Generalized extragradient iterative method for systems of variational inequalities

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R E S E A R C H

Open Access

Generalized extragradient iterative method for

systems of variational inequalities

Lu Chuan Ceng

1

, Mu Ming Wong

2,3*

and Abdul Latif

4

* Correspondence: [email protected] 2_{Scientific Computing Key} Laboratory of Shanghai Universities, Chung Li 32023, Taiwan Shanghai, China Full list of author information is available at the end of the article

Abstract

The purpose of this article is to investigate the problem of finding a common element of the solution sets of two different systems of variational inequalities and the set of fixed points a strict pseudocontraction mapping defined in the setting of a real Hilbert space. Based on the well-known extragradient method, viscosity

approximation method and Mann iterative method, we propose and analyze a generalized extra-gradient iterative method for computing a common element. Under very mild assumptions, we obtain a strong convergence theorem for three sequences generated by the proposed method. Our proposed method is quite general and flexible and includes the iterative methods considered in the earlier and recent literature as special cases. Our result represents the modification, supplement, extension and improvement of some corresponding results in the references.

Mathematics Subject Classification (2000):Primary 49J40; Secondary 65K05; 47H09.

Keywords:systems of variational inequalities, generalized extragradient iterative method, strict pseudo-contraction mappings, inverse-strongly monotone mappings, strong convergence

1. Introduction

LetHbe a real Hilbert space with inner product 〈·, ·〉and norm║· ║. LetC be a nonempty closed convex subset ofH and S : C® C be a self-mapping on C. We denote by Fix(S) the set of fixed points of Sand byPCthe metric projection ofHonto C. Moreover, we also denote byRthe set of all real numbers. For a given nonlinear mappingA:C® H, consider the following classical variational inequality problem of findingx*ÎCsuch that

Ax∗,x−x∗≥0, ∀x ∈ C. (1:1)

The set of solutions of problem (1.1) is denoted byVI(A, C). It is now well known that the variational inequalities are equivalent to the fixed-point problems, the origin of which can be traced back to Lions and Stampacchia [1]. This alternative formulation has been used to suggest and analyze Picard successive iterative method for solving variational inequalities under the conditions that the involved operator must be strongly monotone and Lipschitz continuous. Related to the variational inequalities, we have the problem of finding fixed points of nonexpansive mappings or strict pseudo-contractions, which is the current interest in functional analysis. Several authors

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considered some approaches to solve fixed point problems, optimization problems, var-iational inequality problems and equilibrium problems; see, for example, [2-32] and the references therein.

For finding an element of Fix(S)∩VI(A,C) under the assumption that a setC⊂His nonempty, closed and convex, a mapping S :C ®Cis nonexpansive and a mapping

A:C®His a-inverse strongly monotone, Takahashi and Toyoda [20] introduced the following iterative algorithm:

x0=x∈Cchosen arbitrarily,

xn+1=αnxn+ (1−αn)SPC(xn−λnAxn), ∀n≥0,

where {an} is a sequence in (0, 1), and {ln} is a sequence in (0, 2a). It was proven in [20] that ifFix(S)∩ VI(A,C)=∅then the sequence {xn} converges weakly to somezÎ Fix(S) ∩VI(A, C). Recently, Nadezhkina and Takahashi [19] and Zeng and Yao [32] proposed some so-called extragra-dient method motivated by the idea of Korpelevich [33] for finding a common element of the set of fixed points of a nonexpansive map-ping and the set of solutions of a variational inequality. Further, these iterative meth-ods were extended in [27] to develop a general iterative method for finding a element of Fix(S)∩VI(A, C).

LetA1,A2 :C®Hbe two mappings. In this article, we consider the following

pro-blem of finding (x*,y*)ÎC × Csuch that

λ1A1y∗+x∗−y∗,x−x∗

≥0, ∀x∈C,

λ2A2x∗+y∗−x∗,x−y∗

≥0, ∀x∈C, (1:2)

which is called a general system of variational inequalities, where l1 >0 andl2 >0

are two constants. It was introduced and considered by Ceng et al. [7]. In particular, if

A1 =A2 =A, then problem (1.2) reduces to the following problem of finding (x*, y*)Î C × Csuch that

λ1Ay∗+x∗−y∗,x−x∗

≥0, ∀x∈C,

λ2Ax∗+y∗−x∗,x−y∗

≥0, ∀x∈C, (1:3)

which was defined by Verma [22] (see also [21]) and it is called a new system of var-iational inequalities. Further, if x* =y*additionally, then problem (1.3) reduces to the classical variational inequality problem (1.1). We remark that in [34], Ceng et al. pro-posed a hybrid extragradient method for finding a common element of the solution set of a variational inequality problem, the solution set of problem (1.2) and the fixed-point set of a strictly pseudocontractive mapping in a real Hilbert space. Recently, Ceng et al. [7] transformed problem (1.2) into a fixed point problem in the following way:

Lemma 1.1.[7]. For givenx,y∈C, (x,y)is a solution of problem(1.2)if and only ifx is a fixed point of the mapping G :C® C defined by

G(x) =PC

PC(x−λ2A2x)−λ1A1PC(x−λ2A2x)

, ∀x∈C, (1:4)

where ¯y=PC(x¯−λ2A2x¯)

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Utilizing Lemma 1.1, they proposed and analyzed a relaxed extragradient method for solving problem (1.2). Throughout this article, the set of fixed points of the mappingG

is denoted by Γ. Based on the extragradient method [33] and viscosity approximation method [23], Yao et al. [26] introduced and studied a relaxed extragradient iterative algorithm for finding a common solution of problem (1.2) and the fixed point problem of a strictly pseudocontraction in a real Hilbert space H.

Theorem 1.1. [[26], Theorem 3.2].Let C be a nonempty bounded closed convex sub-set of a real Hilbert space H. Let the mapping Ai: C® H beαi-inverse strongly mono-tone for i = 1, 2. Let S: C® C be a k-strict pseudocontraction mapping such that

Ω:= Fix(S)∩Γ =∅.Let Q : C® C be a r-contraction mapping withρ∈

0,1

2 For

given x0 ÎC arbitrarily, let the sequences{xn}, {yn}and{zn}be generated iteratively by ⎧

⎪ ⎨ ⎪ ⎩

zn=PC(xn−λ2A2xn),

yn=αnQxn+ (1−αn)PC(zn−λ1A1zn),

xx+1=βnxn+γnPC(zn−λ1A1zn) +δnSyn, ∀n≥0,

(1:5)

where λi∈(0, 2αi)for i = 1, 2,and {an}, {bn}, {gn}, {δn}are four sequences in[0, 1] such that

(i)bn+gn+δn=1 and(gn+δn)k≤gn<(1 - 2r)δnfor all n≥ 0; (ii)_x→∞limαn= 0and∞_n₌₁αn=∞

(iii)0<lim inf_n→∞ βn≤lim sup

n→∞ βn<1andlim infn→∞ δn>0

(iv)lim

n→∞

_γn

+1

1−βn+1− γn

1−βn = 0

Then the sequence {xn}generated by(1.5) converges strongly to x* = PΩQx* and(x*,

y*)is a solution of the general system of variational inequalities(1.2),where y* =PC(x* - l2A2x*).

