### 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

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:

*x*0=*x*∈*C*chosen 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

*λ*1*A*1*y*∗+*x*∗−*y*∗,*x*−*x*∗

≥0, ∀*x*∈*C*,

*λ*2*A*2*x*∗+*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

*λ*1*Ay*∗+*x*∗−*y*∗,*x*−*x*∗

≥0, ∀*x*∈*C*,

*λ*2*Ax*∗+*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 given*x*,*y*∈*C*, (*x*,*y*)is a solution of problem(1.2)if and only if*x*
is a fixed point of the mapping G :C® C defined by

*G*(*x*) =*PC*

*PC*(*x*−*λ*2*A*2*x*)−*λ*1*A*1*PC*(*x*−*λ*2*A*2*x*)

, ∀*x*∈*C*, (1:4)

where ¯*y*=*PC*(*x*¯−*λ*2*A*2*x*¯)

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 be*αi*-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 with*ρ*∈

0,1

2 For

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

⎪ ⎨ ⎪ ⎩

*zn*=*PC*(*xn*−*λ*2*A*2*xn*),

*yn*=*αnQxn*+ (1−*αn*)*PC*(*zn*−*λ*1*A*1*zn*),

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

(1:5)

where *λi*∈(0, 2*αi*)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→∞}*lim

*αn*= 0and∞

_{n}_{=1}

*αn*=∞

(iii)0*<*lim inf_{n→∞}*βn*≤lim sup

*n*→∞ *βn<*1andlim inf*n*→∞ *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

*μ*1*B*1*y*∗+*x*∗−*y*∗,*x*−*x*∗

≥0, ∀*x*∈*C*,

*μ*2*B*2*x*∗+*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*−*μ*2*B*2*x*)−*μ*1*B*1*PC*(*x*−*μ*2*B*2*x*)

, ∀*x*∈*C*, (1:7)

where *y*=*PC*(*x*−*μ*2*B*2*x*)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, 2*αi*)fori= 1, 2. Throughout this article, the set of fixed points of the mapping

Fis denoted by Γ0.

Assume that Ai : C® H is*αi*-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 that*Ω*:= Fix(*S*)∩*Γ* ∩*Γ*0=∅. LetQ: C® Cbe a r-contraction

mapping with *ρ*∈

0,1

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*−*μ*2B2x*n*)−*μ*1B1P*C*(*xn*−*μ*2B2x*n*)

,

*yn*=*αnQxn*+ (1−*αn*)*PC*

*PC*(*zn*−*λ*2A2z*n*)−*λ*1A1P*C*(*zn*−*λ*2A2z*n*)

,

*xn*+1=*βnxn*+*γnPC*

*PC*(*zn*−*λ*2A2z*n*)−*λ*1A1P*C*(*zn*−*λ*2A2z*n*)

+*δnSyn*, ∀*n*≥0,

(1:8)

where*λi*∈(0, 2*αi*)and_{μi}_{∈}(0, 2*βi*)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*) and*y*∗ =*PC*(*x*∗−*μ*2*B*2*x*∗)

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*

*PC*(*xn*−*μ*2*B*2*xn*)−*μ*1*B*1*PC*(*xn*−*μ*2*B*2*xn*)

=*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* *x*for some subsequence{*xni*}of{*xn*}

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*−*Ay*2, ∀*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*−*Sy*2≤*x*+*y*2+*k*(*I*−*S*)*x*−(*I*−*S*)*y*2, ∀*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*−*y*2−1−*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*−*y*2≥ *x*−*PCx*2+*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*+*y*2≤ *x*2+ 2*y*,*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*−*p*2≤*x*−*p*2+*kx*−*Sx*2 for all*x*∈*C*and*p*∈Fix(*S*). (2:6)

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 inf*n*→∞ *βn*≤lim sup*n*→∞ *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*=0*δn*=∞, or equivalently,
∞

*n*=0

(1−*δn*) := lim

*n*→∞
*n*

*j*=0

(1−*δj*) = 0;

(ii)lim sup

*n→∞* *σn*≤0or
(ii)∞_{n}_{=0}*δnσn*is convergent.

