Comput. Appl. Math. vol.29 número3

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ISSN 0101-8205 www.scielo.br/cam

Some results on variational inequalities and

generalized equilibrium problems with applications

XIAOLONG QIN1, SUN YOUNG CHO2 and SHIN MIN KANG3

1_{Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China} 2_{Department of Mathematics, Gyeongsang National University, Jinju 660-701, Korea}

3_{Department of Mathematics and the RINS, Gyeongsang National University} Jinju 660-701, Korea

E-mail: smkang@gnu.ac.kr

Abstract. An iterative algorithm is considered for variational inequalities, generalized equilib-rium problems and fixed point problems. Strong convergence of the proposed iterative algorithm is obtained in the framework Hilbert spaces.

Mathematical subject classification: 47H05, 47H09, 47J25, 47N10.

Key words:generalized equilibrium problem, variational inequality, fixed point, nonexpansive mapping.

1 Introduction and preliminaries

Let H be a real Hilbert space, whose inner product and norm are denoted by h∙,∙iandk ∙ k, respectively. LetC be a nonempty closed and convex subset of

H andPCbe the projection ofH ontoC.

Let f,S,A,T be nonlinear mappings. Recall the following definitions:

(1) f :C →Cis said to beα-contractive if there exists a constantα ∈(0,1) such that

kf x− f yk ≤αkx−yk, ∀x,y∈C.

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(2) S :C →C is said to be nonexpansive if

kSx−Syk ≤ kx −yk, ∀x,y ∈C.

Throughout this paper, we use F(S) to denote the set of fixed points of the mappingS.

(3) A:C → His said to be monotone if

hAx −Ay,x−yi ≥0, ∀x,y ∈C.

(4) A:C → H is said to be inverse-strongly monotone if there existsδ >0 such that

hx−y,Ax−Ayi ≥δkAx−Ayk2, ∀x,y ∈C.

Such a mapping Ais also calledδ-inverse-strongly monotone. We know that if S: C → C is nonexpansive, then A = I − S is 1₂ -inverse-strongly monotone; see [1, 21] for more details.

(5) A set-valued mapping T: H → 2H _{is said to be monotone if for all} x,y ∈ H, f ∈ T x and g ∈ T y =⇒ hx − y, f −gi ≥ 0. A mono-tone mappingT: H →2H _{is maximal if the graph of} _G₍_T₎_of_T _{is not}

properly contained in the graph of any other monotone mapping. It is known that a monotone mappingT is maximal if and only if for(x, f)∈

H×H,hx−y, f −gi ≥0 for every(y,g)∈G(T)implies that f ∈T x. Let A be a monotone mapping of C into H and let NCv be the normal cone toC atv∈C, i.e.,NCv= {w∈ H : hv−u, wi ≥0, ∀u∈C}and define

Tv=

  

Av+NCv, v ∈C,

∅, v /∈C.

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ThenT is maximal monotone and 0∈Tvif and only ifhAv,u−vi ≥0, ∀u∈C; see [16] for more details.

Recall that the classical variational inequality is to find anx ∈Csuch that

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In this paper, we use V I(C,A) to denote the solution set of the variational inequality (1.1). For givenz ∈ H andu ∈C, we see that

hu−z, v−ui ≥0, ∀v∈C (1.2)

holds if and only ifu = PCz.It is known that projection operatorPC satisfies

hPCx −PCy,x −yi ≥ kPCx−PCyk2, ∀x,y ∈ H.

One can see that the variational inequality (1.1) is equivalent to a fixed point problem. An elementu ∈ C is a solution of the variational inequality (1.1) if and only ifu ∈C is a fixed point of the mappingPC(I −λA)u,whereλ >0 is a constant and I is the identity mapping. This can be seen from the following.

u∈Cis a solution of the variational inequality (1.1), this is,

hAu,y−ui ≥0, ∀y∈C,

which is equivalent to

h(u−λAu)−u,u−yi ≥0, ∀y ∈C,

whereλ > 0 is a constant. This implies from (1.2) that u = PC(I −λA)u, that is,u is a fixed point of the mapping PC(I −λA). This alternative equiv-alent formulation has played a significant role in the studies of the variational inequalities and related optimization problems.

Let A : C → H be a δ-inverse-strongly monotone mapping and F be a bifunction of C ×C into R, where R denotes the set of real numbers. We consider the following generalized equilibrium problem:

Find x ∈Csuch that F(x,y)+ hAx,y−xi ≥0, ∀y ∈C. (1.3)

In this paper, the set of such anx ∈Cis denoted by E P(F,A), i.e.,

E P(F,A)= {x ∈C: F(x,y)+ hAx,y−xi ≥0, ∀y ∈C}.

Next, we give some special cases of the generalized equilibrium problem (1.3).

(I) If A ≡ 0, the zero mapping, then the problem (1.3) is reduced to the following equilibrium problem:

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In this paper, the set of such anx ∈Cis denoted by E P(F), i.e.,

E P(F)= {x ∈C :F(x,y)≥0, ∀y∈C}.

(II) If F ≡ 0, then the problem (1.3) is reduced to the classical variational inequality (1.1).

In 2005, Iiduka and Takahashi [8] considered the classical variational inequal-ity (1.1) and a single nonexpansive mapping. To be more precise, they obtained the following results.

