A New Approximation Scheme Combining the Viscosity Method with Extragradient Method for Mixed Equilibrium Problems

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Volume 2009, Article ID 257089,25pages doi:10.1155/2009/257089

Research Article

A New Approximation Scheme Combining

the Viscosity Method with Extragradient Method

for Mixed Equilibrium Problems

Jian-Wen Peng

1

_{and Soon-Yi Wu}

2

1_{College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China}

2_{Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan}

Correspondence should be addressed to Jian-Wen Peng,[email protected]

Received 8 November 2009; Revised 2 December 2009; Accepted 6 December 2009

Recommended by Simeon Reich

We introduce a new approximation scheme combining the viscosity method with extragradient method for finding a common element of the set of solutions of a mixed equilibrium problem and the set of fixed points of a finite family of nonexpansive mappings and the set of the variational inequality for a monotone, Lipschitz continuous mapping. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. Based on this result, we also get some new and interesting results. The results in this paper generalize, extend, and improve some well-known results in the literature.

Copyrightq2009 J.-W. Peng and S.-Y. Wu. 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

LetHbe a real Hilbert space with inner product·,·and induced norm · and letCbe a nonempty closed convex subset ofH. Letϕ : C → R∪ {∞}be a function and letF be a bifunction fromC×CtoR, whereRis the set of real numbers. Ceng and Yao1and Bigi et al.2considered the following mixed equilibrium problem:

Find x∈C such thatFx, yϕy≥ϕx, ∀y∈C. 1.1

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Ifϕ0, then the mixed equilibrium problem1.1becomes the following equilibrium problem:

Findingx∈C such thatFx, y≥0, ∀y∈C. 1.2

The set of solutions of1.2is denoted byEPF.

If Fx, y 0 for all x, y ∈ C, the mixed equilibrium problem 1.1 becomes the following minimization problem:

Findingx∈C such thatϕy≥ϕx, ∀y∈C. 1.3

The set of solutions of1.3is denoted by Argminϕ.

The problem 1.1 is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; see, for instance,1–4.

Recall that a mappingSof a closed convex subsetCinto itself is nonexpansive5if there holds that

Sx−Sy≤x−y ∀x, y∈C. 1.4

We denote the set of fixed points of S by FixS. Ceng and Yao1 introduced an iterative scheme for finding a common element of the set of solutions of problem1.1and the set of common fixed points of a finite family of nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem.

Some methods have been proposed to solve the problem 1.2; see, for instance,

3,4,6–12and the references therein. Recently, Combettes and Hirstoaga6introduced an iterative scheme of finding the best approximation to the initial data when EPFis nonempty and proved a strong convergence theorem. Takahashi and Takahashi 7 introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem1.2and the set of fixed points of a nonexpansive mapping in a Hilbert space and proved a strong convergence theorem. Su et al.8introduced the following iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem1.2and the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for anα-inverse strongly monotone mapping in a Hilbert space. Starting with an arbitraryx1 ∈ H, define sequences{xn}and {un}by

Fun, y_r1 n

y−un, un−xn≥0, ∀y∈C,

xn1αnfxn 1−αnSPCun−λnAun, ∀n∈N.

1.5

They proved that under certain appropriate conditions imposed on{αn},{rn}, and{λn}, the

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wherez PFixS∩EPF∩VIC,Afz. Tada and Takahashi9introduced two iterative schemes for finding a common element of the set of solutions of problem1.2and the set of fixed points of a nonexpansive mapping in a Hilbert space and obtained both strong convergence theorem and weak convergence theorem.

On the other hand, for solving the variational inequality problem in the finite-dimensional EuclideanRn, Korpelevich13introduced the following so-called extragradient method:

x1x∈C,

ynPCxn−λAxn,

xn1PCxn−λAyn

1.6

for everyn 0,1,2, . . . ,whereλ ∈ 0,1/k. She showed that if VIC, Ais nonempty, then the sequences{xn}and{yn}, generated by1.6, converge to the same pointz ∈ VIC, A.

The idea of the extragradient iterative process introduced by Korpelevich was successfully generalized and extended not only in Euclidean but also in Hilbert and Banach spaces; see, for example, the recent papers of He et al.14, G´arciga Otero and Iuzem15, and Solodov and

Svaiter16, Solodov17. Moreover, Zeng and Yao18and Nadezhkina and Takahashi19

introduced iterative processes based on the extragradient method for finding the common element of the set of fixed points of nonexpansive mappings and the set of solutions of variational inequality problem for a monotone, Lipschitz continuous mapping. Yao and Yao20introduced an iterative process based on the extragradient method for finding the common element of the set of fixed points of nonexpansive mappings and the set of solutions of variational inequality problem for anα-inverse strongly monotone mapping. Plubtieng and Punpaeng11introduced an iterative process based on the extragradient method for finding the common element of the set of fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of variational inequality problem for α-inverse strongly monotone mappings. Chang et al. 12 introduced some iterative processes based on the extragradient method for finding the common element of the set of fixed points of a infinite family of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of variational inequality problem for anα-inverse strongly monotone mapping. Peng et al.21introduced a new approximation scheme combining the viscosity method with parallel method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a finite family of strict pseudocontractions and obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces.

