Generalized extragradient iterative method for systems of variational inequalities

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.[PUBLICATION ABSTRACT]

Ceng et al. Journal of Inequalities and Applications 2012, 2012:88 http://www.journalofinequalitiesandapplications.com/content/2012/1/88

RESEARCH Open Access

Generalized extragradient iterative method for systems of variational inequalities

Lu Chuan Ceng1, Mu Ming Wong2,3* and Abdul Latif4

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

Let H be a real Hilbert space with inner product , and norm . Let C be a nonempty closed convex subset of H and S : C C be a self-mapping on C. We denote by Fix(S) the set of fixed points of S and by PC the metric projection of H onto

C. Moreover, we also denote by R the set of all real numbers. For a given nonlinear mapping A : C H, consider the following classical variational inequality problem of finding x* C such that

Ax, x x 0, x C. (1:1)

The set of solutions of problem (1.1) is denoted by VI(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

* Correspondence: mailto:[email protected]

Web End [email protected]

2Scientific Computing Key Laboratory of Shanghai Universities, Chung Li 32023, Taiwan Shanghai, ChinaFull list of author information is available at the end of the article

2012 Ceng et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Ceng et al. Journal of Inequalities and Applications 2012, 2012:88 http://www.journalofinequalitiesandapplications.com/content/2012/1/88

considered some approaches to solve fixed point problems, optimization problems, variational 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 set C H is nonempty, closed and convex, a mapping S : C C is nonexpansive and a mapping A : C H is a-inverse strongly monotone, Takahashi and Toyoda [20] introduced the following iterative algorithm:

x0 = x C chosen arbitrarily,xn+1 = nxn + (1 n)SPC(xn nAxn), n 0,

where {an} is a sequence in (0, 1), and {ln} is a sequence in (0, 2a). It was proven in[20] that if Fix(S) VI(A, C) = then the sequence {xn} converges weakly to some z

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 mapping and the set of solutions of a variational inequality. Further, these iterative methods were extended in [27] to develop a general iterative method for finding a element of Fix(S) VI(A, C).

Let A1, A2 : C H be two mappings. In this article, we consider the following problem of finding (x*, y*) C C such that

0, x C,

2A2x + y x, x y 0, x C,

which is called a general system of variational inequalities, where l1 >0 and l2 >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*)

C C such that

0, x C,

2Ax + y x, x y 0, x C,

which was defined by Verma [22] (see also [21]) and it is called a new system of variational 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. proposed 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 2A2x) 1A1PC(x 2A2x) , x C, (1:4)

where y = PC(x 2A2x)

In particular, if the mapping Ai : C H is

i-inverse strongly monotone for i = 1, 2, then the mapping G is nonexpansive provided i (0, 2

i) fori = 1, 2.

Page 2 of 19

1A1y + x y, x x

(1:2)

1Ay + x y, x x

(1:3)

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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 mapping G is denoted by . 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 subset of a real Hilbert space H. Let the mapping Ai : C H be

i-inverse strongly monotone for i = 1, 2. Let S : C C be a k-strict pseudocontraction mapping such that

:= Fix(S) = . Let Q : C C be a r-contraction mapping with

0, 12 For

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

zn = PC(xn 2A2xn),yn = nQxn + (1 n)PC(zn 1A1zn),xx+1 = nxn + nPC(zn 1A1zn) + nSyn, n 0,

(1:5)

where i (0, 2

i) for i = 1, 2, and {an}, {bn}, {gn}, {n} are four sequences in [0, 1] such that(i) bn + gn + n = 1 and (gn + n)k gn <(1 - 2r)n for all n 0;

(ii) lim

x

n = 0 and

n=1 n = (iii)0 < lim inf

n

n lim sup

n

n < 1 and lim inf

n

n > 0

(iv) lim

n

n+1

1 n+1

n1 n

= 0

Then the sequence {xn} generated by (1.5) converges strongly to x* = PQx* and (x*, y*) is a solution of the general system of variational inequalities (1.2), where y* = PC(x*

- l2A2x*).

Let B1, B2 : C H be two mappings. In this article, we also consider another general system of variational inequalities, that is, finding (x*, y*) C C such that

1B1y + x y, x x

0, x C,

2B2x + y x, x y 0, x C,

(1:6)

where 1 >0 and 2 >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 mapping F : C C defined by

F(x) = PC

PC(x 2B2x) 1B1PC(x 2B2x) , x C, (1:7)

where y = PC(x 2B2x) In particular, if the mapping Bi : C H is

i-inverse strongly monotone for i = 1, 2, then the mapping F is nonexpansive provided i (0, 2

i) for i = 1, 2. Throughout this article, the set of fixed points of the mapping F is denoted by 0.