Let B1,B2 :C®Hbe two mappings. In this article, we also consider another general

system of variational inequalities, that is, finding (x*, y*)ÎC × Csuch that

μ1B1y∗+x∗−y∗,x−x∗

≥0, ∀x∈C,

μ2B2x∗+y∗−x∗,x−y∗

≥0, ∀x∈C, (1:6)

whereμ1>0 andμ2>0 are two constants.

Utilizing Lemma 1.1, we know that for given x,y∈C, (x,y)is a solution of problem (1.6) if and only if x is a fixed point of the mappingF:C®Cdefined by

F(x) =PC

PC(x−μ2B2x)−μ1B1PC(x−μ2B2x)

, ∀x∈C, (1:7)

where y=PC(x−μ2B2x)In particular, if the mapping Bi : C ® H is βi-inverse strongly monotone for i = 1, 2, then the mapping F is nonexpansive provided μi∈(0, 2αi)fori= 1, 2. Throughout this article, the set of fixed points of the mapping

Fis denoted by Γ0.

Assume that Ai : C® H isαi-inverse strongly monotone and and Bi : C ® H is

βi-inverse strongly monotone for i= 1, 2. Let S: C®Cbe a k-strict pseudocontrac-tion mapping such thatΩ:= Fix(S)∩Γ ∩Γ0=∅. LetQ: C® Cbe a r-contraction

mapping with ρ∈

0,1

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this area, we propose and analyze the following iterative scheme for computing a com-mon element of the solution set Γ of one general system of variational inequalities (1.2), the solution setΓ0of another general system of variational inequalities (1.6), and

the fixed point set Fix(S) of the mappingS: ⎧

⎪ ⎨ ⎪ ⎩

zn=PC

PC(xn−μ2B2xn)−μ1B1PC(xn−μ2B2xn)

,

yn=αnQxn+ (1−αn)PC

PC(zn−λ2A2zn)−λ1A1PC(zn−λ2A2zn)

,

xn+1=βnxn+γnPC

+δnSyn, ∀n≥0,

(1:8)

whereλi∈(0, 2αi)and_μi_∈(0, 2βi)fori= 1, 2, and {an},{bn}, {gn}, {δn}⊂[0, 1] such that bn +gn+δn = 1 for alln≥ 0. Furthermore, it is proven that the sequences {xn}, {yn} and {zn} generated by (1.8) converge strongly to the same pointx* =PΩQx* under very mild conditions, and (x*, y*) and(x∗,¯y∗)are a solution of general system of varia-tional inequalities (1.2) and a solution of general system of variavaria-tional inequalities (1.6), respectively, wherey* =PC(x* -l2A2x*) andy∗ =PC(x∗−μ2B2x∗)

Our result represents the modification, supplement, extension and improvement of the above Theorem 1.1 in the following aspects.

(a) our problem of finding an element of Fix(S)∩Γ ∩Γ0 is more general and more

complex than the problem of finding an element of Fix(S)∩ Γin the above Theorem 1.1.

(b) Algorithm (1.8) for finding an element of Fix(S)∩Γ∩Γ0 is also more general and

more flexible than algorithm (1.5) for finding an element of Fix(S)∩ Γin the above Theorem 1.1. Indeed, wheneverB1=B2= 0, we have

zn=PC

=xn, ∀n≥0.

In this case, algorithm (1.8) reduces essentially to algorithm (1.5).

(c) Algorithm (1.8) is very different from algorithm (1.5) in the above Theorem YLK because algorithm (1.8) is closely related to the viscosity approximation method with the r-contractionQ:C® Cand involves the Picard successive iteration for the gen-eral system of variational inequalities (1.6).

(d) The techniques of proving strong convergence in our result are very different from those in the above Theorem 1.1 because our techniques depend on the norm inequality in Lemma 2.2 and the inverse-strong monotonicity of mappingsAi,Bi:C® H for i= 1, 2, the demiclosed-ness principle for strict pseudocontractions, and the transformation of two general systems of variational inequalities (1.2) and (1.6) into the fixed-point problems of the nonexpansive self-mappings G: C®CandF: C® C

(see the above Lemma 1.1, respectively.

2. Preliminaries

Let H be a real Hilbert space whose inner product and norm are〈·, ·〉and║ ·║, respectively. Let Cbe a nonempty closed convex subset ofH. We write ®to indicate that the sequence {xn} converges strongly toxand⇀ to indicate that the sequence {xn} converges weakly tox. Moreover, we useωw(xn) to denote the weakω-limit set of the sequence {xn}, that is,

ωw(xn) :=

x:xni xfor some subsequence{xni}of{xn}

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Recall that a mapping A: C ® H is calleda-inverse strongly monotone if there exists a constanta>0 such that

Ax−Ay,x−y≥αAx−Ay2, ∀x,y∈C.

It is obvious that any a-inverse strongly monotone mapping is Lipschitz continuous. A mapping S:C®Cis called a strict pseudocontraction [35] if there exists a constant 0 ≤k <1 such that

Sx−Sy2≤x+y2+k(I−S)x−(I−S)y2, ∀x,y∈C. (2:1)

In this case, we also say that S is ak-strict pseudocontraction. Meantime, observe that (2.1) is equivalent to the following

Sx−Sy,x−y≤x−y2−1−k

2 (I−S)x−(I−S)y

2

, ∀x,y∈C. (2:2)

It is easy to see that if S is a k-strictly pseudocontractive mapping, then I - S is 1−k

2 -inverse strongly monotone and hence 2

1−k-Lipschitz continuous; for further

detail, we refer to [30] and the references therein. It is clear that the class of strict pseudocontractions strictly includes the one of nonexpansive mappings which are map-pings S:C® Csuch that║Sx - Sy║≤║x - y║for allx, yÎ C.