Then* _{n}*lim

_{→∞}

*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 are*αi*-inverse strongly
monotone and* _{βi}*-inverse strongly monotone, respectively. Let S :C ® C be a k-strict
pseudocontraction mapping such that

*Ω*:= Fix(

*S*)∩

*Γ*∩

*Γ*0=∅. Let Q:C®C be a

r-contraction mapping with*ρ*∈

0,1

⎧ ⎪ ⎨ ⎪ ⎩

*zn*=*PC*

*PC*(*xn*−*μ*2B2x*n*)−*μ*1B1P*C*(*xn*−*μ*2B2x*n*)

,

*yn*=*αnQxn*+ (1−*αn*)*PC*

*PC*(*zn*−*λ*2A2z*n*)−*λ*1A1P*C*(*zn*−*λ*2A2z*n*)

,

*xn*+1=*βnxn*+*γnPC*

*PC*(*zn*−*λ*2A2z*n*)−*λ*1A1P*C*(*zn*−*λ*2A2z*n*)

+*δnSyn*, ∀*n*≥0,

(3:2)

where *λi*∈(0, 2*αi*)and_{μi}_{∈}(0, 2*βi*)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→∞}*lim

*αn*= 0and∞

*n*=0*αn*=∞

(iii)0*<*lim inf_{n}_{→∞} *βn*≤lim sup

*n→∞* *βn<*1andlim inf*n*→∞ *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*)and*y*∗=*PC*(*x*∗−*μ*2*B*2*x*∗).

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 and*βi*-inverse
strongly monotone, respectively, we have for allx, y ÎC

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

=*x*−*y*2−2*λiAix*−*Aiy*,*x*−*y*

+*λ*2* _{i}Aix*−

*Aiy*2

≤*x*−*y*2−2*λiαiAix*−*Aiy*
2

+*λ*2* _{i}Aix*−

*Aiy*2

=*x*−*y*2−*λi*(2*αi*−*λi*)*Aix*−*Aiy*2

≤*x*−*y*2,

and

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

≤*x*−*y*2−*μi*(2*βi*−*μi*)*Bix*−*Biy*
2

≤*x*−*y*2.

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*−*λ*2*A*2*zn*),∀*n*≥0. It follows that

*un*+1−*un*=

*xn*+2−*βn*+1x*n*+1

1−*βn*+1 −

*xn*+1−*βnxn*

1−*βn*

= *γn*+1P*C*(*z*˜*n*+1−*λ*1A1˜*zn*+1) +*δn*+1Sy*n*+1
1−*βn*+1 −

*γnPC*(*z*˜*n*−*λ*1A1*z*˜*n*) +*δnSyn*

1−*βn*

= *γn*+1

*PC*(*z*˜*n*+1−*λ*1A1*z*˜*n*+1)−*PC*(˜*zn*−*λ*1A1˜*zn*)

+*δn*+1(*Syn*+1−*Syn*)

1−*βn*+1

+
_{γ}

*n*+1

1−*βn*+1−

*γn*

1−*βn*

*PC*(˜*zn*−*λ*1*A1*˜*zn*) +

_{δ}

*n*+1

1−*βn*+1−

*δn*

1−*βn*

*Syn*.

From Lemma 3.1 and (3.2), we get
* _{γn}*+1

*PC*(*z*˜*n*+1−*λ*1*A*1*z*˜*n*+1)−*PC*(˜*zn*−*λ*1*A*1*z*˜*n*)

+*δn*+1(*Syn*+1−*Syn*)
≤*γn*+1(*yn*+1−*yn*) +*δn*+1(*Syn*+1−*Syn*)+*γn*+1*PC*(*z*˜*n*+1−*λ*1*A*1*z*˜*n*+1)−*yn*+1

+*yn*−*PC*(*z*˜*n*−*λ*1*A*1˜*zn*)

≤(*γn*+1+*δn*+1)*yn*+1−*yn*+*γn*+1*αn*+1*Qxn*+1−*PC*(˜*zn*+1−*λ*1*A*1*z*˜*n*+1)

+*γn*+1*αnQxn*−*PC*(*z*˜*n*−*λ*1*A*1˜*zn*).

(3:4)

Note that

*PC*(˜*zn*+1−*λ*1*A*1˜*zn*+1)−*PC*(*z*˜*n*−*λ*1*A*1˜*zn*)≤(*z*˜*n*+1−*λ*1*A*1˜*zn*+1)−(˜*zn*−*λ*1*A*1*z*˜*n*)

≤ ˜*zn*+1− ˜*zn*

=*PC*(z*n*+1−*λ*2*A*2*zn*+1)−*PC*(z*n*−*λ*2*A*2*zn*)

≤(z*n*+1−*λ*2*A*2*zn*+1)−(z*n*−*λ*2*A*2*zn*)

≤ *zn*+1−*zn*,

(3:5)

and

*zn*+1−*zn*=*PC*

*PC*(*xn*+1−*μ*2*B*2*xn*+1)−*μ*1*B*1*PC*(*xn*+1−*μ*2*B*2*xn*+1)

−*PC*

*PC*(*xn*−*μ*2*B*2*xn*)−*μ*1*B*1*PC*(*xn*−*μ*2*B*2*xn*)
≤*PC*(*xn*+1−*μ*2*B*2*xn*+1)−*μ*1*B*1*PC*(*xn*+1−*μ*2*B*2*xn*+1)

−*PC*(*xn*−*μ*2*B*2*xn*)−*μ*1*B*1*PC*(*xn*−*μ*2*B*2*xn*)
≤*PC*(*xn*+1−*μ*2*B*2*xn*+1)−*PC*(*xn*−*μ*2*B*2*xn*)
≤(*xn*+1−*μ*2*B*2*xn*+1)−(*xn*−*μ*2*B*2*xn*)
≤ *xn*+1−*xn*.

