Strong Convergence of an Iterative Method for Equilibrium Problems and Variational Inequality Problems

Volume 2009, Article ID 362191,21pages doi:10.1155/2009/362191

Research Article

Strong Convergence of an Iterative

Method for Equilibrium Problems and

Variational Inequality Problems

HongYu Li

_{and HongZhi Li}

1_{Department of Mathematics, TianJin Polytechnic University, TianJin 300160, China}

2_{Department of Mathematics, Agricultural University of Hebei, BaoDing 071001, China}

Correspondence should be addressed to HongYu Li,lhy [email protected]

Received 26 August 2008; Revised 11 November 2008; Accepted 9 January 2009

Recommended by Massimo Furi

We introduce an iterative method for finding a common element of the set of solutions of equilibrium problems, the set of solutions of variational inequality problems, and the set of fixed points of finite many nonexpansive mappings. We prove strong convergence of the iterative sequence generated by the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for the minimization problem.

Copyrightq2009 H. Li and H. Li. 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 norm · , respectively. Suppose thatCis nonempty, closed convex subset ofHandFis a bifunction fromC×CtoR, where

Ris the set of real number. The equilibrium problem is to find ax∈Csuch that

Fx, y≥0, ∀y∈C. 1.1

The set of such solutions is denoted by EPf. Numerous problems in physics, optimization, and economics reduce to find a solution of equilibrium problem. Some methods have been proposed to solve the equilibrium problems in Hilbert space, see, for instance, Blum and Oettli1, Combettes and Hirstoaga2, and Moudafi3.

A mappingA:C → His called monotone ifAu−Av, u−v ≥0.Ais called relaxed

u, v-cocoercive, if there exist constantsu >0 andv >0 such that

whenu 0, Ais calledv-strong monotone; whenv 0, Ais called relaxedu-cocoercive. LetA:C → Hbe a monotone operator, the variational inequality problem is to find a point

u∈C, such that

Au, v−u ≥0, ∀v∈C. 1.3

The set of solutions of variational inequality problem is denoted by VIC,A. The variational inequality problem has been extensively studied in literatures, see, for example,4,5and references therein.

LetB be a strong positive bounded linear operator onH with coefficientγ, that is, there exists a constantγ >0 such thatBx, x ≥γx2_{,for allx∈H. 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∈FTAx, x − x, b, 1.4

whereTis a nonexpansive mapping onHandbis a point onH.

A mappingTfromCinto itself is called nonexpansive, ifTx−Ty ≤ x−y,∀x, y∈C. The set of fixed points ofTis denoted byFT. Let{T_i}N_i1be a finite family of nonexpansive mappings andFN_i1FTi/∅, define the mappings

Un,1 λn,1T1

1−λn,1

I,

Un,2 λn,2T2Un,1

1−λ_n,2

I,

.. .

Un,N−1 λn,N−1TN−1Un,N−2

1−λ_n,N−1

I,

WnUn,Nλn,NTNUn,N−1

1−λn,NI,

1.5

where{λn,i}Ni1 ⊂0,1for alln≥1. Such a mappingWnis calledW-mapping generated by T1, T2, . . . , TNand{λn,i}Ni1. We know thatWnis nonexpansive andFWn

_N

i1FTi, see6.

LetS : C → C be a nonexpansive mapping andf : C → Cis a contractive with coefficientα∈0,1. Marino and Xu7considered the following general iterative scheme:

xn1 αnγfxn1−αnBSxn. 1.6

They proved that{xn}converges strongly tozPFSI−Bγfz, wherePFSis the metric

By combining equilibrium problems and1.6, Plutbieng and Pumpaeng8proposed the following algorithm:

Fun, y_r1 n

y−un, un−xn≥0, ∀y∈H,

xn1αnγfxnI−αnBSun.