For every pointxÎH, there exists a unique nearest point inC, denoted byPCxsuch that

x−PCx ≤x−y, ∀x∈C.

The mapping PC is called the metric projection ofH ontoC. We know thatPC is a firmly nonexpansive mapping ofHontoC; that is, there holds the following relation

PCx−PCy,x−y

≥PCx−PCy 2

, ∀x,y∈H.

Consequently,PCis nonexpansive and monotone. It is also known thatPC is charac-terized by the following properties: PCxÎCand

x−PCx,PCx−y

≥0, (2:3)

x−y2≥ x−PCx2+y−PCx 2

, ∀x∈H, y∈C. (2:4)

See [36] for more details.

In order to prove our main result in the next section, we need the following lemmas. The following lemma is an immediate consequence of an inner product.

Lemma 2.1.In a real Hilbert space H, there holds the inequality

x+y2≤ x2+ 2y,x+y, ∀x,y∈H. (2:5)

Recall thatS :C®Cis called a quasi-strict pseudocontraction if the fixed point set ofS, Fix(S), is nonempty and if there exists a constant 0≤k <1 such that

Sx−p2≤x−p2+kx−Sx2 for allx∈Candp∈Fix(S). (2:6)

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The following lemma was proved by Suzuki [37].

Lemma 2.2.[37]Let{xn}and{yn}be bounded sequences in a Banach space X and let {bn}be a sequence in [0, 1]with 0<lim infn→∞ βn≤lim supn→∞ βn<1. Suppose xn+1 = (1

-bn)yn+ bnxn for all integers n ≥ 0 andlim sup

n→∞ (yn+1−yn− xn+1−xn)≤0. Then,

lim

n→∞yn−xn= 0.

Lemma 2.3.[[17], Proposition 2.1]Assume C is a nonempty closed convex subset of a real Hilbert space H and let S: C®C be a self-mapping on C.

(a) If S is a k-strict pseudocontraction, then S satisfies the Lipschitz condition

Sx−Sy≤ 1 +k

1−kx−y, ∀x,y∈C. (2:7)

(b) if S is a k-strict pseudocontraction, then the mapping I - S is demiclosed (at 0).

That is, if {xn} is a sequence in C such that xnx˜ and (I - S)xn ® 0, then

(I−S)˜x= 0,i.e.,˜x∈Fix(S)

(c) if S is a k-quasi-strict pseudocontraction, then the fixed point set Fix(S)of S is closed and convex so that the projection PFix(S)is well defined.

Lemma 2.4.[24]Let {an}be a sequence of nonnegative numbers satisfying the condi-tion

an+1≤(1−δn)an+δnσn, ∀n≥0,

where {δn}, {sn}are sequences of real numbers such that (i){δn}⊂[0, 1]and

_∞

n=0δn=∞, or equivalently, ∞

n=0

(1−δn) := lim

n→∞ n

j=0

(1−δj) = 0;

(ii)lim sup

n→∞ σn≤0or (ii)∞_n₌₀δnσnis convergent.

Then_nlim_→∞an= 0.

3. Strong convergence theorems

We are now in a position to state and prove our main result.

Lemma 3.1. [[26], Lemma 3.1]Let C be a nonempty closed convex subset of a real Hilbert space H. Let S: C®C be a k-strict pseudocontraction mapping. Letgandδbe two nonnegative real numbers. Assume (g+δ)k≤g. Then

_γ(x−y) +δ(Sx−Sy)≤(γ +δ)x−y, ∀x,y∈C. (3:1)

Theorem 3.1. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Assume that for i = 1, 2,the mappings Ai, Bi : C®H areαi-inverse strongly monotone and_βi-inverse strongly monotone, respectively. Let S :C ® C be a k-strict pseudocontraction mapping such thatΩ:= Fix(S)∩Γ ∩Γ0=∅. Let Q:C®C be a

r-contraction mapping withρ∈

0,1

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⎧ ⎪ ⎨ ⎪ ⎩

zn=PC

,

(3:2)

where λi∈(0, 2αi)and_μi_∈(0, 2βi)for i = 1, 2, and{an}, {bn}, {gn}, {δn} are four sequences in[0, 1]such that:

(i)bn+gn+δn= 1and(gn+δn)k≤gn<(1-2r)δnfor all n≥0;

(ii) _x→∞limαn= 0and∞

n=0αn=∞

(iii)0<lim inf_n_→∞ βn≤lim sup

n→∞ βn<1andlim infn→∞ δn>0

(iv) lim

n→∞

_γn

+1

1−βn+1 − γn

1−βn = 0

Then, the sequences {xn}, {yn}, {zn}generated by(3.2) converge strongly to the same point x* =PΩQx*, and(x*,y*) and(x∗,y¯∗)are a solution of general system of

varia-tional inequalities (1.2) and a solution of general system of variational inequalities

(1.6), respectively, where y* =PC(x*-l2A2x*)andy∗=PC(x∗−μ2B2x∗).

Proof. Let us show that the mappingsI -liAiand I -μiBiare nonexpansive fori= 1, 2. Indeed, since for i = 1, 2,Ai, Bi are αi-inverse strongly monotone andβi-inverse strongly monotone, respectively, we have for allx, y ÎC

(I−λiAi)x−(I−λiAi)y2=(x−y)−λi(Aix−Aiy)2

=x−y2−2λiAix−Aiy,x−y

+λ2_iAix−Aiy2

≤x−y2−2λiαiAix−Aiy 2

+λ2_iAix−Aiy 2

=x−y2−λi(2αi−λi)Aix−Aiy2

≤x−y2,

and

(I−μiBi)x−(I−μiBi)y2=(x−y)−μi(Bix−Biy)2

≤x−y2−μi(2βi−μi)Bix−Biy 2

≤x−y2.

This shows that bothI -liAiand I -μiBiare nonexpansive fori= 1, 2. We divide the rest of the proof into several steps.

Step 1. lim

x→∞xn+1−xn= 0.

Indeed, first, we can write (3.2) as xn+1 = bnxn + (1 - bn)un, ∀n ≥ 0, where

un=

xn+1−βnxn

1−βn . Set˜zn=PC(zn−λ2A2zn),∀n≥0. It follows that

un+1−un=

xn+2−βn+1xn+1

1−βn+1 −

xn+1−βnxn

1−βn

= γn+1PC(z˜n+1−λ1A1˜zn+1) +δn+1Syn+1 1−βn+1 −

γnPC(z˜n−λ1A1z˜n) +δnSyn

1−βn

= γn+1

PC(z˜n+1−λ1A1z˜n+1)−PC(˜zn−λ1A1˜zn)

+δn+1(Syn+1−Syn)

1−βn+1

+ _γ

n+1

1−βn+1−

γn

1−βn

PC(˜zn−λ1A1˜zn) +

_δ

n+1

1−βn+1−

δn

1−βn

Syn.