Theorem IT. Let C be a closed convex subset of a real Hilbert space H . Let A be an α-inverse-strongly monotone mapping of C into H and S be a nonexpansive mapping of C into itself such that F(S)∩V I(C,A)6= ∅. Suppose that x1=x ∈C and{xn}is given by

xn+1=αnx +(1−αn)S PC(xn−λnAxn), ∀n ≥1, (1.5) where{αn}is a sequence in[0,1)and{λn}is a sequence in[0,2α]. If{αn}and

{λn}are chosen so that{λn} ⊂ [a,b]for some a,b with0<a<b<2α,

lim

n→∞αn=0, ∞

X n=1

αn = ∞,

∞

X n=1

|αn+1−αn|<∞and

∞

X n=1

|λn+1−λn|<∞,

then{xn}converges strongly to PF(S)∩V I(C,A)x.

On the other hand, we see that the problem (1.3) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, mini-max problems, the Nash equilibrium problem in noncooperative games and others; see, for instance, [2, 5, 9]. Recently, many authors considered iterative methods for the problems (1.3) and (1.4), see [3-7, 11-15, 18, 20, 22, 24] for more details.

To study the equilibrium problems (1.3) and (1.4), we may assume that F

satisfies the following conditions:

(A1) F(x,x)=0 for all x ∈C;

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(A3) for each x,y,z∈C,

lim sup

t↓0

F(t z+(1−t)x,y)≤ F(x,y);

(A4) for each x ∈ C, y 7→ F(x,y)is convex and weakly lower semi-con-tinuous.

PutF(x,y) = F(x,y)+ hAx,y −xifor each x,y ∈ C. It is not hard to see thatF also confirms (A1)-(A4).

In 2007, Takahashi and Takahashi [20] introduced the following iterative

method _

   

F(yn,u)+ 1

rnhu−yn,yn−xni ≥0, ∀u∈C,

xn+1=αnf(xn)+(1−αn)T yn, n≥1,

(1.6)

where f is a α-contraction, T is a nonexpansive mapping. They considered the problem of approximating a common element of the set of fixed points of a single nonexpansive mapping and the set of solutions of the equilibrium problem (1.4). Strong convergence theorems of the iterative algorithm (1.6) are established in a real Hilbert space.

Recently, Takahashi and Takahashi [22] further considered the generalized equilibrium problem (1.3). They obtained the following result in a real Hilbert space.

Theorem TT. Let C be a closed convex subset of a real Hilbert space H and F :C×C →Rbe a bi-function satisfying(A1)-(A4). Let A be anα -inverse-strongly monotone mapping of C into H and S be a non-expansive mapping of C into itself such that F(S)∩E P(F,A) 6= ∅. Let u ∈ C and x1 ∈ C and let

{zn} ⊂C and{xn} ⊂C be sequences generated by

    

F(zn,y)+ hAxn,y−zni + 1

rhy−zn,zn−xni ≥0, ∀y∈C,

xn+1=βnxn+(1−βn)S[αnu+(1−αn)zn], ∀n ≥1,

(1.7)

where{αn} ⊂ [0,1],{βn} ⊂ [0,1]and{rn} ⊂ [0,2α]satisfy

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lim

n→∞(λn−λn+1)=0, nlim→∞αn =0, and ∞

X n=1

αn = ∞.

Then,{xn}converges strongly to z=PF(S)∩E P(F,A)u.

Very recently, Chang, Lee and Chan [5] introduced a new iterative method for solving equilibrium problem (1.4), variational inequality (1.1) and the fixed point problem of nonexpansive mappings in the framework of Hilbert spaces. More precisely, they proved the following theorem.

Theorem CLC. Let H be a real Hilbert space, C be a nonempty closed convex subset of H and F be a bifunction satisfying the conditions (A1)-(A4). Let A:C → H be anα-inverse-strongly monotone mapping and{Si :C →C}be a family of infinitely nonexpansive mappings with F∩V I(C,A)∩E P(F)6= ∅, where F := ∩∞_i₌₁F(Si)and f :C →C be aξ-contractive mapping. Let{xn},

{yn} {kn}and{un}be sequences defined by

              

F(un,y)+ 1

rnhy−un,un−xni ≥0, ∀y ∈C, xn+1=αnf(xn)+βnxn+γnWnkn, ∀n ≥1, kn= PC(yn−λnAyn),

yn = PC(un−λnAun),

(1.8)

where{Wn :C →C}is the sequence defined by(1.9),{αn},{βn}and{γn}are sequences in[0,1],{λn}is a sequence in[a,b] ⊂(0,2α)and{rn}is a sequence in(0,∞). If the following conditions are satisfied:

(1) αn+βn+γn =1;

(2) limn→∞αn=0;P∞_n₌₁αn = ∞;

(3) 0<lim infn→∞βn ≤lim supn→∞βn <1;

(4) lim infn→∞rn >0;

P∞

n=1|rn+1−rn|<∞;

(5) limn→∞|λn+1−λn| =0,

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In this paper, motivated and inspired by the research going on in this direc-tion, we introduce a general iterative method for finding a common element of the set of solutions of generalized equilibrium problems, the set of solutions of variational inequalities, and the set of common fixed points of a family of nonexpansive mappings in the framework of Hilbert spaces. The results pre-sented in this paper improve and extend the corresponding results of Ceng and Yao [3, 4], Chang Lee and Chan [5], Iiduka and Takahashi [8], Qin, Shang and Zhou [12], Su, Shang and Qin [18], Takahashi and Takahashi [20, 22], Yao and Yao [25] and many others.

In order to prove our main results, we need the following definitions and lemmas.

A spaceX is said to satisfy Opial condition [10] if for each sequence{xn}in

X which converges weakly to pointx ∈ X, we have

lim inf

n→∞ kxn−xk<lim infn→∞ kxn−yk, ∀y∈ X,y6=x.