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2. Preliminaries

LetHbe a real Hilbert space with inner product·,·and norm · . LetCbe a nonempty closed convex subset ofH. Let symbols → and denote strong and weak convergence, respectively. In a real Hilbert spaceH, it is well known that

λx 1−λy2_λ_x2 ₁₋_λ_y2₋_λ₁₋_λ_x₋_y2 _2.1

for allx, y∈Handλ∈0,1.

For anyx∈H, there exists a unique nearest point inC, denoted byPCx, such that

x−PCx ≤ x−yfor ally∈C. The mappingPCis called the metric projection ofHonto C. We know thatPCis a nonexpansive mapping fromHontoC. It is also known thatPCx∈C

and

x−PCx, PCx−y≥0 2.2

for allx∈Handy∈C.

It is easy to see that2.2is equivalent to

x−y2

≥ x−PCx2y−PCx2 2.3

for allx∈Handy∈C.

A mappingAofCintoHis called monotone if

Ax−Ay, x−y≥0 2.4

for allx, y∈C. A mappingAofCintoHis calledα-inverse strongly monotone if there exists a positive real numberαsuch that

x−y, Ax−Ay≥αAx−Ay2 _2.5

for all x, y ∈ C. A mappingA : C → H is calledk-Lipschitz continuous if there exists a positive real numberksuch that

Ax−Ay≤kx−y 2.6

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LetAbe a monotone mapping ofCintoH. In the context of the variational inequality problem the characterization of projection2.2implies the following:

u∈VIC, A ⇒uPCu−λAu, λ >0,

uPCu−λAu for someλ >0⇒u∈VIC, A.

2.7

It is also known thatHsatisfies Opial’s condition22, that is, for any sequence{xn} ⊂ Hwithxn x, the inequality

lim inf

n→ ∞ xn−x<lim infn→ ∞ xn−y 2.8

holds for everyy∈Hwithx /y.

A set-valued mappingT :H → 2His called monotone if for allx, y∈H,f∈Txand

g∈Tyimplyx−y, f−g ≥0. A monotone mappingT :H → 2His maximal if its graphGT ofTis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingTis maximal if and only if forx, f∈H×H,x−y, f−g ≥0 for every y, g∈GTimpliesf∈Tx. Let A be a monotone,k-Lipschitz continuous mapping ofCinto

Hand letNCvbe normal cone toCatv∈C, that is,NCv{w∈H:v−u, w ≥0,∀u∈C}.

Define

Tv

⎧ ⎨ ⎩

AvNCv ifv∈C,

∅ ifv /∈C. 2.9

ThenT is maximal monotone and 0∈Tvif and only ifv∈VIC, A see23.

For solving the mixed equilibrium problem, let us give the following assumptions for the bifunctionF,ϕand the setC:

A1Fx, x 0 for allx∈C;

A2Fis monotone, that is,Fx, y Fy, x≤0 for anyx, y∈C; A3for eachy∈C,x→Fx, yis weakly upper semicontinuous; A4for eachx∈C,y→Fx, yis convex;

A5for eachx∈C,y→Fx, yis lower semicontinuous;

B1for eachx∈Handr >0, there exist a bounded subsetDx⊆Candyx∈Csuch that

for anyz∈C\Dx,

Fz, yxϕyx1_ryx−z, z−x< ϕz; 2.10

B2Cis a bounded set;

for anyz∈C\Dx,

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for anyz∈C\Dx,

Fz, yx1_ryx−z, z−x<0. 2.12

We will use the following results in the sequel.

Lemma 2.1see21,24. LetCbe a nonempty closed convex subset ofH. LetF be a bifunction fromC×CtoRsatisfyingA1–A5and letϕ:C → R∪ {∞}be a proper lower semicontinuous and convex function. Assume that eitherB1orB2holds. Forr >0andx∈H,define a mapping Tr :H → Cas follows:

Trx

z∈C:Fz, yϕy1

r

y−z, z−x≥ϕz, ∀y∈C

2.13

for allx∈H. Then the following conclusions hold:

1for eachx∈H,Trx/∅;

2Tr is single-valued;

3Tr is firmly nonexpansive, that is, for anyx, y∈H,

Trx−Try2≤Trx−Try, x−y; 2.14

4FixTr MEPF, ϕ;

5MEPF, ϕis closed and convex.

Lemma 2.2see25,26. Assume that{αn}is a sequence of nonnegative real numbers such that

αn1≤

1−γnαnδn, 2.15

whereγnis a sequence in0,1and{δn}is a sequence such that

i∞n1γn∞;

iilim sup_n_{→ ∞}δn/γn≤0or∞n1|δn|<∞.