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From Lemma 3.1 and (3.2), we get _γn+1

PC(zñ+1−λ1A1zñ+1)−PC(˜zn−λ1A1zñ)

+δn+1(Syn+1−Syn) ≤γn+1(yn+1−yn) +δn+1(Syn+1−Syn)+γn+1PC(z˜n+1−λ1A1z˜n+1)−yn+1

+yn−PC(z˜n−λ1A1˜zn)

≤(γn+1+δn+1)yn+1−yn+γn+1αn+1Qxn+1−PC(˜zn+1−λ1A1z˜n+1)

+γn+1αnQxn−PC(z˜n−λ1A1˜zn).

(3:4)

Note that

PC(˜zn+1−λ1A1˜zn+1)−PC(zñ−λ1A1˜zn)≤(zñ+1−λ1A1˜zn+1)−(˜zn−λ1A1zñ)

≤ ˜zn+1− ˜zn

=PC(zn+1−λ2A2zn+1)−PC(zn−λ2A2zn)

≤(zn+1−λ2A2zn+1)−(zn−λ2A2zn)

≤ zn+1−zn,

(3:5)

and

zn+1−zn=PC

PC(xn+1−μ2B2xn+1)−μ1B1PC(xn+1−μ2B2xn+1)

−PC

PC(xn−μ2B2xn)−μ1B1PC(xn−μ2B2xn) ≤PC(xn+1−μ2B2xn+1)−μ1B1PC(xn+1−μ2B2xn+1)

−PC(xn−μ2B2xn)−μ1B1PC(xn−μ2B2xn) ≤PC(xn+1−μ2B2xn+1)−PC(xn−μ2B2xn) ≤(xn+1−μ2B2xn+1)−(xn−μ2B2xn) ≤ xn+1−xn.

(3:6)

Then it follows from (3.5) and (3.6) that

yn+1−yn

≤ PC(˜zn+1−λ1A1z˜n+1)−PC(˜zn−λ1A1˜z)+αn+1Qxn+1PC(z˜n+1−λ1A1˜zn+1)

+αnQxn−PC(z˜n−λ1A1˜zn)

≤ xn+1−xn+αnQxn−PC(˜zn−λ1A1z˜n)+αn+1Qxn+1−PC(˜zn+1−λ1A1˜zn+1).

(3:7)

Therefore, from (3.3), (3.4) and (3.7), we have

un+1−un ≤ xn+1−xn+

1 + γn+1

1−βn+1 αn

Qxn−PC(˜zn−λ1A1z˜n)

+

1 + γn+1

1−βn+1 αn+1

Qxn+1−PC(z˜n+1−λ1A1z˜n+1)

+ γn+1 1−βn+1 −

γn

1−βn

PC(z˜n−λ1A1˜zn+Syn.

This implies that

lim sup

n→∞ (un+1−un − xn+1−xn)≤0.

Hence by Lemma 2.2 we get limn®∞‖un-xn‖= 0. Consequently,

lim

n→∞xn+1−xn= limn→∞(1−βn)un−xn= 0. (3:8)

Step 2. lim

n→∞A1z˜n−A1y ∗_{= lim}

n→∞A2zn−A2x ∗_{= lim}

n→∞B1˜xn−B1˜y ∗_{= lim}

n→∞B2xn−B2x ∗_{= 0}_.

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x∗=PC

PC

x∗−μ2B2x∗

−μ1B1PC

x∗−μ2B2x∗

.

Put y* = PC(x* -l2A2x*) andy∗=PC

x∗−μ2B2x∗. Thenx* =PC(y* - l1A1y*) and

x∗=PC

y∗−μ1B1y∗

. Thus it follows that PC(˜zn−λ1A1˜zn)−PC(y∗−λ1A1y∗)2

≤(˜zn−λ1A1z˜n)−

y∗−λ1A1y∗2

≤z˜n−y∗2−λ1

2α1−λ1 A1˜zn−A1y∗2

=PC(zn−λ2A2zn)−PC(x∗−λ2A2x∗)2−λ1

2α1−λ1 A1z˜n−A1y∗2

≤(zn−λ2A2zn)−

x∗−λ2A2x∗2−λ1

2α1−λ1 A1˜zn−A1y∗

2

≤zn−x∗

2

−λ2

α2−λ2 A2zn−A2x∗

2

−λ1

2α1−λ1 A1˜zn−A1y∗

2

, (3:9)

and

zn−x∗

2

=PC

˜

xn−μ1B1˜xn

−PC

y∗−μ1B1y∗ 2

≤x˜n−μ1B1x˜n

−y∗−μ1B1y∗2

≤x˜n−y∗

2 −μ1

2β1−μ1

B1x˜n−B1y∗ 2

=PC(xn−μ2B2xn)−PC

x∗−μ2B2x∗ 2

−μ1

2β1−μ1

B1˜xn−B1y∗ 2

≤xn−x∗

2 −μ2

β2−μ2 B2xn−B2x2−μ1

2β1−μ1

B1x˜n−B1y∗ 2

(3:10)

It follows from (3.2), (3.9) and (3.10) that yn−x∗

2

≤αnQxn−x∗

2

+(1−αn)PC(˜zn−λ1A1z˜n)−PC

y∗−λ1A1y∗2

≤αnQxn−x∗2+PC(˜zn−λ1A1z˜n)−PC(y∗−λ1A1y∗)2

≤αnQxn−x∗

2

+zn−x∗

2

−λ2

α2−λ2 A2zn−A2x∗

2

−λ1

2α1−λ1 A1z˜n−A1y∗

2

≤αnQxn−x∗

2

+xn−x∗

2

−μ2

β2−μ2

B2xn−B2x∗

2

−μ1

2β1−μ

B1x˜n−B1y∗

2

−λ2

α2−λ2 A2zn−A2x∗

2

−λ1

2α1−λ1 A1z˜n−A1y∗

2

.

