Iterative algorithms for a system of generalized variational inequalities in Hilbert spaces

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R E S E A R C H

Open Access

Iterative algorithms for a system of

generalized variational inequalities in Hilbert

spaces

Mingliang Zhang

*

*_{Correspondence:} [email protected] School of Mathematics and Information Science, Henan University, Kaifeng, 475000, China

Abstract

In this paper, a new system of generalized nonlinear variational inequalities involving three operators is introduced. A three-step iterative algorithm is considered for the system of generalized nonlinear variational inequalities. Strong convergence theorems of the three-step iterative algorithm are established.

MSC: 47H05; 47H09; 47J25

Keywords: variational inequality; projection method; relaxed cocoercive mapping; convergence

1 Introduction

Variational inequalities are among the most interesting and intensively studied classes of mathematical problems and have wide applications in the ﬁelds of optimization and con-trol, economics, transportation equilibrium and engineering sciences. There exists a vast amount of literature (see, for instance, [–]) on the approximation solvability of nonlin-ear variational inequalities as well as operator equations.

Iterative algorithms have played a central role in the approximation solvability, espe-cially of nonlinear variational inequalities as well as of nonlinear equations, in several fields such as applied mathematics, mathematical programming, mathematical finance, control theory and optimization, engineering sciences and others. Projection methods have played a significant role in the numerical resolution of variational inequalities based on their convergence analysis. However, the convergence analysis does require some sort of strong monotonicity besides the Lipschitz continuity. There have been some recent de-velopments where convergence analysis for projection methods under somewhat weaker conditions such as cocoercivity [] and partial relaxed monotonicity [] is achieved.

Recently, Changet al.[] introduced a two-step iterative algorithm for a system of non-linear variational inequalities and established strong convergence theorems. Huang and Noor [] introduced the so-called explicit two-step iterative algorithm for a system of nonlinear variational inequalities involving two diﬀerent nonlinear operators and estab-lished strong convergence theorems.

In this paper, we consider, based on the projection method, the approximate solvability of a new system of generalized nonlinear variational inequalities involving three diﬀerent nonlinear operators in the framework of Hilbert spaces. The results presented in this paper

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extend and improve the corresponding results announced in Huang and Noor [], Chang

et al.[], Verma [–] and many others.

LetHbe a real Hilbert space, whose inner product and norm are denoted by·,·and · , respectively. LetCbe a nonempty closed convex subset ofHandPC be the metric

projection fromHontoC.

Given nonlinear operatorsT,f :C→H andg:C→C, we consider the problem of ﬁndingu∈Csuch that

g(u) –f(u) +λTu,v–g(u)≥, ∀v∈C, (.)

whereλ>  is a constant. The variational inequality (.) is called the generalized varia-tional inequality involving three operators.

We see that an elementu∈Cis a solution to the generalized variational inequality (.) if and only ifu∈Cis a ﬁxed point of the mapping

I–g+PC(f –λT),

whereIis the identity mapping. This equivalence plays an important role in this work. Iff =g, then the generalized variational inequality (.) is equivalent to the following. Findu∈Csuch that

Tu,v–g(u)≥, ∀v∈C. (.)

Further, ifg=I, then the problem (.) is reduced to ﬁndingu∈Csuch that

Tu,v–u ≥, ∀v∈C, (.)

which is known as the classical variational inequality originally introduced and studied by Stampacchia [].

LetT:C→Hbe a mapping. Recall the following deﬁnitions. ()Tis said to be monotone if

Tu–Tv,u–v ≥, ∀u,v∈C.

()Tis calledδ-strongly monotone if there exists a constantδ>  such that

Tx–Ty,x–y ≥δx–y, ∀x,y∈C.

This implies that

Tx–Ty ≥δx–y, ∀x,y∈C,

that is,Tisδ-expansive.

()Tis said to beγ-cocoercive if there exists a constantγ >  such that

Tx–Ty,x–y ≥γTx–Ty, ∀x,y∈C.

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()T is said to be relaxedγ-cocoercive if there exists a constantγ >  such that

Tx–Ty,x–y ≥(–γ)Tx–Ty, ∀x,y∈C.

