ≤ x−y

for eachx, y ∈ C. We denote byFT the set of fixed points of T. We know that FT is nonempty ifCis bounded; for more detail see1. A one-parameter familyS{Tt:t∈Ê

_}

fromCofXinto itself is said to be a nonexpansive semigroup onCif it satisfies the following conditions:

iT0xxfor allx∈C;

iiTst Ts◦Ttfor alls, t∈Ê

_;

iiifor eachx∈Cthe mappingt→Ttxis continuous;

ivTtx−Tty ≤ x−yfor allx, y∈Candt∈Ê

_.

We denote byFSthe set of all common fixed points of S, that is,FS : ∩t∈ÊFTt

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a self-mapping f : C → C is a contraction if there exists a constant α ∈ 0,1such that fx−fy ≤αx−yfor eachx, y∈C. As in3, we use the notation_Cto denote the collection of all contractions onC, that is,_C {f :C → Ca contraction}. Note that each f∈Chas a unique fixed point inC.

In the last ten years, the iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example,3–5. LetHbe a real Hilbert space, whose inner product and norm are denoted by ·,·and · , respectively. Let

Abe a strongly positive bounded linear operator onH: that is, there is a constantγ >0 with property

Ax, x ≥γx2 ∀x∈H. 1.1

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∈F

1

2 Ax, x − x, b, 1.2

whereCis the fixed point set of a nonexpansive mappingTonHandbis a given point inH. In 2003, Xu3proved that the sequence{xn}generated by

x0∈Cchosen arbitrarily,

xn1 I−αnATxnαnu, ∀n≥0,

1.3

converges strongly to the unique solution of the minimization problem1.2provided that the sequence {αn} satisfies certain conditions. Using the viscosity approximation method,

Moudafi 6 introduced the iterative process for nonexpansive mappings see 3, 7 for further developments in both Hilbert and Banach spaces and proved that if H is a real Hilbert space, the sequence{xn}generated by the following algorithm:

xn1 αnfxn 1−αnTxn, ∀n≥0,

1.4

wheref:C → Cis a contraction mapping with constantα∈0,1and{αn} ⊂0,1satisfies

certain conditions, converges strongly to a fixed point ofTinCwhich is unique solutionx∗

of the variational inequality:

f−Ix∗, y−x∗≤0, ∀y∈FT. 1.5

In 2006, Marino and Xu8 combined the iterative method1.3with the viscosity approximation method1.4considering the following general iterative process:

xn1αnγfxn I−αnATxn, ∀n≥0,

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where 0< γ < γ/α. They proved that the sequence{xn}generated by1.6converges strongly

to a unique solutionx∗of the variational inequality:

γf−Ax∗, y−x∗≤0, ∀y∈FT, 1.7

which is the optimality condition for the minimization problem:

min

x∈C

1

2 Ax, x −hx, 1.8

whereCis the fixed point set of a nonexpansive mappingTandhis a potential function for

γf i.e.,hx γfxforx ∈ H. Kim and Xu9studied the sequence generated by the following algorithm:

ynαnxn 1−αnTxn,

xn1βnu

1−βn

yn, ∀n≥0,

1.9

and proved strong convergence of scheme 1.9 in the framework of uniformly smooth Banach spaces. Later, yao, et al.10introduced a new iteration process by combining the modified Mann iteration9and the viscosity approximation method introduced by Moudafi

6. LetCbe a closed convex subset of a Banach space, and letT :C → Cbe a nonexpansive mapping such thatFT/∅andf∈C. Define{xn}in the following way:

x1∈Cchosen arbitrarily;

ynαnxn 1−αnTxn;

xn1βnfxn

1−βn

yn, ∀n≥0,

1.10

where{αn}and{βn}are two sequences in0,1. They proved under certain diﬀerent control

conditions on the sequences{αn}and{βn}that{xn}converges strongly to a fixed point ofT.