(3:6)

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

*yn*+1−*yn*

≤ *PC(˜zn*+1−*λ*1*A*1*z*˜*n*+1*)*−*PC(˜zn*−*λ*1*A*1˜*z)*+*αn*+1*Qxn*+1*PC*(*z*˜*n*+1−*λ*1*A*1˜*zn*+1)

+*αnQxn*−*PC*(*z*˜*n*−*λ*1*A*1˜*zn*)

≤ *xn*+1−*xn*+*αnQxn*−*PC(˜zn*−*λ*1*A*1*z*˜*n)*+*αn*+1*Qxn*+1−*PC(˜zn*+1−*λ*1*A*1˜*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*−*λ*1*A*1*z*˜*n)*

+

1 + *γn*+1

1−*βn*+1 *αn*+1

*Qxn*+1−*PC(z*˜*n*+1−*λ*1*A*1*z*˜*n*+1*)*

+ *γn*+1
1−*βn*+1 −

*γn*

1−*βn*

*PC*(*z*˜*n*−*λ*1*A*1˜*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*= lim*n→∞(*1−*βn)un*−*xn*= 0. (3:8)

Step 2. lim

*n*→∞*A*1*z*˜*n*−*A*1*y*
∗_{= lim}

*n*→∞*A*2*zn*−*A*2*x*
∗_{= lim}

*n*→∞*B*1˜*xn*−*B*1˜*y*
∗_{= lim}

*n*→∞*B*2*xn*−*B*2*x*
∗_{= 0}_{.}

*x*∗=*PC*

*PC*

*x*∗−*μ*2*B*2*x*∗

−*μ*1*B*1*PC*

*x*∗−*μ*2*B*2*x*∗

.

Put y* = PC(x* -l2A2x*) and*y*∗=*PC*

*x*∗−*μ*2*B*2*x*∗. Thenx* =PC(y* - l1A1y*) and

*x*∗=*PC*

*y*∗−*μ*1*B*1*y*∗

. Thus it follows that
*PC*(˜*zn*−*λ*1A1˜*zn*)−*PC*(*y*∗−*λ*1*A1y*∗)2

≤*(˜zn*−*λ*1A1*z*˜*n)*−

*y*∗−*λ*1A1*y*∗2

≤*z*˜*n*−*y*∗2−*λ*1

2*α*1−*λ*1 *A1*˜*zn*−*A1y*∗2

=*PC*(*zn*−*λ*2A2z*n*)−*PC*(*x*∗−*λ*2A2*x*∗)2−*λ*1

2*α*1−*λ*1 *A1z*˜*n*−*A1y*∗2

≤(*zn*−*λ*2A2z*n*)−

*x*∗−*λ*2A2x∗2−*λ*1

2*α*1−*λ*1 *A1*˜*zn*−*A1y*∗

2

≤*zn*−*x*∗

2

−*λ*2

_{}

*α*2−*λ*2 *A2zn*−*A2x*∗

2

−*λ*1

2*α*1−*λ*1 *A1*˜*zn*−*A1y*∗

2

, (3:9)

and

*zn*−*x*∗

2

=*PC*

˜

*xn*−*μ*1*B*1˜*xn*

−*PC*

*y*∗−*μ*1*B*1*y*∗
2

≤*x*˜*n*−*μ*1*B*1*x*˜*n*

−*y*∗−*μ*1*B*1*y*∗2

≤*x*˜*n*−*y*∗

2
−*μ*1

2*β*1−*μ*1

_{}

*B*1*x*˜*n*−*B*1*y*∗
2

=*PC(xn*−*μ*2*B*2*xn)*−*PC*

*x*∗−*μ*2*B*2*x*∗
2

−*μ*1

2*β*1−*μ*1

_{}

*B*1˜*xn*−*B*1*y*∗
2

≤*xn*−*x*∗

2
−*μ*2

*β*2−*μ*2 *B*2*xn*−*B*2*x*2−*μ*1

2*β*1−*μ*1

_{}

*B*1*x*˜*n*−*B*1*y*∗
2

(3:10)