Assume that Ai : C H is

i-inverse strongly monotone and and Bi : C H is

i-inverse strongly monotone for i = 1, 2. Let S : C C be a k-strict pseudocontraction mapping such that := Fix(S) 0 = . Let Q : C C be a r-contraction mapping with

0, 1 2

. Motivated and inspired by the research work going on in

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this area, we propose and analyze the following iterative scheme for computing a common element of the solution set of one general system of variational inequalities(1.2), the solution set 0 of another general system of variational inequalities (1.6), and the fixed point set Fix(S) of the mapping S:

zn = PC

PC(xn 2B2xn) 1B1PC(xn 2B2xn)

,

yn = nQxn + (1 n)PC

PC(zn 2A2zn) 1A1PC(zn 2A2zn)

,

xn+1 = nxn + nPC

PC(zn 2A2zn) 1A1PC(zn 2A2zn) + nSyn, n 0,

(1:8)

where i (0, 2

i) and i (0, 2

i) for i = 1, 2, and {an},{bn}, {gn}, {n} [0, 1] such that bn + gn + n = 1 for all n 0. Furthermore, it is proven that the sequences {xn}, {yn} and {zn} generated by (1.8) converge strongly to the same point x* = PQx* 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 variational inequalities(1.6), respectively, where y* = PC(x* - l2A2x*) and y = PC(x 2B2x)

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) 0 is more general and more complex than the problem of finding an element of Fix(S) in the above Theorem1.1.(b) Algorithm (1.8) for finding an element of Fix(S)0 is also more general and more flexible than algorithm (1.5) for finding an element of Fix(S) in the above Theorem 1.1. Indeed, whenever B1 = B2 = 0, we have

zn = PC

PC(xn 2B2xn) 1B1PC(xn 2B2xn) = 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-contraction Q : C C and involves the Picard successive iteration for the general 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 mappings Ai, 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 C and F: 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 C be a nonempty closed convex subset of H. We write to indicate that the sequence {xn} converges strongly to x and to indicate that the sequence {xn}

converges weakly to x. Moreover, we use w(xn) to denote the weak -limit set of the sequence {xn}, that is,

w(xn) :=

x : xni x for some subsequence{xni} of {xn} .

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Recall that a mapping A : C H is called a-inverse strongly monotone if there exists a constant a >0 such that

Ax Ay, x y

Ax

Ay

2,

x, y C.

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

Sx

Sy

x, y C. (2:1)

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

Sx Sy, x y

x

y

2

x

(I

S)x (I S)y

2,

+ y

2

+ k

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

21 k

2

1 k

2

(I

S)x (I S)y

2,

-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 mappings S : C C such that Sx - Sy x - y for all x, y C.

For every point x H, there exists a unique nearest point in C, denoted by PCx such that

x PCx

x y

, x C.

The mapping PC is called the metric projection of H onto C. We know that PC is a firmly nonexpansive mapping of H onto C; that is, there holds the following relation

PCx PCy, x y

P

Cx PCy

2,

x, y H.

Consequently, PC is nonexpansive and monotone. It is also known that PC is characterized by the following properties: PCx C and

x PCx, PCx y 0, (2:3)

x y

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 PCx 2 +

y PCx

2,

2

x 2 + 2

y, x + y , x, y H. (2:5)

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

Sx

p

+ k x Sx 2 for all x C and p Fix(S). (2:6)

We also say that S is a k-quasi-strict pseudocontraction if condition (2.6) holds.

2

x

p

2

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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 inf

n

n lim sup

n

n < 1. Suppose xn+1 = (1 -

bn)yn + bnxn for all integers n 0 and lim sup

n

(

y

n+1

yn

xn+1 xn ) 0. Then,

lim

n

y

= 0.

n

xn

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 k

x

, 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 xn x 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 condition

an+1 (1 n)an + nn, n 0,

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

n=0 n = , or equivalently,

y

n=0(1 n) := lim n

n

j=0(1 j) = 0;

(ii) lim sup

n

n 0 or

(ii)

n=0 nn is convergent. Then lim

n

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. Let g and be two nonnegative real numbers. Assume (g + )k g. Then

(x y) + (Sx Sy)

, 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

i-inverse strongly

monotone and

( + )

x

y

i-inverse strongly monotone, respectively. Let S : C C be a k-strict pseudocontraction mapping such that := Fix(S) 0 = . Let Q : C C be a r-contraction mapping with

0, 12

For given x0 C arbitrarily, let the sequences

{xn}, {yn} and {zn} be generated iteratively by

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zn = PC

PC(xn 2B2xn) 1B1PC(xn 2B2xn)

,

yn = nQxn + (1 n)PC

PC(zn 2A2zn) 1A1PC(zn 2A2zn)

,

xn+1 = nxn + nPC

PC(zn 2A2zn) 1A1PC(zn 2A2zn) + nSyn, n 0,

(3:2)

where i (0, 2

i) and i (0, 2

i) for i = 1, 2, and {an}, {bn}, {gn}, {n} are four sequences in [0, 1] such that:(i) bn + gn + n = 1 and (gn + n)k gn <(1 - 2r)n for all n 0;

(ii) lim

x

n = 0 and

n=0 n =

(iii) 0 < lim inf

n

n lim sup

n

n < 1 and lim inf

n

n > 0

(iv) lim

= 0

n

n+1

1 n+1

n1 n

Then, the sequences {xn}, {yn}, {zn} generated by (3.2) converge strongly to the same point x* = PQx*, 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 2B2x).