Recently, Chen and Song11studied the sequence generated by the algorithm in a uniformly convex Banach space, as follows:

xn1 αnfxn 1−αn

1

tn

_t_n

0

Tsxnds, ∀n∈Æ,

1.11

and they proved that the sequence{xn}defined by1.11converges strongly to the unique

solution of the variational inequality:

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In 2010, Sunthrayuth and Kumam12introduced the a general iterative scheme generated by

xn1 αnγfxn βnxn

1−βn

I−αnA

1

tn

_t_n

0

Tsxnds, ∀n≥0,

1.13

for the approximation of common fixed point of a one-parameter nonexpansive semigroup in a Banach space under some appropriate control conditions. They proved strong convergence theorems of the iterative scheme which solve some variational inequality. Very recently, Kumam and Wattanawitoon13studied and introduced a new composite explicit viscosity iteration method of fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces. They proved strong convergence theorems of the composite iterative schemes which solve some variational inequalities under some appropriate con-ditions. In the same year, Sunthrayuth et al.14introduced a general composite iterative scheme for nonexpansive semigroups in Banach spaces. They established some strong con-vergence theorems of the general iteration scheme under diﬀerent control conditions.

In this paper, motivated by Yao et al.10, Sunthrayuth, and Kumam12and Kumam and Wattanawitoon13we introduce a new general iterative algorithm 3.23for finding a common point of the set of solution of some variational inequality for nonexpansive semigroups in Banach spaces which admit a weakly continuous duality mapping and then proved the strong convergence theorem generated by the proposed iterative scheme. The results presented in this paper improve and extend some others from Hilbert spaces to Banach spaces and some others as special cases.

2. Preliminaries

Throughout this paper, we writexn xresp.,xn∗xto indicate that the sequence{xn}

weaklyresp., weak∗converges tox; as usualxn → xwill symbolize strong convergence;

also, a mappingIdenote the identity mapping. LetX be a real Banach space, and letX∗be its dual space. LetU {x ∈ X :x 1}. A Banach spaceX is said to be uniformly convex if, for each ∈ 0,2, there exists aδ > 0 such that for eachx, y ∈ U,x−y ≥ implies xy/2≤1−δ. It is known that a uniformly convex Banach space is reflexive and strictly convexsee also15. A Banach space is said to be smooth if the limit limt→0xty − x/t

exists for eachx, y∈U. It is also said to be uniformly smooth if the limit is attained uniformly forx, y∈U.

Let ϕ : 0,∞ : Ê

_→

Ê

_{be a continuous strictly increasing function such that}

ϕ0 0 andϕt → ∞ast → ∞. This functionϕis called a gauge function . The duality mappingJϕ:X → 2X

∗

associated with a gauge functionϕis defined by

Jϕx

f∗∈X∗:x, f∗xϕx, f∗ ϕx, ∀x∈X, 2.1

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Browder16initiated the study of certain classes of nonlinear operators by means of the duality mappingJϕ. Following Browder16, we say that Banach spaceX has a weakly

continuous duality mapping if there exists a gauge functionϕfor which the duality mapping

Jϕxis single-valued and continuous from the weak topology to the weak∗ topology; that

is, for each{xn}withxn x, the sequence{Jxn}converges weakly∗toJϕx. It is known

thatlp_{has a weakly continuous duality mapping with a gauge function} _ϕ_t _tp−1 _{for all}

1 < p < ∞. SetΦt ₀tϕτdτ, for allt ≥ 0; thenJϕx ∂Φx, where∂denotes the

subdiﬀerential in the sense of convex analysisrecall that the subdiﬀerential of the convex functionφ:X → Êatx∈Xis the set∂φx {x

∗ _∈_X_;_φ_y_≥_φ_x _x∗_{, y}₋_x_,_{for all}_y_∈

X}.

In a Banach space having a weakly continuous duality mapping Jϕ with a gauge

function ϕ, we defined an operator A is to be strongly positive see17if there exists a constantγ >0 with the property

Ax, Jϕx

≥γxϕx, 2.2

aI−bA sup

x≤1

_aI₋_bA_{x, J}_ϕ_x_, _a_∈₀_,₁_{, b}_∈₋₁_,₁_. _2.3

IfX:His a real Hilbert space, then the inequality2.2reduces to1.1.

The first part of the next lemma is an immediate consequence of the subdiﬀerential inequality and the proof of the second part can be found in18.