(3:11)

Utilizing the convexity of║·║2, we have

xn+1−x∗ 2

=βn

xn−x∗

+(1−βn) 1 1−βn

PC(˜zn−λ1A1˜zn)−x∗) +δn

Syn−x∗

2

≤βnxn−x∗2+(1−βn)

γn

1−βn

PC(˜zn−λ1A1˜zn)−x∗

+ δn 1−βn

Syn−x∗

2

=βnxn−x∗2+(1−βn)

γn

yn−x∗

+δn

Syn−x∗

1−βn +

αnγn

1−βn(PC(˜zn−λ1A1z˜n)−Qxn)

2

≤βnxn−x∗2+(1−βn)

γn

yn−x∗

+δn

Syn−x∗

1−β∗

2

+Mαn

≤βnxn−x∗2+(1−βn)yn−x∗2+Mαn,

(10)

whereM >0 is some appropriate constant. So, from (3.11) and (3.12) we have

xn+1−x∗2≤xn−x∗2−μ2

β2−μ2 (1−βn)B2xn−B2x∗2

−μ1

2β1−μ1 (1−βn)B1x˜n−B1y∗2−λ2

α2−λ2

(1−βn)A2zn−A2x∗2

−λ1

2α1−λ1

(1−βn)A1˜zn−A1y∗ 2

+

M+Qxn−x∗

2 αn.

Therefore,

λ1

2α1−λ1

(1−βn)A1z˜n−A1y∗2+λ2

α2−λ2

(1−βn)A2zn−A2x∗2

+μ1

2β₁−μ1 (1−βn)B1x˜n−B1y∗ 2

+μ2

β2−μ2 (1−βn)B2xn−B2x∗ 2

≤xn−x∗2−xn+1−x∗2+

M+Qxn−x∗2

αn

≤xn−x∗+xn+1−x∗xn−xn+1+

M+Qxn−x∗2

αn.

Since lim inf

n→∞λ1(2α1−λ1) (1−βn) >0, lim inf_n_→∞λ2(2α2−λ2) (1−βn) >0, lim inf_n_→∞μ1

2β1−μ1

(1−βn) >0, lim inf_n_→∞μ2β2−μ2

(1−βn) >0,

║xn- xn+1║®0 andan® 0, we have

lim

n→∞A1˜zn−A1y ∗_{= lim}

n→∞A2zn−A2x ∗_{= lim}

n→∞B1x˜n−B1y ∗_{= lim}

n→∞B2xn−B2x ∗_{= 0.}

Step 3. limn®∞║zn- yn║= limn®∞║xn- zn║= limn®∞║Syn- yn║= 0. Indeed, setvn=PC(z˜n−λ1A1˜zn). Noting thatPCis firmly nonexpansive, we have

z_˜n−y∗2=PC(zn−λ2A2zn)−PC

x∗−λ2A2x∗2

≤ (zn−λ2A2zn)−

x∗−λ2A2x∗

,z˜n−y∗

=1₂zn−x∗−λ2

A2zn−A2x∗2+˜zn−y∗2

−zn−x∗

−λ2

A2zn−A2x∗

−z˜n−y∗2

≤1 2

zn−x∗2+z˜n−y∗2−(zn− ˜zn)−λ2

A2zn−A2x∗

−x∗−y∗2

=1 2

zn−x∗2+˜zn−y∗2−zn− ˜z−

x∗−y∗2

+2λ2zn− ˜zn−

x∗−y∗,A2zn−A2x∗ −λ22A2zn−A2x∗2

,

and

vn−x∗ 2

=PC(z˜n−λ1A1˜zn)−PC

y∗−λ1A1y∗ 2

≤˜zn−λ1A1˜zn−

y∗−λ1A1y∗

,vn−x∗

= 1₂z˜n−λ1A1˜zn−

y∗−λ1A1y∗2+vn−x∗2

−z˜n−λ1A1z˜n−

y∗−λ1A1y∗

−vn−x∗ 2

≤1 2

˜

zn−y∗2+vn−x∗2−z˜n−vn+

x∗−y∗2

+2λ1

A1z˜n−A1y∗,˜zn−vn+

x∗−y∗−λ2₁A1z˜n−A1y∗2

≤1 2

zn−x∗ 2

+vn−x∗ 2

−˜zn−vn+

x∗−y∗2

+2λ1

A1z˜n−A1y∗,˜zn−vn+

x∗−y∗

due to (3.9). Thus, we have

(11)

and

vn−x∗

2

≤zn−x∗

2

−z˜n−vn+

x∗−y∗2+2λ1A1z˜n−A1y∗z˜n−vn+

x∗−y∗.

It follows that

yn−x∗≤αnQxn−x∗ 2

+(1−αn)vn−x∗ 2

≤αnQxn−x∗2+vn−x∗2 ≤αnQxn−x∗

2

+zn−x∗ 2

−z˜n−vn+

x∗−y∗2

+ 2λ1A1˜zn−A1y∗˜zn−vn+

x∗−y∗.

(3:14)

Utilizing (3.2), (3.10), (3.12) and (3.13), we have

xn+1−x∗ 2

≤βnxn−x∗

2

+(1−βn)yn−x∗

2

+Mαn

≤βnxn−x∗

2

+(1−βn)

αnQxn−x∗

2

+(1−αn)PC(z˜n−λ1A1˜zn)−PC

y∗−λ1A1y∗2

+Mαn

≤βnxn−x∗2+(1−βn)

αnQxn−x∗2+(1−αn)(˜zn−λ1A1z˜n)−y∗−λ1A1y∗2

2

+Mαn

≤βnxn−x∗2+(1−βn)

αnQxn−x∗2+(1−αn)˜zn−y∗2

+Mαn

≤βnxn−x∗2+(1−βn)

αnQxn−x∗2+(1−βn)˜zn−y∗2

+Mαn

≤βnxn−x∗2+(1−βn) αnQxn−x∗2+(1−βn)

˜

zn−x∗2−zn− ˜zn−

x∗−y∗2

+2λ2zn− ˜zn−x∗−y∗,A2zn−A2x∗+Mαn

≤βnxn−x∗2+(1−βn) αnQxn−x∗2+(1−βn)

xn−x∗2−zn− ˜zn−x∗−y∗2 +2λ2zn− ˜zn−

x∗−y∗A2zn−A2x∗+Mαn =xn−x∗2−(1−βn)zn− ˜zn−(x∗−y∗)2

+ 2(1−βn)λ2zn− ˜zn−(x∗−y∗)A2zn−A2x∗+

M+ (1−βn)Qxn−x∗2

αn.