1.7

They proved that if the sequences{α_n}and{r_n} satisfy some appropriate conditions, then sequence{xn}convergence to the unique solutionzof the variational inequality

B−γfz, x−z≥0, ∀x∈FS∩EPF. 1.8

Motivated by 8, Colao et al.9 introduced an iterative method for equilibrium problem and finite family of nonexpansive mappings

Fun, y_r1 n

y−un, un−xn≥0, ∀y∈H,

xn1αnγfxnβxn1−βI−αnBWnun,

1.9

and proved that{x_n} converges strongly to a pointx∗ ∈ F∩EPFand x∗ also solves the variational inequality1.8. For equilibrium problems, also see10,11.

On the other hand, letA : C → Cbe aα-cocoercive mapping, for finding common element of the solution of variational inequality problems and the set of fixed point of nonexpansive mappings, Takahashi and Toyoda12introduced iterative scheme

xn1 αnxn1−αnSPCI−λnAxn. 1.10

They proved that{xn}converges weakly toz∈FS∩VIC,A. Inspired by1.10and13, Y. Yao and J.-C. Yao14given the following iterative process:

ynPCI−λnAxn,

xn1αnuβnxnγnSPCI−λnAyn,

1.11

and proved that {xn} converges strongly to z ∈ FS∩VIC,A. By combining viscosity approximation method and1.10, Chen et al.15introduced the process

and studied the strong convergence of sequence{xn}generated by1.12. Motivated by1.6,

1.11, and1.12, Qin et al.16introduced the following general iterative process

ynPCI−snAxn,

xn1αnγfWnxnI−αnBWnPCI−tnAyn,

1.13

and established a strong convergence theorem of{xn}to an element ofNi1FTi∩VIC,A.

The purpose of this paper is to introduce the iterative process:x1∈Hand

Fun, y _r1 n

y−un, un−xn≥0, ∀y∈C,

ynbnun1−bnWnPCI−snAun,

xn1αnγfWnxnβxn1−βI−αnBWnPCI−tnAyn,

1.14

whereWnis defined by1.5,Aisu, v-cocoercive, andBis a bounded linear operator. We should show that the sequences{xn}converge strongly to an element ofNi1FTi∩VIC,A∩

EPF. Our result extends the corresponding results of Qin et al.16and Colao et al.9, and many others.

2. Preliminaries

LetHbe a real Hilbert space andCa nonempty, closed convex subset ofH. We denote strong convergence of{xn}toxbyxn → xand weak convergence byxn x. LetPC : C → H is a mapping such that for every pointx ∈ H, there exists a unique P_Cx ∈ C satisfying

x−PCx ≤ x−y, for ally∈C.PCis called the metric projection ofHontoC. It is known thatPC is a nonexpansive mapping fromHontoC. It is also known thatPCx∈Cand

x−PCx, PCx−y≥0, ∀x∈H, y∈C, 2.1

x−y, Pcx−PCy≥PCx−PCy2, ∀x, y∈H. 2.2

Let A : C → H be a monotone mapping of C into H, then u ∈ VIC,A if and only if

uPCu−λAu,for allλ >0. The following result is useful in the rest of this paper.

Lemma 2.1see17. Assume{an}is a sequence of nonegative real number such that

an1≤

1−αnanδn, ∀n≥0, 2.3

where{αn}is a sequence in0,1, and{δn}is a sequence inRsuch that

1∞_n0αn∞,

2lim sup_{n→ ∞}δn/αn≤0or∞n0|δn|<∞.

Lemma 2.2see18. Let{xn},{un}be bounded sequences in Banach spaceEsatisfyingxn1 τnxn 1−τnunfor alln≥0andlim infn→ ∞un1−un − xn1−xn≤0. Letτnbe a sequence

in0,1with0<lim infn→ ∞τn≤lim sup_{n→ ∞}τn<1. Then,limn→ ∞xn−un0.

Lemma 2.3. For allx, y∈H, there holds the inequality

xy ≤ x2_{2y, xy. 2.4}

Lemma 2.4see7. Assume thatAis a strong positive linear bounded operator on a Hilbert space Hwith coefficientγ >0and0< ρ≤ A−1_{. ThenI−ρA ≤1−ργ.}

For solving the equilibrium problem for a bifunctionF :C×C → R, we assume that

Fsatisfies the following conditions:

A1Fx, x 0 for allx∈C;

A2Fis monotone:Fx, y Fy, x≤0 for allx, y∈C;

A3for allx, y, z∈C, lim sup_t↓0Ftz 1−tx, y≤Fx, y;

A4for allx∈C, Fx, · is convex and lower semicontinuous.