Lemma 2.1see18. Assume that a Banach spaceXhas a weakly continuous duality mappingJϕ

with gaugeϕ.

iFor allx, y∈X, the following inequality holds:

Φ xy ≤Φx y, Jϕ

xy. 2.4

In particular, for allx, y∈X,

_x_y 2

≤ x22y, Jxy. 2.5

iiAssume that a sequence{xn}inXconverges weakly to a pointx∈X. Then the following

identity holds:

lim sup

n→ ∞ Φ

_x_n₋_y _{lim sup}

n→ ∞ Φxn−x Φ

_y₋_x _, _∀_{x, y}_∈_X. _2.6

Lemma 2.2see17. Assume that a Banach spaceXhas a weakly continuous duality mappingJϕ

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Lemma 2.3see11. LetCbe a closed convex subset of a uniformly convex Banach spaceXand letS {Tt:t∈Ê

_}_{be a nonexpansive semigroup on}_C_{such that}_F_S_/_∅_{. Then, for each}_{r >}₀

andh≥0,

lim

t→ ∞_xsup_∈_C_∩_B

r

1

t

_t

0

Tsx ds−Th

1

t

_t

0

Tsx ds 0. 2.7

Lemma 2.4see19. Assume that{an}is a sequence of nonnegative real numbers such that

an1≤1−μn

anδn, 2.8

where{μn}is a sequence in0,1and{δn}is a sequence inÊsuch that

i∞_n₀μn ∞;

iilim sup_n_{→ ∞}δn/μn≤0 or∞n0|δn|<∞.

Then, limn→ ∞an0.

3. Main Results

LetXbe a Banach space which admits a weakly continuous duality mappingJϕwith gauge

ϕsuch thatϕis invariant on0,1, and letCbe a nonempty closed convex subset ofXsuch thatC±C⊂ C. LetS {Tt :t∈ Ê

_}_{be a nonexpansive semigroup from}_C_{into itself, let}

f be a contraction mapping with a coeﬃcientα ∈ 0,1, letAbe a strongly positive linear bounded operator with a coeﬃcientγ >0 such that 0 < γ < γϕ1/α, and lett∈ 0,1such thatt≤ϕ1A−1_{which satisfies}_t _→ _{0. Define the mapping}_Tf

t :C → Cby

T_tf :tγf I−tA1 λt

_λ_t

0

Tsds 3.1

to be a contraction mapping. Indeed, for eachx, y∈C,

_Tf

tx−T

f

ty

tγfx−fy I−tA

1

λt

_λ_t

0

Tsx−Tsyds

≤tγ fx−fy I−tA

1

λt

_λ_t

0

_T_s_x₋_T_s_y _ds

≤tγα x−y ϕ11−tγ x−y

≤1−tϕ1γ−γα x−y .

3.2

Thus, by Banach contraction mapping principle, there exists a unique fixed pointxt∈C, that

is,

xttγfxt I−tA

1

λt

_λ_t

0

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Remark 3.1. We note that spacelp has a weakly continuous duality mapping with a gauge

functionϕt tp−1_{for all 1}_{< p <}_{∞. This shows that}_ϕ_{is invariant on}₀_,₁_.

Lemma 3.2. LetX be a uniformly convex Banach space which admits a weakly continuous duality

mappingJϕwith gaugeϕsuch thatϕis invariant on0,1, and letCbe a nonempty closed convex

subset ofX such thatC±C ⊂ C. LetS {Tt : t ∈ Ê

_} _{be a nonexpansive semigroup from}_C

into itself such thatFS/∅, letf be a contraction mapping with a coeﬃcientα∈0,1, letAbe a strongly positive linear bounded operator with a coeﬃcientγ >0 such that 0< γ < γϕ1/α, and let

t ∈ 0,1such thatt ≤ ϕ1A−1_{which satisfies}_t _→ _{0. Then the net}_{_x

t}defined by3.3with

{λt}0<t<1 is a positive real divergent sequence, converges strongly ast → 0 to a common fixed point

x∗, in whichx∗∈FS, and is the unique solution of the variational inequality:

γfx∗−Ax∗, Jϕx−x∗

≤0, ∀x∈FS. 3.4

Proof. Firstly, we show the uniqueness of a solution of the variational inequality 3.4. Suppose thatx, x ∗∈FSare solutions of3.4; then

≤0,

γfx−Ax, J ϕx∗−x

≤0. 3.5

Adding up3.5, we obtain

0≥γfx∗−Ax∗−γfx−Ax, Jϕx−x∗

Ax−x∗, Jϕx−x∗

−γfx−fx∗, Jϕx−x∗

≥γx−x∗ϕx−x∗ −γ fx−fx∗ Jϕx−x∗

≥γΦx−x∗−γαΦx−x∗

γ−γαΦx−x∗

≥ϕ1γ−γαΦx−x∗,

3.6

which is a contradiction, we must havex x∗, and the uniqueness is proved. Here in after, we usexto denote the unique solution of the variational inequality3.4.