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

2

≤*αnQxn*−*x*∗

2

+*(*1−*αn)PC(˜zn*−*λ*1A1*z*˜*n)*−*PC*

*y*∗−*λ*1A1y∗2

≤*αnQxn*−*x*∗2+*PC(˜zn*−*λ*1A1*z*˜*n)*−*PC*(*y*∗−*λ*1A1y∗)2

≤*αnQxn*−*x*∗

2

+*zn*−*x*∗

2

−*λ*2

_{}

*α*2−*λ*2 *A2zn*−*A2x*∗

2

−*λ*1

2*α*1−*λ*1 *A1z*˜*n*−*A1y*∗

2

≤*αnQxn*−*x*∗

2

+*xn*−*x*∗

2

−*μ*2

*β*2−*μ*2

_{}

*B2xn*−*B2x*∗

2

−*μ*1

2*β*1−*μ*

_{}

*B1x*˜*n*−*B1y*∗

2

−*λ*2

_{}

*α*2−*λ*2 *A2zn*−*A2x*∗

2

−*λ*1

2*α*1−*λ*1 *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*−*λ*1*A*1˜*zn)*−*x*∗) +*δn*

*Syn*−*x*∗

2

≤*βnxn*−*x*∗2+*(*1−*βn)*

*γn*

1−*βn*

*PC(˜zn*−*λ*1*A*1˜*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*−*λ*1*A*1*z*˜*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,*

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

*xn*+1−*x*∗2≤*xn*−*x*∗2−*μ*2

*β*2−*μ*2 *(*1−*βn)B*2*xn*−*B*2*x*∗2

−*μ*1

2*β*1−*μ*1 *(*1−*βn)B*1*x*˜*n*−*B*1*y*∗2−*λ*2

*α*2−*λ*2

*(*1−*βn)A*2*zn*−*A*2*x*∗2

−*λ*1

2*α*1−*λ*1

*(*1−*βn)A*1˜*zn*−*A*1*y*∗
2

+

*M*+*Qxn*−*x*∗

2
*αn*.

Therefore,

*λ*1

2*α*1−*λ*1

*(*1−*βn)A*1*z*˜*n*−*A*1*y*∗2+*λ*2

_{}

*α*2−*λ*2

*(*1−*βn)A*2*zn*−*A*2*x*∗2

+*μ*1

2*β*_{1}−*μ*1 *(*1−*βn)B*1*x*˜*n*−*B*1*y*∗
2

+*μ*2

*β*2−*μ*2 *(*1−*βn)B*2*xn*−*B*2*x*∗
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*→∞*λ*1*(*2*α*1−*λ*1*) (*1−*βn) >*0, lim inf_{n}_{→∞}*λ*2*(*2*α*2−*λ*2*) (*1−*βn) >*0, lim inf_{n}_{→∞}*μ*1

2*β*1−*μ*1

*(*1−*βn) >*0, lim inf_{n}_{→∞}*μ*2*β*2−*μ*2

*(*1−*βn) >*0,

║xn- xn+1║®0 andan® 0, we have

lim

*n*→∞*A*1˜*zn*−*A*1*y*
∗_{= lim}

*n*→∞*A*2*zn*−*A*2*x*
∗_{= lim}

*n*→∞*B*1*x*˜*n*−*B*1*y*
∗_{= lim}

*n*→∞*B*2*xn*−*B*2*x*
∗_{= 0.}

Step 3. limn®∞║zn- yn║= limn®∞║xn- zn║= limn®∞║Syn- yn║= 0.
Indeed, set*vn*=*PC(z*˜*n*−*λ*1*A*1˜*zn)*. Noting thatPCis firmly nonexpansive, we have

