A Hybrid Extragradient Viscosity Approximation Method for Solving Equilibrium Problems and Fixed Point Problems of Infinitely Many Nonexpansive Mappings

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Volume 2009, Article ID 374815,32pages doi:10.1155/2009/374815

Research Article

A Hybrid Extragradient Viscosity

Approximation Method for Solving Equilibrium

Problems and Fixed Point Problems of Infinitely

Many Nonexpansive Mappings

Chaichana Jaiboon and Poom Kumam

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

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

Received 25 December 2008; Accepted 4 May 2009

Recommended by Wataru Takahashi

We introduce a new hybrid extragradient viscosity approximation method for finding the common element of the set of equilibrium problems, the set of solutions of fixed points of an infinitely many nonexpansive mappings, and the set of solutions of the variational inequality problems for β-inverse-strongly monotone mapping in Hilbert spaces. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality, which is the optimality condition for a minimization problem. Results obtained in this paper improve the previously known results in this area.

Copyrightq2009 C. Jaiboon 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.

1. Introduction

LetH be a real Hilbert space, and let Cbe a nonempty closed convex subset ofH. Recall that a mappingT of H into itself is called nonexpansivesee1if Tx−Ty ≤ x−y for allx, y ∈H. We denote byFT {x ∈ C: Tx x}the set of fixed points ofT. Recall also that a self-mappingf :H → His acontractionif there exists a constantα∈0,1such thatfx−fy ≤αx−y, for all x, y ∈H.In addition, letB:C → Hbe a nonlinear mapping. LetPC be the projection ofHontoC. The classical variational inequality which is

denoted byV IC, Bis to findu∈Csuch that

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For a givenz∈H,u∈Csatisfies the inequality

u−z, v−u ≥0, ∀v∈C, 1.2

if and only ifuPCz. It is well known thatPCis a nonexpansive mapping ofHontoCand

satisfies

x−y, PCx−PCy≥PCx−PCy2, ∀x, y∈H. 1.3

Moreover,PCxis characterized by the following properties:PCx∈Cand for allx∈H, y∈C,

x−PCx, y−PCx≤0, 1.4

x−y2

≥ x−PCx2y−PCx2. 1.5

It is easy to see that the following is true:

u∈V IC, B⇐⇒uPCu−λBu, λ >0. 1.6

One can see that the variational inequality1.1is equivalent to a fixed point problem. The variational inequality has been extensively studied in literature; see, for instance, 2–

6. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems. Recall the following.

1A mappingBofCintoHis called monotone if

Bx−By, x−y≥0, ∀x, y∈C. 1.7

2A mappingB is calledβ-strongly monotonesee7,8if there exists a constant β >0 such that

Bx−By, x−y≥βx−y2_, _∀_{x, y}_∈_C. 1.8

3A mappingBis calledk-Lipschitz continuous if there exists a positive real number ksuch that

Bx−By≤kx−y, ∀x, y∈C. 1.9

4A mapping Bis called β-inverse-strongly monotonesee 7, 8 if there exists a constantβ >0 such that

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Remark 1.1. It is obvious that anyβ-inverse-strongly monotone mappingBis monotone and 1/β-Lipschitz continuous.

5An operatorAis strongly positive onH if there exists a constantγ > 0 with the property

Ax, x ≥γx2_, _∀_x_∈_H. _1.11

6A set-valued mappingT :H → 2His called monotone if for allx, y∈H,f ∈Tx, andg ∈Tyimplyx−y, f−g ≥0. A monotone mappingT :H → 2His maximal if the graph ofGT ofT is 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 everyy, g∈GTimpliesf∈Tx. LetBbe a monotone map ofCintoH,and letNCv

be the normal cone toCatv∈C, that is,NCv{w∈H:u−v, w ≥0, for allu∈C},.

Tv

⎧ ⎨ ⎩

BvNCv, v∈ C,

∅, v /∈ C. 1.12

ThenT is the maximal monotone and 0∈Tvif and only ifv∈V IC, B; see9.

7 Let F be a bifunction ofC×Cinto R, whereR is the set of real numbers. The equilibrium problem forF:C×C → Ris to findx∈Csuch that

Fx, y ≥0, ∀y∈C. 1.13

The set of solutions of 1.13 is denoted by EPF. Given a mapping T : C → H, let Fx, y Tx, y− x for all x, y ∈ C. Then, z ∈ EPF if and only if Tz, y−z ≥ 0 for all y ∈ C.Numerous problems in physics, saddle point problem, fixed point problem, variational inequality problems, optimization, and economics are reduced to find a solution of 1.13. Some methods have been proposed to solve the equilibrium problem; see, for instance, 10–16. Recently, Combettes and Hirstoaga 17introduced an iterative scheme of finding the best approximation to the initial data whenEPFis nonempty and proved a strong convergence theorem.

In 1976, Korpelevich18introduced the following so-called extragradient method:

x0x∈C,

ynPCxn−λBxn,

xn1PCxn−λByn

1.14

for alln≥0,whereλ∈0,1/k, Cis a closed convex subset ofRn,andBis a monotone and k-Lipschitz continuous mapping ofCintoRn. He proved that ifV IC, Bis nonempty, then the sequences{xn}and{yn}, generated by1.14, converge to the same pointz∈V IC, B.

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the set of solution of variational inequalities forβ-inverse-strongly monotone, Takahashi and Toyoda19introduced the following iterative scheme:

x0∈C chosen arbitrary,

xn1αnxn 1−αnSPCxn−λnBxn, ∀n≥0,

1.15

whereBisβ-inverse-strongly monotone,{αn}is a sequence in0, 1, and{λn}is a sequence

in0,2β. They showed that ifFS∩V IC, Bis nonempty, then the sequence{xn}generated

by1.15converges weakly to somez∈FS∩V IC, B. Recently, Iiduka and Takahashi20

proposed a new iterative scheme as follows:

x0x∈C chosen arbitrary,

xn1αnx 1−αnSPCxn−λnBxn, ∀n≥0,

1.16

whereBisβ-inverse-strongly monotone,{αn}is a sequence in0, 1, and{λn}is a sequence

in0,2β. They showed that ifFS∩V IC, Bis nonempty, then the sequence{xn}generated

by1.16converges strongly to somez∈FS∩V IC, B.