Next, we show that{xt}is bounded. Indeed, for eachp∈FS, we have

_x_t₋_p _t

γfxt−Ap

I−tA

1

λt

_λ_t

0

Tsxt−p

ds

≤t γfxt−Ap I−tA1

λt

0

_T_s_x_t₋_p _ds

≤tγ fxt−f

p t γfp−Ap ϕ11−tγ xt−p

≤1−tϕ1γ−γα xt−p t γf

p−Ap .

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It follows that

_x_t₋_p _≤ 1

ϕ1γ−γα γf

p−Ap . 3.8

Hence,{xt}is bounded, so are{fxt}and{A1/λt

_λ_t

0 Tsxtds}.

Next, we show thatxt−Thxt → 0 ast → 0. We note that

xt− 1

λt

_λ_t

0

Tsxtds

t γfxt−A

1

λt

_λ_t

0

Tsxtds

. 3.9

Moreover, we note that

xt−Thxt ≤

xt− 1

λt

_λ_t

0

Tsxtds

1

λt

_λ_t

0

Tsxtds−Th

1

λt

_λ_t

0

Tsxtds

Th

1

λt

_λ_t

0

Tsxtds

−Thxt

≤2

xt− 1

λt

_λ_t

0

Tsxtds

1

λt

_λ_t

0

Tsxtds−Th

1

λt

_λ_t

0

Tsxtds

,

3.10

for allh≥0. Define the setK {z∈C:z−p ≤ γfp−Ap/ϕ1γ−γα}; thenKis a nonempty bounded closed convex subset ofCwhich isTs-invariant for eachh≥0. Since {xt} ⊂KandKis bounded, there existsr >0 such thatK⊂Br, and it follows byLemma 2.3

that

lim

λt→ ∞

1

λt

_λ_t

0

Tsxtds−Th

1

λt

_λ_t

0

Tsxtds

0, 3.11

for eachh ≥ 0. From3.9-3.10, lettingt → 0 and noting3.11then, for eachh ≥ 0, we obtain

xt−Thxt −→0. 3.12

Assume that{tn}∞n1 ⊂0,1is such thattn → 0 asn → ∞. Putxn :xtn andλn :λtn. We

will show that{xn}contains a subsequence converging strongly tox ∈ FS. Since{xn}is

bounded sequence and Banach spaceXis a uniformly convex, hence it is reflexive, and there exists a subsequence{xnj}of{xn}which converges weakly to somex∈Casj → ∞. Again,

sinceJϕis weakly sequentially continuous, we have byLemma 2.1that

lim sup

j→ ∞ Φ

_x

nj−z

lim sup

j→ ∞ Φ

_x

nj−x

Φz−x. 3.13

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It follows thatHz Hx Φz−x, for allz∈C. From3.12, we have

HThx lim sup

j→ ∞ Φ

_x

nj−Thx

lim sup

j→ ∞

Φ Thxnj−Thx

≤lim sup

j→ ∞

Φ xnj−x

Hx.

3.14

On the other hand, we note that

HThx lim sup

j→ ∞ Φ

_x

nj−x

ΦThx−x

Hx ΦThx−x.