The following result is in Blum and Oettli1.

Lemma 2.5see1. LetC be a nonempty closed convex subset of a Hilbert spaceE, let F be a bifunction fromC×CintoRsatisfying (A1)–(A4), letr >0, and letx∈H. Then there existsz∈C such that

Fz, y 1_ry−z, z−x ≥0, ∀y∈C. 2.5

We also know the following lemmas.

Lemma 2.6see19. Let Cbe a nonempty closed convex subset of Hilbert space H, let F be a bifunction fromC×CtoRsatisfying (A1)–(A4), letr > 0, and letx ∈ H, define a mappingTr :

H → Cas follows:

Trx z∈C:Fz, y 1_ry−z, z−x ≥0,∀y∈C

, 2.6

for allx∈H. Then, the following holds:

1Tr is single-valued;

2T_r is firmly nonexpansive-type mapping, that is, for allx, y∈H,

_Trx−Try2_≤T

rx−Try, x−y; 2.7

3FTr EPF;

A monotone operatorT :H → 2His said to be maximal monotone if its graphGT

{u, v:v∈Tu}is not properly contained in the graph of any other monotone operators. Let

Abe a monotone mapping ofCintoHand letNCvbe the normal cone forCat a point

v∈C, that is

NCv x∈H:v−y, x ≥0,∀y∈C. 2.8

Define

Tv ⎧ ⎨ ⎩

AvNCv, v∈C,

∅, v /∈C. 2.9

It is known that in this caseTis maximal monotone, and 0∈Tvif and only ifv∈VIC,A.

3. Strong Convergence Theorem

Theorem 3.1. LetHbe a real Hilbert space andCbe a nonempty closed convex subset ofH. {Ti}Ni1 a finite family of nonexpansive mappings fromCinto itself andF :C×C → Ra bifunction satisfying (A1)–(A4). LetA:C → Hbe relaxedu, v-cocoercive andμ-Lipschitzian. Letf :C → Cbe an α-contraction with0≤α <1andBa strong positive linear bounded operator with coefficientγ >0, γis a constant with0< γ < γ/α. Let sequences{αn}, {bn}be in0,1and{rn}be in0,∞,βis a

constant in0,1. AssumeC0N_i1FTi∩VIC,A∩EPF/∅and

ilim_{n→ ∞}α_n0, ∞_n1α_n∞;

iilimn→ ∞|rn1−rn|0, lim infn→ ∞rn>0;

iii{sn},{tn} ∈a, bfor somea, bwith0≤a≤b≤2v−uμ2/μ2andlimn→ ∞|sn1−sn|

limn→ ∞|tn1−tn|0;

ivlim_{n→ ∞}|λ_n1−λn|limn→ ∞|bn1−bn|0.

Then the sequence {xn} generated by 1.14 converges strongly to x∗ ∈ C0 and x∗ solves the variational inequalityx∗PC0I−B−γfx∗, that is,

γfx∗_−Bx∗_{, x−x}∗_{≤0, ∀x∈C}

0. 3.1

Proof. Without loss of generality, we can assumeαn≤1−βB−1. Then fromLemma 2.4we know

_{1−βI−αnB 1−βI−} αn

1−βB

≤1−β

1− αn 1−βγ

1−β−αnγ. 3.2

SinceAis relaxedu, v-cocoercive and μ-Lipschitzian andiiiholds, we know from14

that for allx, y∈Candn≥1, the following holds:

_I−snA

x−I−snAy≤ x−y,

_I−tnA

x−I−tnAy≤ x−y.

We divide the proof into several steps.

Step 1. {x_n}is bounded.