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example,21–24and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. 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, 1.17

whereAis a linear bounded operator,Cis the fixed point set of a nonexpansive mapping SonH, andbis a given point inH. Moreover, it is shown in25that the sequence {xn}

defined by the scheme

xn1 nγfxn 1−nASxn 1.18

converges strongly to z PFSI − A γfz. Recently, Plubtieng and Punpaeng 26

proposed the following iterative algorithm:

Fun, y _r1 n

y−un, un−xn≥0, ∀y∈H,

xn1nγfxn I−nASun.

1.19

They prove that if the sequences{n}and{rn}of parameters satisfy appropriate condition,

then the sequences{xn}and{un}both converge to the unique solutionzof the variational

inequality

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which is the optimality condition for the minimization problem

min

x∈FS∩EPF

1

2Ax, x −hx, 1.21

wherehis a potential function forγfi.e.,hx γfxforx∈H.

Furthermore, for finding approximate common fixed points of an infinite countable family of nonexpansive mappings {Tn} under very mild conditions on the parameters.

Wangkeeree27introduced an iterative scheme for finding a common element of the set of solutions of the equilibrium problem1.13and the set of common fixed points of a countable family of nonexpansive mappings onC. Starting with an arbitrary initialx1 ∈ C, define a sequence{xn}recursively by

Fun, y _r1 n

y−un, un−xn≥0, ∀y∈C,

ynPCun−λnBun,

xn1 αnfxn βnxnγnSnPCun−λnByn , ∀n≥1,

1.22

where{αn},{βn},and{γn}are sequences in0,1. It is proved that under certain appropriate

conditions imposed on {αn},{βn},{γn}, and {rn}, the sequence {xn} generated by 1.22

strongly converges to the unique solution q ∈ ∩∞_n₁FSn∩V IC, B∩EPF, where p

P∩∞

n1FSn∩V IC,B∩EPFfqwhich extend and improve the result of Kumam14.

Definition 1.2see21. Let{Tn}be a sequence of nonexpansive mappings ofCinto itself,

and let{μn}be a sequence of nonnegative numbers in0,1. For eachn≥1, define a mapping

WnofCinto itself as follows:

Un,n1I,

Un,nμnTnUn,n1

1−μn I,

Un,n−1μn−1Tn−1Un,n1−μn−1 I,

.. .

Un,kμkTkUn,k1

1−μk I,

Un,k−1μk−1Tk−1Un,k1−μk−1 I,

.. .

Un,2μ2T2Un,3

1−μ2 I,

WnUn,1μ1T1Un,2

1−μ1 I.

1.23

Such a mappingWnis nonexpansive fromCtoC,and it is called theW-mapping generated

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On the other hand, Colao et al.28introduced and considered an iterative scheme for finding a common element of the set of solutions of the equilibrium problem1.13and the set of common fixed points of infinitely many nonexpansive mappings onC. Starting with an arbitrary initialx0∈C, define a sequence{xn}recursively by

Fun, y _r1

ny−un, un−xn ≥0, ∀y∈H,

xn1nγfxn βxn1−β I−nA Wnun,

1.24

where{n}is a sequence in0,1. It is proved28that under certain appropriate conditions

imposed on{n}and{rn}, the sequence{xn}generated by1.24strongly converges toz ∈

∩∞

n1FTn∩EPF, wherezis an equilibrium point forF and is the unique solution of the variational inequality1.20, that is,zP∩∞

n1FTn∩EPFI−A−γfz.

In this paper, motivated by Wangkeeree27, Plubtieng and Punpaeng26, Marino and Xu25, and Colao, et al.28, we introduce a new iterative scheme in a Hilbert spaceH which is mixed by the iterative schemes of1.18,1.19,1.22, and1.24as follows.

Letfbe a contraction ofHinto itself,Aa strongly positive bounded linear operator on Hwith coeﬃcientγ >0,andBaβ-inverse-strongly monotone mapping ofCintoH; define sequences{xn},{yn},{kn},and{un}recursively by

x1 x∈C chosen arbitrary,

Fun, y _r1

ny−un, un−xn ≥0, ∀y∈C,

knαnun 1−αnPCun−λnByn ,

xn1nγfxn βnxn1−βn I−nA Wnkn, ∀n≥1,

1.25

where {Wn}is the sequence generated by 1.23,{n},{αn},and {βn} ⊂ 0,1 and {rn} ⊂

0,∞satisfying appropriate conditions. We prove that the sequences{xn},{yn},{kn}and

{un}generated by the above iterative scheme1.25converge strongly to a common element

of the set of solutions of the equilibrium problem1.13, the set of common fixed points of infinitely family nonexpansive mappings, and the set of solutions of variational inequality

1.1for a β-inverse-strongly monotone mapping in Hilbert spaces. The results obtained in this paper improve and extend the recent ones announced by Wangkeeree 27, Plubtieng and Punpaeng26, Marino and Xu25, Colao, et al.28, and many others.

2. Preliminaries

We now recall some well-known concepts and results.

LetHbe a real Hilbert space, whose inner product and norm are denoted by·,·and

· , respectively. We denote weak convergence and strong convergence by notationsand

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A spaceHis said to satisfy Opial’s condition29if for each sequence{xn}inHwhich

converges weakly to pointx∈H, we have

lim inf

n→ ∞ xn−x<lim infn→ ∞ xn−y, ∀y∈H, y /x. 2.1

Lemma 2.1see25. LetCbe a nonempty closed convex subset ofH, letf be a contraction of H into itself withα ∈ 0,1, and letAbe a strongly positive linear bounded operator on Hwith coeﬃcientγ >0. Then , for0< γ < γ/α,

x−y,A−γf x−A−γf y≥γ−αγ x−y2_{, x, y}_∈_H. 2.2

That is,A−γfis strongly monotone with coeﬃcientγ−γα.

Lemma 2.2see25. Assume thatA is a strongly positive linear bounded operator onH with coeﬃcientγ >0and0< ρ≤ A−1. ThenI−ρA ≤1−ργ.