Proof. Let us show that the mappings I - liAi and I - iBi are nonexpansive for i = 1,2. Indeed, since for i = 1, 2, Ai, Bi are

i-inverse strongly monotone and

i-inverse

strongly monotone, respectively, we have for all x, y C

(I

iAi)x (I iAi)y

2

(x

= y) i(Aix Aiy)

2

=

x

y

2

2i

Aix Aiy, x y + 2i

A

ix Aiy

2i

i

A

ix Aiy

2

x

y

2

A

ix Aiy

2

+ 2i

A

ix Aiy

2

=

x

y

2

i(2

i i)

2

x

y

2,

and

(I

iBi)x (I iBi)y

2

=

(x

y) i(Bix Biy)

x

y

2

i i)

B

ix Biy

2.

2

i(2

2

x y

This shows that both I - liAi and I - iBi are nonexpansive for i = 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 nxn

1 n

. Set zn = PC(zn 2A2zn), n 0. It follows that

un+1 un =

xn+2 n+1xn+1

1 n+1

xn+1 nxn1 n= n+1PC(zn+1 1A1zn+1) + n+1Syn+1

1 n+1

nPC(zn 1A1zn) + nSyn 1 n

= n+1

PC(zn+1 1A1zn+1) PC(zn 1A1zn)

+ n+1(Syn+1 Syn)

1 n+1

n+1

+ 1 n+1

n1 n

PC(zn 1A1zn) + n+11 n+1 n1 n

Syn.

(3:3)

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From Lemma 3.1 and (3.2), we get

n+1

PC(zn+1 1A1zn+1) PC(zn 1A1zn)

+ n+1(Syn+1 Syn)

C(zn+1 1A1zn+1) yn+1

+

yn PC(zn 1A1zn)

(n+1 + n+1)

n+1(yn+1 yn) + n+1(Syn+1 Syn)

+ n+1

P

(3:4)

y

n+1

yn

+ n+1n+1

Qx

.

n+1

PC(zn+1 1A1zn+1)

+ n+1n

Qx

n

PC(zn 1A1zn)

Note that

P

C(zn+1 1A1zn+1) PC(zn 1A1zn)

(z

n+1

1A1zn+1) (zn 1A1zn)

zn+1 zn =

P

C(zn+1 2A2zn+1) PC(zn 2A2zn)

(3:5)

(z

n+1

2A2zn+1) (zn 2A2zn)

zn+1 zn ,

and

zn+1 zn =

P

C

PC(xn+1 2B2xn+1) 1B1PC(xn+1 2B2xn+1)

PC

PC(xn 2B2xn) 1B1PC(xn 2B2xn)

P

C(xn+1 2B2xn+1) 1B1PC(xn+1 2B2xn+1)

PC(xn 2B2xn) 1B1PC(xn 2B2xn)

(3:6)

P

C(xn+1 2B2xn+1) PC(xn 2B2xn)

(x

n+1

2B2xn+1) (xn 2B2xn)

xn+1 xn .

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

y

n+1

yn

PC (zn+1 1A1zn+1) PC (zn 1A1z) + n+1

Qx

n+1PC(zn+1 1A1zn+1)

+ n

Qx

n

PC(zn 1A1zn)

xn+1 xn + n Qxn PC (zn 1A1zn) + n+1 Qxn+1 PC (zn+1 1A1zn+1) .

(3:7)

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

un+1 un xn+1 xn +

1 + n+11 n+1

n Qxn PC (zn 1A1zn)

+

1 + n+11 n+1

n+1 Qxn+1 PC (zn+1 1A1zn+1)

n+1

+ 1 n+1

n1 n

P

C(zn 1A1zn

+

Sy

n

.

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 n) un xn = 0. (3:8)

Step 2. lim

n

A

1 zn A1y

= lim

n

A

2zn A2x

= lim

n

B

1 xn B1y

= lim

n

B

2xn B2x

= 0.

Indeed, let x* . Utilizing Lemma 1.1 we have x* = Sx*, x* = PC[PC(x* - l2A2x*) -l1A1PC(x* - l2A2x*)] and

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x = PC

PC

x 2B2x 1B1PC

x 2B2x .