*z*_{˜}*n*−*y*∗2=*PC(zn*−*λ*2*A*2*zn)*−*PC*

*x*∗−*λ*2*A*2*x*∗2

≤ (*zn*−*λ*2*A*2*zn)*−

*x*∗−*λ*2*A*2*x*∗

,*z*˜*n*−*y*∗

=1_{2}*zn*−*x*∗−*λ*2

*A*2*zn*−*A*2*x*∗2+˜*zn*−*y*∗2

−*zn*−*x*∗

−*λ*2

*A*2*zn*−*A*2*x*∗

−*z*˜*n*−*y*∗2

≤1 2

_{}

*zn*−*x*∗2+*z*˜*n*−*y*∗2−*(zn*− ˜*zn)*−*λ*2

*A*2*zn*−*A*2*x*∗

−*x*∗−*y*∗2

=1 2

_{}

*zn*−*x*∗2+˜*zn*−*y*∗2−*zn*− ˜*z*−

*x*∗−*y*∗2

+2*λ*2*zn*− ˜*zn*−

*x*∗−*y*∗,*A*2*zn*−*A*2*x*∗ −*λ*22*A*2*zn*−*A*2*x*∗2

,

and

*vn*−*x*∗
2

=*PC(z*˜*n*−*λ*1*A*1˜*zn)*−*PC*

*y*∗−*λ*1*A*1*y*∗
2

≤˜*zn*−*λ*1*A*1˜*zn*−

*y*∗−*λ*1*A*1*y*∗

,*vn*−*x*∗

= 1_{2}*z*˜*n*−*λ*1*A*1˜*zn*−

*y*∗−*λ*1*A*1*y*∗2+*vn*−*x*∗2

−*z*˜*n*−*λ*1*A*1*z*˜*n*−

*y*∗−*λ*1*A*1*y*∗

−*vn*−*x*∗
2

≤1 2

_{}

˜

*zn*−*y*∗2+*vn*−*x*∗2−*z*˜*n*−*vn*+

*x*∗−*y*∗2

+2*λ*1

*A*1*z*˜*n*−*A*1*y*∗,˜*zn*−*vn*+

*x*∗−*y*∗−*λ*2_{1}*A*1*z*˜*n*−*A*1*y*∗2

≤1 2

_{}

*zn*−*x*∗
2

+*vn*−*x*∗
2

−˜*zn*−*vn*+

*x*∗−*y*∗2

+2*λ*1

*A*1*z*˜*n*−*A*1*y*∗,˜*zn*−*vn*+

*x*∗−*y*∗

due to (3.9). Thus, we have

and

*vn*−*x*∗

2

≤*zn*−*x*∗

2

−*z*˜*n*−*vn*+

*x*∗−*y*∗2+2*λ*1*A*1*z*˜*n*−*A*1*y*∗*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

+ 2*λ*1*A*1˜*zn*−*A*1*y*∗˜*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*−*λ*1*A*1˜*zn)*−*PC*

*y*∗−*λ*1*A*1*y*∗2

+*Mαn*

≤*βnxn*−*x*∗2+*(*1−*βn)*

*αnQxn*−*x*∗2+*(*1−*αn)(*˜*zn*−*λ*1*A*1*z*˜*n)*−*y*∗−*λ*1*A*1*y*∗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

+2*λ*2*zn*− ˜*zn*−*x*∗−*y*∗,*A*2*zn*−*A*2*x*∗+*Mαn*

≤*βnxn*−*x*∗2+*(*1−*βn) αnQxn*−*x*∗2+*(*1−*βn)*
_{}

*xn*−*x*∗2−*zn*− ˜*zn*−*x*∗−*y*∗2
+2*λ*2*zn*− ˜*zn*−

*x*∗−*y*∗*A*2*zn*−*A*2*x*∗+*Mαn*
=*xn*−*x*∗2−(1−*βn*)*zn*− ˜*zn*−(*x*∗−*y*∗)2

+ 2(1−*βn*)*λ*2*zn*− ˜*zn*−(*x*∗−*y*∗)*A*2*zn*−*A*2*x*∗+

*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*)*λ*2*zn*− ˜*zn*−(*x*∗−*y*∗)*A*2*zn*−*A*2*x*∗.

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+ 2*λ*1*A*1˜*zn*−*A*1*y*∗˜*zn*−*vn*+ (x∗−*y*∗)+*Mαn*

≤*βnxn*−*x*∗2+ (1−*βn*)

*αnQxn*−*x*∗2+*xn*−*x*∗2

−*z*˜*n*−*vn*+ (x∗−*y*∗)2+ 2*λ*1*A*1˜*zn*−*A*1*y*∗˜*zn*−*vn*+ (x∗−*y*∗)+*Mαn*

≤*xn*−*x*∗

2

−(1−*βn*)˜*zn*−*vn*+ (x∗−*y*∗)

2

+ 2*λ*1(1−*βn*)*A*1*z*˜*n*−*A*1*y*∗*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+ 2*λ*1(1−*βn*)*A*1*z*˜*n*−*A*1*y*∗

×*z*˜*n*−*vn*+ (x∗−*y*∗)+

*M*+*Qxn*−*x*∗2

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*−*μ*2*B*2*xn*)−*PC*(*x*∗−*μ*2*B*2*x*∗)
2

≤(*xn*−*μ*2*B*2*xn*)−(*x*∗−*μ*2*B*2*x*∗),*x*˜*n*−*y*∗

=1 2

_{}

*xn*−*x*∗−*μ*2(*B*2*xn*−*B*2*x*∗)
2

+*x*˜*n*−*y*∗

2

−(*xn*−*x*∗)−*μ*2(*B*2*xn*−*B*2*x*∗)−(*x*˜*n*−*y*∗)

2

≤1 2

_{}

*xn*−*x*∗2+*x*˜*n*−*y*∗2−(*xn*− ˜*xn*)−*μ*2(*B*2*xn*−*B*2*x*∗)−(*x*∗−*y*∗)2

=1 2

_{}

*xn*−*x*∗2+˜*xn*−*y*∗2−*xn*− ˜*xn*−(*x*∗−*y*∗)2
+2*μ*2

*xn*− ˜*xn*−(*x*∗−*y*∗),*B*2*xn*−*B*2*x*∗

−*μ*2

2*B*2*xn*−*B*2*x*∗2

,

and

*zn*−*x*∗2=*PC*(*x*˜*n*−*μ*1*B*1*x*˜*n*)−*PC*(*y*∗−*μ*1*B*1*y*∗)2
≤*x*˜*n*−*μ*1*B*1*x*˜*n*−(*y*∗−*μ*1*B*1*y*∗),*zn*−*x*∗