For solving the equilibrium problem for a bifunctionF :C×C → R, let us assume thatFsatisfies the following conditions:

A1Fx, x 0 for allx∈C;

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

A3for eachx, y, z∈C,limt↓0Ftz 1−tx, y≤Fx, y;

A4for eachx∈C, y→Fx, yis convex and lower semicontinuous. The following lemma appears implicitly in30.

Lemma 2.3see30. LetCbe a nonempty closed convex subset ofHand letFbe a bifunction of C×CintoRsatisfying (A1)–(A4). Letr >0andx∈H. Then, there existsz∈Csuch that

Fz, y 1_ry−z, z−x≥0 ∀y∈C. 2.3

The following lemma was also given in17.

Lemma 2.4see17. Assume thatF :C×C → Rsatisfies (A1)–(A4). Forr > 0andx ∈H, define a mappingTr :H → Cas follows:

Trx

z∈C:Fz, y 1 r

y−z, z−x≥0, ∀y∈C

2.4

for allz∈H. Then, the following holds:

1Tr is single-valued;

2Tr is firmly nonexpansive, that is, for anyx, y∈H,

_T_r_x₋_T_r_y2_≤_T

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3FTr EPF;

4EPFis closed and convex.

For eachn, k ∈ N, let the mappingUn,kbe defined by1.23. Then we can have the

following crucial conclusions concerningWn. You can find them in31. Now we only need

the following similar version in Hilbert spaces.

Lemma 2.5see31. LetC be a nonempty closed convex subset of a real Hilbert spaceH. Let T1, T2, . . . be nonexpansive mappings of C into itself such that ∩∞_n₁FTn is nonempty, and let

μ1, μ2, . . .be real numbers such that0 ≤ μn ≤ b < 1 for everyn ≥ 1. Then, for everyx ∈ Cand

k∈N, the limitlimn→ ∞Un,kxexists.

UsingLemma 2.5, one can define a mappingWofCinto itself as follows:

Wx lim

n→ ∞Wnxnlim→ ∞Un,1x 2.6

for everyx ∈ C. Such aW is called the W-mapping generated byT1, T2, . . . and μ1, μ2, . . .. Throughout this paper, we will assume that 0≤μn≤b <1 for everyn≥1. Then, we have the

following results.

Lemma 2.6see31. LetC be a nonempty closed convex subset of a real Hilbert spaceH. Let T1, T2, . . . be nonexpansive mappings of C into itself such that ∩∞n1FTn is nonempty, and let μ1, μ2, . . .be real numbers such that0≤μn≤b <1for everyn≥1. Then,FW ∩∞n1FTn. Lemma 2.7see32. If{xn}is a bounded sequence inC, thenlimn→ ∞Wxn−Wnxn0.

Lemma 2.8see33. Let{xn}and{zn}be bounded sequences in a Banach spaceX,and let{βn}be

a sequence in0,1with0<lim infn→ ∞βn≤lim sup_n_{→ ∞}βn<1.Supposexn1 1−βnznβnxn

for all integersn≥0andlim sup_n_{→ ∞}yn1−zn − xn1−xn≤0.Then,limn→ ∞zn−xn0.

Lemma 2.9see34. Assume that{an}is a sequence of nonnegative real numbers such that

an1≤1−lnanσn, n≥0, 2.7

where{ln}is a sequence in0,1and{σn}is a sequence inRsuch that

1∞_n₁ln∞;

2lim sup_n_{→ ∞}σn/ln≤0or∞n1|σn|<∞.

Thenlimn→ ∞an0.

Lemma 2.10. LetHbe a real Hilbert space. Then for allx, y∈H,

1xy2≤ x22y, xy;

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3. Main Results

In this section, we prove the strong convergence theorem for infinitely many nonexpansive mappings in a real Hilbert space.

Theorem 3.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH, letFbe a bifunction fromC×CtoRsatisfying (A1)–(A4), let{Tn}be an infinitely many nonexpansive ofCinto itself,

and let B be an β-inverse-strongly monotone mapping ofC into H such that Θ : ∩∞_n₁FTn∩

EPF∩V IC, B/∅. Letfbe a contraction ofHinto itself withα∈0,1,and letAbe a strongly positive linear bounded operator onH with coeﬃcient γ > 0 and0 < γ < γ/α. Let {xn},{yn},

{kn},and{un}be sequences generated by1.25, where{Wn}is the sequence generated by1.23,

{n},{αn},and{βn}are three sequences in0,1,and{rn}is a real sequence in0,∞satisfying the

following conditions:

ilimn→ ∞n0,∞n1n∞;

iilimn→ ∞αn0and∞n1αn∞;

iii0<lim infn→ ∞βn≤lim sup_n_{→ ∞}βn<1;

ivlim infn→ ∞rn>0andlimn→ ∞|rn1−rn|0;

v{λn/β} ⊂τ,1−δfor someτ, δ∈0,1andlimn→ ∞λn0.

Then,{xn}and{un}converge strongly to a pointz∈Θwhich is the unique solution of the variational

inequality

A−γf z, z−x≥0, ∀x∈Θ. 3.1

Equivalently, one haszPΘI−Aγfz.

Proof. Note that from the conditioni, we may assume, without loss of generality, thatn ≤

1−βnA−1for alln∈N. FromLemma 2.2, we know that if 0≤ρ≤ A−1, thenI−ρA ≤

1−ργ. We will assume thatI−A ≤1−γ. First, we show thatI−λnBis nonexpansive. Indeed,

from theβ-inverse-strongly monotone mapping definition onBand conditionv, we have

I−λnBx−I−λnBy2x−y−λnBx−By2

x−y2₋_2λ

nx−y, Bx−Byλ2nBx−By2

≤x−y2₋_2λ

nβBx−By2λ2nBx−By2

x−y2_λ

nλn−2β Bx−By2

≤x−y2_,

3.2

which implies that the mapping I−λnB is nonexpansive. On the other hand, sinceAis a

strongly positive bounded linear operator on H, we have

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Observe that

1−βn I−nA x, x1−βn−nAx, x

≥1−βn−nA

≥0,

3.4

and this show that1−βnI−nAis positive. It follows that

₁₋_β_n _I₋_n_A_sup₁₋_β_n _I₋_n_A _{x, x}_:_x_∈_H,_x₁

sup1−βn−nAx, x:x∈H,x1

≤1−βn−nγ.