()Tis said to be relaxed (γ,δ)-cocoercive if there exist two constantsγ,δ>  such that

Tx–Ty,x–y ≥(–γ)Tx–Ty+δx–y, ∀x,y∈C.

LetTi:C×C×C→H, fi:C→Handgi:C→C be nonlinear mappings for each i= , , . Consider a system of generalized nonlinear variational inequality (SGNVI) as follows.

Find (x*_,_y*_,_z*₎_∈_C_×_C_×_C_{such that for all}_s_,_t_,_r_{> ,}

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

sT(y*,z*,x*) +g(x*) –f(y*),x–g(x*) ≥, ∀x∈C,

tT(z*,x*,y*) +g(y*) –f(z*),x–g(x*) ≥, ∀x∈C,

rT(x*,y*,z*) +g(z*) –f(x*),x–g(x*) ≥, ∀x∈C.

(.)

One can easily see SGNVI (.) is equivalent to the following projection problem:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

g(x*) =PC(f(y*) –sT(y*,z*,x*)), ∀s> ,

g(y*) =PC(f(z*) –tT(z*,x*,y*)), ∀t> ,

g(z*) =PC(f(x*) –rT(x*,y*,z*)), ∀r> .

(.)

Next, we consider some special classes of SGNVI (.) as follows. (I) Ifg=g=g=I, then SGNVI (.) is reduced to the following.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

sT(y*,z*,x*) +x*–f(y*),x–x* ≥, ∀x∈C,

tT(z*,x*,y*) +y*–f(z*),x–x* ≥, ∀x∈C,

rT(x*,y*,z*) +z*–f(x*),x–x* ≥, ∀x∈C.

(.)

We see that the problem (.) is equivalent to the following projection problem:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x*₌_P

C(f(y*) –sT(y*,z*,x*)), ∀s> ,

y*₌_P

C(f(z*) –tT(z*,x*,y*)), ∀t> ,

z*₌_P

C(f(x*) –rT(x*,y*,z*)), ∀r> .

(.)

(II) Iff=f=f=I, then SGNVI (.) is reduced to the following.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

sT(y*,z*,x*) +g(x*) –y*,x–g(x*) ≥, ∀x∈C,

tT(z*,x*,y*) +g(y*) –z*,x–g(x*) ≥, ∀x∈C,

rT(x*,y*,z*) +g(z*) –x*,x–g(x*) ≥, ∀x∈C.

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We see that the problem (.) is equivalent to the following projection problem: ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

g(x*) =PC[y*–sT(y*,z*,x*)], ∀s> ,

g(y*) =PC[z*–tT(z*,x*,y*)], ∀t> ,

g(z*) =PC[x*–rT(x*,y*,z*)], ∀r> .

(.)

(III) Ifg=g=g=f=f=f=I, then SGNVI (.) is reduced to the following.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

sT(y*,z*,x*) +x*–y*,x–x* ≥, ∀x∈C,

tT(z*,x*,y*) +y*–z*,x–x* ≥, ∀x∈C,

rT(x*,y*,z*) +z*–x*,x–x* ≥, ∀x∈C.

(.)

One can easily get that the problem (.) is equivalent to the following projection prob-lem:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x*₌_P

C(y*–sT(y*,z*,x*)), ∀s> ,

y*₌_P

C(z*–tT(z*,x*,y*)), ∀t> ,

z*=PC(x*–rT(x*,y*,z*)), ∀r> .

(.)

(IV) IfT,TandT are univariate mappings, then SGNVI (.) is reduced to the

fol-lowing.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

sTy*+g(x*) –f(y*),x–g(x*) ≥, ∀x∈C,

tTz*+g(y*) –f(z*),x–g(x*) ≥, ∀x∈C,

rTx*+g(z*) –f(x*),x–g(x*) ≥, ∀x∈C.

(.)

One can easily see that the problem (.) is equivalent to the following projection prob-lem:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

g(x*) =PC(f(y*) –sTy*), ∀s> ,

g(y*) =PC(f(z*) –tTz*), ∀t> ,

g(z*) =PC(f(x*) –rTx*), ∀r> .

(.)

2 Preliminaries

In this section, to study the approximate solvability of the problems (.), (.), (.), (.) and (.), we introduce the following three-step algorithms.