Takep ∈C0, notice thatun Trnxnand formLemma 2.62thatTrn is nonexpansive,

we have

_{un−pTrnxn−Trnp≤xn−p.} 3.4

SincepWnPCp−snAp, we have

_yn−pbnun

1−bnWnPCun−snAun−p

≤bnun−p1−bnWnPCun−snAun−p

≤bnun−p1−bnun−snAun−p−snAp

≤bnun−p1−bnun−p

≤xn−p.

3.5

Then we have

_{xn1−pαnγf}

Wnxnβxn1−βI−αnBWnPCI−tnAyn−p

αnγfWnxn−Bpβxn−p

1−βI−α_nBW_nP_CI−t_nAy_n−p

αnγfWnxn−Bpβxn−p1−β−αnγyn−p.

3.6

Thus From3.5we have

_{xn1−p≤αnγf}

Wnxn−fpαnγfp−Bpβxn−p

1−β−α_nγx_n−p

≤αnγαxn−pαnγfp−Bp 1−αnγxn−p

1−αnγ−αγxn−pαnγfp−Bp

1−αnγ−αγxn−pαnγ−αγ·

_γf

p−Bp γ−αγ

≤max xn−p,

_γf

p−Bp γ−αγ

≤max x1−p,

_γf

p−Bp γ−αγ

,

3.7

Step 2. limn→ ∞xn1−xn0.

Letx_n1βxn 1−βzn,for alln≥0, where

zn ₁₋1_βαnγfWnxn1−βI−αnBWnPCyn−tnAyn. 3.8

Then we have

_zn1−zn 1

1−βγ

αn1f

Wn1xn1

−αnfWnxn

1−βI−α_n1B

Wn1PCyn1−tn1Ayn1

−1−βI−α_nBW_nP_Cy_n−t_nAy_n

γ

1−β

αn1f

Wn1xn1

−αnfWnxn

Wn1PCyn1−tn1Ayn1

−WnPCyn−tnAyn

− 1

1−β

αn1BWn1PCyn1−tn1Ayn1

−αnBWnPCyn−tnAyn

≤Wn1PCyn1−tn1Ayn1

−Wn1PCyn−tn1Ayn

Wn1PCyn−tn1Ayn−Wn1PCyn−tnAyn

Wn1PC

yn−tnAyn−WnPCyn−tnAynK1,

3.9

where

K1 αn1

1−β

γfWn1xn1BWn1PCyn1−tn1Ayn1

αn

1−β

γfWnxnBWnPCyn−tnAyn.

3.10

Next we estimateWn1PCyn1−tn1Ayn1−Wn1PCyn−tn1Ayn,Wn1PCyn−tn1Ayn−

Wn1PCyn−tnAynandWn1PCyn−tnAyn−WnPCyn−tnAyn. At first

_Wn1PC

yn−tn1Ayn−Wn1PCyn−tnAyn≤tnAyn−tn1Ayntn1−tn·Ayn.

PutvnPCyn−tnAyn, we have

_Wn1PC

yn−tnAyn−WnPCyn−tnAyn

Wn1vn−Wnvn

Un1,Nvn−Un,Nvn

λn1,NTNUn1,N−1vn1−λn1,Nvn−λn,NTNUn,N−1vn−1−λn,Nvn

≤λn1,NTNUn1,N−1vn−λn,NTNUn,N−1vnλn1,N−λn,N ·vn

≤λn1,NUn1,N−1vn−Un,N−1vnλn1,N−λn,N· TNUn,N−1vn

λn1,N−λn,N ·vn

≤Un1,N−1vn−Un,N−1vnλn1,N−λn,NTNUn,N−1vnvn.

3.12

By recursion we get

_Wn1PC

yn−tnAyn−WnPCyn−tnAyn≤M · N

i1

_{λn1,i−λn,i,} 3.13

for someM >0. Similarly, we also get

_Wn1PC

un−snAun−WnPCun−snAun≤M · N

i1

_{λn1,i−λn,i. 3.14}

Since

Fun, y_r1 n

y−un, un−xn≥0, ∀y∈C,

Fun1, y

_r1

n1

y−un1, un1−xn1

≥0, ∀y∈C.

3.15

Putyun1in the first inequality andyunin the second one, we have

Fun, un1

1

rn

un1−un, un−xn≥0,

Fun1, un_r1 n1

un−un1, un1−xn1

≥0.