3.5

LetQPΘ, whereΘ:∩∞n1FTn∩EPF∩V IC, B. Note thatf is a contraction ofHinto itself withα∈0,1. Then, we have

QI−Aγf x−QI−Aγf y P_ΘI−Aγf x−P_ΘI−Aγf y

≤I−Aγf x−I−Aγf y

≤ I−Ax−yγfx−fy

≤1−γ x−yγαx−y

1−γγα x−y

1−γ−γα x−y, ∀x, y∈H.

3.6

Since 0 < 1−γ −γα < 1, it follows that QI−Aγf is a contraction ofH into itself. Therefore by the Banach Contraction Mapping Principle, which implies that there exists a unique elementz∈Hsuch thatzQI−Aγfz PΘI−Aγfz.

We will divide the proof into five steps.

Step 1. We claim that{xn}is bounded. Indeed, pick anyp∈Θ. From the definition ofTr, we

note thatunTrnxn. If follows that

_u_n₋_p_T_r_n_x_n₋_T_r_n_p_≤_x_n₋_p_. _3.7

SinceI−λnBis nonexpansive andpPCp−λnBpfrom1.6, we have

yn−pPCun−λnBun−PCp−λnBp

≤un−λnAun−p−λnBp

I−λnAun−I−λnBp

≤un−p≤xn−p.

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Putvn PCun−λnByn. Sincep ∈ V IC, B, we havep PCp−λnBp. Substitutingx

un−λnAynandypin1.5, we can write

vn−p2 ≤un−λnByn−p2−un−λnByn−vn2

un−p2−2λnByn, un−pλ2nByn2

− un−vn22λnByn, un−vn−λ2nByn2

un−p2− un−vn22λnByn, p−vn

un−p2− un−vn22λnByn−Bp, p−yn

2λnBp, p−yn2λnByn, yn−vn.

3.9

Using the fact that B is β-inverse-strongly monotone mapping, and p is a solution of the variational inequality problemV IC, B, we also have

Byn−Bp, p−yn≤0, Bp, p−yn ≤0. 3.10

It follows from3.9and3.10that

vn−p2≤un−p2− un−vn22λnByn, yn−vn

un−p2−un−yn yn−vn22λnByn, yn−vn

≤un−p2−un−yn2−yn−vn2

−2un−yn, yn−vn2λnByn, yn−vn

un−p2−un−yn2−yn−vn22un−λnByn−yn, vn−yn.

3.11

Substitutingxbyun−λnBunandyvnin1.4, we obtain

un−λnBun−yn, vn−yn≤0. 3.12

It follows that

un−λnByn−yn, vn−ynun−λnBun−yn, vn−yn

λnBun−λnByn, vn−yn

≤λnBun−λnByn, vn−yn

≤λnBun−Bynvn−yn

≤ λ_βnun−ynvn−yn.

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Substituting3.13into3.11, we have

vn−p2≤un−p2−un−yn2−yn−vn22un−λnByn−yn, vn−yn

≤un−p2−un−yn2−yn−vn22λ_βnun−ynvn−yn

≤un−p2−un−yn2−yn−vn2λ

2

n

β2un−yn 2_v

n−yn2

un−p2−un−yn2λ

2

n

β2un−yn 2

un−p2

λ2

n

β2 −1

un−yn2

≤un−p2≤xn−p2.

3.14

Settingknαnun 1−αnvn, we can calculate

_x_n₁₋_p_n

γfxn−Ap βnxn−p 1−βn I−nA Wnkn−p

≤1−βn−nγ kn−pβnxn−pnγfxn−Ap

≤1−βn−nγ αnun−p 1−αnvn−p

βnxn−pnγfxn−Ap

≤1−βn−nγ αnxn−p 1−αnxn−p

βnxn−pnγfxn−Ap

1−βn−nγ xn−pβnxn−pnγfxn−Ap

1−nγ xn−pnγfxn−fp nγfp −Ap

≤1−nγ xn−pnγαxn−pnγfp −Ap

1−γ−γα n xn−pγ−γα n

γfp −Ap γ−γα .

3.15

By induction,

_x_n₋_p_≤_max_x₁₋_p_,γfp −Ap γ−γα

, n∈N. 3.16

Hence,{xn}is bounded, so are{un},{vn},{Wnkn},{fxn},{Bun},{yn},and{Byn}.

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Observing thatunTrnxnandun1Trn1xn1,we get

Fun, y _r1 n

y−un, un−xn≥0 ∀y∈H 3.17

Fun1, y 1 rn1

y−un1, un1−xn1

≥0 ∀y∈H. 3.18

Puttingyun1in3.17andyunin3.18, we have

Fun, un1 _r1

nun1−un, un−xn ≥0

Fun1, un _r1

n1

un−un1, un1−xn1 ≥0.

3.19

So, fromA2we have

un1−un,un_r−xn

n −

un1−xn1 rn1

≥0, 3.20

and hence

un1−un, un−un1un1−xn− rn

rn1

un1−xn1

≥0. 3.21

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

for alln∈N.Then, we have

un1−un2≤

un1−un, xn1−xn

1− rn

rn1

un1−xn1

≤ un1−un

xn1−xn1−_rrn n1

un1−xn1

,

3.22

and hence

un1−un ≤ xn1−xn_r1 n1|

rn1−rn|un1−xn1

≤ xn1−xnM_c1|rn1−rn|,

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whereM1sup{un−xn:n∈N}. Note that

vn1−vn ≤PCun1−λn1Byn1 −PCun−λnByn

≤un1−λn1Byn1−

un−λnByn

un1−λn1Bun1−un−λn1Bun

λn1

Bun1−Byn1−Bun λnByn

≤ un1−λn1Bun1−un−λn1Bun

λn1

Bun1Byn1Bun λnByn

≤ un1−unλn1

Bun1Byn1Bun λnByn,

kn1−knαn1un1 1−αn1vn1−αnun−1−αnvn

αn1un1−un αn1−αnun

1−αn1vn1−vn αn−αn1vn

≤αn1un1−un 1−αn1vn1−vn|αn−αn1|unvn

αn1un1−un 1−αn1

×un1−unλn1

Bun1Byn1Bun

λnByn|αn−αn1|unvn

un1−un 1−αn1λn1

Bun1Byn1Bun

1−αn1λnByn|αn−αn1|unvn

≤ xn1−xn M1

c |rn1−rn| 1−αn1λn1

×Bun1Byn1Bun

1−αn1λnByn|αn−αn1|unvn.