Algorithm . For any (x,y,z)∈C×C×C, compute the sequences{xn},{yn}and{zn}

by the following iterative process: ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

xn+= ( –αn)xn+αn(xn–g(xn) +PC(f(yn) –sT(yn,zn,xn))), n≥,

g(zn+) =PC(f(xn+) –rT(xn+,yn,zn)), n≥,

g(yn+) =PC(f(zn+) –tT(zn+,xn,yn)), n≥,

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Ifg=g=g=I, then Algorithm . is reduced to the following.

Algorithm . For any (x,y,z)∈C×C×C, compute the sequences{xn},{yn}and{zn}

by the following iterative process: ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

xn+= ( –αn)xn+αnPC(f(yn) –sT(yn,zn,xn)), n≥,

zn+=PC(f(xn+) –rT(xn+,yn,zn)), n≥,

yn+=PC(f(zn+) –tT(zn+,xn,yn)), n≥,

wherer,s,t>  are three constants and{αn}is a sequence in [, ].

Iff=f=f=I, the identity mapping, then Algorithm . is reduced to the following.

Algorithm . For any (x,y,z)∈C×C×C, compute the sequences{xn},{yn}and{zn}

by the following iterative process:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

xn+= ( –αn)xn+αn(xn–g(xn) +PC(yn–sT(yn,zn,xn))), n≥,

g(zn+) =PC(xn+–rT(xn+,yn,zn)), n≥,

g(yn+) =PC(zn+–tT(zn+,xn,yn)), n≥,

Iff=f=f=g=g=g=I, the identity mapping, then Algorithm . is reduced to the

following.

Algorithm . For any (x,y,z)∈C×C×C, compute the sequences{xn},{yn}and{zn}

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

xn+= ( –αn)xn+αnPC(yn–sT(yn,zn,xn)), n≥,

zn+=PC(xn+–rT(xn+,yn,zn)), n≥,

yn+=PC(zn+–tT(zn+,xn,yn)), n≥,

(IV) IfT,TandTare univariate mappings, then Algorithm . is reduced to the

fol-lowing.

Algorithm . For any (x,y,z)∈C×C×C, compute the sequences{xn},{yn}and{zn}

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

xn+= ( –αn)xn+αn(xn–g(xn) +PC(f(yn) –sTyn)), n≥,

g(zn+) =PC(f(xn+) –rTxn+), n≥,

g(yn+) =PC(f(zn+) –tTzn+), n≥,

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Lemma .[] Assume that{an}is a sequence of nonnegative real numbers such that

an+≤( –λn)an+bn, ∀n≥n,

where nis a nonnegative integer,{λn}is a sequence in(, )with

∞

n=λn=∞and bn=

◦(λn),thenlimn→∞an= .

Deﬁnition . A mappingT:C×C×C→His said to be relaxed (γ,δ)-cocoercive if there exist constantsγ,δ>  such that for allx,x∈C,

T(x,y,z) –T x,y,z,x–x

≥(–γ)T(x,y,z) –T x,y,z+δx–x, ∀y,y,z,z∈C.

Deﬁnition . A mappingT:C×C×C→His said to beβ-Lipschitz continuous in the

ﬁrst variable if there exists a constantβ>  such that for allx,x∈C,

T(x,y,z) –T x,y,z≤βx–x, ∀y,y,z,z∈C.

3 Main results

Theorem . Let C be a nonempty,closed and convex subset of a real Hilbert space H.Let

Ti:C×C×C→H be a relaxed(γi,δi)-cocoercive andβi-Lipschitz continuous mapping in the ﬁrst variable,fi:C→H be a relaxed(ηi,ρi)-cocoercive andλi-Lipschitz continu-ous mapping and gi:C→C be a relaxed(η¯i,ρ¯i)-cocoercive andλ¯i-Lipschitz continuous mapping for each i= , , .Suppose that(x*_,_y*_,_z*₎_∈_C_×_C_×_{C is a solution to the}

prob-lem(.).Let{xn},{yn}and{zn}be the sequences generated by Algorithm..Assume that the following conditions are satisﬁed:

(a) ∞_n_=αn=∞; (b) ≤θ,θ< ;

(c) (θ+θ)(θ+θ)(θ+θ)≤( –θ)( –θ)( –θ),

where

θ=

 – sδ+ sγβ+sβ, θ=

 – ρ+λ+ ηλ,

θ=

 – ρ¯+λ¯+ η¯λ¯, θ=

 – tδ+ tγβ+tβ,

θ=

 – ρ+λ+ ηλ, θ=

 – ρ¯+λ¯+ η¯λ¯,

θ=

 – rδ+ rγβ+rβ, θ=

 – ρ+λ+ ηλ,

and

θ=

 – ρ¯+λ¯+ η¯λ¯.

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Proof In view of (x*_,_y*_,_z*_{) being a solution to the problem (.), we see that}

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x*_{= ( –}_α

n)x*+αn(x*–g(x*) +PC(f(y*) –sT(y*,z*,x*))), n≥,

g(z*) =PC(f(x*) –rT(x*,y*,z*)), n≥,

g(y*) =PC(f(z*) –tT(z*,x*,y*)), n≥.

It follows from Algorithm (.) that

xn+–x*

=( –αn)xn+αn xn–g(xn) +PC f(yn) –sT(yn,zn,xn)

–x*

≤( –αn)xn–x*+αn xn–g(xn) +PC f(yn) –sT(yn,zn,xn)

– x*–g x*

+PC f y*

–sT y*,z*,x*

≤( –αn)xn–x*+αnxn–x*– g(xn) –g x*+αnf(yn) –f y*

–s T(yn,zn,xn) –T y*,z*,x*

≤( –αn)xn–x*+αnxn–x*– g(xn) –g x*

+αnyn–y*– f(yn) –f y*

+αn yn–y*

–s T(yn,zn,xn) –T y*,z*,x*. (.)

By the assumption thatTis relaxed (γ,r)-cocoercive andβ-Lipschitz continuous in the

ﬁrst variable, we obtain that

yn–y*

–s T(yn,zn,xn) –T y*,z*,x*

=yn–y*– s

T(yn,zn,xn) –T y*,z*,x*

,yn–y*

+sT(yn,zn,xn) –T y*,z*,x*

≤yn–y*– s (–γ)T(yn,zn,xn) –T y*,z*,x*+δyn–y*

+sT(yn,zn,xn) –T y*,z*,x* 

= ( – sδ)yn–y*



+ sγ+sT(yn,zn,xn) –T y*,z*,x* 

≤( – sδ)yn–y*



+ sγ+s

β_yn–y*



=θ_yn–y*



, (.)

whereθ=

 – sδ+ sγβ+sβ. On the other hand, it follows from the assumption

thatfis relaxed (η,ρ)-cocoercive andλ-Lipschitz continuous that

_y_n–y*– f(yn) –f y* 

=yn–y*



– f(yn) –f y*

,yn–y*

+f(yn) –f y* 

≤yn–y*



–  (–η)f(yn) –f y* 

+ρyn–y*



+λ_yn–y*



=  – ρ+λyn–y*



+ ηf(yn) –f y* 

≤θ_yn–y*



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whereθ=

 – ρ+λ+ ηλ. In a similar way, we can obtain that

xn–x*– g(xn) –g x*≤θxn–x*, (.)

whereθ=

 – ρ¯+λ¯+ η¯λ¯. Substituting (.), (.) and (.) into (.), we arrive at

xn+–x*≤

 –αn( –θ)xn–x*+αn(θ+θ)yn–y*. (.)

Next, we estimateyn–y*. From Algorithm ., we see that

g(yn+) –g y*=PC f(zn+) –tT(zn+,xn,yn)

–PC f z*

–tT z*,x*,y*

≤ f(zn+) –tT(zn+,xn,yn)

– f z*

–tT z*,x*,y*

≤ zn+–z*

– f(zn+) –f z*

+ zn+–z*

–t T(zn+,xn,yn) –T z*,x*,y*. (.)