Lemma 1.1 ([2]). Let C be a nonempty closed convex subset of H and F :

C×C → Rbe a bifunction satisfying(A1)-(A4). Then, for any r > 0 and x ∈ H , there exists z∈C such that

F(z,y)+1

rhy−z,z−xi ≥0, ∀y∈C.

Lemma 1.2 ([2], [7]). Suppose that all the conditions in Lemma1.1are sat-isfied. For any give r>0define a mapping Tr : H →C as follows:

Trx =

z ∈C :F(z,y)+1

rhy−z,z−xi ≥0, ∀y ∈C

, ∀x ∈ H,

then the following conclusions hold:

(1) Tr is single-valued;

(2) Tr is firmly nonexpansive, i.e., for any x,y∈ H ,

kTrx−Tryk2 ≤ hTrx−Try,x −yi; (3) F(Tr)=E P(F);

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Lemma 1.3 ([23]). Assume that{αn}is a sequence of nonnegative real num-bers such that

αn+1≤(1−γn)αn+δn,

where{γn}is a sequence in(0,1)and{δn}is a sequence such that

(1) P∞

n=1γn = ∞;

(2) lim sup_n_→∞δn/γn≤0 or P∞_n₌₁|δn|<∞. Then limn→∞αn=0.

Definition 1.4 ([19]). Let{Si :C →C}be a family of infinitely nonexpan-sive mappings and{γi}be a nonnegative real sequence with 0≤γi <1,∀i≥1.

Forn ≥1 define a mappingWn:C →Cas follows:

Un,n+1=I,

Un,n=γnSnUn,n+1+(1−γn)I, Un,n−1=γn−1Sn−1Un,n+(1−γn−1)I,

.. .

Un,k =γkSkUn,k+1+(1−γk)I, Un,k−1=γk−1Sk−1Un,k+(1−γk−1)I,

.. .

Un,2=γ2S2Un,3+(1−γ2)I, Wn =Un,1=γ1S1Un,2+(1−γ1)I.

(1.9)

Such a mappingWnis nonexpansive fromCtoC and it is called aW-mapping generated bySn,Sn−1, . . . ,S1andγn, γn−1, . . . , γ1.

Lemma 1.5 ([19]). Let C be a nonempty closed convex subset of a Hilbert space H ,{Si :C → C}be a family of infinitely nonexpansive mappings with

∩∞

i=1F(Si) 6= ∅and{γi}be a real sequence such that0< γi ≤l <1,∀i ≥1. Then

(1) Wnis nonexpansive and F(Wn)= ∩∞

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(2) for each x ∈ C and for each positive integer k, the limitlimn→∞Un,k exists.

(3) the mapping W :C →C defined by

W x := lim

n→∞Wnx =nlim→∞Un,1x, x ∈C,

is a nonexpansive mapping satisfying F(W)= ∩∞

i=1F(Si)and it is called the W -mapping generated by S1,S2, . . .andγ1, γ2, . . . .

Lemma 1.6 ([5]). Let C be a nonempty closed convex subset of a Hilbert space H ,{Si :C → C}be a family of infinitely nonexpansive mappings with

∩∞

i=1F(Si) 6= ∅and{γi}be a real sequence such that0< γi ≤l <1,∀i ≥1. If K is any bounded subset of C, then

lim

n→∞sup_x_∈_KkW x−Wnxk =0.

Throughout this paper, we always assume that 0< γi ≤l <1,∀i ≥1.

Lemma 1.7 ([17]). Let{xn}and{yn}be bounded sequences in a Hilbert space H and{βn}be a sequence in[0,1]with

0<lim inf

n→∞ βn≤lim sup_n_→∞ βn<1.

Suppose that xn+1=(1−βn)yn+βnxnfor all n≥0and

lim sup

n→∞

kyn+1−ynk − kxn+1−xnk

≤0.

Thenlimn→∞kyn−xnk =0.

2 Main results

Theorem 2.1. Let C be a nonempty closed convex subset of a Hilbert space H and F be a bifunction from C×C toRwhich satisfies(A1)-(A4). Let A1:C→

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Assume that:= F P∩E P(F,A3)∩V I 6= ∅, where F P = ∩∞_i₌₁F(Si)and V I = V I(C,A1)∩V I(C,A2). Let f : C → C be an α-contraction. Let x1∈C and{xn}be a sequence generated by

              

F(un,y)+ hA3xn,y−uni + 1

rnhy−un,un−xni ≥0, ∀y∈C, zn =PC(un−λnA2un),

yn =PC(zn−ηnA1zn),

xn+1=αnf(xn)+βnxn+γnWnyn, ∀n≥1,

(2.1)

where{Wn:C →C}is the sequence generated in(1.9),{αn},{βn}and{γn}are sequences in(0,1)such thatαn+βn+γn=1for each n≥1and{rn},{λn}and

{ηn}are positive number sequences. Assume that the above control sequences satisfy the following restrictions:

(R1) 0<a ≤ηn ≤b<2δ1,0<a′≤λn ≤b′ <2δ2,0<aˉ ≤rn ≤ ˉb<2δ3,

∀n≥1;

(R2) limn→∞αn=0 and P∞

n=1αn= ∞;

(R3) 0<lim infn→∞βn ≤lim supn→∞βn <1;

(R4) limn→∞(λn−λn+1)=limn→∞(ηn−ηn+1)=limn→∞(rn−rn+1)=0. Then the sequence{xn}converges strongly to z∈,which solves uniquely the following variational inequality:

h(I − f)z,z−xi ≤0, ∀x ∈.