. Then x* = PC(y* - l1A1y*) and

x = PC

Put y* = PC(x* - l2A2x*) and y = PC

x 2B2x

y 1B1y

. Thus it follows that

P

C(zn 1A1zn) PC(y 1A1y)

2

(z

n

1A1zn)

y 1A1y

2

z

2

1

n

y

2 1 1 A 1

zn A1y

P

= C(zn 2A2zn) PC(x 2A2x)

2

1

2

2 1 1 A 1

zn A1y

2

(3:9)

(z

n

2A2zn)

x 2A2x

2

1

2 1 1 A 1

zn A1y

2

z

2

2

n

x

2 2 A2zn A2x

2

1

2 1 1 A 1

zn A1y

2,

and

z

2

P

C

= xn 1B1xn PC

y 1B1y

2

n

x

x

n

1B1xn

y 1B1y

2

x

2

1

2 1 1 B 1

xn B1y

2

n

y

(3:10)

P

= C (xn 2B2xn) PC

x 2B2x

2

1

2 1 1 B 1

xn B1y

2

x

2

2

n

x

2 2

B2xn B2x 2 1 2 1 1 B 1

xn B1y

2

It follows from (3.2), (3.9) and (3.10) that

y

n

x

2

n

Qx

2

+ (1 n)

n

x

y 1A1y

2

n

Qx

2

P

C (zn 1A1zn) PC

n

x

2

+

P

+ C (zn 1A1zn) PC(y 1A1y)

2

n

Qx

z

n

n

x

x

2

2

2 2 A2zn A2x

2

1

2 1 1 A 1

zn A1y

2

(3:11)

n

Qx

2

+

x

n

n

x

x

2

2

2 2 B2xn B2x

2

1

2 1

B

1 xn B1y

2

2

2 2 A2zn A2x

2

1

2 1 1 A1zn A1y

2

.

Utilizing the convexity of 2, we have

x

n+1

x

2

n

= xn x + (1 n)11 n

PC (zn 1A1zn) x) + n

Syn x

2

n

x

2

+ (1 n)

n1 n

n

x

PC (zn 1A1zn) x + n1 n

Syn x

2

2

+ (1 n)

n

= n

x

yn x + n

Syn x

n

2 (3:12)

n

x

+ nn 1 1 n

(PC (zn 1A1zn) Qxn)

n

x

2

+ (1 n)

n

yn x + n

Syn x 1

2

+ Mn

n

x

n

x

2

+ (1 n)

n

x

yn x

2

+ Mn,

Ceng et al. Journal of Inequalities and Applications 2012, 2012:88 http://www.journalofinequalitiesandapplications.com/content/2012/1/88

Page 10 of 19

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

x

n+1

x

2

x

2

2

B

2xn B2x

n

x

2 2

(1 n)

2

1

21 1

(1 n)

B

1 xn B1y

2

2

2 2 (1 n)

A

2zn A2x

2

1

21 1 (1 n)

A

1 zn A1y

2

Qx

n

+

M +

x

2

n.

Therefore,

1

2 1 1 (1 n)

A

1 zn A1y

2

A

2zn A2x

2

+ 2

2 2 (1 n)

(1 n)

B

1 xn B1y

2

(1 n)

B

2xn B2x

2

+ 1

2 1 1

+ 2

2 2

x

2

x

n+1

n

x

x

2

Qx

n

+

M +

x

2

n

x

+

x

n+1

n

x

x

xn xn+1 +

M +

Qx

n

x

2

n.

Since lim inf (2

) (1 ) > 0, lim inf (2

) (1 ) > 0, lim inf

2

(1

) > 0, lim inf

(1

) > 0,

xn - xn+1 0 and an 0, we have

lim

n A

1 zn A1y

= lim

n

A

2zn A2x

= lim

n

B

1 xn B1y

= lim

n

B

2xn B2x

= 0.

Step 3. limn zn - yn = limn xn - zn = limn Syn - yn = 0. Indeed, set vn = PC (zn 1A1zn). Noting that PC is firmly nonexpansive, we have

z

n

y

2

P

= C (zn 2A2zn) PC

x 2A2x

2

(zn 2A2zn)

x 2A2x

, zn y

= 12

zn x 2

A2zn A2x

2

2

+

z

n

y

z

n

x

2

A2zn A2x

zn y

2

12

zn x

2

+

z

n

y

2

(z

n

zn) 2

A2zn A2x

x y

2

= 12

zn x

2

+

z

n

y

2

z

n

z

x y

2

+22 zn zn

x y

, A2zn A2x 22

A

2zn A2x

2

,

and

v

n

x

2

P

= C (zn 1A1zn) PC

y 1A1y

2

zn 1A1zn

y 1A1y

, vn x

= 12

zn 1A1zn

y 1A1y

2

+

v

n

x

2

z

n

1A1zn

y 1A1y

vn x

2

12

zn y

2

+

v

n

x

2

z

n

vn +

x y

2

+21

A1zn A1y, zn vn +

x y

21

A

1 zn A1y

2

12

zn x

2

+

v

n

x

2

z

n

vn +

x y

2

+21

A1zn A1y, zn vn +

x y

due to (3.9). Thus, we have

z

n

y

2

zn z

2 z

n

zn

x y

2+2

2 zn zn

x y , A2zn A2x

22

A

2zn A2x

2,

(3:13)

Ceng et al. Journal of Inequalities and Applications 2012, 2012:88 http://www.journalofinequalitiesandapplications.com/content/2012/1/88

Page 11 of 19

and

v

n

x

2

zn x

2 z

A

n

vn +

x y

2+2

1 1 zn A1y

z

x y .

n

vn +

It follows that

y

n

x

n

Qx

2

+ (1 n)

v

n

n

x

x

2

n

Qx

2

+

v

n

n

x

x

2

(3:14)

n

Qx

2

+

n

x

x

2

z

n

+ 21

A

1 zn A1y

z

z

n

vn +

x y

2

n

vn +

x y

.