By the assumption thatT is relaxed (γ,r)-cocoercive andβ-Lipschitz continuous in

the ﬁrst variable, we obtain that

zn+–z*

–t T(zn+,xn,yn) –T z*,x*,y* 

=zn+–z* 

– tT(zn+,xn,yn) –T z*,x*,y*

,zn+–z*

+tT(zn+,xn,yn) –T z*,x*,y* 

≤zn+–z* 

– t (–γ)T(zn+,xn,yn) –T z*,x*,y* 

+δzn+–z* 

+tT(zn+,xn,yn) –T z*,x*,y* 

= ( – tδ)zn+–z*+ tγ+tT(zn+,xn,yn) –T z*,x*,y*

≤( – tδ)zn+–z*+ tγ+t

β_zn+–z*

=θ_zn+–z*, (.)

where θ=

 – tδ+ tγβ+tβ. It follows from the assumption thatf is relaxed

(η,ρ)-cocoercive andλ-Lipschitz continuous that

zn+–z*

– f(zn+) –f z* 

=zn+–z* 

– f(zn+) –f z*

,zn–z*

+f(zn+) –f z* 

≤zn+–z* 

–  (–η)f(zn+) –f z* 

+ρzn+–z* 

+λ_zn+–z* 

=  – ρ+λzn+–z* 

+ ηf(zn+) –f z* 

=θ_zn+–z* 

, (.)

whereθ=

 – ρ+λ+ ηλ. Substituting (.) and (.) into (.), we see that

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On the other hand, we have

yn+–y*≤yn+–y*– g(yn+) –g y*+g(yn+) –g y*. (.)

From the proof of (.), we arrive at

yn+–y*

– g(yn+) –g y*≤θyn+–y*, (.)

whereθ=

 – ρ¯+λ¯+ η¯λ¯. Substituting (.) and (.) into (.), we see that

yn+–y*≤θyn+–y*+ (θ+θ)zn+–z*.

It follows from the condition (b) that

yn+–y*≤

θ+θ

 –θ

zn+–z*.

That is,

yn–y*≤ θ+θ

 –θ

zn–z*. (.)

Finally, we estimatezn–z*. It follows from Algorithm . that

g(zn+) –g z*

=PC f(xn+) –rT(xn+,yn,zn)

–PC

f x*

–rT x*,y*,z*

≤ xn+–x*

– f(xn+) –f x*

+ xn+–x*

–r T(xn+,yn,zn) –T x*,y*,z*. (.)

In a similar way, we can show that

xn+–x*

–r T(xn+,yn,zn) –T x*,y*,z*≤θxn+–x* (.)

and

xn+–x*

– f(xn+) –f x*≤θxn+–x*, (.)

whereθ=

 – rδ+ rγβ+rβ andθ=

 – ρ+λ+ ηλ. Substituting (.)

and (.) into (.), we arrive at

g(zn+) –g z*≤(θ+θ)xn+–x*. (.)

Note that

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On the other hand, we have

zn+–z*– g(zn+) –g z*≤θzn+–z*, (.)

θ=

 – ρ¯+λ¯+ η¯λ¯. Substituting (.) and (.) into (.), we arrive at

zn+–z*≤θzn+–z*+ (θ+θ)xn+–x*.

It follows from the condition (b) that

zn+–z*≤

θ+θ

 –θ

xn+–x*.

That is,

zn–z*≤ θ+θ

 –θ

xn–x*. (.)

Combining (.), (.) with (.), we obtain that

xn+–x*≤

 –αn

 –θ– (θ+θ)

θ+θ

 –θ

θ+θ

 –θ

xn–x*.

Since∞_n_=αn=∞and the condition (c), we can conclude the desired conclusion easily

from Lemma .. This completes the proof.

Remark . Theorem . includes the corresponding results in Huang and Noor []

Changet al.[], and Verma [–] as special cases.

From Theorem ., we can get the following results immediately.