Proof. First, we show, for eachn≥1,that the mappingsI −ηnA1,I −λnA2

and I −rnA3 are nonexpansive. Indeed, for ∀x,y ∈ C, we obtain from the restriction (R1) that

k(I −ηnA1)x−(I −ηnA1)yk2

= k(x−y)−ηn(A1x −A1y)k2

= kx−yk2−2ηnhx−y,A1x−A1yi +ηn2kA1x −A1yk 2

≤ kx−yk2−2ηnδ1kA1x −A1yk2+η2nkA1x−A1yk 2

= kx−yk2+ηn(ηn−2δ1)kA1x −A1yk2

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which implies that the mapping I −ηnA1 is nonexpansive, so are I −λnA2

andI −rnA3for eachn ≥ 1. Note thatun can be re-written asun =Tr_n(I −

rnA3)xnfor eachn ≥1. Take x∗ _∈ _._{Noticing that}_x∗ ₌ _PC₍_I ₋_η

nA1)x∗ = PC(I −λnA2)x∗=Trn(I −rnA3)x

∗_{, we have}

kun−x∗k = kTr_n(I −rnA3)xn−Tr_n(I −rnA3)x∗k ≤ kxn−x∗k. (2.2)

On the other hand, we have

kzn−x∗k = kPC(un−λnA2un)−PC(x∗−λnA2x∗)k

≤ k(un−λnA2un)−(x∗−λnA2x∗)k

≤ kun−x∗k.

(2.3)

It follows from (2.2) and (2.3) that

kzn−x∗k ≤ kxn−x∗k, (2.4)

which yields that

kyn−x∗k = kPC(zn−ηnA1zn)−PC(x∗−ηnA1x∗)k

≤ k(zn−ηnA1zn)−(x∗−ηnA1x∗)k

≤ kzn−x∗k

≤ kxn−x∗k.

(2.5)

From the algorithm (2.1) and (2.5), we arrive at

kxn+1−x∗k = kαnf(xn)+βnxn+γnWnyn−x∗k

≤αnkf(xn)−x∗k +βnkxn−x∗k +γnkWnyn−x∗k

≤αnkf(xn)− f(x∗)k +αnkf(x∗)−x∗k

+βnkxn−x∗k +γnkyn−x∗k

≤ααnkxn−x∗k +αnkf(x∗)−x∗k

+βnkxn−x∗k +γnkxn−x∗k

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By simple inductions, we obtain that

kxn−x∗k ≤max

kx1−x∗k,kf(x

∗₎₋_x∗_k

1−α

,

which gives that the sequence{xn}is bounded, so are{yn},{zn}and{un}. With-out loss of generality, we can assume that there exists a bounded set K ⊂ C

such that

xn,yn,zn,un∈ K, ∀n ≥1. (2.6)

Notice that un+1 = Trn+1(I −rn+1A3)xn+1 and un = Trn(I −rnA3)xn, we see from Lemma 1.2 that

F(un+1,y)+

1

rn+1

hy−un+1,un+1−(I−rn+1A3)xn+1i ≥0, ∀y ∈C, (2.7)

and

F(un,y)+ 1

rnhy−un,un−(I −rnA3)xni ≥0, ∀y ∈C. (2.8)

Lety =un in (2.7) andy =un+1in (2.8). By adding up these two inequalities

and using the assumption (R2), we obtain that

un+1−un,

un−(I −rnA3)xn

rn −

un+1−(I −rn+1A3)xn+1 rn+1

≥0.

Hence, we have

un+1−un,un−un+1+un+1−(I −rnA3)xn

− rn

rn+1

(un+1−(I −rn+1A3)xn+1)

≥0.

This implies that

kun+1−unk2≤

un+1−un, (I −rn+1A3)xn+1−(I −rnA3)xn

+

1− rn

rn+1

(un+1−(I −rn+1A3)xn+1)

≤ kun+1−unk

k(I −rn+1A3)xn+1−(I −rnA3)xnk

+ |1− rn

rn+1

|kun+1−(I −rn+1A3)xn+1)k

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It follows that

kun+1−unk

≤ k(I −rn+1A3)xn+1−(I −rnA3)xnk

+|rn+1−rn|

rn+1

kun+1−(I −rn+1A3)xn+1k

= k(I −rn+1A3)xn+1−(I −rn+1A3)xn

+(I −rn+1A3)xn−(I −rnA3)xnk

+|rn+1−rn|

rn+1

kun+1−(I −rn+1A3)xn+1k

≤ kxn+1−xnk + |rn+1−rn|M1,

(2.9)

whereM1is an appropriate constant such that

M1=sup

n≥1

kA3xnk +kun+1−(I −rn+1A3)xn+1k ˉ

a

.

From the nonexpansivity ofPC, we also have

kzn+1−znk

= kPC(un+1−λn+1A2un+1)−PC(un−λnA2un)k

≤ kun+1−λn+1A2un+1−(un−λnA2un)k

= k(I −λn+1A2)un+1−(I −λn+1A2)un+(λn−λn+1)A2unk

≤ kun+1−unk + |λn−λn+1|kA2unk.

(2.10)

Substituting (2.9) into (2.10), we arrive at

kzn+1−znk ≤ kxn+1−xnk + |rn+1−rn|M1+ |λn−λn+1|kA2unk. (2.11)

In a similar way, we can obtain that

kyn+1−ynk ≤ kzn+1−znk + |ηn−ηn+1|kA1znk. (2.12)

Combining (2.11) with (2.12), we see that

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whereM2is an appropriate constant such that

M2=max

sup

n≥1

{kA1znk},sup

n≥1

{kA2unk},M1

.