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

x

n+1

x

2

n

x

2

+ (1 n)

y

n

n

x

x

2

+ Mn

n

x

2

+ (1 n)

2

n

x

n

Qx

n

x

+ (1 n)

P

C (zn 1A1zn) PC

y 1A1y

2

+ Mn

n

xn x

2

+ (1 n)

(

zn 1A1zn)

n

Qxn x

y 1A1y

2 2

+ Mn

2

+ (1 n)

2

+ (1 n)

n

x

2

+ (1 n)

z

n

n

x

n

Qx

n

x

y

2

+ Mn

n

x

2

+ (1 n)

2

+ (1 n)

z

n

n

x

n

Qx

n

x

y

2

+ Mn

n

x

2

+ (1 n) n

Qx

n

n

x

x

2

+ (1 n)

zn x

2

z

n

zn

x y

2

+22

zn zn

x y

, A2zn A2x

+ Mn

n

x

2

+ (1 n) n

Qx

n

n

x

x

2

+ (1 n)

xn x

2

z

n

zn

x y

2

+22

z

n

zn

x y

A

2zn A2x

+ Mn

2

(1 n)

z

n

=

x

n

x

zn (x y)

2

+ 2(1 n)2

z

A

2zn A2x

n

zn (x y)

+ M + (1 n)

Qx

n

x

2

n.

It follows that

(1 n)

z

n

zn (x y)

2

x

n

x

x

xn+1 xn +

n+1

x

M +

Qx

n

x

+ 2(1 n)2

z

A

2zn A2x

.

2

n

n

zn (x y)

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

lim

n

z

n

zn (x y)

= 0. (3:15)

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

x

n+1

x

2

n

x

2

+ (1 n)

2

+

z

n

n

x

n

Qx

n

x

x

2

z

2

+ 21

A

n

vn + (x y)

1 zn A1y

z

n

vn + (x y)

+ Mn

n

x

2

+ (1 n)

2

+

x

n

n

x

n

Qx

n

x

x

2

z

2

+ 21

A

n

vn + (x y)

1 zn A1y

z

n

vn + (x y)

x

n

+ Mn

x

2

(1 n)

z

2

n

vn + (x y)

+ 21(1 n)

A

1 zn A1y

z

+ (M +

Qx

n

n

vn + (x y)

x

2)

n.

So, we obtain

(1 n)

z

n

vn + (x y)

2

x

n

x

2

x

n+1

x

2

+ 21(1 n)

1 zn A1y

z

+

n

vn + (x y)

n

x

2

n.

A

M +

Qx

Ceng et al. Journal of Inequalities and Applications 2012, 2012:88 http://www.journalofinequalitiesandapplications.com/content/2012/1/88

Page 12 of 19

Hence,

lim

n

z

n

vn + (x y)

= 0.

This together with yn - n anQxn - n 0, implies that

lim

n z

n

yn + (x y)

= 0. (3:16)

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

lim

n z

n

yn

= 0.

On the other hand, by firm nonexpansiveness of PC, we have

x

n

y

2

P

= C(xn 2B2xn) PC(x 2B2x)

2

(xn 2B2xn) (x 2B2x), xn y

= 1

2

xn x 2(B2xn B2x)

2

+

x

n

y

2

(x

n

x) 2(B2xn B2x) (xn y)

2

1

2

xn x

2

+

x

n

y

2

(x

n

xn) 2(B2xn B2x) (x y)

2

= 1

2

xn x

2

+

x

n

y

2

x

n

xn (x y)

B

2xn B2x

2

+22

xn xn (x y), B2xn B2x 22

2

,

and

z

2

P

= C(xn 1B1xn) PC(y 1B1y)

2

n

x

xn 1B1xn (y 1B1y), zn x

= 1

2

xn 1B1xn (y 1B1y)

2

2

+

z

n

x

x

n

1B1xn (y 1B1y) (zn x)

2

1

2

x

n

y

2

2

x

+

z

n

x

n

zn + (x y)

21

B

2

+ 21

B1xn B1y, xn zn + (x y)

1 xn B1y

2

1

2

xn x

2

+

z

n

x

2

x

n

zn + (x y)

2

+ 21

B1xn B1y, xn zn + (x y)

.