3.24

Setting

zn xn1−βnxn

1−βn

nγfxn 1−βn I−nA Wnkn

(15)

we havexn1 1−βnznβnxn, n≥1. It follows that

zn1−zn n1γfxn1

1−βn1 I−n1A Wn1kn1 1−βn1

− nγfxn

1−βn I−nA Wnkn

1−βn

n1 1−βn1

γfxn1− n

1−βnγfxn Wn1kn1−Wnkn

n

1−βnAWnkn−

n1 1−βn1

AWn1kn1

n1 1−βn1

γfxn1−AWn1kn1 n

1−βn

AWnkn−γfxn

Wn1kn1−Wn1knWn1kn−Wnkn.

3.26

It follows from3.24and3.26that

zn1−zn − xn1−xn ≤ n1 1−βn1

_γf_x_n₁AWn1kn1

n

1−βn

AWnknγfxn Wn1kn1−Wn1kn

Wn1kn−Wnkn − xn1−xn

≤ n1 1−βn1

γfxn1AWn1kn1

n

1−βn

AWnknγfxn kn1−kn

Wn1kn−Wnkn − xn1−xn

≤ n1 1−βn1

γfxn1AWn1kn1

n

1−βn

AWnknγfxn M_c1|rn1−rn|

1−αn1λn1

Bun1Byn1Bun

Wn1kn−Wnkn.

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SinceTiandUn,iare nonexpansive, we have

Wn1kn−Wnknμ1T1Un1,2kn−μ1T1Un,2kn ≤μ1Un1,2kn−Un,2kn

μ1μ2T2Un1,3kn−μ2T2Un,3kn

≤μ1μ2Un1,3kn−Un,3kn

.. .

≤μ1μ2· · ·μnUn1,n1kn−Un,n1kn

≤M2

n

i1 μi,

3.28

whereM2≥0 is a constant such thatUn1,n1kn−Un,n1kn ≤M2for alln≥0. Combining3.27and3.28, we have

zn1−zn − xn1−xn ≤ n1

1−βn1

_γf_x_n₁AWn1kn1

n

1−βn

AWnknγfxn M_c1|rn1−rn|

1−αn1λn1

Bun1Byn1Bun

M2

n

i1 μi,

3.29

which implies thatnoting thati,ii,iii,iv,v, and 0< μi≤b <1,for alli≥1

lim sup

n→ ∞ zn1−zn − xn1−xn≤0. 3.30

Hence, byLemma 2.8, we obtain

lim

n→ ∞zn−xn0. 3.31

It follows that

lim

n→ ∞xn1−xnnlim→ ∞

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Applying3.32andii,iv, andvto3.23and3.24, we obtain that

lim

n→ ∞un1−unnlim→ ∞kn1−kn0. 3.33

Sincexn1nγfxn βnxn 1−βnI−nAWnkn, we have

xn−Wnkn ≤ xn−xn1xn1−Wnkn

xn−xn1nγfxn βnxn1−βn I−nA Wnkn−Wnkn

xn−xn1nγfxn−AWnkn βnxn−Wnkn

≤ xn−xn1nγfxnAWnkn βnxn−Wnkn,

3.34

that is

xn−Wnkn ≤ 1

1−βnxn−xn1

n

1−βn

_γf_x_n_AW_n_k_n _. _3.35

Byi,iii, and3.32it follows that

lim

n→ ∞Wnkn−xn0. 3.36

Step 3. We claim that the following statements hold:

ilimn→ ∞un−kn0;

iilimn→ ∞xn−un0.

For anyp∈Θ:∩∞_n₁FTn∩EPF∩V IC, Band3.14, we have

kn−p2αnun−p 1−αnvn−p2

≤αnun−p2 1−αnvn−p2

≤αnun−p2 1−αn

_u_n₋_p2

λ2

n

β2 −1

_u_n₋_y_n2

un−p2 1−αn

λ2

n

β2 −1

_u_n₋_y_n2

≤xn−p2 1−αn

λ2

n

β2 −1

un−yn2.

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Observe that

xn1−p21−βnI−nAWnkn−p βnxn−p nγfxn−Ap2

1−βnI−nAWnkn−p βnxn−p2n2γfxn−Ap2

2βnnxn−p, γfxn−Ap

2n1−βn I−nA Wnkn−p , γfxn−Ap

≤1−βn−nγWnkn−pβnxn−p2

2

nγfxn−Ap2

2βnnxn−p, γfxn−Ap

2n1−βn I−nA Wnkn−p , γfxn−Ap

≤1−βn−nγkn−pβnxn−p2cn

≤1−βn−nγ 2kn−p2βn2xn−p2

21−βn−nγ βnkn−pxn−pcn

≤1−βn−nγ 2kn−p2βn2xn−p2

1−βn−nγ βnkn−p2xn−p2

cn

1−nγ 2−21−nγ βnβ2n

kn−p2β2nxn−p2

1−nγ βn−β2n

kn−p2xn−p2

cn

1−nγ 2kn−p2−1−nγ βnkn−p2

1−nγ βnxn−p2cn

1−nγ 1−βn−nγ kn−p21−nγ βnxn−p2cn,

3.38

where

cn2nγfxn−Ap22βnnxn−p, γfxn−Ap

2n1−βn I−nA Wnkn−p , γfxn−Ap.