Letting

xn+1=(1−βn)vn+βnxn, ∀n≥1, (2.14)

we see that

vn+1−vn =

αn+1f(xn+1)+γn+1Wn+1yn+1

1−βn+1

−αnf(xn)+γnWnyn 1−βn

= αn+1f(xn+1) 1−βn+1

+(1−αn+1−βn+1)Wn+1yn+1 1−βn+1

−

αnf(xn)

1−βn

+(1−αn−βn)Wnyn 1−βn

= αn+1 1−βn+1

f(xn+1)−Wn+1yn+1

− αn

1−βn

f(xn)−Wnyn

+Wn+1yn+1−Wnyn.

It follows that

kvn+1−vnk ≤

αn+1

1−βn+1

kf(xn+1)−Wn+1yn+1k

+ αn 1−βn

kf(xn)−Wnynk + kWn+1yn+1−Wnynk.

(2.15)

kWn+1yn+1−Wnynk

= kWn+1yn+1−W yn+1+W yn+1−W yn+W yn−Wnynk

≤ kWn+1yn+1−W yn+1k + kW yn+1−W ynk + kW yn−Wnynk

≤sup

x∈K

kWn+1x −W xk + kW x −Wnxk + kyn+1−ynk,

(2.16)

whereK is the bounded subset of C defined by (2.6). Substituting (2.13) into (2.16), we arrive at

kWn+1yn+1−Wnynk ≤sup x∈K

kWn+1x−W xk+ |W x−Wnxk + kxn+1−xnk

+ |rn+1−rn| + |λn−λn+1| + |ηn−ηn+1|

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which combines with (2.15) yields that

kvn+1−vnk − kxn+1−xnk

≤ αn+1 1−βn+1

kf(xn+1)−Wn+1yn+1k +

αn

1−βn

kf(xn)−Wnynk

+sup

x∈K

kWn+1x−W xk + kW x −Wnxk

+ |rn+1−rn| + |λn−λn+1| + |ηn−ηn+1|

M2.

In view of the restriction (R2), (R3) and (R4), we obtain from Lemma 1.6 that

lim sup

n→∞

kvn+1−vnk − kxn+1−xnk

≤0.

Hence, we obtain from Lemma 1.7 that

lim

n→∞kvn−xnk =0.

In view of (2.14), we have

kxn+1−xnk =(1−βn)kvn−xnk.

Thanks to the restriction (R3), we see that

lim

n→∞kxn+1−xnk =0. (2.17)

For anyx∗_∈_{, we see that}

kxn+1−x∗k2= kαn(f(xn)−x∗)+βn(xn−x∗)+γn(Wnyn−x∗)k2

≤αnkf(xn)−x∗k2+βnkxn−x∗k2+γnkWnyn−x∗k2

≤αnkf(xn)−x∗k2+βnkxn−x∗k2+γnkyn−x∗k2.

(2.18)

Note that

kyn−x∗k2= kPC(zn−ηnA1zn)−x∗k2

≤ k(I −ηnA1)zn−(I −ηnA1)x∗k2

= k(zn−x∗)−ηn(A1zn−A1x∗)k2

= kzn−x∗k2−2ηnhzn−x∗,A1zn−A1x∗i

+η_n2kA1zn−A1x∗k2

≤ kzn−x∗k2−2ηnδ1kA1zn−A1x∗k2

+ηn2kA1zn−A1x

∗_k2

= kxn−x∗k2+ηn(ηn−2δ1)kA1zn−A1x∗k2.

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kxn+1−x∗k2≤αnkf(xn)−x∗k2+kxn−x∗k2+γnηn(ηn−2δ1)kA1zn−A1x∗k2.

This implies that

γnηn(2δ1−ηn)kA1zn−A1x∗k2

≤αnkf(xn)−x∗k2+ kxn−x∗k2− kxn+1−x∗k2

≤αnkf(xn)−x∗k2+(kxn−x∗k + kxn+1−x∗k)kxn−xn+1k.

By virtue of the restrictions (R1) and (R2), we obtain from (2.17) that

lim

n→∞kA1zn−A1x

∗_{k =}₀_. ₍₂_.₂₀₎

Next, we show that

lim

n→∞kA2un−A2x

∗_{k =}₀_. ₍₂_.₂₁₎

Indeed, by using (2.18), we obtain that

kxn+1−x∗k2≤αnkf(xn)−x∗k2+βnkxn−x∗k2+γnkzn−x∗k2. (2.22)

kzn−x∗k2= kPC(un−λnA2un)−x∗k2

≤ k(I −λnA2)un−(I −λnA2)x∗k2

= k(un−x∗)−λn(A2un−Aux∗)k2

= kun−x∗k2−2λnhun−x∗,A2un−A2x∗i +λ2nkA2un−A2x

∗

k2 ≤ kun−x∗k2−2λnδ2kA2un−A2x∗k2+λ2nkA2un−A2x

∗_k2

= kun−x∗k2+λn(λn−2δ2)kA2un−A2x∗k2.

(2.23)

kxn+1−x∗k2≤αnkf(xn)−x∗k2+kxn−x∗k2+γnλn(λn−2δ2)kA2un−A2x∗k2.