3.15

Combining 3.14with3.15, we obtainΦThx−x ≤ 0. This implies thatThx x, that is,x∈FS. In fact, sinceΦt ₀tϕτdτ, for allt ≥0 andϕ :Ê

_→

Ê

_{is the gauge}

function, then for 1≥k≥0,ϕky≤ϕyand

Φkt

_kt

0

ϕτdτ k

_t

0

ϕkydy≤k

_t

0

ϕydykΦt. 3.16

ByLemma 2.1, we have

Φxn−x Φ

tn

γfxn−Ax

I−tA

1

λn

_λ_t

0

Tsxnds−x

≤Φ I−tnA

1

λn

_λ_n

0

Tsxn−xds

tn

γfxn−Ax, J ϕxn−x

≤Φϕ11−tnγ

xn−x

tn

γfxn−γfx, Jϕxn−x

tn

γfx−Ax, J ϕxn−x

≤ϕ11−tnγ

Φxn−x tnγαxn−x Jϕxn−x

tn

ϕ11−tnγ

Φxn−x tnγαΦxn−x tn

≤1−tn

ϕ1γ−γαΦxn−x tn

.

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This implies that

Φxn−x≤ 1

ϕ1γ−γα

. 3.18

In particular, we have

Φ xnj−x

≤ 1

ϕ1γ−γα

γfx−Ax, J ϕ

xnj−x

. 3.19

Since the mapping Jϕ is single-valued and weakly continuous, it follows from3.19 that

Φxnj−x → 0 asj → ∞. This implies thatxnj → xasj → ∞.

Next, we show thatxsolves the variational inequality3.4, for eachx∈FS. From

3.3, we derive that

γf−Axt−1

tI−tA

1

λt

0

Tsxtds−xt

. 3.20

Now, we observe that

1

λt

_λ_t

0

I−Tsx ds− 1 λt

_λ_t

0

I−Tsxtds, Jϕx−xt

x−xt, Jϕx−xt

−

1

λt

_λ_t

0

Tsx−Tsxtds, Jϕx−xt

≥ x−xt Jϕx−xt − 1

λt

0

Tsx−Tsxtds Jϕx−xt

≥Φx−xt− x−xt Jϕx−xt

Φx−xt−Φx−xt 0.

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It follows from3.20that

γf−Axt, Jϕx−xt

−1

t

I−tA

1

λt

0

Tsxtds−xt

, Jϕx−xt

−1

t

I−tA

1

λt

0

Tsxtds− 1

λt

0

xtds

, Jϕx−xt

−1 t 1 λt λt 0

I−Tsx ds− 1 λt

λt

0

I−Tsxtds, Jϕx−xt

A 1 λt λt 0

Ts−Ixtds

, Jϕx−xt

≤ A 1 λt _λ_t 0

Ts−Ixtds

, Jϕx−xt

.

3.22

Now, replacingtandλtwithtnj andλnj, respectively, in3.22, and lettingj → ∞, and we

notice thatTs−Ixnj → Ts−Ix0 forx∈FS, we obtain that γf−Ax, J ϕx−x ≤

0. That is,xis a solution of the variational inequality3.4. By uniqueness, asxx∗, we have shown that each cluster point of the net{xt}is equal tox∗. Then, we conclude thatxt → x∗

ast → 0. This proof is complete.

Theorem 3.3. LetXbe a uniformly convex Banach space which admits a weakly continuous duality

mappingJϕwith the gauge functionϕsuch thatϕis invariant in0,1, and letCbe a nonempty closed

convex subset ofXsuch thatC±C⊂C. LetS{Tt:t∈Ê

_}_{be a nonexpansive semigroup from}

Cinto itself such thatFS/∅, letfbe a contraction mapping with a coeﬃcientα∈0,1, and let

Abe a strongly positive linear bounded operator with a coeﬃcientγ >0 such that 0< γ < γϕ1/α. Let{αn}∞n0,{βn}∞n0,{γn}∞n0 be the sequences in0,1and let{tn}∞n0 be a positive real divergent

sequence. Assume that the following conditions hold:

C1limn→ ∞αn0 and

_∞

n0αn∞,

C2limn→ ∞γn0,

C3βnoαn,

Then the sequence{xn}defined by

x0∈Cchosen arbitrarily;

zn γnxn

1−γn

1

tn

0

Tsxnds;

ynαnγfzn I−αnAzn;

xn1 βnxn

1−βn

yn, ∀n≥0,

3.23

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Proof. From the condition C1, we may assume, with no loss of generality, that αn ≤

ϕ1A−1_{for each}_n_≥_{0. From}_{Lemma 2.2}_{, we have}_I₋_α

nA ≤ϕ11−αnγ.