3.16

Adding both inequality, byA2we have

un1−un,

1

rn

un−xn−_r1 n1

un1−xn1

therefore, we have

un1−un, un−un1

xn1−xn

rn1−rn rn1

un1−xn1

≥0, 3.18

which implies that

_un1−un2_≤u

n1−un,xn1−xnrn_r1−rn n1

un1−xn1

≤un1−un · xn1−xn|rn_r1−rn| n1

_un1−xn1. 3.19

Hence we have

_{un1−un≤xn1−xn}|rn1−rn|

rn1

_un1−xn1, 3.20

so, by3.20and the propertyI−tnAx−I−tnAy ≤ x−y, we arrive at

_Wn1PC

yn1−tn1Ayn1

−Wn1PCyn−tn1Ayn

≤yn1−yn

bn1un1

1−b_n1

Wn1PCI−sn1A

un1

−bnun−1−bnWnPCI−snAun

bn1

un1−unbn1−bnun

1−bn1

Wn1PCun1−sn1Aun1

−Wn1PCun−sn1Aun

1−bn1

Wn1PC

un−sn1Aun

−Wn1PC

un−snAun

×1−bn1

Wn1PC

un−snAun−WnPCun−snAun

bn−bn1

WnPCun−snAun

≤bn1un1−un|bn1−bn| · un

1−b_n1un1−un1−bn1

|sn−sn1| · Aun

1−bn1Wn1PCun−snAun−WnPCun−snAun

bn−bn1· WnPCun−snAun

un1−un1−bn1Wn1PCun−snAun−WnPCun−snAunK2,

where

K2:

1−b_n1sn−sn1 ·Aunbn1−bnWnPCun−snAunvn. 3.22

Therefore, by3.14and3.20we get

_Wn1PC

yn1−tn1Ayn1

−Wn1PCyn−tn1Ayn

≤xn1−xn|rn_r1−rn| n1

_un1−xn1

1−b_n1

M ·N

i1

_{λn1,i−λn,iK2.}

3.23

Now submitting3.11,3.13, and3.23into3.9, we have

_{zn1−znxn1−xn}|rn1−rn| rn1

_un1−xn1

1−b_n1

M ·N

i1

_{λn1,i−λn,iK2}

tn1−tn ·AynM·

N

i1

_{λn1,i−λn,iK1.}

3.24

Thus conditionsii,iii, andivimply that

lim sup n→ ∞

_{zn1−zn−xn1−xn≤}0. 3.25

Then,Lemma 2.2yields

lim

n→ ∞xn1−xnnlim→ ∞1−βxn−zn0. 3.26

Note that

_yn−p2_b

nun1−bnWnPCI−snAun−p2

≤bnun−p21−bnun−snAun−p−snAp2

bnun−p21−bn

×un−p2−2snun−p, Aun−Aps2nAun−Ap2

≤un−p21−bn

×2snuAun−Ap2−2snvun−p2sn2Aun−Ap2

un−p21−bn2snus2n−2s_μn2v

Aun−Ap2,

3.27

_vn−p2_P

Cyn−tnAyn−PCp−tnAp2

≤yn−p−tnAyn−Ap2

yn−p2−2tnyn−p, Ayn−Apt2nAyn−Ap2

≤yn−p22tnuAyn−Ap−2tnvyn−p2t2nAyn−Ap2

≤yn−p2

2tnut2n−2_μtn2v

Ayn−Ap,

3.28

hence by 2t_nut2

n−2tnv/μ2<0, we know

_vn−p2_≤y

n−p2≤un−p21−bn2snus2n−2s_μn2v

Aun−Ap2. 3.29

ByLemma 2.3, we have

_xn1−p2_α

nγfWnxnβxn1−βI−αnBWnvn−p2

1−βWnvn−pβxn−pαnγfWnxn−BWnvn2

≤1−βWnvn−pβxn−p22αnγfWnxn−BWnvn, xn1−p

≤1−βWnvn−p2βxn−p22αnγfWnxn−BWnvn, xn1−p

≤1−βvn−p2βxn−p22αnM1,

for someM1>0. Submitting3.28into3.30, we have

_xn1−p2_≤

1−βyn−p2

2tnut2n−2_μtn2v

Ayn−Ap

βxn−p22αnM1

≤xn−p2 1−β

2tnut2n−2_μtn2v

Ayn−Ap2αnM1,

3.31

which implies

1−β

2tnv

μ2 −2tnu−t 2

n

Ayn−Ap

≤xn−p2−xn1−p22αnM1

xn−xn1xn1−p2−xn1−p22αnM1

xn−xn122

xn−xn1, xn1−p

2αnM1

≤xn−xn122xn−xn1· xn1−p2αnM1,

3.32

hence from conditionsi,iii, and3.26, we have

lim

n→ ∞Ayn−Ap0. 3.33

Similarly, submitting3.29into3.30, we also have

lim

n→ ∞Aun−Ap0. 3.34

Step 4. limn→ ∞Wnvn−vn0. By2.2we have

_vn−p2_P

Cyn−tnAyn−PCp−tnAp2

≤yn−tnAyn−p−tnAp, vn−p

1

2yn−tnAyn−

p−tnAp2vn−p2

−yn−vn−tnAyn−Ap2

≤ 1

2yn−p

2_v

n−p2−yn−vn−tnAyn−Ap2

1

2yn−p

2_v

n−p2−yn−vn2−t2nAyn−Ap2

2tnyn−vn, Ayn−Ap,

hence

_vn−p2_≤x

n−p2−yn−vn2−t2nAyn−Ap2

2t_ny_n−v_n ·Ay_n−Ap.

3.36

Submitting3.36into3.30, we have

_xn1−p2_≤

1−βxn−p2−yn−vn2−t2nAyn−Ap22tnyn−vn·Ayn−Ap

βxn−p22αnM1.

3.37

This implies that

1−βyn−vn2 ≤xn−p2−xn1−p2−t2nAyn−Ap2

2tnyn−vn · Ayn−Ap2αnM1.

3.38

Hence byStep 3we get

lim

n→ ∞yn−vn0. 3.39

Putyn bnun 1−bnWnqn, whereqnPCI−snAun. We have

_Wnqn−p2_≤P

CI−snAun−PCI−snAp2

≤un−snAun−p−snAp, Wnqn−p

1

2un−snAun−

p−snAp2Wnqn−p2

−un−Wnqn−snAun−Ap2

≤ 1

2un−p

2_W

nqn−p2−un−Wnqn2

−s2

nAyn−Ap22snun−Wnqn, Aun−Ap.

3.40

Hence

_Wnqn−p2_≤u

n−p2−un−Wnqn2−s2nAyn−Ap2

2snun−Wnqn ·Aun−Ap,

so, we get

_yn−p2_≤b

nun−p21−bnWnqn−p2

≤un−p2−1−bnun−Wnqn2

1−b_n−s2

nAyn−Ap22snun−Wnqn ·Aun−Ap.

3.42

Submitting into3.30, we have

_xn1−p2_≤

1−βun−p2−1−bnun−Wnqn2

1−b_n−s2

nAyn−Ap22snun−Wnqn· Aun−Ap

βxn−p22αnM1

≤xn−p2−1−β1−bnun−Wnqn2K3,

3.43

where

K3:

1−b_n−s2_nAy_n−Ap22s_nu_n−W_nq_n· Au_n−Ap2α_nM1, 3.44

which implies

lim

n→ ∞un−Wnqn0. 3.45

So, we have

lim

n→ ∞yn−unnlim→ ∞

1−b_nu_n−W_nq_n0, 3.46

which together with3.39gives

lim

n→ ∞un−vn0. 3.47

SinceunTrnxnandTrnis firmly nonexpansive, we have

_un−p2_T

rnxn−Trnp

≤Trnxn−Trnp, xn−p

1

2un−p

2_x

n−p2−Trnxn−xn 2_,

which implies

_un−p2_≤x

n−p2−un−xn2, 3.49

which together with3.30gives

_xn1−p2 _≤

1−βv_n−p2βx_n−p22α_nM1

≤1−βv_n−u_n2u_n−p22v_n−u_n, u_n−p

βxn−p22αnM1

≤1−βvn−un2 1−βxn−p2−un−xn2

21−βvn−un· un−pβxn−p22αnM1.