=1 2

_{}
˜

*xn*−*μ*1*B*1˜*xn*−(*y*∗−*μ*1*B*1*y*∗)2+*zn*−*x*∗2

−*x*˜*n*−*μ*1*B*1˜*xn*−(*y*∗−*μ*1*B*1*y*∗)−(*zn*−*x*∗)2

≤1
2*x*˜*n*−*y*

∗2

+*zn*−*x*∗2−*x*˜*n*−*zn*+ (*x*∗−*y*∗)2
+ 2*μ*1

*B*1˜*xn*−*B*1*y*∗,*x*˜*n*−*zn*+ (*x*∗−*y*∗)

−*μ*2

1*B*1*x*˜*n*−*B*1*y*∗2
≤1

2
_{}

*xn*−*x*∗

2

+*zn*−*x*∗

2

−*x*˜*n*−*zn*+ (*x*∗−*y*∗)

2

+ 2*μ*1

*B*1˜*xn*−*B*1*y*∗,*x*˜*n*−*zn*+ (*x*∗−*y*∗)

.

Thus, we have
*x*_{˜}*n*−*y*∗

2

≤*xn*−*x*∗

2

−*xn*− ˜*xn*−(*x*∗−*y*∗)

2

+2*μ*2

*xn*− ˜*xn*−(*x*∗−*y*∗),*B*2*xn*−*B*2*x*∗

−*μ*2

2*B*2*xn*−*B*2*x*∗
2

, (3:17)

and

*zn*−*x*∗2≤*xn*−*x*∗2−*x*˜*n*−*zn*+ (*x*∗−*y*∗)2+2*μ*1*B*1*x*˜*n*−*B*1*y*∗*x*˜*n*−*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

+ 2*μ*2

*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

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) μ*2*xn*− ˜*xn*−

*x*∗−*y*∗*B*2*xn*−*B*2*x*∗.

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+ 2*μ*1*B*1*x*˜*n*−*B*1*y*∗*x*˜*n*−*zn*+

*x*∗−*y*∗] +*Mαn*

≤*xn*−*x*∗2−*(*1−*βn)x*˜*n*−*zn*+

*x*∗−*y*∗2

+ 2*μ*1*(*1−*βn)B*1*x*˜*n*−*B*1*y*∗*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+ 2*μ*1*(*1−*βn)B*1˜*xn*−*B*1*y*∗
×*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*−*λ*1*A*1˜*zn)*−*xn*

≤ *xn*+1−*xn*+*γnyn*−*xn*+*γnαnQxn*−*PC(z*˜*n*−*λ*1*Anz*˜*n)*,

so we obtain that

lim

*n*→∞*Syn*−*xn*= 0 and *n*lim→∞*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 subsequence*xni*

of {xn} such that*xni* *v*and

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 with*xni* *v*,

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*−*λ*2*A*2*zn)*−*λ*1*A*1*PC(zn*−*λ*2*A*2*zn)*−*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*−*μ*2*B*2*xn)*−*μ*1*B*1*PC(xn*−*μ*2*B*2*xn)*−*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*→∞

*Qx*∗−*x*∗,*xn*−*x*∗

= 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*−*λ*1*A*1*z*˜*n)*−*Qxn)*=2

≤*βn*

*xn*−*x*∗

+*γn*

*yn*−*x*∗

+*δn*

*Syn*−*x*∗2+ 2*γnαn*

*PC(z*˜*n*− |*λ*1*A*1*z*˜*n)*−*Qxn*,*xn*+1−*x*∗

≤*βnxn*−*x*∗2+*(*1−*βn)*

1
−
1−β*n*

*γn*

*yn*−*x*∗

+*δn*

*Syn*−*x*∗ 2
+ 2*γnαn*

*PC(*˜*zn*−*λ*1*A*1*z*˜*n)*−*x*∗,*xn*+1−*x*∗

+ 2*γnαn*

*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+ 2*γnαnPC(˜zn*−*λ*1*A*1*z*˜*n)*−*x*∗*xn*+1−*x*∗