It follows that

(1−βn)zn− ˜zn−(x∗−y∗)2≤xn−x∗xn+1−x∗xn+1−xn+

M+Qxn−x∗2

αn

+ 2(1−βn)λ2zn− ˜zn−(x∗−y∗)A2zn−A2x∗.

Note that║xn+1 - xn║® 0,an® 0 and ║A2zn- A2x*║®0. Then we immediately

deduce that

lim

n→∞zn− ˜zn−(x

∗₋_y∗₎_{= 0.}

(3:15)

In the meantime, utilizing (3.10), (3.12) and (3.14) we have

xn+1−x∗2≤βnxn−x∗2+ (1−βn)

αnQxn−x∗2+zn−x∗2

−z˜n−vn+ (x∗−y∗)2+ 2λ1A1˜zn−A1y∗˜zn−vn+ (x∗−y∗)+Mαn

≤βnxn−x∗2+ (1−βn)

αnQxn−x∗2+xn−x∗2

−z˜n−vn+ (x∗−y∗)2+ 2λ1A1˜zn−A1y∗˜zn−vn+ (x∗−y∗)+Mαn

≤xn−x∗

2

−(1−βn)˜zn−vn+ (x∗−y∗)

2

+ 2λ1(1−βn)A1z˜n−A1y∗z˜n−vn+ (x∗−y∗)+ (M+Qxn−x∗2)αn.

So, we obtain

(1−βn)z˜n−vn+ (x∗−y∗)2≤xn−x∗2−xn+1−x∗2+ 2λ1(1−βn)A1z˜n−A1y∗

×z˜n−vn+ (x∗−y∗)+

M+Qxn−x∗2

(12)

Hence,

lim

n→∞˜zn−vn+ (x∗−y∗)= 0.

This together with║yn-υn║≤an║Qxn-υn║®0, implies that

lim

n→∞˜zn−yn+ (x∗−y∗)= 0. (3:16)

Thus, from (3.15) and (3.16) we conclude that

lim

n→∞zn−yn= 0.

On the other hand, by firm nonexpansiveness of PC, we have x_˜n−y∗

2

=PC(xn−μ2B2xn)−PC(x∗−μ2B2x∗) 2

≤(xn−μ2B2xn)−(x∗−μ2B2x∗),x˜n−y∗

=1 2

xn−x∗−μ2(B2xn−B2x∗) 2

+x˜n−y∗

2

−(xn−x∗)−μ2(B2xn−B2x∗)−(x˜n−y∗)

2

≤1 2

xn−x∗2+x˜n−y∗2−(xn− ˜xn)−μ2(B2xn−B2x∗)−(x∗−y∗)2

=1 2

xn−x∗2+˜xn−y∗2−xn− ˜xn−(x∗−y∗)2 +2μ2

xn− ˜xn−(x∗−y∗),B2xn−B2x∗

−μ2

2B2xn−B2x∗2

,

and

zn−x∗2=PC(xñ−μ1B1xñ)−PC(y∗−μ1B1y∗)2 ≤xñ−μ1B1xñ−(y∗−μ1B1y∗),zn−x∗

=1 2

˜

xn−μ1B1˜xn−(y∗−μ1B1y∗)2+zn−x∗2

−x˜n−μ1B1˜xn−(y∗−μ1B1y∗)−(zn−x∗)2

≤1 2x˜n−y

∗2

+zn−x∗2−x˜n−zn+ (x∗−y∗)2 + 2μ1

B1˜xn−B1y∗,x˜n−zn+ (x∗−y∗)

−μ2

1B1x˜n−B1y∗2 ≤1

2

xn−x∗

2

+zn−x∗

2

−x˜n−zn+ (x∗−y∗)

2

+ 2μ1

B1˜xn−B1y∗,x˜n−zn+ (x∗−y∗)

.

Thus, we have x_˜n−y∗

2

≤xn−x∗

2

−xn− ˜xn−(x∗−y∗)

2

+2μ2

xn− ˜xn−(x∗−y∗),B2xn−B2x∗

−μ2

2B2xn−B2x∗ 2

, (3:17)

and

zn−x∗2≤xn−x∗2−xñ−zn+ (x∗−y∗)2+2μ1B1xñ−B1y∗xñ−zn+ (x∗−y∗). (3:18)

Consequently, from (3.10), (3.11), (3.12) and (3.17) we have xn+1−x∗2≤βnxn−x∗2+(1−βn)yn−x∗2+Mαn

≤βnxn−x∗

2

+(1−βn)

αnQxn−x∗

2

+zn−x∗

2

+Mαn

≤βnxn−x∗2+(1−βn)(αnQxn−x∗2+x˜n−y∗2) +Mαn

≤βnxn−x∗

2

+(1−βn)

αnQxn−x∗

2

+xn−x∗

2

−xn− ˜xn−

x∗−y∗2

+ 2μ2

xn− ˜xn−

x∗−y∗,B2xn−B2x∗

] +Mαn

≤xn−x∗

2

−(1−βn)xn− ˜xn−

x∗−y∗2

+ 2(1−βn) μnxn− ˜xn−x∗−y∗B2xn−B2x∗+

M+Qxn−x∗2

(13)

It follows that

(1−βn)xn− ˜xn−

x∗−y∗2≤xn−x∗+xn+1−x∗xn+1−xn+

M+Qxn−x∗2

αn

+ 2(1−βn) μ2xn− ˜xn−

x∗−y∗B2xn−B2x∗.

Note that║xn+1- xn║® 0,an ®0 and ║B2xn- B2x*║® 0. Then we immediately

deduce that

lim

n→∞xn− ˜xn−

x∗−y∗= 0. (3:19)

Furthermore, utilizing (3.11), (3.12) and (3.18) we have xn+1−x∗

2

≤βnxn−x∗ 2

+(1−βn)yn−x∗ 2

+Mαn

≤βnxn−x∗ 2

+(1−βn)

αnQxn−x∗ 2

+zn−x∗ 2

+Mαn

≤βnxn−x∗2+(1−βn)

αnQxn−x∗2+xn−x∗2 −x˜n−zn+

x∗−y∗2+ 2μ1B1x˜n−B1y∗x˜n−zn+

x∗−y∗] +Mαn

≤xn−x∗2−(1−βn)x˜n−zn+

x∗−y∗2

+ 2μ1(1−βn)B1x˜n−B1y∗x˜n−zn+

x∗−y∗+M+Qxn−x∗2

αn.