3.50

So

1−βun−xn2≤1−βvn−un2xn−p2−xn1−p2

21−βvn−un· un−p2αnM1.

3.51

Now3.47and conditioniimply that

lim

n→ ∞un−xn0. 3.52

Since

_{xn−Wnvn≤xn−xn1xn1−Wnvn}

xn−xn1αnγfWnxnβxn1−βI−αnBWnvn−Wnvn

xn−xn1αnγfWnxn−BWnvnβxn−Wnvn

≤xn−xn1αnγfWnxnBWnvnβxn−Wnvn.

3.53

Then we get

_xn−Wnvn≤ 1

1−βxn−xn1

αn 1−β

γfWnxnBWnvn, 3.54

hence

lim

Note that

_{Wnvn−vn≤Wnvn−xnxn−unun−vn,} 3.56

thus from3.47–3.55, we have

lim

n→ ∞vn−Wnvn0. 3.57

Step 5. lim sup_{n→ ∞}γfx∗−Bx∗, xn1−x∗ ≤0, wherex∗is the unique solution of variational

inequalityγfx∗−Bx∗, x−x∗ ≤0,for allx∈C0.

Take a subsequence{xnj}of{xn1}, such that

lim sup n→ ∞

γfx∗_−Bx∗_{, x}

n1−x∗

lim

j→ ∞

γfx∗_−Bx∗_{, x}

nj−x∗

. 3.58

Since{xn}is bounded, without loss of generality, we assume{xnj}itself converges weakly to

a pointp. We should provep∈C0

_N

i1FTi∩VIC,A∩EPF.

First, let

Tv ⎧ ⎨ ⎩

AvNCv, v∈C,

∅, v /∈C, 3.59

with the same argument as used in14, we can derivep∈T−1_{0, sinceTis maximal monotone,}

we knowp∈VIC,A.

Next, fromA2, for ally∈Cwe have

1

rn

y−un, un−xn≥Fy, un, 3.60

in particular

y−unj,

unj−xnj rnj

≥Fy, unj

. 3.61

ConditionA4implies thatFis weakly semicontinuous, then from3.52and letj → ∞we have

Fy, p≤0, ∀y∈C. 3.62

Replacingybyyt:ty 1−tpwitht∈0,1, usingA1andA4, we get

Divide by t in both side yieldsFty 1−tp, y ≥ 0, let t → 0, by A3 we conclude

Fp, y≥0,for ally∈C. Therefore,p∈EPF.

Finally, fromxn−vn → 0 we know thatvnj pj → ∞. Assumep /∈

_N

i1FTi,

that is,p /Wnp,for alln∈N. Since Hilbert space satisfies Opial’s condition, we have

lim inf

j→ ∞ vnj−p≤lim infj→ ∞ vnj−Wnjp

≤lim inf

j→ ∞ vnj−WnjvnjWnjvnj−Wnjp

≤lim inf

j→ ∞ vnj−p,

3.64

this is a contradiction, thusp∈N_i1FT_i, therefore,p∈C0

_N

i1FTi∩VIC,A∩EPF. So

we know

lim sup n→ ∞

γfx∗_−Bx∗_{, x}

n1−x∗

lim

j→ ∞

γfx∗_−Bx∗_{, x}

nj−x∗

γfx∗_−Bx∗_{, p−x}∗_≤0.

3.65

Step 6. The sequence{xn}converges strongly tox∗.