Then,limn→ ∞αn0.

Lemma 2.3. In a real Hilbert spaceH, there holds the following inequality:

_x_y2

≤ x2₂_{y, x}_y _2.16

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Lemma 2.4see21. Let{xn}and{wn}be bounded sequences in a Banach space, and let{βn}be a sequence of real numbers such that0<lim infn→ ∞βn≤lim supn→ ∞βn<1for alln0,1,2, . . . . Suppose thatxn1 1−βnwnβnxnfor alln0,1,2, . . .andlim supn→ ∞wn1−wnxn1−

xn ≤0. Then,limn→ ∞wn−xn0.

Let {Ti}Ni1 be a finite family of nonexpansive mappings of C into itself and let

λ1, λ2, . . . , λN be real numbers such that 0 ≤ λi ≤ 1 for every i 1,2, . . . , N. We define a

mappingWofCinto itself as follows:

U1λ1T1 1−λ1I,

U2λ2T2U1 1−λ2I,

.. .

UN−1λN−1TN−1UN−2 1−λN−1I,

W:UN λNTNUN−1 1−λNI.

2.17

Such a mappingWis called theW-mapping generated byT1, T2, . . . , TNandλ1, λ2, . . . , λN. It

is easy to see that nonexpansivity of eachTi ensures the nonexpansivity ofW.The concept

of W-mappings was introduced in 27, 28. It is now one of the main tools in studying

convergence of iterative methods for approaching a common fixed point of nonlinear mappings; more recent progresses can be found in10,29,30and the references cited therein.

Lemma 2.5see29. LetCbe a nonempty closed convex set of a strictly convex Banach space. Let T1, T2, . . . , TN be nonexpansive mappings of C into itself such that iN1FTi/∅ and let

λ1, λ2, . . . , λNbe real numbers such that0< λi<1for everyi1,2, . . . , N−1and0< λN≤1. Let Wbe theW-mapping generated byT1, T2, . . . , TNandλ1, λ2, . . . , λN. ThenFixW Ni1FixTi.

Lemma 2.6 see 10. Let C be a nonempty convex subset of a Banach space. Let {Ti}Ni1 be a

finite family of nonexpansive mappings ofCinto itself and let{λn,1}{λn,2}, . . . ,{λn,N}be sequences in 0,1 such that λn,i → λi i 1, . . . , N. Moreover for every integer n ≥ 1, let W and Wn be the W-mappings generated by T1, T2, . . . , TN and λ1, λ2, . . . , λN and T1, T2, . . . , TN and

{λn,1}{λn,2}, . . . ,{λn,N}, respectively. Then for everyx∈C, it follows that

lim

n→ ∞Wnx−Wx0. 2.18

3. Strong Convergence Theorems

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Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction fromC×CtoR satisfyingA1–A5and letϕ : C → R∪ {∞}be a proper lower semicontinuous and convex function. LetAbe a monotone and k-Lipschitz continuous mapping of C into H. Let T1, T2, . . . , TN be a finite family of nonexpansive mappings of C into H such that

Ω N

n1FixTi∩VIC, A∩MEPF, ϕ/∅. Let{λn,1},{λn,2}, . . . ,{λn,N}be sequences inε1, ε2

with0< ε1 ≤ε2 <1. LetWnbe theW-mapping generated byT1, T2, . . . , TNandλn,1, λn,2, . . . , λn,N. Assume that eitherB1orB2holds. Letfbe a contraction ofHinto itself and let{xn},{un}, and

{yn}be sequences generated by

x1x∈C,

Fun, yϕy−ϕun _r1 n

ynPCun−γnAun,

xn1αnfxn βnxn1−αn−βnWnPCun−γnAyn

3.1

for everyn 1,2, . . .where {γn},{rn},{αn},{λn1},{λn2}, . . . ,{λnN}, and {βn}are sequences of numbers satisfying the conditions:

C1limn→ ∞αn0and∞n1αn∞;

C21>lim sup_n_{→ ∞}βn≥lim infn→ ∞βn>0;

C3limn→ ∞γn0;

C4lim infn→ ∞rn>0andlimn→ ∞|rn1−rn|0;

C5limn→ ∞|λn,i−λn−1,i|0for alli1,2, . . . , N.

Then,{xn},{un}, and{yn}converge strongly towPΩfw.