Firstly, we show that{xn}is bounded. Letp∈FS; we get

_z_n₋_p _γ_n

xn−p

1−γn

1

tn

0

Tsxn−p

ds

≤γn xn−p

1−γn

1

tn

_t_n

0

_T_s_x_n₋_p _ds

≤γn xn−p

1−γn xn−p

xn−p ,

_y_n₋_p _α_n

γfzn−Ap

I−αnA

zn−p

≤αnγ fzn−f

p αn γf

p−Ap I−αnA zn−p

≤1−αn

ϕ1γ−γα xn−p αn γf

p−Ap .

3.24

It follows that

_x_n₁₋_p _β_n

xn−p

1−βn

yn−p

≤βn xn−p

1−βn yn−p

≤βn xn−p

1−βn

1−αn

ϕ1γ−γα xn−p αn γf

p−Ap

1−αn

ϕ1γ−γα1−βn xn−p αn

ϕ1γ−γα1−βn

γfp−Ap

ϕ1γ−γα

≤max

_x_n₋_p _, γfp−Ap

ϕ1γ−γα

.

3.25

By induction onn, we have

_x_n₋_p _≤_max _x₀₋_p _, γfp−Ap

ϕ1γ−γα

, ∀n≥0. 3.26

Thus,{xn}is bounded. Since{xn}is bounded, then1/tn

_t_n

0 Tsxnds−p ≤ xn−pand

{A1/tn

_t_n

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Next, we show that limn→ ∞xn−Thxn0, for allh≥0. From3.23, we note that

xn1− 1

tn

_t_n

0

Tsxnds

≤ xn1−yn yn−zn

zn− 1

tn

_t_n

0

Tsxnds

≤βn xn−yn αn γfzn−Azn γn

xn−

1

tn

0

Tsxnds

.

3.27

By the conditionsC1–C3, then3.27, we obtain

lim

n→ ∞

xn1− 1

tn

0

Tsxnds

0. 3.28

Moreover, we note that

xn1−Thxn1 ≤

xn1− 1

tn

_t_n

0

Tsxnds

1

tn

_t_n

0

Tsxnds−Th

1

tn

_t_n

0

Tsxnds

Th

1

tn

0

Tsxnds

−Thxn1

≤2

xn1−1

tn

0

Tsxnds

1

tn

0

Tsxnds−Th

1

tn

0

Tsxnds

.

3.29

Define the setK {z ∈ C :z−p ≤ x0−pγfp−Ap/ϕ1γ−γα}. ThenK is a

nonempty bounded closed convex subset ofC, which isTs—invariant for eachs ≥0 and contains{xn}; it follows fromLemma 2.3that

lim

n→ ∞

1

tn

_t_n

0

Tsxnds−Th

1

tn

_t_n

0

Tsxnds

0, ∀h≥0. 3.30

Then, for allh≥0, from3.28and3.30, into3.29, we obtain limn→ ∞xn1−Thxn10,

and hence

lim

n→ ∞xn−Thxn0, ∀h≥0. 3.31

Next, we show that lim sup_n_{→ ∞} γfx∗−Ax∗, Jϕxn−x∗ ≤ 0. We can take subsequence

{xnj} ⊂ {xn}such that

lim

j→ ∞

γfx∗−Ax∗, Jϕ

xnj−x∗

lim sup

n→ ∞

γfx∗−Ax∗, Jϕxn−x∗

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By the assumption thatXis uniformly convex, hence it is reflexive and{xn}is bounded; then

there exists a subsequence{xnj}which converges weakly to somex∈Casj → ∞. SinceJϕ

is weakly continuous, fromLemma 2.1, we have

lim sup

j→ ∞ Φ

_x

nj−z

lim sup

j→ ∞ Φ

_x

nj−x

Φz−x, ∀z∈C. 3.33

LetHz lim sup_j_{→ ∞}Φxnj−z, for allz∈C.

It follows thatHz Hx Φz−x, for allz∈C. From3.31, we have

HThx lim sup

j→ ∞ Φ

_x

nj−Thx

lim sup

j→ ∞

Φ Thxnj−Thx

≤lim sup

j→ ∞ Φ

_x

nj−x

Hx.

3.34

HThx lim sup

j→ ∞ Φ

_x

nj−x

ΦThx−x

Hx ΦThx−x.