Corollary . Let C be a nonempty,closed and convex subset of a real Hilbert space H.Let Ti:C×C×C→H be a relaxed(γi,δi)-cocoercive andβi-Lipschitz continuous mapping in the ﬁrst variable and fi:C→H be a relaxed(ηi,ρi)-cocoercive andλi-Lipschitz contin-uous mapping for each i= , , .Suppose that(x*_,_y*_,_z*₎_∈_C_×_C_×_{C is a solution to the}

problem(.).Let{xn},{yn}and{zn}be the sequences generated by Algorithm..Assume that the following conditions are satisﬁed:

(a) ∞_n_=αn=∞;

(b) (θ+θ)(θ+θ)(θ+θ)≤,

where

θ=

 – sδ+ sγβ+sβ, θ=

 – ρ+λ+ ηλ,

θ=

 – tδ+ tγβ+tβ, θ=

 – ρ+λ+ ηλ,

and

θ=

 – rδ+ rγβ+rβ, θ=

 – ρ+λ+ ηλ.

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Corollary . Let C be a nonempty,closed and convex subset of a real Hilbert space H.Let Ti:C×C×C→H be a relaxed(γi,δi)-cocoercive andβi-Lipschitz continuous mapping in the ﬁrst variable and gi:C→C be a relaxed(η¯i,ρ¯i)-cocoercive andλ¯i-Lipschitz contin-uous mapping for each i= , , .Suppose that(x*_,_y*_,_z*₎_∈_C_×_C_×_{C is a solution to the}

problem(.).Let{xn},{yn}and{zn}be the sequences generated by Algorithm..Assume that the following conditions are satisﬁed:

(a) ∞_n_=αn=∞; (b) ≤θ,θ< ;

(c) θθθ≤( –θ)( –θ)( –θ),

where

θ=

 – sδ+ sγβ+sβ, θ=

 – ρ¯+λ¯+ η¯λ¯,

θ=

 – tδ+ tγβ+tβ, θ=

 – ρ¯+λ¯+ η¯λ¯,

and

θ=

 – rδ+ rγβ+rβ, θ=

 – ρ¯+λ¯+ η¯λ¯.

Then the sequences{xn},{yn}and{zn}converge strongly to x*,y*and z*,respectively.

Corollary . Let C be a nonempty,closed and convex subset of a real Hilbert space H.Let Ti:C×C×C→H be a relaxed(γi,δi)-cocoercive andβi-Lipschitz continuous mapping in the ﬁrst variable for each i= , , .Suppose that(x*_,_y*_,_z*₎_∈_C_×_C_×_{C is a solution}

to the problem(.).Let{xn},{yn}and{zn}be the sequences generated by Algorithm.. Assume that the following conditions are satisﬁed:

(a) ∞_n_=αn=∞; (b) θθθ≤,

where

θ=

 – sδ+ sγβ+sβ, θ=

 – tδ+ tγβ+tβ,

and

θ=

 – rδ+ rγβ+rβ.

Corollary . Let C be a nonempty,closed and convex subset of a real Hilbert space H.Let Ti:C→H be a relaxed(γi,δi)-cocoercive andβi-Lipschitz continuous mapping,fi:C→H be a relaxed(ηi,ρi)-cocoercive andλi-Lipschitz continuous mapping and gi:C→C be a relaxed(η¯i,ρ¯i)-cocoercive andλ¯i-Lipschitz continuous mapping for each i= , , .Suppose that(x*_,_y*_,_z*₎_∈_C_×_C_×_{C is a solution to the problem}_{(.).}_Let_{_x

n},{yn}and{zn}be the sequences generated by Algorithm..Assume that the following conditions are satisﬁed:

(a) ∞_n_=αn=∞; (b) ≤θ,θ< ;

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where

θ=

 – sδ+ sγβ+sβ, θ=

 – ρ+λ+ ηλ,

θ=

 – ρ¯+λ¯+ η¯λ¯, θ=

 – tδ+ tγβ+tβ,

θ=

 – ρ+λ+ ηλ, θ=

 – ρ¯+λ¯+ η¯λ¯,

θ=

 – rδ+ rγβ+rβ, θ=

 – ρ+λ+ ηλ,

and

θ=

 – ρ¯+λ¯+ η¯λ¯.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author is grateful to the editors, Kejifazhan (NO. 112400430123), and Jichuheqianyanjishuyanjiu (NO. 112400430123), Henan.

Received: 17 October 2012 Accepted: 23 November 2012 Published: 28 December 2012 References

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Cite this article as:Zhang:Iterative algorithms for a system of generalized variational inequalities in Hilbert spaces.