Proof. We show thatP_Ωf is a contraction ofCinto itself. In fact, there existsa ∈0,1such thatfx−fy ≤ax−yfor allx, y∈C. So, we have

PΩfx−PΩfy≤fx−fy≤ax−y 3.2

for all x, y ∈ C.SinceH is complete, there exists a unique elementu0 ∈ Csuch thatu0

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Puttn PCun−γnAynfor everyn1,2, . . . .Letu∈Ωand let{Trn}be a sequence of mappings defined as inLemma 2.1. ThenuPCu−γnAu. FromunTrnxn∈C, we have

un−uTrnxn−Trnu ≤ xn−u. 3.3

From2.3, the monotonicity ofA, andu∈VIC, A, we have

tn−u2

≤un−γnAyn−u2−un−γnAyn−tn2

un−u2− un−tn22γnAyn, u−tn

un−u2− un−tn22γnAyn−Au, u−ynAu, u−ynAyn, yn−tn

≤ un−u2− un−tn22γnAyn, yn−tn

≤ un−u2−un−yn2−2un−yn, yn−tn −yn−tn22γnAyn, yn−tn

un−u2−un−yn2−yn−tn22un−γnAyn−yn, tn−yn.

3.4

Further, Sinceyn PCun−γnAunandAisk-Lipschitz continuous, we have

un−γnAyn−yn, tn−ynun−γnAun−yn, tn−ynγnAun−γnAyn, tn−yn

≤γnAun−γnAyn, tn−yn≤γnkun−yntn−yn.

3.5

So, it follows fromC3that the following inequality holds forn≥n0, wheren0is a positive

integer:

tn−u2≤ un−u2−un−yn2−yn−tn22γnkun−yntn−yn

≤ un−u2−un−yn2−yn−tn2γn2k2un−yn2tn−yn2

un−u2

γn2k2−1un−yn2

≤ un−u2.

3.6

PutM0max{x1−u,1/1−afu−u}.It is obvious thatx1−u ≤M0.Suppose

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From3.3,3.6andxn1αnfxn βnxn 1−αn−βnWntn, we haveuWnuand

xn1−uαnfxn βnxn1−αn−βnWntn−u

≤αnfxn−fuαnfu−uβnxn−u1−αn−βnWntn−u

≤αnaxn−uαnfu−uβnxn−u1−αn−βnWntn−u

≤αnaxn−uαnfu−uβnxn−u1−αn−βntn−u

≤αnaxn−uαnfu−uβnxn−u1−αn−βnxn−u

1−aαn

_f_u₋_u

1−a 1−1−aαnxn−u

≤1−aαnM0 1−1−aαnM0M0

3.7

for everyn 1,2, . . .. Therefore,{xn}is bounded. From3.3and3.6, we also obtain that

{tn}and{un}are bounded.

Fromyn PCun−γnAunand the monotonicity and the Lipschitz continuity ofA, we

have

yn−u2PCun−γnAun−PCu−γnAu2

≤un−γnAun−u−γnAu2

un−u2−2γnAun−Au, un−uγn2Aun−Au2

≤ un−u2γn2k2un−u2

1γ_n2k2un−u2.

3.8

Hence, we obtain that {yn}is bounded. It follows from the Lipschitz continuity ofA that

{Axn},{Aun}, and{Ayn} are bounded. Since f andWn are nonexpansive, we know that

{fxn}and{Wntn}are also bounded. From the definition oftn, we get

tn1−tnPCun1−γn1Ayn1

−PCun−γnAyn

≤un1−γn1Ayn1

−un−γnAyn

≤un1−γn1Aun1

−un−γn1Aunγn1

Aun1−Ayn1−AunγnAyn

≤ un1−unγn1Aun1−Aunγn1Aun1−Ayn1−AunγnAyn

≤ un1−unkγn1un1−unγn1Aun1−Ayn1−AunγnAyn ≤ un1−unγn1γnM1,

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whereM1is an approximate constant such that

M1≥sup

n≥1

kun1−unAun1−Ayn1−AunAyn. _3.10

On the other hand, fromunTrnxnandun1Trn1xn1, we have

y−un, un−xn≥0, ∀y∈C, 3.11

Fun1, y

ϕy−ϕun1

1

rn1

y−un1, un1−xn1

≥0, ∀y∈C. 3.12

Puttingyun1in3.11andyunin3.12, we have

Fun, un1 ϕun1−ϕun _r1

nun1−un, un−xn ≥0,

Fun1, un ϕun−ϕun1

1

rn1

un−un1, un1−xn1 ≥0.

3.13

So, from the monotonicity ofF, we get

un1−un,un_r−xn n −

un1−xn1

rn1

≥0, 3.14

and hence

un1−un, un−un1un1−xn−_rrn n1

un1−xn1

≥0. 3.15

Without loss of generality, let us assume that there exists a real numberbsuch thatrn> b >0

for alln∈N.Then,

un1−un2≤

un1−un, xn1−xn

1− rn

rn1

un1−xn1

≤ un1−un

xn1−xn

1− rn

rn1

un1−xn1

,

3.16

and hence

un1−un ≤ xn1−xn_r1 n1|

rn1−rn|un1−xn1

≤ xn1−xn1_b|rn1−rn|M2,

3.17

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It follows from3.9and3.7that