3.39

It follows from conditionithat

lim

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Substituting3.37into3.38, and usingv, we have

_x_n₁₋_p2_≤

1−nγ 1−βn−nγ

_x_n₋_p2

1−αn

λ2

n

β2 −1

_u_n₋_y_n2

1−nγ 2xn−p2

1−nγ 1−βn−nγ 1−αn

λ2

n

β2 −1

un−yn2cn

≤xn−p2 1−αn

λ2

n

β2 −1

un−yn2cn.

3.41

It follows that

1−αnδun−yn2≤1−αn

1−λ

2

n

β2

un−yn2

≤xn−p2−xn1−p2cn

xn−p−xn1−p xn−pxn1−p cn

≤ xn−xn1xn−pxn1−p cn.

3.42

Since limn→ ∞cn0 and from3.32, we obtain

lim

n→ ∞un−yn0. 3.43

Note that

kn−vnαnun−vn. 3.44

Since limn→ ∞αn0, we have

lim

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AsBis 1/β-Lipschitz continuous, we obtain

_v_n₋_y_n_P_C

un−λnByn −PCun−λnBun

≤un−λnByn −un−λnBun

λnBun−Byn

≤ λn

βun−yn,

3.46

then, we get

lim

n→ ∞vn−yn0. 3.47

from

un−kn ≤un−ynyn−vnvn−kn. 3.48

Applying3.43,3.45, and3.47, we have

lim

n→ ∞un−kn0. 3.49

For anyp∈Θ, note thatTr is firmly nonexpansiveLemma 2.4, then we have

un−p2Trnxn−Trnp 2

≤Trnxn−Trnp, xn−p

un−p, xn−p

1

2

un−p2xn−p2− xn−un2

,

3.50

and hence

_u_n₋_p2_≤_x

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which together with3.38gives xn1−p2 ≤

1−nγ 1−βn−nγ kn−p21−nγ βnxn−p2cn

1−nγ 1−nγ−βn

×kn−un2un−p22kn−un, un−p

≤1−nγ 1−nγ−βn kn−un2

1−nγ 1−nγ−βn un−p2

21−nγ 1−nγ−βn kn−unun−p

1−nγ 1−nγ−βn xn−p2− xn−un2

1−nγ 1−nγ−βn xn−p2

−1−nγ 1−nγ−βn xn−un2

1−nγ 2xn−p2−1−nγ 1−nγ−βn xn−un2

1−nγ 1−nγ−βn kn−un2

21−nγ 1−nγ−βn kn−unun−pcn

1−2nγnγ 2xn−p2

−1−nγ 1−nγ−βn xn−un2

≤xn−p2nγ 2xn−p2

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− 1−nγ 1−nγ−βn xn−un2

21−nγ 1−nγ−βn kn−unun−pcn.

3.52

So

1−nγ 1−nγ−βn xn−un2

≤xn−p2−xn1−p2

nγ 2xn−p2

xn−p−xn1−p xn−pxn1−p

nγ 2xn−p21−nγ 1−nγ−βn kn−un2

≤ xn−xn1xn−pxn1−p

nγ 2xn−p2

21−nγ 1−nγ−βn kn−unun−pcn.

3.53

Usingn → 0,cn → 0 asn → ∞,3.32, and3.49, we obtain

lim

n→ ∞xn−un0. 3.54

Since lim infn→ ∞rn>0, we obtain

lim

n→ ∞

xn−un

rn

lim

n→ ∞

1

rnxn−un0. 3.55

Observe that

Wnun−un ≤ Wnun−WnknWnkn−xnxn−un

≤ un−knWnkn−xnxn−un.

3.56

Applying3.36,3.49, and3.54to the last inequality, we obtain

lim

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Let W be the mapping defined by2.6. Since {un} is bounded, applyingLemma 2.7and

3.57, we have

Wun−un ≤ Wun−WnunWnun−un → 0 asn → ∞. 3.58

Step 4. We claim that lim sup_n_{→ ∞}A−γfz, z−xn ≤0,wherezis the unique solution of

the variational inequalityA−γfz, z−x ≥0, for allx∈Θ.

SincezP_ΘI−Aγfzis a unique solution of the variational inequality3.1, to show this inequality, we choose a subsequence{uni}of{un}such that

lim

i→ ∞

A−γf z, z−uni

lim sup

n→ ∞

A−γf z, z−un. 3.59

Since{uni}is bounded, there exists a subsequence{un_ij}of{uni}which converges weakly to

w ∈C. Without loss of generality, we can assume thatuni w.FromWun−un → 0,we

obtainWuni w. Next, We show thatw∈Θ, whereΘ:∩∞n1FTn∩EPF∩V IC, B. First, we show thatw∈EPF. SinceunTrnxn, we have

Fun, y _r1 n

y−un, un−xn≥0, ∀y∈C. 3.60

If follows fromA2that

1 rn

y−un, un−xn≥ −Fun, y ≥Fy, un , 3.61

and hence

y−uni,

uni−xni

rni

≥Fy, uni . 3.62

Sinceuni −xni/rni → 0 anduni w,it follows byA4thatFy, w≤0 for ally∈H.Fort

with 0< t≤1 andy∈H,letytty 1−tw.Sincey∈Handw∈H,we haveyt∈Hand

henceFyt, w≤0.So, fromA1andA4we have

0Fyt, yt ≤tFyt, y 1−tFyt, w ≤tFyt, y , 3.63

and henceFyt, y≥0. FromA3, we haveFw, y≥0 for ally∈Hand hencew∈EPF.

Next, we show that w ∈ ∩∞_n₁FTn. By Lemma 2.6, we have FW ∩∞_n₁FTn.

Assumew /∈FW.Sinceuni wandw /Ww,it follows by the Opial’s condition that

lim inf

i→ ∞ uni−w<lim infi→ ∞ uni−Ww

≤lim inf

i→ ∞ {uni−WuniWuni−Ww}

≤lim inf

i→ ∞ uni−w,

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which derives a contradiction. Thus, we havew ∈FW ∩_n∞₁FTn. By the same argument

as that in the proof of35, Theorem 2.1, Pages 10–11, we can show thatw∈V IC, B.Hence w∈Θ. SincezPΘI−Aγfz, it follows that

lim sup

n→ ∞

A−γf z, z−xnlim sup

n→ ∞

A−γf z, z−un

lim

i→ ∞

A−γf z, z−uni

A−γf z, z−w≤0.