This in turn gives that

γnλn(2δ2−λn)kA2un−A2x∗k2

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In view of the restrictions (R1) and (R2), we obtain from (2.17) that (2.21) holds. On the other hand, we see from (2.22) that

kxn+1−x∗k2≤αnkf(xn)−x∗k2+βnkxn−x∗k2+γnkun−x∗k2. (2.24)

It follows that

kxn+1−x∗k2≤αnkf(xn)−x∗k2+βnkxn−x∗k2

+γnkxn−x∗−rn(A3xn−A3x∗)k2

≤αnkf(xn)−x∗k2+βnkxn−x∗k2

+γn(kxn−x∗k2+rn2kA3xn−A3x

∗

k2 −2rnhA3xn−A3x∗,xn−x∗i)

≤αnkf(xn)−x∗k2+βnkxn−x∗k2+γnkxn−x∗k2

−rnγn(2δ3−rn)kA3xn−A3x∗k2.

This implies that

rnγn(2δ3−rn)kA3xn−A3x∗k2

≤αnkf(xn)−x∗k2+(kxn−x∗k + kxn+1−x∗k)kxn−xn+1k.

In view of the restrictions (R1), (R2) and (R3), we see from (2.17) that

lim

n→∞kA3xn−A3x

∗_{k =}₀_. ₍₂_.₂₅₎

On the other hand, we see from Lemma 1.2 that

kun−x∗k2= kTrn(I −rnA3)xn−Trn(I −rnA3)x

∗_k2

≤ h(I −rnA3)xn−(I −rnA3)x∗,un−x∗i

= 1

2(k(I −rnA3)xn−(I −rnA3)x

∗

k2+ kun−x∗k2 − k(I −rnA3)xn−(I −rnA3)x∗−(un−x∗)k2) ≤ 1

2(kxn−x

∗_k2_{+ k}_un₋_x∗_k2_{− k}_xn₋_un₋_rn₍_A3xn₋_A3x∗_)k2₎

= 1

2(kxn−x

∗_k2_{+ k}_un₋_x∗_k2_{− k}_xn₋_un_k2₋_r2

nkA3xn−A3x

∗_k2

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This in turn implies that

kun−x∗k2≤ kxn−x∗k2− kxn−unk2−r_n2kA3xn−A3x∗k2 +2rnhA3xn−A3x∗,xn−uni

≤ kxn−x∗k2− kxn−unk2 +2rnkA3xn−A3x∗kkxn−unk.

(2.26)

Combining (2.24) with (2.26), we arrive at

kxn+1−x∗k2≤αnkf(xn)−x∗k2+ kxn−x∗k2−γnkxn−unk2

+2rnkA3xn−A3x∗kkxn−unk.

It follows that

γnkxn−unk2≤αnkf(xn)−x∗k2+(kxn−x∗k + kxn+1−x∗k)kxn−xn+1k

+2rnkA3xn−A3x∗kkxn−unk.

Thanks to the restrictions (R2) and (R3), we see from (2.17) and (2.25) that

lim

n→∞kxn−unk =0. (2.27)

In view of the firm nonexpansivity of PC, we see that

kzn−x∗k2= kPC(I −λnA2)un−PC(I −λnA2)x∗k2

≤ h(I −λnA2)un−(I −λnA2)x∗,zn−x∗i

= 1 2

k(I −λnA2)un−(I −λnA2)x∗k2+ kzn−x∗k2

− k(I −λnA2)un−(I −λnA2)x∗−(zn−x∗)k2

≤ 1 2

kun−x∗k2+ kzn−x∗k2− kun−zn−λn(A2un−A2x∗)k2

= 1 2

kun−x∗k2+ kzn−x∗k2− kun−znk2 +2λnhun−zn,A2un−A2x∗i −λ2nkA2un−A2x

∗_k2

,

which implies that

kzn−x∗k2≤ kun−x∗k2− kun−znk2

+2λnhun−zn,A2un−A2x∗i −λ2nkA2un−A2x

∗_k2

≤ kxn−x∗k2− kun−znk2+2λnkun−znkkA2un−A2x∗k.

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kxn+1−x∗k2≤αnkf(xn)−x∗k2+ kxn−x∗k2−γnkun−znk2

+2λnkun−znkkA2un−A2x∗k,

from which it follows that

γnkun−znk2≤αnkf(xn)−x∗k2+ kxn−x∗k2− kxn+1−x∗k2

+2λnkun−znkkA2un−A2x∗k

≤αnkf(xn)−x∗k2+(kxn−x∗k + kxn+1−x∗k)kxn−xn+1k

+2λnkun−znkkA2un−A2x∗k.

In view of the restrictions (R2) and (R3), we obtain from (2.17) and (2.21) that

lim

n→∞kun−znk =0. (2.29)

In a similar way, we can obtain that

lim

n→∞kyn−znk =0. (2.30)

Note that

kWnyn−xnk ≤ kxn−xn+1k + kxn+1−Wnynk

≤ kxn−xn+1k +αnkf(xn)−Wnynk +βnkxn−Wnynk.

It follows that

(1−βn)kWnyn−xnk ≤ kxn−xn+1k +αnkf(xn)−Wnynk.

In view of the restrictions (R2) and (R3), we obtain from (2.17) that

lim

n→∞kWnyn−xnk =0. (2.31)

Notice that

kWnyn−ynk ≤ kyn−znk + kzn−unk + kun−xnk + kxn−Wnynk.