tn1−tn ≤ xn1−xn1_b|rn1−rn|M2

γn1γnM1, 3.18

Define a sequence{vn}such that

xn1βnxn1−βnvn, ∀n≥1. 3.19

Then, we have

vn1−vn xn2−βn1xn1

1−βn1 −

xn1−βnxn

1−βn

αn1fxn1

1−αn1−βn1

Wn1tn1

1−βn1 −

αnfxn 1−αn−βnWntn

1−βn

αn1

1−βn1

fxn1−

αn

1−βnfxn Wn1tn1−Wntn

αn

1−βnWntn− αn1

1−βn1

Wn1tn1,

3.20

Next we estimateWn1tn1−Wntn. It follows from the definition ofWnthat

Wn1tn−Wntnλn1,NTNUn1,N−1tn 1−λn1,Ntn−λn,NTNUn,N−1tn−1−λn,Ntn

≤ |λn1,N−λn,N|tnλn1,NTNUn1,N−1tn−λn,NTNUn,N−1tn

≤ |λn1,N−λn,N|tnλn1,NTNUn1,N−1tn−TNUn,N−1tn

|λn1,N−λn,N|TNUn,N−1tn

≤2M3|λn1,N−λn,N|λn1,NUn1,N−1tn−Un,N−1tn,

3.21

whereM3is an approximate constant such that

M3≥max

sup

n≥1

{tn},sup n≥1

{TkUn,k−1tn} |k1,2, . . . , N

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Since 0< λni ≤1 for alln≥1 andi1,2, . . . , N, we have

Un1,N−1tn−Un,N−1tn

λn1,N−1TN−1Un1,N−2tn 1−λn1,N−1tn−λn,N−1TN−1Un,N−2tn−1−λn,N−1tn

≤ |λn1,N−1−λn,N−1|tnλn1,N−1TN−1Un1,N−2tn−λn,N−1TN−1Un,N−2tn

≤ |λn1,N−1−λn,N−1|tnλn1,N−1TN−1Un1,N−2tn−TN−1Un,N−2tn

|λn1,N−1−λn,N−1|TN−1Un,N−2tn

≤2M3|λn1,N−1−λn,N−1|Un1,N−2tn−Un,N−2tn.

3.23

It follows that

Un1,N−1tn−Un,N−1tn

≤2M3|λn1,N−1−λn,N−1|2M3|λn1,N−2−λn,N−2|Un1,N−3tn−Un,N−3tn

≤2M3

N−1

i2

|λn1,i−λn,i|Un1,1tn−Un,1tn

2M3

N−1

i2

|λn1,i−λn,i|λn1,1T1tn 1−λn1,1tn−λn,1T1tn−1−λn,1tn

≤2M3

N−1

i1

|λn1,i−λn,i|.

3.24

Substituting3.24into3.21yields that

Wn1tn−Wntn ≤2M3|λn1,N−λn,N|2λn1,NM3

N−1

i1

|λn1,i−λn,i|

≤2M3

N

i1

|λn1,i−λn,i|.

3.25

Hence, we have

Wn1tn1−Wntn ≤ Wn1tn1−Wn1tnWn1tn−Wntn

≤ tn1−tn2M3

N

i1

|λn1,i−λn,i|.

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From3.20,3.26, and3.18, we have

vn1−vn − xn1−xn

≤ αn1

1−βn1

_f_x_n₁_W_n₁_t_n₁ αn

1−βn

_f_x_n_W_n_t_n

Wn1tn1−Wntn − xn1−xn

≤ αn1

1−βn1

_f_x_n₁Wn1tn1₁₋αn_β

n

_f_x_nWntn

tn1−tn2M3

N

i1

|λn1,i−λn,i| − xn1−xn

≤ αn1

1−βn1

_f_x_n₁Wn1tn1

αn

1−βn

_f_x_nWntn2M3

N

i1

|λn1,i−λn,i|

1

b|rn1−rn|M2

γn1γnM1.

3.27

It follows fromC1–C5that

lim sup

n→ ∞ vn1−vn − xn1−xn≤0. 3.28

Hence byLemma 2.4, we have limn→ ∞vn−xn0. Consequently

lim

n→ ∞xn1−xnnlim→ ∞

1−βnvn−xn0. 3.29

Sincexn1 αnfxn βnxn 1−αn−βnWntn, we have

xn−Wntn ≤ xn1−xnxn1−Wntn

≤ xn1−xnαnfxn−Wntnβnxn−Wntn,

3.30

and thus

xn−Wntn ≤ 1

1−βn

xn1−xnαnfxn−Wntn. 3.31

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Sincexn1 αnfxn βnxn 1−αn−βnWntn,foru∈ Ω, it follows from3.3and

3.6that

xn1−u2αnfxn βnxn 1−αn−βnWntn−u2

≤αnfxn−u2βnxn−u21−αn−βnWntn−u2

≤αnfxn−u2βnxn−u21−αn−βntn−u2

≤αnfxn−u2βnxn−u21−αn−βnun−u2

γn2k2−1un−yn2

≤αnfxn−u2 1−αnxn−u21−αn−βnγn2k2−1un−yn2

3.32

from which it follows that

un−yn2 ≤ αn

1−αn−βn1−γn2k2

fxn−u2− xn−u2

1

xn−u2− xn1−u2

≤ αn

1

1−αn−βn1−γn2k2xn−u − xn1−uxn1−xn.