Thus, we have

x

n

y

2

x

2 x

n

xn (x y)

2+2

B

2xn B2x

2,

n

x

2 xn xn (x y), B2xn B2x

22

(3:17)

and

z

2

x

n

x

2 x

2+21 B

1 xn B1y

x

. (3:18)

n

x

n

zn + (x y)

n

zn + (x y)

Consequently, from (3.10), (3.11), (3.12) and (3.17) we have

x

n+1

x

2

n

x

y

n

n

x

x

2

+ Mn

2

+ (1 n)

2

+ (1 n)

n

x

n

x

n

Qx

n

x

2

+ (1 n) (n

Qx

n

2

+

z

n

x

2

+ Mn

n

x

n

x

x

2

2) +

Mn

+

x

n

y

2

n

x

2

+ (1 n)

2

+

x

n

n

x

n

Qx

n

x

x

2

x

n

xn

x y

+ 22

xn xn

x y

, B2xn B2x

] + Mn

x

n

x

2

(1 n)

x

n

xn

x y

2

+ 2 (1 n) n

x

x y

B

2xn B2x

+

n

xn

n

x

2

n.

M +

Qx

Ceng et al. Journal of Inequalities and Applications 2012, 2012:88 http://www.journalofinequalitiesandapplications.com/content/2012/1/88

Page 13 of 19

It follows that

(1 n)

x

n

xn

x y

2

x

n

x

xn+1 xn +

+

x

n+1

x

2

n

M +

Qx

n

x

+ 2 (1 n) 2

x

.

n

xn

x y

B

2xn B2x

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

lim

n x

n

xn

x y

= 0. (3:19)

Furthermore, utilizing (3.11), (3.12) and (3.18) we have

x

n+1

x

2

n

x

2

+ (1 n)

y

n

n

x

x

2

+ Mn

n

x

2

+ (1 n)

2

+

n

x

n

Qx

n

x

x

2

+ Mn

n

x

2

+ (1 n)

2

+

x

n

z

n

n

x

n

Qx

n

x

x

2

x

n

zn +

x y

2

+ 21

B

1 xn B1y

x

n

zn +

x y

] +

Mn

x

n

x

2

(1 n)

x

n

zn +

x y

2

+ 21 (1 n)

B

1 xn B1y

x

Qx

n

n

zn +

x y

+

M +

x

2

n.

So, we get

(1 n)

x

n

zn + (x y)

2

x

n

x

2

x

2

+ 21 (1 n)

B

n+1

x

1 xn B1y

x

n

zn +

x y

Qx

n

+

M +

x

2

n.

Hence,

lim

n

= 0. (3:20)

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

lim

n

xn zn = 0.

Since

x

n

zn +

x y

xn+1 xn + n PC (zn 1A1zn) xn

xn+1 xn + n

y

n

n

Syn xn

xn

+ nn Qxn PC (zn 1Anzn) ,

so we obtain that

lim

n

Sy

n

xn

= 0 and lim

n

Sy

= 0.

n

yn

Step 4. lim supn Qx* - x*, xn - x* 0 where x* = PQx*.

Indeed, as H is reflexive and {xn} is bounded, we may assume, without loss of generality, 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, implies that yni v. Again from Step 3 it is known that yn - Syn 0 as n . Thus it is clear from

Ceng et al. Journal of Inequalities and Applications 2012, 2012:88 http://www.journalofinequalitiesandapplications.com/content/2012/1/88

Page 14 of 19

Lemma 2.3 (ii) that Fix(S). Next, we prove that 0. As a matter of fact, observe that

y

n

G

yn

n

Qx

n

G

yn

+ (1 n)

P

C [PC (zn 2A2zn) 1A1PC (zn 2A2zn) G

yn

= n

Qx

n

G

yn

+ (1 n)

G y

n

n

Qx

n

G

yn

+ (1 n)

z

n

yn

0,

and

zn F (zn)

n Qxn F (zn) + (1 n) PC [PC (xn 2B2xn) 1B1PC (xn 2B2xn) F (zn) = n Qxn F (zn) + (1 n) F (xn) F (zn)

n Qxn F (zn) + (1 n) xn zn

0,

where G and F are given in (1.4) and (1.7), respectively. According to Lemma 2.3 (ii) we obtain 0. Therefore, . Hence, it follows from (2.3) that

lim sup

n

Qx x, xn x = lim i

Qx x, xni x

= Qx x, x

0.

Step 5. limn xn = x*.

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

x

n+1

x

2

= n

xn x + n

yn x + n

Syn x

yn x + n

Syn x + nn (PC (zn 1A1zn) Qxn) =

n

xn x + n

PC (zn |1A1zn) Qxn, xn+1 x

n

2

x

n

2

+ 2nn

(3:21)

x

2

+ (1 n)

1

n

1 yn x + n

Syn x

2

+ 2nn

PC (zn 1A1zn) x, xn+1 x + 2nn

x Qxn, xn+1 x .