+ 2*γnαn*

*x*∗−*Qxn*,*xn*+1−*x*∗

≤*βnxn*−*x*∗2+*(*1−*βn)*

*(*1−*αn)PC(˜zn*−*λ*1*A*1*z*˜*n)*−*x*∗2+ 2*αn*

*Qxn*−*x*∗,*yn*−*x*∗

+ 2*γnαnPC(˜zn*−*λ*1*A*1*z*˜*n)*−*x*∗*xn*+1−*x*∗+ 2*γnαn*

*x*∗−*Qxn*,*xn*+1−*x*∗

=*βnxn*−*x*∗2+*(*1−*βn)*

*(*1−*αn)G(zn)*−*G*

*x*∗2+ 2*αn*

*Qxn*−*x*∗,*yn*−*x*∗

+ 2*γnαnG(zn)*−*G*

*x*∗*xn*+1−*x*∗+ 2*γnαn*

*x*∗−*Qxn*,*xn*+1−*x*∗

≤*βnxn*−*x*∗2+*(*1−*βn)*

*(*1−*αn)zn*−*x*∗2+ 2*αn*

*Qxn*−*x*∗,*yn*−*x*∗

+ 2*γnαnzn*−*x*∗*xn*+1−*x*∗+ 2*γnαn*

*x*∗−*Qxn*,*xn*+1−*x*∗

.

*xn*+1−*x*∗
2

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

*Qxn*−*x*∗,*yn*−*x*∗

+ 2*γnαnxn*−*x*∗*xn*+1−*x*∗+ 2*γnαn*

*x*∗−*Qxn*,*xn*+1−*x*∗

=[1−*(*1−*βn) αn*]*xn*−*x*∗

2

+ 2*αnγn*

*Qxn*−*x*∗,*yn*−*xn*+1

+ 2*αnδn*

*Qxn*−*x*∗,*yn*−*x*∗

+ 2*αnγnxn*−*x*∗*xn*+1−*x*∗

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

+ 2*αnδn*

*Qxn*−*x*∗,*xn*−*x*∗

+ 2*αnδn*

*Qxn*−*x*∗,*yn*−*xn*

+ 2*αnγnxn*−*x*∗*xn*+1−*x*∗

≤[1−*(*1−*βn) αn*]*xn*−*x*∗

2

+ 2*αnγnQxn*−*x*∗*yn*−*xn*+1

+ 2*αnδnρxn*−*x*∗

2

+ 2*αnδn*

*Qx*∗−*x*∗,*xn*−*x*∗

+ 2*αnδnQxn*−*x*∗*yn*−*xn*+ 2*αnγnxn*−*x*∗*xn*+1−*x*∗

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

+ 2*αnδnρxn*−*x*∗2+ 2*αnδn*

*Qx*∗−*x*∗,*xn*−*x*∗

+ 2*αnδnQxn*−*x*∗*yn*−*xn*+*αnγn*

_{}

*xn*−*x*∗

2

+*xn*+1−*x*∗
2

,

that is,

*xn*+1−*x*∗
2

≤

1−*(*1−2*ρ) δn*−*γn*
1−*αnγn* *αn*

*xn*−*x*∗

2

+[*(*1−2*ρ) δn*−*γn*]*αn*
1−*αnγn*

×

2*γn*

*(*1−2*ρ) δn*−*γn*

*Qxn*−*x*∗*yn*−*xn*+1

+ 2*δn*

*(*1−2*ρ) δn*−*γn*

*Qxn*−*x*∗*yn*−*xn*+

2*δn*

*(*1−2*ρ) δn*−*γn*

*Qx*∗−*x*∗,*xn*−*x*∗

.

Note that lim inf*n→∞(*1−2*ρ) δn*−*γn*

1−*αnγn* *>*0. It follows that

∞

*n*=0

*(*1−2*ρ) δn*−*γn*

1−*αnγn* *αn*=∞. It is obvious that

lim sup

*x*→∞

2*γn*

*(*1−2*ρ) δn*−*γn*

*Qxn*−*x*∗*yn*−*xn*+1+

2*δn*

*(*1−2*ρ) δn*−*γn*

*Qxn*−*x*∗*yn*−*xn*

+ 2*δn*

*(*1−2*ρ) δn*−*γn*

*Qx*∗−*x*∗,*xn*−*x*∗

≤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 be*αi*-inverse strongly monotone for i= 1, 2. Let S
: C®C be a k-strict pseudocontraction such that*Ω*:= Fix(*S*)∩*Γ* =∅.Let Q:C® C

be a r-contraction with*ρ*∈

0,1

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

*yn*=*αnQxn*+ (1−*αn*)*PCPC*(*xn*−*λ*2A2x*n*)−*λ*1A1P*C*(*xn*−*λ*2A2x*n*),

*xn*+1=*βnxn*+*γnPC*

*PC*(*xn*−*λ*2A2x*n*)−*λ*1A1P*C*(*xn*−*λ*2A2x*n*)

+*δnSyn*, ∀*n*≥0,

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

(ii)limn®∞an= 0and

∞

*n*=0*αn*=∞;

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

(iv)lim*n*→∞

*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 given*βi*∈(0,∞), Biis

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

*zn*=*PC*

*PC*(*xn*−*μ*2*B*2*xn*)−*μ*1*B*1*PC*(*xn*−*μ*2*B*2*xn*)

=*xn*, ∀*n*≥0.