3.33

It follows fromC1–C3andxn1−xn → 0 thatun−yn → 0.

By the same argument as in3.6, we also have

tn−u2 ≤ un−u2−un−yn2−yn−tn22γnkun−yntn−yn

≤ un−u2−un−yn2−yn−tn2un−yn2γn2k2tn−yn2

un−u2

γn2k2−1yn−tn2.

3.34

Combining the above inequality and3.32, we have

xn1−u2≤αnfxn−u2βnxn−u21−αn−βntn−u2

≤αnfxn−u2βnxn−u21−αn−βnun−u2γn2k2−1yn−tn2

≤αnfxn−u2 1−αnxn−u21−αn−βnγn2k2−1yn−tn2,

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and thus

tn−yn2 ≤ αn

1

xn−u2− xn1−u2

≤ αn

1

1−αn−βn1−γn2k2xn−u − xn1−uxn1−xn,

3.36

which implies thattn−yn → 0.

Fromun−tn ≤ un−ynyn−tnwe also haveun−tn → 0. AsAisk-Lipschitz

continuous, we haveAyn−Atn → 0.

Foru∈Ω, we have, fromLemma 2.1,

un−u2Trnxn−Trnu2≤ Trnxn−Trnu, xn−u

un−u, xn−u 1

2

un−u2xn−u2− xn−un2

. 3.37

Hence,

un−u2≤ xn−u2− xn−un2. 3.38

By3.3,3.6,3.32, and3.38, we have

xn1−u2 ≤αnfxn−u2βnxn−u21−αn−βntn−u2

≤αnfxn−u2βnxn−u21−αn−βnun−u2

≤αnfxn−u2βnxn−u21−αn−βnxn−u2− xn−un2

≤αnfxn−u2 1−αnxn−u2−1−αn−βnxn−un2.

3.39

Hence,

1−αn−βnxn−un2 ≤αnfxn−u2−αnxn−u2xn−u2− xn1−u2

≤αnfxn−u2−αnxn−u2 xn−uxn1−uxn−xn1. 3.40

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Since

_W_n_y_n₋_y_n_≤_W_n_y_n₋_W_n_t_n_W_n_t_n₋_x_n_x_n₋_u_n_u_n₋_y_n

≤yn−tnWntn−xnxn−unun−yn.

3.41

It follows that

lim

n→ ∞Wnyn−yn0. 3.42

Next we show that

lim sup

n→ ∞

fu0−u0, xn−u0

≤0, _3.43

whereu0PΩfu0. To show this inequality, we can choose a subsequence{xnj}of{xn}such that

lim

n→ ∞

fu0−u0, xnj−u0

lim sup

n→ ∞

fu0−u0, xn−u0

. _3.44

Since{xni} is bounded, there exists a subsequence{xnij} of{xni} which converges

weakly tow. Without loss of generality, we can assume that{xni} w.Fromxn−un → 0,

we obtain thatu_ni w. Fromun−yn → 0, we also obtain thatyni w. Fromun−tn → 0,

we also obtain thattni w. Since{uni} ⊂CandCis closed and convex, we obtainw∈C.

In order to show thatw ∈Ω, we first showw ∈ MEPF, ϕ.Byun Trnxn,we know that

y−un, un−xn≥0, ∀y∈C. 3.45

It follows fromA2that

ϕy−ϕun _r1 n

y−un, un−xn≥Fy, un, ∀y∈C. 3.46

Hence,

ϕy−ϕuni

y−uni,

uni−xni rni

≥Fy, uni

, ∀y∈C. 3.47

It follows fromA4,A5, and the weakly lower semicontinuity ofϕ,uni−xni/rni → 0 anduni wthat

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Fortwith 0< t≤1 andy∈C, letytty 1−tw.Sincey∈Candw∈C, we obtain yt∈Cand henceFyt, w ϕw−ϕyt≤0. So byA4and the convexity ofϕ, we have

0Fyt, ytϕyt−ϕyt

≤tFyt, y 1−tFyt, wtϕy 1−tϕw−ϕyt

≤t Fyt, yϕy−ϕyt!.