3.65

It follows from the last inequality,3.36, and3.54that

lim sup

n→ ∞

γfz−Az, Wnkn−z≤0. 3.66

Step 5. Finally, we show that{xn}and{un}converge strongly tozPΘI−Aγfz. Indeed,

from1.25, we have

xn1−z2

nγfxn βnxn 1−βnI−nAWnkn−z2

1−βnI−nAWnkn−z βnxn−z nγfxn−Az2

1−βnI−nAWnkn−z βnxn−z2n2γfxn−Az2

2βnnxn−z, γfxn−Az

2n1−βn I−nA Wnkn−z, γfxn−Az

≤1−βn−nγWnkn−zβnxn−z2n2γfxn−Az2

2βnnγxn−z, fxn−fz2βnnxn−z, γfz−Az

21−βn γnWnkn−z, fxn−fz21−βn nWnkn−z, γfz−Az

− 22

nAWnkn−z, γfz−Az

≤1−βn−nγWnkn−zβnxn−z2n2γfxn−Az2

2βnnγxn−zfxn−fz2βnnxn−z, γfz−Az

21−βn γnWnkn−zfxn−fz21−βn nWnkn−z, γfz−Az

− 22

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≤1−βn−nγxn−zβnxn−z2n2γfxn−Az2

2βnnγαxn−z22βnnxn−z, γfz−Az

21−βn γnαxn−z221−βn nWnkn−z, γfz−Az

− 22

1−nγ 22βnnγα21−βn γnα

xn−z2n2γfxn−Az2

2βnnxn−z, γfz−Az21−βn nWnkn−z, γfz−Az

− 22

≤1−2γ−αγ nxn−z2γ22nxn−z2n2γfxn−Az2

2βnnxn−z, γfz−Az21−βn nWnkn−z, γfz−Az

22

nAWnkn−zγfz−Az

1−2γ−αγ nxn−z2n

×n

γ2_x

n−z2γfxn−Az2

2AWnkn−zγfz−Az2βnxn−z, γfz−Az

21−βn Wnkn−z, γfz−Az.

3.67

Since{xn},{fxn},and{Wnkn}are bounded, we can take a constantM >0 such that

γ2_x

n−z2γfxn−Az22AWnkn−zγfz−Az≤M 3.68

for alln≥0. It then follows that

xn1−z2≤

1−2γ−αγ nxn−z2nσn, 3.69

where

σn2βnxn−z, γfz−Az21−βn Wnkn−z, γfz−AznM. 3.70

Using i,3.65, and3.66, we get lim sup_n_{→ ∞}σn ≤ 0. ApplyingLemma 2.9to3.69, we

conclude thatxn → zin norm. Finally, noticing

un−zTrnxn−Trnz ≤ xn−z, 3.71

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Corollary 3.228,Theorem 3.1. LetCbe nonempty closed convex subset of a real Hilbert space H, letF be a bifunction fromC×CtoRsatisfying (A1)–(A4), and let{Tn}be an infinitely many

nonexpansive ofCinto itself such thatΘ:∩∞_n₁FTn∩EPF/∅. Letfbe a contraction ofHinto

itself withα∈0,1,and letAbe a strongly positive linear bounded operator onHwith coeﬃcient γ >0and0< γ < γ/α. Let{xn}and{un}are the sequences generated by

x1x∈C chosen arbitrary,

Fun, y _r1

xn1nγfxn βxn1−β I−nA Wnun, ∀n≥1,

3.72

where{Wn}is the sequence generated by1.23,β∈0,1,{n}is a sequences in0,1,and{rn}is

a real sequence in0,∞satisfying the following conditions:

ilimn→ ∞n0;

iilim infn→ ∞rn>0andlimn→ ∞|rn1−rn|0.

Then,{xn} and{un}converge strongly to a pointz ∈Θwhich is the unique solution of the

variational inequality

A−γf z, z−x ≥0, ∀x∈Θ. 3.73

Equivalently, one haszP_ΘI−Aγfz.

Proof. PutB0,{βn}β,and{αn}0 inTheorem 3.1., thenynkn un. The conclusion of

Corollary 3.2can obtain the desired result easily.

Corollary 3.3. LetCbe nonempty closed convex subset of a real Hilbert spaceH, letFbe a bifunction fromC×CtoRsatisfying (A1)–(A4) and letBbe anβ-inverse-strongly monotone mapping ofCinto Hsuch thatΘ:EPF∩V IC, B/∅. Letf be a contraction ofHinto itself withα∈0,1and letAbe a strongly positive linear bounded operator onHwith coeﬃcientγ >0and0< γ < γ/α. Let

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

Fun, y _r1

kn αnun 1−αnPCun−λnByn ,

xn1 nγfxn βnxn1−βn I−nA kn, ∀n≥1,

(27)

where {n}, {αn}, and {βn} are three sequences in 0,1, and {rn} is a real sequence in 0,∞

satisfying the following conditions:

ilimn→ ∞n0and∞n1n∞;

iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1;

ivlim infn→ ∞rn>0andlimn→ ∞|rn1−rn|0;

v{λn/β} ⊂τ,1−δfor someτ, δ∈0,1andlimn→ ∞λn0.

inequality

A−γf z, z−x ≥0, ∀x∈Θ. 3.75

Proof. PutTnxxfor alln∈Nand for allx∈C. ThenWnxxfor allx∈C. The conclusion

follows fromTheorem 3.1.

Corollary 3.4. Let Cbe nonempty closed convex subset of a real Hilbert space H, let {Tn} be an

infinitely many nonexpansive ofCinto itself, and letBbe aβ-inverse-strongly monotone mapping ofCintoHsuch thatΘ : ∩∞_n₁FTn∩V IC, B/∅. Letf be a contraction ofHinto itself with

α∈ 0,1and letAbe a strongly positive linear bounded operator onHwith coeﬃcientγ >0and 0< γ < γ/α. Let{xn},{yn}, and{kn}be sequences generated by

x1x∈C chosen arbitrary, ynPCxn−λnBxn,

knαnxn 1−αnPCxn−λnByn ,

xn1nγfxn βnxn1−βn I−nA Wnkn, ∀n≥1,

3.76

where{Wn}is the sequences generated by1.23, and{n},{αn},{βn}are three sequences in0,1

ilimn→ ∞n0and∞n1n∞;

iv{λn/β} ⊂τ,1−δfor someτ, δ∈0,1andlimn→ ∞λn0.