By Lemma 3.1 and (3.21), we have

x

n+1

x

2

n

x

2

+ (1 n)

n

x

y

n

x

2

C (zn 1A1zn) x

x

+ 2nn

P

n+1

x

+ 2nn

x Qxn, xn+1 x

n

x

n

x

2

+ (1 n)

(1 n)

P

C (zn 1A1zn) x

2

+ 2n

Qxn x, yn x

C (zn 1A1zn) x

x

+ 2nn

+ 2nn

P

n+1

x

= n

x

2

+ (1 n)

x Qxn, xn+1 x

n

x

(1 n)

G

(zn) G

x

2

+ 2n

Qxn x, yn x

x

n+1

+ 2nn

G

(zn) G

x

x

+ 2nn

x Qxn, xn+1 x

n

xn x

2

+ (1 n)

(1 n)

zn x

2

+ 2n

Qxn x, yn x

x Qxn, xn+1 x .

From (3.10), we note that zn - x* xn - x*. Hence, according to 1 - bn = gn + n we have

xn+1 x + 2nn

+ 2nn

z

n

x

Ceng et al. Journal of Inequalities and Applications 2012, 2012:88 http://www.journalofinequalitiesandapplications.com/content/2012/1/88

Page 15 of 19

x

n+1

x

2

n

x

2

+ (1 n) (1 n)

n

x

x

n+1

x

n

x

2

+ 2n (1 n)

Qxn x, yn x

+ 2 nn

x

n

x

x

+ 2nn

x Qxn, xn+1 x

= [1 (1 n) n]

x

n

x

2

+ 2nn

Qxn x, yn xn+1

+ 2 nn

Qxn x, yn x + 2nn

x

x

[1 (1 n) n]

x

2

+ 2nn

Qx

n

x

n+1

n

x

x

n+1

x

y

Qxn x, yn xn + 2nn

x

n

x

n

xn+1

+ 2nn

Qxn x, xn x + 2nn

n

x

x

[1 (1 n) n]

x

2

+ 2nn

Qx

n

n

x

x

y

n

xn+1

2

+ 2nn

+ 2nn

x

Qx

n

+ 2nn

n

x

Qx x, xn x

+ 2 nn

x

y

x

n

n

xn

x

x

n+1

x

[1 (1 n) n]

x

2

+ 2nn

Qx

n

n

x

x

y

n

xn+1

2

+ 2nn

n

x

Qx x, xn x

+ 2nn

x

y

n

2

+ 2nn

Qx

n

x

xn

+ xn+1 x

+ nn

xn x

2

,

that is,

x

n+1

x

2

1 (1 2) n n1 nnn

x

n

x

2

+ [(1 2) n n] n

1 nn

2n(1 2) n n

Qx

n

x

y

n

xn+1

+ 2n

(1 2) n n

Qx

y

n

n

x

xn

+ 2n

(1 2) n n

Qx x, xn x .

Note that lim infn

(1 2) n n 1 nn

> 0. It follows that

n=0(1 2) n n1 nnn = . It is obvious that

lim sup

x

2n(1 2) n n

Qx

n

x

y

+ 2n

(1 2) n n

Qx

n

n

xn+1

x

y

n

xn

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

i-inverse strongly monotone for i = 1, 2. Let S : C C be a k-strict pseudocontraction such that := Fix(S) = . Let Q : C C be a r-contraction with

0, 1 2

+ 2n

(1 2) n n

. For given x0 C arbitrarily, let the sequences {xn},

{yn} be generated iteratively by

yn = nQxn + (1 n)PC

PC(xn 2A2xn) 1A1PC(xn 2A2xn)

,

xn+1 = nxn + nPC

PC(xn 2A2xn) 1A1PC(xn 2A2xn)

+ nSyn, n 0,

where i (0, 2

i) for i = 1, 2, and {an}, {bn}, {gn}, {n} are four sequences in [0, 1] such that:

(i) bn + gn + n = 1 and (gn + n)k gn <(1 - 2r)n for all n 0;

Ceng et al. Journal of Inequalities and Applications 2012, 2012:88 http://www.journalofinequalitiesandapplications.com/content/2012/1/88

Page 16 of 19

n=0 n = ;

(iii) 0 <lim infn bn lim supn bn <1 and lim infn n >0;

(iv) limn

n+1

1 n+1

(ii) limn an = 0 and

n1 n

= 0.

Then the sequences {xn}, {yn} converge strongly to the same point x* = PQx*, 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 if Bi = 0 for i = 1, 2, then for any given

i (0, ), Bi is

i-inverse strongly monotone. In Theorem 3.1, putting Bi = 0 and taking i (0, 2

i)

for i = 1, 2 we have := Fix(S) n 0 = Fix(S) and

zn = PC

PC(xn 2B2xn) 1B1PC(xn 2B2xn) = xn, n 0.