From (2.27), (2.29), (2.30) and (2.31), we arrive at

lim

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Next, we prove that

lim sup

n→∞

h(f −I)z,xn−zi ≤0,

wherez = Pf(z). To see this, we choose a subsequence {xni} of{xn}such that

lim sup

n→∞

h(f −I)z,xn−zi = lim

i→∞h(f −I)z,xni −zi. (2.33)

Since {xn_i} is bounded, there exists a subsequence {xn_{i j}} of {xn_i} which con-verges weakly tow. Without loss of generality, we may assume thatxni ⇀ w. On the other hand, we have

kxn−ynk ≤ kxn−unk + kun−znk + kzn−ynk.

It follows from (2.27), (2.29) and (2.30) that

lim

n→∞kxn−ynk =0. (2.34)

Therefore, we see thatyni ⇀ w. First, we prove thatw ∈ V I(C,A1). For the purpose, letT be the maximal monotone mapping defined by:

T x =

  

A1x+NCx, x ∈C,

∅, x ∈/C.

For any given(x,y) ∈ G(T), hence y− A1x ∈ NC. Since yn ∈ C, by the definition ofNC, we have

hx−yn,y−A1xi ≥0. (2.35)

Notice that

yn =PC(I −ηnA1)zn.

It follows that

hx −yn,yn−(I −ηnA1)zni ≥0

and hence

x −yn, yn−zn ηn

+A1zn

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From the monotonicity of A1, we see that

hx−yni,yi ≥ hx−yni,A1xi

≥ hx−yni,A1xi −

x−yni,

yn_i −zn_i

ηni

+A1zni

= hx −yn_i,A1x −A1yn_ii + hx−yn_i,A1yn_i −A1zn_ii

−

x−yn_i, yni −zni ηni

≥ hx−yn_i,A1yn_i −A1zn_ii −

x−yn_i, yni −zni ηni

.

Since yn_i ⇀ w and A1 is Lipschitz continuous, we obtain from (2.30) that hx−w,yi ≥0.Notice thatT is maximal monotone, hence 0∈Tw. This shows thatw∈ V I(C,A1).It follows from (2.27) and (2.29), we also have

lim

n→∞kxn−znk =0.

Therefore, we obtainzni ⇀ w.Similarly, we can provew ∈ V I(C,A2).That is,w∈V I =V I(C,A2)∩V I(C,A1).

Next, we show thatw ∈ F P = ∩∞_i₌₁F(Si). Suppose the contrary,w /∈ F P, i.e.,Ww6=w. Since yn_i ⇀ w, we see from Opial condition that

lim inf

i→∞ kyni −wk<lim infi→∞ kyni −Wwk

≤lim inf

i→∞

kyn_i −W yn_ik + kW yn_i −Wwk

≤lim inf

i→∞

kyn_i −W yn_ik + kyn_i −wk .

(2.36)

kW yn−ynk ≤ kW yn−Wnynk + kWnyn−ynk

≤sup

x∈K

kW x−Wnxk + kWnyn−ynk.

From Lemma 1.6, we obtain from (2.32) that limn→∞kW yn−ynk =0, which

combines with (2.36) yields that that

lim inf

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This derives a contradiction. Thus, we havew∈ F P.

Next, we show that w ∈ E P(F,A3).It follows from (2.27) that un ⇀ w. Sinceun =Trn(I −r A3)xn, for anyy∈C, we have

F(un,y)+ hA3xn,y−uni + 1

rnhy−un,un−xni ≥0.

From the assumption (A2), we see that

hA3xn,y−uni + 1

rnhy−un,un−xni ≥ F(y,un), ∀y ∈C.

Replacingnbyni, we arrive at

hA3xn_i,y−un_ii +

y−un_i,uni −xni

rni

≥ F(y,un_i), ∀y ∈C. (2.37)

Puttingyt =t y+(1−t)wfor anyt∈(0,1]andy ∈C,we see thatyt ∈C. It follows from (2.37) that

hyt−uni,A3yti ≥ hyt −uni,A3yti − hA3xni,yt−unii

−

yt −uni,

un_i −xn_i rn_i

+F(yt,uni)

= hyt−un_i,A3yt −A3un_ii + hyt−un_i,A3un_i −A3xn_ii

−

yt −uni,

uni −xni

rni

+F(yt,uni).

In view of the monotonicity of A3, (2.27) and the restriction (R1), we obtain from the assumption (A4) that

hyt −w,A3yti ≥ F(yt, w). (2.38)

From the assumptions (A1) and (A4), we see that

0=F(yt,yt)≤t F(yt,y)+(1−t)F(yt, w)

≤t F(yt,y)+(1−t)hyt −w,A3yti

=t F(yt,y)+(1−t)thy−w,A3yti,

from which it follows that

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It follows from the assumption (A3) thatw ∈ E P(F,A3). On the other hand, we see from (2.33) that

lim sup

n→∞

h(f −I)z,xn−zi = h(f −I)z, w−zi ≤0. (2.39)

Finally, we show thatxn→z,asn→ ∞.Note that

kxn+1−zk2= hαnf(xn)+βnxn+γnWnyn−z,xn+1−zi

=αnhf(xn)−z,xn+1−zi +βnhxn−z,xn+1−zi

+γnhWnyn−z,xn+1−zi

≤αnhf(xn)− f(z),xn+1−zi +αnhf(z)−z,xn+1−zi

+βnkxn−zkkxn+1−zk +γnkyn−zkkxn+1−zk

≤ α

2αn(kxn−zk

2

+ kxn+1−zk2)+αnhf(z)−z,xn+1−zi

+(1−αn)kxn−zkkxn+1−zk

≤ α

2αn(kxn−zk

2_{+ k}_xn

+1−zk2)+αnhf(z)−z,xn+1−zi

+(1−αn)

2 (kxn−zk

2_{+ k}_xn

+1−zk2)

≤ 1−αn(1−α)

2 kxn−zk

2

+1

2kxn+1−zk

2

+αnhf(z)−z,xn+1−zi,

which implies that

kxn+1−zk2≤ [1−αn(1−α)]kxn−zk2+2αnhf(z)−z,xn+1−zi.