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

*yn*=*αnQxn*+ (1−*αn*)*PC*

*PC*(*xn*−*λ*2A2x*n*)−*λ*1A1P*C*(*xn*−*λ*2A2x*n*)

,

*xn*+1=*βnxn*+*γnPC*

*PC*(*xn*−*λ*2A2x*n*)−*λ*1A1P*C*(*xn*−*λ*2A2x*n*)

+*δnSyn*, ∀*n*≥0,

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 are*αi*-inverse strongly
monotone and * _{βi}*-inverse strongly monotone, respectively. Let S : C®C be a k-strict
pseudocontraction such that

*Ω*:= Fix(

*S*)∩

*Γ*∩

*Γ*0=∅.For fixed u ÎC and given x0 Î C arbitrarily, let the sequences{xn}, {yn}and{zn}be generated iteratively by

⎧ ⎪ ⎨ ⎪ ⎩

*zn*=*PC*

*PC*(*xn*−*μ*2B2x*n*)−*μ*1B1P*C*(*xn*−*μ*2B2x*n*)

,

*yn*=*αnu*+ (1−*αn*)*PC*

*PC*(*zn*−*λ*2A2*zn*)−*λ*1*A1PC*(*zn*−*λ*2A2z*n*)

,

*xn*+1=*βnxn*+*γnPCPC*(*zn*−*λ*2A2z*n*)−*λ*1A1P*C*(*zn*−*λ*2A2z*n*)+*δnSyn*, ∀*n*≥0,

where *zn*−*x*∗

2

≤*xn*−*x*∗

2

−*x*˜*n*−*zn*+ (x∗−*y*∗)

2

+2*μ*1*B*1*x*˜*n*−*B*1*y*∗˜*xn*−*zn*+ (x∗−*y*∗).

and_{μi}_{∈}(0, 2*βi*)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;

(ii)limn®∞an= 0and

∞

*n*=0*αn*=∞;

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

(iv)lim*n→∞*

_{γn}

+1

1−*βn*+1−
*γn*

1−*βn* = 0.

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 are*αi*-inverse strongly
monotone and* _{βi}*-inverse strongly monotone, respectively. Let S : C® C be a
nonex-pansive mapping such that

*Ω*:= Fix(

*S*)∩

*Γ*∩

*Γ*0=∅.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*

*PC*(*xn*−*μ*2B2x*n*)−*μ*1B1P*C*(*xn*−*μ*2B2x*n*)

,

*yn*=*αnQxn*+ (1−*αn*)*PC*

*PC*(*zn*−*λ*2A2z*n*)−*λ*1A1P*C*(*zn*−*λ*2A2z*n*)

,

*xn*+1=*βnxn*+*γnPC*

*PC*(*zn*−*λ*2A2z*n*)−*λ*1A1P*C*(*zn*−*λ*2A2z*n*)

+*δnSyn*, ∀*n*≥0,

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

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

(ii)limn®∞an= 0and

∞

*n*=0*αn*=∞;

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

(iv)lim*n→∞*

_{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*)and*y*∗=*PC*(*x*∗−*μ*2*B*2*x*∗).

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 are*αi*-inverse strongly
monotone and* _{βi}*-inverse strongly monotone, respectively. Let S : C® C be a
nonex-pansive mapping such that

*Ω*:= Fix(

*S*)∩

*Γ*∩

*Γ*0=∅. For fixed u ÎC and given x0 Î C arbitrarily, let the sequences{xn}, {yn}and{zn}be generated iteratively by

⎧ ⎪ ⎨ ⎪ ⎩

*zn*=*PC*

*PC*(*xn*−*μ*2B2x*n*)−*μ*1B1P*C*(*xn*−*μ*2B2x*n*)

,

*yn*=*αnu*+ (1−*αn*)*PC*

*PC*(*zn*−*λ*2A2*zn*)−*λ*1*A1PC*(*zn*−*λ*2A2z*n*)

,

*xn*+1=*βnxn*+*γnPC*

*PC*(*zn*−*λ*2A2z*n*)−*λ*1A1P*C*(*zn*−*λ*2A2z*n*)

+*δnSyn*, ∀*n*≥0,

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

(i)bn+gn+δn= 1andgn<δnfor all n≥0;

(ii)limn®∞an= 0and

∞

*n*=0*αn*=∞;

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

(iv)lim*n→∞*

_{γn}

+1

1−*βn*+1−
*γn*

1−*βn* = 0.

a solution of general system(1.6)of variational inequalities, respectively, where y* =PC
(x* -l2A2x*)and*y*∗=*PC*(*x*∗−*μ*2*B*2*x*∗).

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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