3.49

Dividing byt, we get

Fyt, yϕy−ϕyt≥0. 3.50

Lettingt → 0, it follows fromA3and the weakly lower semicontinuity ofϕthat

Fw, yϕy−ϕw≥0 3.51

for ally∈Cand hencew∈MEPF, ϕ. Now we show thatw∈VIC, A.Put

Tw1

⎧ ⎨ ⎩

Aw1NCw1 ifw1∈C,

∅ ifw1/∈C,

3.52

whereNCw1is the normal cone toCatw1∈C. We have already mentioned that in this case

the mappingT is maximal monotone, and 0∈Tw1if and only ifw1∈VIC, A. Letw1, g∈

GT. ThenTw1Aw1NCw1and henceg−Aw1∈NCw1. So, we havew1−t, g−Aw1 ≥0

for allt∈C. On the other hand, fromtn PCun−λnAynandw1∈Cwe have

un−λnAyn−tn, tn−w1

≥0, 3.53

and hence

w1−tn,tn_λ−un n Ayn

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Therefore, we have

w1−tni, g

≥ w1−tni, Aw1

≥ w1−tni, Aw1 −

w1−tni,

tni−uni λni

Ayni

w1−tni, Aw1−Ayni−

w1−tni, Aw1−AtniAtni−Ayni−

w1−tni, Aw1−Atni

w1−tni, Atni −Ayni

−

v−tni,

≥w1−tni, Atni−Ayni

−

w1−tni,

.

3.55

Hence we obtainw1−w, g ≥ 0 asi → ∞. SinceT is maximal monotone, we have

w∈T−1_{0 and hence}_w_∈_VI_{C, A}_.

We next show thatw ∈N_i₁FixTi.To see this, we observe that we may assumeby

passing to a further subsequence if necessaryλni,k → λk fork 1,2, . . . , N.LetW be the W-mapping generated byT1, T2, . . . , TNandλ1, λ2, . . . , λN. ByLemma 2.5, we know thatWis

nonexpansive andN_i₁FixTi FixW. it follows fromLemma 2.6that

Wnix−→Wx, ∀x∈C. 3.56

Assumew /∈FixW.Sincexni wandw /Ww, it follows from the Opial condition,3.42, and3.56that

lim inf

i→ ∞ xni−w<lim infi→ ∞ xni−Ww

≤lim inf

i→ ∞ {xni−WnixniWnixni−WxniWxni−Ww}

≤lim inf

i→ ∞ xni−w,

3.57

which is a contradiction. Hence, we havew ∈ FixW N_i₁FixTi. This impliesw ∈ Ω.

Therefore, we have

lim sup

n→ ∞

fu0−u0, xn−u0

lim

j→ ∞

fu0−u0, xnj−u0

fu0−u0, w−u0

≤0. _3.58

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FromLemma 2.3, we have

xn1−u02αnfxn−u0 βnxn−u0 1−αn−βnWntn−u02

≤βnxn−u0 1−αn−βnWntn−u022αnfxn−u0, xn1−u0

≤1−αn−βnWntn−u02βnxn−u022αnfxn−u0, xn1−u0

≤1−αn−βnWntn−u02βnxn−u022αnfxn−fu0, xn1−u0

2αnfu0−u0, xn1−u0

≤1−αn−βntn−u02βnxn−u022αnaxn−u0xn1−u0

≤1−αnxn−u02αna

xn−u02xn1−u02

,

3.59

and thus

xn1−u02≤

1− αn 1−aαn

xn−u02 αn

1−aαn

2fu0−2u0, xn1−u0

. 3.60

It follows fromLemma 2.2,3.58, and3.60that limn→ ∞xn−u0 0. From xn− un → 0 andyn−un → 0, we haveun → u0andyn → u0. The proof is now complete.

4. Applications

By Theorem 3.1, we can obtain some new and interesting strong convergence theorems as

follows.

Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction fromC×CtoR satisfyingA1–A5and letϕ : C → R∪ {∞}be a proper lower semicontinuous and convex function. LetT1, T2, . . . , TNbe a finite family of nonexpansive mappings

ofCintoHsuch thatΣ N_n₁FixTi∩MEPF, ϕ/∅. Let{λn,1},{λn,2}, . . . ,{λn,N}be sequences in ε1, ε2 with 0 < ε1 ≤ ε2 < 1. Let Wn be the W-mapping generated by T1, T2, . . . , TN and λn,1, λn,2, . . . , λn,N. Assume that eitherB1 or B2holds. Letf be a contraction of H into itself

and let{xn},{un}, and{yn}be sequences generated by

x1x∈C,

xn1αnfxn βnxn1−αn−βnWnun

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for everyn1,2, . . .where{rn},{αn},{λn1},{λn2}, . . .,{λnN}, and{βn}are sequences of numbers satisfying the following conditions:

Then,{xn},{un}, and{yn}converge strongly towPΣfw.

Proof. PuttingA0, byTheorem 3.1we obtain the desired result.