From the restriction (R2), we obtain from Lemma 1.3 that limn→∞kxn−zk =

0. This completes the proof.

Corollary 2.2. Let C be a nonempty closed convex subset of a Hilbert space H and F be a bifunction from C ×C toRwhich satisfies(A1)-(A4). Let A1 :

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F P = ∩∞_i₌₁F(Si)and V I = V I(C,A1)∩V I(C,A2). Let f :C →C be an

α-contraction. Let x1∈C and{xn}be a sequence generated by

              

F(un,y)+ 1

rnhy−un,un−xni ≥0, ∀y ∈C, zn = PC(un−λnA2un),

yn = PC(zn−ηnA1zn),

xn+1=αnf(xn)+βnxn+γnWnyn, ∀n ≥1,

∀n≥1;

n=1αn= ∞;

h(I − f)z,z−xi ≤0, ∀x ∈.

Proof. Putting A3≡0, we see that

hA3x−A3y,x−yi ≥δkA3x−A3yk2, ∀x,y ∈C

for allδ ∈ (0,∞).We can conclude from Theorem 2.1 the desired conclusion

easily. This completes the proof.

Remark 2.3. If A1≡ A2andλn ≡ηn, then Corollary 2.2 is reduced to

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Corollary 2.4. Let C be a nonempty closed convex subset of a Hilbert space H . Let A1:C → H be aδ1-inverse-strongly monotone mapping, A2:C → H be aδ2-inverse-strongly monotone mapping, A3:C → H be aδ3-inverse-strongly monotone mapping and{Si : C → C}be a family of infinitely nonexpansive mappings. Assume that := F P ∩ E P(F,A3)∩ V I 6= ∅, where F P = ∩_i∞₌₁F(Si) and V I = V I(C,A1)∩V I(C,A2). Let f : C → C be an α -contraction. Let x1∈C and{xn}be a sequence generated by

            

zn = PC(xn−rnA3xn), zn = PC(un−λnA2un), yn =PC(zn−ηnA1zn),

∀n≥1;

n=1αn= ∞;

h(I − f)z,z−xi ≤0, ∀x ∈.

Proof. PuttingF ≡0, we see that

hA3xn,y−uni + 1

rnhy−un,un−xni ≥0, ∀y ∈C

is equivalent to

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This implies that

un =PC(xn−rnA3xn).

From the proof of Theorem 2.1, we can conclude the desired conclusion

immediately. This completes the proof.

Remark 2.5. Corollary 2.4 includes Theorem 3.1 of Yao and Yao [25] as a special case, see [25] for more details.

As some applications of our main results, we can obtain the following results. Recall that a mappingT:C →Cis said to be ak-strict pseudo-contraction if there exists a constantk∈ [0,1)such that

kT x −T yk2≤ kx −yk2+kk(I −T)x−(I −T)yk2, ∀x,y ∈C. Note that the class ofk-strict pseudo-contractions strictly includes the class of nonexpansive mappings.

PutA= I−T, whereT:C →Cis ak-strict pseudo-contraction. ThenAis

1−k

2 -inverse-strongly monotone; see [1] for more details.

Corollary 2.6. Let C be a nonempty closed convex subset of a Hilbert space H and F be a bifunction from C ×C to R which satisfies (A1)-(A4). Let T1:C → C be a k1-inverse-strongly monotone mapping, T2:C → C be a k2-inverse-strongly monotone mapping, T3:C → C be a k3-inverse-strongly monotone mapping and {Si: C → C} be a family of infinitely nonexpansive mappings. Assume that:= F P∩E P(F,I −T3)∩V I 6= ∅, where F P = ∩_i∞₌₁F(Si)and V I = F(T1)∩F(T2). Let f:C →C be anα-contraction. Let x1∈C and{xn}be a sequence generated by

              

F(un,y)+ h(I −T3)xn,y−uni + 1

rnhy−un,un−xni ≥0, ∀y ∈C, zn =(1−λnun)+λnT2un,

yn=(1−ηnzn)+ηnT1zn,

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(R1) 0<a≤ηn≤b< (1−k1),0<a′≤λn ≤b′ < (1−k2),0<aˉ ≤rn≤

ˉ

b< (1−k3),∀n≥1;

n=1αn= ∞;

(R4) limn→∞(λn−λn+1)=limn→∞(ηn−ηn+1)=limn→∞(rn−rn+1)=0. Then the sequence{xn}converges strongly to z ∈ , which solves uniquely the following variational inequality:

h(I − f)z,z−xi ≤0, ∀x ∈.

Proof. Taking Aj = I −Tj, wee see that Aj :C → H is aδj-strict

pseudo-contraction with δj = 1−kj

2 and F(Tj) = V I(C,Aj) for j = 1,2. From

Theorem 2.1, we can obtain the desired conclusion easily. This completes the

proof.

3 Conclusion

The iterative process (2.1) presented in this paper which can be employed to approximate common elements in the solution set of the generalized equilibrium problem (1.3), in the solution set of the classical variational inequality (1.1) and in the common fixed point set of a family nonexpansive mappings is general. It is of interest to improve the main results presented in this paper to the frame-work of real Banach spaces.

Acknowledgments. The authors are extremely grateful to the editor and the referees for useful suggestions that improve the contents of the paper.

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