3.35

Combining3.34with3.35, we obtainΦThx−x≤0. This implies thatThxx; that is,x∈FS.

Since the duality mapJϕis single-valued and weakly continuous, we get that

lim sup

n→ ∞

γfx∗−Ax∗, Jϕxn−x∗

lim

j→ ∞

xnj−x∗

≤0,

3.36

as required. Hence,

lim sup

n→ ∞

γfx∗−Ax∗, Jϕxn1−x∗

≤0. 3.37

Sincexn1−ynβnxn−yn, by conditionC3, we obtain that limn→ ∞xn1−yn0. It

follows from3.37, that

lim sup

n→ ∞

yn−x∗

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Finally, we show thatxn → x∗asn → ∞. Now, fromLemma 2.1, we have

Φ yn−x∗ Φ αn

γfzn−Ax∗

I−αnAzn−x∗

Φ αn

γfzn−γfx∗

αn

γfx∗−Ax∗ I−αnAzn−x∗

≤Φ αn

γfzn−γfx∗

I−αnAzn−x∗

αn

yn−x∗

≤1−αn

ϕ1γ−γαΦxn−x∗ αn

yn−x∗

.

3.39

Φxn1−x∗ Φ βnxn−x∗

1−βn

yn−x∗

≤1−βn

Φ yn−x∗ βn

xn−x∗, Jϕxn1−x∗.

3.40

It follows from3.40that

Φxn1−x∗≤1−αn

ϕ1γ−γαΦxn−x∗ αn

yn−x∗

βn

xn−x∗, Jϕxn1−x∗

≤1−αn

ϕ1γ−γαΦxn−x∗αnγfx∗−Ax∗, Jϕ

yn−x∗

βn

αnM

,

3.41

whereMsup_n_≥₀{xn−x∗ϕxn1−x∗}.

Put μn : αnϕ1γ −γα and δn : αn γfx∗ −Ax∗, Jϕyn −x∗ βn/αnM.

Then3.41reduces to formulaΦxn1 −x∗ ≤ 1−μnΦxn−x∗ δn. By conditions

C1andC3and noting3.38, it is easy to see that∞_n₀μn∞and lim sup_n_{→ ∞}δn/μn

lim sup_n_{→ ∞}1/ϕ1γ−γα γfx∗−Ax∗, Jϕyn−x∗βn/αnM≤0. ApplyingLemma 2.4,

we obtainΦxn−x∗ → 0 asn → ∞this implies thatxn → x∗asn → ∞. This completes

the proof.

Takingγn0 in3.23, we can get the following corollary easily.

Corollary 3.4. LetXbe a uniformly convex Banach space which admits a weakly continuous duality

mappingJϕwith the gauge functionϕsuch thatϕinvariant in0,1,Cbe a nonempty closed convex

subset ofX such thatC±C ⊂ C. LetS {Tt : t ∈ Ê

_} _{be a nonexpansive semigroup from}_C

into itself such thatFS/∅,f be a contraction mapping with a coeﬃcientα ∈ 0,1andAbe a strongly positive linear bounded operator with a coeﬃcientγ > 0 such that 0 < γ < γϕ1/α. Let {αn}∞n0,{βn}∞n0be the sequences in0,1and{tn}∞n0be a positive real divergent sequence. Assume

the following conditions are hold:

C1limn→ ∞αn0 and∞n0αn ∞;

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x0∈Cchosen arbitrarily,

ynαnγf

1

tn

_t_n

0

Tsxnds

I−αnA

1

tn

_t_n

0

Tsxnds,

xn1 βnxn

1−βn

yn, ∀n≥0,

3.42

converges strongly to the common fixed pointx∗, in whichx∗ ∈ FSis the unique solution of the variational inequality:

≤0, ∀x∈FS. 3.43

A strong mean convergence theorem for nonexpansive mapping was first established by Baillon20and it was generalized to that for nonlinear semigroups by Reich et al.21–

23. It is clear that Theorem 3.3are valid for nonexpansive mappings. Thus, we have the following mean ergodic theorem of viscosity iteration process for nonexpansive mappings in Hilbert spaces.