Then, {xn} converges strongly to a point z ∈ Θ which is the unique solution of the variational

inequality

A−γf z, z−x ≥0, ∀x∈Θ. 3.77

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Proof. PutFx, y 0 for allx, y ∈Candrn 1 for alln∈NinTheorem 3.1. Then, we have

unPCxnxn. So, byTheorem 3.1, we can conclude the desired conclusion easily.

IfA I, γ ≡1 andγn 1−n−βninTheorem 3.1, then we can obtain the following

result immediately.

Corollary 3.5. LetCbe nonempty closed convex subset of a real Hilbert spaceH, letFbe a bifunction fromC×CtoRsatisfying (A1)–(A4), let{Tn}be an infinitely many nonexpansive ofCinto itself, and

letBbe anβ-inverse-strongly monotone mapping ofCintoHsuch thatΘ:∩_n∞₁FTn∩EPF∩

V IC, B/∅. Letfbe a contraction ofHinto itself withα∈0,1. Let{xn},{yn},{kn},and{un}

be sequences generated by

Fun, y _r1

knαnun 1−αnPCun−λnByn ,

xn1nfxn βnxnγnWnkn, ∀n≥1,

3.78

where{Wn}is the sequences generated by1.23,{n},{αn}, and {βn}are three sequences in0,1

and{rn}is a real sequence in0,∞satisfying the following conditions:

inβnγn1;

iilimn→ ∞n0and∞n1n∞;

iiilimn→ ∞αn0and∞n1αn∞;

iv0<lim infn→ ∞βn≤lim supn→ ∞βn<1;

vlim infn→ ∞rn>0andlimn→ ∞|rn1−rn|0,

vi{λn/β} ⊂τ,1−δfor someτ, δ∈0,1andlimn→ ∞λn0.

Then,{xn} and {un} converge strongly to a pointz ∈ Θ which is the unique solution of the

variational inequality

z−fz, z−x ≥0, ∀x∈Θ. 3.79

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Corollary 3.6. LetCbe nonempty closed convex subset of a real Hilbert spaceH, letFbe a bifunction fromC×CtoRsatisfying (A1)–(A4) and let{Tn}be an infinite family of nonexpansive ofCinto

itself such thatΘ:∩∞_n₁FTn∩EPF/∅. Letf be a contraction ofHinto itself withα∈0,1.

Let{xn}and{un}be sequences generated by

Fun, y _r1

xn1nfxn βnxnγnWnun, ∀n≥1,

3.80

where {n}, {αn}, and {βn} are three sequences in 0,1, and {rn} is a real sequence in 0,∞

inβnγn1;

iilimn→ ∞n0and∞n1n∞;

ivlim infn→ ∞rn>0andlimn→ ∞|rn1−rn|0.

inequality

A−γf z, z−x ≥0, ∀x∈Θ. 3.81

Proof. Put B 0 and {αn} 0 in Corollary 3.5. then yn kn un. The conclusion of

Corollary 3.6can obtain the desired result easily.

4. Application for Optimization Problem

In this section, we shall utilize the results presented in the paper to study the following optimization problem:

min hx,

x∈C. 4.1

wherehxis a convex and lower semicontinuous functional defined on a closed subsetCof a Hilbert spaceH. We denote byTthe set of solution of4.1. LetFbe a bifunction fromC×C toRdefined byFx, y hy−hx. We consider the following equilibrium problem, that is, to findx∈Csuch that

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It is obvious thatEPF T, whereEPFdenotes the set of solution of equilibrium problem

4.2. In addition, it is easy to see thatFx, ysatisfies the conditionsA1–A4inSection 1. Therefore, from theCorollary 3.6, we know the following iterative sequence{xn}defined by

x1 ∈C chosen arbitrary,

hy −hun _r1 n

y−un, un−xn

xn1nfxn βnxnγnun,

≥0, ∀y∈C 4.3

where{n},{βn}, and{γn}are three sequences in0,1,and{rn}is a real sequence in0,∞

satisfying the following conditions:

inβnγn1;

iilimn→ ∞n0 and∞n1n∞;

ivlim infn→ ∞rn>0 and limn→ ∞|rn1−rn|0.

Then,{xn}converges strongly to a pointzPTfzof optimization problem4.1.

In special case, we pickfx 0 for allx∈Handβn0,rn1 ,n1/2 for alln∈N,

thenxn11/2un, and from4.3we obtain a special iterative scheme

hy −hun

y−un, un−1

2un−1

≥0, ∀y∈C, n≥2,

hy −hu1 y−u1, u1−x1 ≥0, ∀y∈C.

4.4

Then,{un}converges strongly to a solutionzPT0 of optimization problem4.1. In fact, the

zis the minimum norm point on theT.

Therefore, we consider a special from of optimization problem 4.1 which is as follows:i.e., is takinghx x

min x,

x∈C. 4.5

In fact, the solution of optimization problem4.4is named the minimum norm point on the closed convex setC. From iterative algorithm4.4we obtain the following iterative algorithm4.5, and{un}is defined by

y− un

y−un, un− 1

2un−1

≥0, ∀y∈C, n≥2, y− u1

y−u1, u1−x1

≥0, ∀y∈C

4.6

for any initial guessx1∈H. Then,{un}converges strongly to a minimum norm point on the

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Acknowledgments

The first author was supported by the Faculty of Applied Liberal Arts RMUTR Research Fund and King Mongkut’s Diamond scholarship for fostering special academic skills by KMUTT. The second author was supported by the Thailand Research Fund and the Commission on Higher Education under Grant no. MRG5180034. Moreover, the authors would like to thank Professor Somyot Plubiteng for providing valuable suggestions, and they also would like to thank the referee for the comments.

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