Theorem 4.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction fromC×CtoRsatisfyingA1–A5. LetAbe a monotone andk-Lipschitz continuous mapping ofCintoH. LetT1, T2, . . . , TNbe a finite family of nonexpansive mappings ofCintoHsuch thatΛ N_n₁FixTi∩VIC, A∩EPF/∅. Let{λn,1},{λn,2}, . . . ,{λn,N}be sequences inε1, ε2

with0< ε1 ≤ε2 <1. LetWnbe theW-mapping generated byT1, T2, . . . , TNandλn,1, λn,2, . . . , λn,N. Assume that eitherB4orB2holds. Letfbe a contraction ofHinto itself and let{xn},{un}, and

x1x∈C,

Fun, y_r1 n

4.2

for everyn 1,2, . . .where {γn},{rn},{αn},{λn1},{λn2}, . . .,{λnN}, and {βn}are sequences of numbers satisfying the following conditions:

C3limn→ ∞γn0;

Then,{xn},{un}, and{yn}converge strongly towPΛfw.

Proof. Puttingϕ0, byTheorem 3.1we obtain the desired result.

Theorem 4.3. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Letϕ : C →

R∪{∞}be a proper lower semicontinuous and convex function. LetAbe a monotone andk-Lipschitz continuous mapping ofCintoH. LetT1, T2, . . . , TN be a finite family of nonexpansive mappings of

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λn,1, λn,2, . . . , λn,N. Assume that eitherB3orB2holds. Letfbe a contraction ofHinto itself and let{xn},{un}, and{yn}be sequences generated by

x1x∈C,

ϕy−ϕun _r1 n

4.3

for everyn 1,2, . . . .where{γn},{rn},{αn},{λn1},{λn2}, . . .,{λnN}, and{βn}are sequences of numbers satisfying the following conditions:

C3limn→ ∞γn0;

C5limn→ ∞|λn,i−λn−1,i|0for alli1,2, . . . , N. Then,{xn},{un}, and{yn}converge strongly towPΘfw.

Proof. LetFx, y 0 for allx, y∈C, byTheorem 3.1we obtain the desired result.

Theorem 4.4. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction fromC×CtoR satisfyingA1–A5and letϕ : C → R∪ {∞}be a proper lower semicontinuous and convex function. LetAbe a monotone andk-Lipschitz continuous mapping ofC intoH. LetSbe a nonexpansive mapping ofCintoHsuch thatFixS∩VIC, A∩MEPF, ϕ/∅. Assume that eitherB1orB2holds. Letfbe a contraction ofHinto itself and let{xn},{un}, and

x1x∈C,

xn1αnfxn βnxn1−αn−βnSPCun−γnAyn

4.4

for every n 1,2, . . . where {γn}, {rn}, {αn} and {βn} are sequences of numbers satisfying the following conditions:

C3limn→ ∞γn0;

C4lim infn→ ∞rn>0andlimn→ ∞|rn1−rn|0.

Then,{xn},{un}, and{yn}converge strongly towPFixS∩VIC,A∩MEPF,ϕfw.

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Theorem 4.5. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone and k-Lipschitz continuous mapping of C into H. Let T1, T2, . . . , TN be a finite family of nonexpansive mappings of C into H such that Γ N_n₁FixTi∩ VIC, A/∅. Let

{λn,1},{λn,2}, . . . ,{λn,N}be sequences inε1, ε2with0 < ε1 ≤ε2 < 1. LetWnbe theW-mapping generated byT1, T2, . . . , TNandλn,1, λn,2, . . . , λn,N. Let{xn}and{yn}be sequences generated by

x1x∈C,

ynPCxn−γnAxn,

xn1αnfxn βnxn1−αn−βnWnPCxn−γnAyn

4.5

for everyn1,2, . . .where{γn},{αn},{λn1},{λn2}, . . .,{λnN}, and{βn}are sequences of numbers satisfying the following conditions:

C3limn→ ∞γn0;

C5limn→ ∞|λn,i−λn−1,i|0for alli1,2, . . . , N. Then,{xn}and{yn}converge strongly towPΓfw.

Proof. Letϕ0 and letFx, y 0 for allx, y ∈C. Thenun PCxn xn. ByTheorem 3.1we

obtain the desired result.

Remark 4.6. 1Since the α-inverse-strongly-monotonicity ofA has been weakened by the monotonicity and Lipschitz continuity of A. Theorems 3.1, 4.2, and 4.4 generalize and improveTheorem 3.1in11,Theorem 3.1in12, andTheorem 3.1in8and the main results

in31.Theorem 4.5improvesTheorem 3.1in20.

2 It is easy to see that Theorems 3.1, 4.2, and 4.4 also generalize and improve Theorems3.1, and4.2in9.

3It is clear thatTheorem 4.5generalizes, extends, and improvesTheorem 3.1in18

andTheorem 3.1in19.

4Theorem 3.1improves and extendsTheorem 3.1in1.

Acknowledgments

The authors would like to express their thanks to the referee for helpful suggestions. This research was supported by the National Center of Theoretical SciencesSouthof Taiwan, the National Natural Science Foundation of ChinaGrants 10771228 and 10831009, the Natural Science Foundation of ChongqingGrant no. CSTC, 2009BB8240, and the Research Project of Chongqing Normal UniversityGrant no. 08XLZ05.

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