Corollary 3.5. Let H be a real Hilbert space, and letCbe a nonempty closed convex subset ofH

such thatC±C⊂ C. LetT be a nonexpansive mapping fromCinto itself such thatFT/∅,f be a contraction mapping with a coeﬃcientα ∈0,1, and letAbe a strongly positive linear bounded operator with a coeﬃcientγ > 0 such that 0 < γ < γϕ1/α. Let{αn}∞n0,{βn}∞n0, and{γn}∞n0 be

the sequences in0,1and let{tn}∞n0be a positive real divergent sequence. Assume that the following

conditions are hold:

C1 limn→ ∞αn0 and∞n0αn∞;

C2limn→ ∞γn0;

C3βnoαn.

x0∈Cchosen arbitrarily,

znγnxn

1−γn

1

n1

n

j0

Tjxn,

ynαnγfzn I−αnAzn,

xn1 βnxn

1−βn

yn, ∀n≥0,

3.44

converges strongly to the common fixed pointx∗, in whichx∗ ∈ FTis the unique solution of the variational inequality:

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Acknowledgments

The authors are grateful for the reviewers for the careful reading of the paper and for the suggestions which improved the quality of this work. They would like to thank the National Research University Project of Thailand’s Oﬃce of the Higher Education Commission for financial support under NRU-CSEC project no. 54000267.

References

1 F. E. Browder, “Fixed-point theorems for noncompact mappings in Hilbert space,” Proceedings of the

National Academy of Sciences of the United States of America, vol. 53, pp. 1272–1276, 1965.

2 F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National

Academy of Sciences of the United States of America, vol. 54, pp. 1041–1044, 1965.

3 H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical

Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.

4 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.

5 H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and

Applications, vol. 116, no. 3, pp. 659–678, 2003.

6 A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical

Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.

7 S. Plubtieng and T. Thammathiwat, “A viscosity approximation method for equilibrium problems, fixed point problems of nonexpansive mappings and a general system of variational inequalities,”

Journal of Global Optimization, vol. 46, no. 3, pp. 447–464, 2010.

8 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.

9 T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations,” Nonlinear Analysis:

Theory, Methods & Applications, vol. 61, no. 1-2, pp. 51–60, 2005.

10 Y. Yao, R. Chen, and J.-C. Yao, “Strong convergence and certain control conditions for modified Mann iteration,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 6, pp. 1687–1693, 2008.

11 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.

12 P. Sunthrayuth and P. Kumam, “A general iterative algorithm for the solution of variational inequalities for a nonexpansive semigroup in Banach spaces,” Journal of Nonlinear Analysis and

Optimization: Theory and Applications, vol. 1, no. 1, pp. 139–150, 2010.

13 P. Kumam and K. Wattanawitoon, “A general composite explicit iterative scheme of fixed point solutions of variational inequalities for nonexpansive semigroups,” Mathematical and Computer

Modelling, vol. 53, no. 5-6, pp. 998–1006, 2011.

14 P. Sunthrayuth, K. Wattanawitoon, and P. Kumam, “Convergence theorems of a general composite iterative method for nonexpansive semigroups in Banach spaces,” Mathematical Analysis, vol. 2011, Article ID 576135, 24 pages, 2011.

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

16 F. E. Browder, “Convergence theorems for sequences of nonlinear operators in Banach spaces,”

Mathematische Zeitschrift, vol. 100, pp. 201–225, 1967.

17 R. Wangkeeree, N. Petrot, and R. Wangkeeree, “The general iterative methods for nonexpansive mappings in Banach spaces,” Journal of Global Optimization. In press.

18 T.-C. Lim and H. K. Xu, “Fixed point theorems for asymptotically nonexpansive mappings,” Nonlinear

Analysis: Theory, Methods & Applications, vol. 22, no. 11, pp. 1345–1355, 1994.

19 H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.

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21 W. Kaczor, T. Kuczumow, and S. Reich, “A mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate sense,” Journal of Mathematical Analysis and

Applications, vol. 246, no. 1, pp. 1–27, 2000.

22 S. Reich, “Almost convergence and nonlinear ergodic theorems,” Journal of Approximation Theory, vol. 24, no. 4, pp. 269–272, 1978.

23 S. Reich, “A note on the mean ergodic theorem for nonlinear semigroups,” Journal of Mathematical

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