So, we get

(1−βn)x˜n−zn+ (x∗−y∗)2≤xn−x∗2−xn+1−x∗2+ 2μ1(1−βn)B1˜xn−B1y∗ ×x˜n−zn+

x∗−y∗+M+Qxn−x∗2

αn.

Hence,

lim

n→∞x˜n−zn+

x∗−y∗= 0. (3:20)

Thus, from (3.19) and (3.20) we conclude that

lim

n→∞xn−zn= 0.

Since _δn

Syn−xn≤ xn+1−xn+γnPC(z˜n−λ1A1˜zn)−xn

≤ xn+1−xn+γnyn−xn+γnαnQxn−PC(z˜n−λ1Anz˜n),

so we obtain that

lim

n→∞Syn−xn= 0 and nlim→∞Syn−yn= 0.

Step 4. lim supn®∞〈Qx* -x*,xn- x*〉≤0 wherex* =PΩQx*.

Indeed, as His reflexive and {xn} is bounded, we may assume, without loss of gener-ality, that there exists a subsequencexni

of {xn} such thatxni vand

lim sup

n→∞

Qx∗−x∗,xn−x∗

= lim sup

i→∞

Qx∗−x∗,xni−x

.

From Step 3 it is known that║xn - yn║®0 as n®∞. This together withxni v,

(14)

Lemma 2.3 (ii) thatυ ÎFix(S). Next, we prove thatυ ÎΓ ∩Γ0. As a matter of fact, observe that

yn−G

yn

≤αnQxn−G

yn+(1−αn)PC[PC(zn−λ2A2zn)−λ1A1PC(zn−λ2A2zn)−G

yn

=αnQxn−G

yn+(1−αn)G

yn

≤αnQxn−G

yn+(1−αn)zn−yn

→0,

and

zn−F(zn)

≤αnQxn−F(zn)+(1−αn)PC[PC(xn−μ2B2xn)−μ1B1PC(xn−μ2B2xn)−F(zn)

=αnQxn−F(zn)+(1−αn)F(xn)−F(zn)

≤αnQxn−F(zn)+(1−αn)xn−zn

→0,

whereGandFare given in (1.4) and (1.7), respectively. According to Lemma 2.3 (ii) we obtainυ ÎΓ∩Γ0. Therefore,υÎΩ. Hence, it follows from (2.3) that

lim sup

n→∞

= lim

i→∞

Qx∗−x∗,xni−x∗

=Qx∗−x∗,υ−x∗

≤0.

Step 5. limn®∞xn=x*.

Indeed, from (3.2) and the convexity of || · ||2, we have

xn+1−x∗2=βnxn−x∗+γnyn−x∗+δnSyn−x∗+γnαn(PC(z˜n−λ1A1z˜n)−Qxn)=2

≤βn

xn−x∗

+γn

yn−x∗

+δn

Syn−x∗2+ 2γnαn

PC(z˜n− |λ1A1z˜n)−Qxn,xn+1−x∗

≤βnxn−x∗2+(1−βn)

1 − 1−βn

γn

yn−x∗

+δn

Syn−x∗ 2 + 2γnαn

PC(˜zn−λ1A1z˜n)−x∗,xn+1−x∗

+ 2γnαn

x∗−Qxn,xn+1−x∗

.

(3:21)

By Lemma 3.1 and (3.21), we have

xn+1−x∗2

≤βnxn−x∗2+(1−βn)yn−x∗2+ 2γnαnPC(˜zn−λ1A1z˜n)−x∗xn+1−x∗

+ 2γnαn

≤βnxn−x∗2+(1−βn)

(1−αn)PC(˜zn−λ1A1z˜n)−x∗2+ 2αn

Qxn−x∗,yn−x∗

+ 2γnαnPC(˜zn−λ1A1z˜n)−x∗xn+1−x∗+ 2γnαn

=βnxn−x∗2+(1−βn)

(1−αn)G(zn)−G

x∗2+ 2αn

+ 2γnαnG(zn)−G

x∗xn+1−x∗+ 2γnαn

≤βnxn−x∗2+(1−βn)

(1−αn)zn−x∗2+ 2αn

+ 2γnαnzn−x∗xn+1−x∗+ 2γnαn

.

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xn+1−x∗ 2

≤βnxn−x∗2+(1−βn) (1−αn)xn−x∗2+ 2αn(1−βn)

+ 2γnαnxn−x∗xn+1−x∗+ 2γnαn

=[1−(1−βn) αn]xn−x∗

2

+ 2αnγn

Qxn−x∗,yn−xn+1

+ 2αnδn

+ 2αnγnxn−x∗xn+1−x∗

≤[1−(1−βn) αn]xn−x∗2+ 2αnγnQxn−x∗yn−xn+1

+ 2αnδn

Qxn−x∗,xn−x∗

+ 2αnδn

Qxn−x∗,yn−xn

+ 2αnγnxn−x∗xn+1−x∗

≤[1−(1−βn) αn]xn−x∗

2

+ 2αnγnQxn−x∗yn−xn+1

+ 2αnδnρxn−x∗

2

+ 2αnδn

+ 2αnδnQxn−x∗yn−xn+ 2αnγnxn−x∗xn+1−x∗

≤[1−(1−βn) αn]xn−x∗2+ 2αnγnQxn−x∗yn−xn+1

+ 2αnδnρxn−x∗2+ 2αnδn

+ 2αnδnQxn−x∗yn−xn+αnγn

xn−x∗

2

+xn+1−x∗ 2

,

that is,

xn+1−x∗ 2

≤

1−(1−2ρ) δn−γn 1−αnγn αn

xn−x∗

2

+[(1−2ρ) δn−γn]αn 1−αnγn

×

2γn

(1−2ρ) δn−γn

Qxn−x∗yn−xn+1

+ 2δn

(1−2ρ) δn−γn

Qxn−x∗yn−xn+

2δn

(1−2ρ) δn−γn

.

Note that lim infn→∞(1−2ρ) δn−γn

1−αnγn >0. It follows that

∞

n=0

(1−2ρ) δn−γn

1−αnγn αn=∞. It is obvious that

lim sup

x→∞

2γn

(1−2ρ) δn−γn

Qxn−x∗yn−xn+1+

2δn

(1−2ρ) δn−γn

Qxn−x∗yn−xn

+ 2δn

(1−2ρ) δn−γn

≤0.

Therefore, all conditions of Lemma 2.4 are satisfied. Consequently, in terms of Lemma 2.4 we immediately deduce that xn®x*. This completes the proof.□

Next we present some applications of Theorem 3.1 in several special cases.