From the definition of{xn}and Lemmas2.3, and2.4, we have

_xn1−x∗2

1−βI−αnBWnvn−x∗βxn−x∗αnγfWnxn−Bx∗2

≤1−βI−αnBWnvn−x∗βxn−x∗2

2αnγfWnxn−Bx∗, xn1−x∗

≤1−β1−βI−αnB 1−β

Wnvn−x∗

2

βxn−x∗2

2αnγfWnxn−fx∗, xn1−x∗

2αnγfx∗−Bx∗, xn1−x∗

≤ 1−β−αnγ

2

1−β Wnvn−x

∗2_βx

n−x∗2

ααnγxn−x∗2xn1−x∗2

2α_nγfx∗−Bx∗, x_n1−x∗

≤

_1−β−α

nγ2

1−β βααnγ

xn−x∗2

ααnγxn1−x∗22αnγfx∗−Bx∗, xn1−x∗

1−2γ−αγα_n α

2

nγ2 1−β

xn−x∗2

ααnγxn1−x∗22αnγfx∗−Bx∗, xn1−x∗

,

which implies that

_xn1−x∗2 _≤1−2γ−αγαn

1−ααnγ

α2

nγ2

1−β1−ααnγ

xn−x∗2

2αn

1−ααnγ

γfx∗_−Bx∗_{, x}

n1−x∗

1−2αnγ−αγ 1−αα_nγ

xn−x∗2

αn

1−ααnγ

αnγ2

1−βxn−x

∗2

2γfx∗−Bx∗, xn1−x∗

.

3.67

Since from conditioniwe have∞_n12αnγ−αγ/1−ααnγ ∞and

lim sup n→ ∞

αnγ2

1−βxn−x

∗2_2γfx∗_−Bx∗_{, x}

n1−x∗

≤0, 3.68

so, byLemma 2.1, we concludexn → x∗. This completes the proof.

PuttingF ≡ 0 andbn β 0 for alln ≥ 1 in Theorem 3.1, we obtain the following corollary.

Corollary 3.2. LetHbe a real Hilbert space andCbe a nonempty closed convex subset ofH.{T_i}N_i1a finite family of nonexpansive mappings fromCinto itself. LetA:C → Hbe relaxedu, v-cocoercive andμ-Lipschitzian. Letf :C → Cbe anα-contraction with0≤α <1andBa strong positive linear bounded operator with coefficientγ >0,γbe a constant with0< γ < γ/α. Let sequences{α_n}, {b_n} in0,1andβbe a constant in0,1. AssumeC0

_N

i1FTi∩VIC,A/∅and

ilim_{n→ ∞}α_n0, ∞_n1α_n∞;

ii{sn}, {tn} ∈a, bfor somea, bwith0 ≤a≤b≤2v−uμ2/μ2andlimn→ ∞|sn1− sn|limn→ ∞|tn1−tn|0;

iiilimn→ ∞|λn1−λn|0.

Then the sequence{xn}generated byx1 ∈Cand

ynbnxn1−bnWnPCI−snAxn,

xn1αnγf

Wnxnβxn1−βI−αnBWnPCI−tnAyn,

3.69

converges strongly tox∗∈C0andx∗solves the variational inequalityx∗PC0I−B−γfx∗, that is,

γfx∗_−Bx∗_{, x−x}∗_{≤0, ∀x∈C}

0. 3.70

Corollary 3.3. LetH be a real Hilbert space andCbe a nonempty closed convex subset ofH. Sa finite family of nonexpansive mappings fromCinto itself andF:C×C → Ra bifunction satisfying (A1)–(A4). Letf :C → Cbe anα-contraction with0≤α <1andBa strong positive linear bounded operator with coefficientγ >0, γis a constant with0< γ < γ/α. Let sequences{αn}, {bn}in0,1

and{r_n}in0,∞. AssumeC0FS∩EPF/∅and

ilim_{n→ ∞}α_n0, ∞_n1α_n∞;

iilimn→ ∞|rn1−rn|0, lim infn→ ∞rn>0.

Then the sequence{x_n}generated byx1 ∈Cand

Fun, y_r1 n

y−un, un−xn≥0, ∀y∈H,

xn1αnγfSxn1−αnBSun,

3.71

converges strongly tox∗∈C0andx∗solves the variational inequalityx∗PC0I−B−γfx∗, that is,

γfx∗_−Bx∗_{, x−x}∗_{≤0, ∀x∈C}

0. 3.72

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