Corollary 3.1.Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let the mapping Ai:C®H beαi-inverse strongly monotone for i= 1, 2. Let S : C®C be a k-strict pseudocontraction such thatΩ:= Fix(S)∩Γ =∅.Let Q:C® C

be a r-contraction withρ∈

0,1

2 . For given x0Î C arbitrarily, let the sequences{xn}, {yn}be generated iteratively by

yn=αnQxn+ (1−αn)PCPC(xn−λ2A2xn)−λ1A1PC(xn−λ2A2xn),

PC(xn−λ2A2xn)−λ1A1PC(xn−λ2A2xn)

where λi∈(0, 2αi)for i = 1,2,and {an}, {bn}, {gn}, {δn}are four sequences in[0, 1] such that:

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(ii)limn®∞an= 0and

∞

n=0αn=∞;

(iii)0<lim infn®∞bn≤lim supn®∞bn<1andlim infn®∞δn>0;

(iv)limn→∞

γn+1

1−βn+1− γn

1−βn = 0.

Then the sequences{xn}, {yn}converge strongly to the same point x* =PΩQx*, and(x*, y*)is a solution of general system (1.2)of variational inequalities, where y* = PC (x* -l2A2x*).

Proof. It is easy to see that ifBi= 0 fori= 1, 2, then for any givenβi∈(0,∞), Biis

βi-inverse strongly monotone. In Theorem 3.1, puttingBi= 0 and taking μi∈(0, 2βi) for i= 1, 2 we haveΩ:= Fix(S)∩ΓnΓ0= Fix(S)∩Γand

zn=PC

=xn, ∀n≥0.

In this case, algorithm (3.2) reduces to the following algorithm

,

Therefore, in terms of Theorem 3.1 we immediately obtain the desired result. □

Remark3.1. Compared with Theorem YLK (i.e., [[26], Theorem 3.2]), Corollary 3.1 coincides essentially with Theorem YLK. Therefore, Theorem 3.1 includes Theorem YLK as a special case.

Corollary 3.2.Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Assume that for i= 1, 2,the mappings Ai, Bi: C® H areαi-inverse strongly monotone and _βi-inverse strongly monotone, respectively. Let S : C®C be a k-strict pseudocontraction such thatΩ := Fix(S)∩Γ ∩Γ0=∅.For fixed u ÎC and given x0 Î C arbitrarily, let the sequences{xn}, {yn}and{zn}be generated iteratively by

⎧ ⎪ ⎨ ⎪ ⎩

zn=PC

,

yn=αnu+ (1−αn)PC

,

xn+1=βnxn+γnPCPC(zn−λ2A2zn)−λ1A1PC(zn−λ2A2zn)+δnSyn, ∀n≥0,

where zn−x∗

2

≤xn−x∗

2

−x˜n−zn+ (x∗−y∗)

2

+2μ1B1x˜n−B1y∗˜xn−zn+ (x∗−y∗).

and_μi_∈(0, 2βi)and{an}, {bn}, {gn}, {δn}are four sequences in[0, 1]such that:

(i)bn+gn+δn= 1and(gn+δn)k≤gn<δnfor all n≥0;

∞

n=0αn=∞;

(iii)0<lim infn®∞bn≤ lim supn®∞bn<1andlim infn®∞δn>0;

(iv)limn→∞

_γn

+1

1−βn+1− γn

1−βn = 0.

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Corollary 3.3.Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Assume that for i= 1, 2,the mappings Ai, Bi: C® H areαi-inverse strongly monotone and_βi-inverse strongly monotone, respectively. Let S : C® C be a nonex-pansive mapping such thatΩ:= Fix(S)∩Γ ∩Γ0=∅.Let Q:C®C be ar-contraction

with ρ∈

0,1

2 . For given x0 Î C arbitrarily, let the sequences {xn}, {yn}and {zn} be

generated iteratively by

⎧ ⎪ ⎨ ⎪ ⎩

zn=PC

,

(i)bn+gn+δn= 1andgn<(1 - 2r)δnfor all n≥0;

∞

n=0αn=∞;

(iii)0<lim infn®∞bn≤lim supn®∞bn<1andlim infn®∞gn>0;

(iv)limn→∞

_γn

+1

1−βn+1− γn

1−βn = 0.

Then the sequences {xn}, {yn}, {zn} converge strongly to the same point x* = PΩQx*,

and(x*,y*)and(x∗,y¯∗)are a solution of general system (1.2)of variational inequalities and a solution of general system(1.6)of variational inequalities, respectively, where y* =PC(x* -l2A2x*)andy∗=PC(x∗−μ2B2x∗).

Corollary 3.4.Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Assume that for i= 1, 2,the mappings Ai, Bi: C® H areαi-inverse strongly monotone and_βi-inverse strongly monotone, respectively. Let S : C® C be a nonex-pansive mapping such thatΩ:= Fix(S)∩Γ ∩Γ0=∅. For fixed u ÎC and given x0 Î C arbitrarily, let the sequences{xn}, {yn}and{zn}be generated iteratively by

⎧ ⎪ ⎨ ⎪ ⎩

zn=PC

,

yn=αnu+ (1−αn)PC

,

(i)bn+gn+δn= 1andgn<δnfor all n≥0;

∞

n=0αn=∞;

(iii)0<lim infn®∞bn≤ lim supn®∞bn<1andlim infn®∞gn>0;

(iv)limn→∞

_γn

+1

1−βn+1− γn

1−βn = 0.

(18)

a solution of general system(1.6)of variational inequalities, respectively, where y* =PC (x* -l2A2x*)andy∗=PC(x∗−μ2B2x∗).

Acknowledgements

In this research, the first author was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), Leading Academic Discipline Project of Shanghai Normal University (DZL707), Ph.D. Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405). The second author was partially supported by the NSC100-2115-M-033-001. The third author gratefully acknowledge the financial support from the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah.

Author details 1

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China2Scientific Computing Key Laboratory of Shanghai Universities, Chung Li 32023, Taiwan Shanghai, China3_{Department of Applied Mathematics,} Chung Yuan Christian University, Chung Li 32023, Taiwan4_{Department of Mathematics, King Abdulaziz University, P.O.} Box 80203, Jeddah 21589, Saudi Arabia

Authors’contributions

All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 16 August 2011 Accepted: 16 April 2012 Published: 16 April 2012

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doi:10.1186/1029-242X-2012-88

Cite this article as:Cenget al.:Generalized extragradient iterative method for systems of variational inequalities.

Journal of Inequalities and Applications20122012:88.

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