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

yn = nQxn + (1 n)PC PC(xn 2A2xn) 1A1PC(xn 2A2xn)

,

xn+1 = nxn + nPC

PC(xn 2A2xn) 1A1PC(xn 2A2xn)

+ nSyn, n 0,

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

i -inverse strongly

monotone and

i -inverse strongly monotone, respectively. Let S : C C be a k-strict pseudocontraction such that := Fix(S) 0 = . For fixed u C and given x0

C arbitrarily, let the sequences {xn}, {yn} and {zn} be generated iteratively by

zn = PC

PC(xn 2B2xn) 1B1PC(xn 2B2xn)

,

yn = nu + (1 n)PC

PC(zn 2A2zn) 1A1PC(zn 2A2zn)

,

xn+1 = nxn + nPC

PC(zn 2A2zn) 1A1PC(zn 2A2zn) + nSyn, n 0,

where

z

n

x

2

x

2 x

n

2+21 B

n

x

zn + (x y)

1 xn B1y

x

.

n

zn + (x y)

and i (0, 2

i) and {an}, {bn}, {gn}, {n} are four sequences in [0, 1] such that:

(i) bn + gn + n = 1 and (gn + n)k gn< n for all n 0;

(ii) limn an = 0 and

n=0 n = ;

(iii) 0 <lim infn bn lim supn bn <1 and lim infn n >0;

(iv) limn

n+1

1 n+1

n1 n

= 0.

Then the sequences {xn}, {yn}, {zn} converge strongly to the same point x* = Pu, 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 2B2x).

Ceng et al. Journal of Inequalities and Applications 2012, 2012:88 http://www.journalofinequalitiesandapplications.com/content/2012/1/88

Page 17 of 19

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

i -inverse strongly

monotone and

i -inverse strongly monotone, respectively. Let S : C C be a nonexpansive mapping such that := Fix(S) 0 = . Let Q : C C be a r-contraction with

0, 1 2

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

generated iteratively by

zn = PC

PC(xn 2B2xn) 1B1PC(xn 2B2xn)

,

yn = nQxn + (1 n)PC

PC(zn 2A2zn) 1A1PC(zn 2A2zn)

,

xn+1 = nxn + nPC

PC(zn 2A2zn) 1A1PC(zn 2A2zn) + nSyn, n 0,

where i (0, 2

i) and i (0, 2

i) for i = 1, 2, and {an}, {bn}, {gn}, {n} are four sequences in [0, 1] such that:

(i) bn + gn + n = 1 and gn <(1 - 2r)n for all n 0;

(ii) limn an = 0 and

n=0 n = ;

(iii) 0 <lim infn bn lim supn bn <1 and lim infn gn >0;

(iv) limn

n+1

1 n+1

n1 n

= 0.

Then the sequences {xn}, {yn}, {zn} converge strongly to the same point x* = PQx*, 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 2B2x).

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

i -inverse strongly

monotone and

i -inverse strongly monotone, respectively. Let S : C C be a nonexpansive mapping such that := Fix(S) 0 = . For fixed u C and given x0

C arbitrarily, let the sequences {xn}, {yn} and {zn} be generated iteratively by

zn = PC

PC(xn 2B2xn) 1B1PC(xn 2B2xn)

,

yn = nu + (1 n)PC

PC(zn 2A2zn) 1A1PC(zn 2A2zn)

,

xn+1 = nxn + nPC

PC(zn 2A2zn) 1A1PC(zn 2A2zn) + nSyn, n 0,

where i (0, 2

i) and i (0, 2

i) for i = 1, 2, and {an}, {bn}, {gn}, {n} are four sequences in [0, 1] such that:

(i) bn + gn + n = 1 and gn < n for all n 0;

(ii) limn an = 0 and

n=0 n = ;

(iii) 0 <lim infn bn lim supn bn <1 and lim infn gn >0;

(iv) limn

n+1

1 n+1

n1 n

= 0.

Then the sequences {xn}, {yn}, {zn} converge strongly to the same point x* = Pu, and (x*, y*) and (x, y) are a solution of general system (1.2) of variational inequalities and

Ceng et al. Journal of Inequalities and Applications 2012, 2012:88 http://www.journalofinequalitiesandapplications.com/content/2012/1/88

a solution of general system (1.6) of variational inequalities, respectively, where y* = PC (x* - l2A2x*) and y = PC(x 2B2x).

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

1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China 2Scientific Computing Key Laboratory of Shanghai Universities, Chung Li 32023, Taiwan Shanghai, China 3Department of Applied Mathematics, Chung Yuan Christian University, Chung Li 32023, Taiwan 4Department 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

References1. Lions, JL, Stampacchia, G: Variational inequalities. Commun Pure Appl Math. 20, 493512 (1967). doi:10.1002/ cpa.3160200302

2. Bnouhachem, A, Noor, MA, Hao, Z: Some new extragradient iterative methods for variational inequalities. Nonlinear Anal. 70, 13211329 (2009). doi:10.1016/j.na.2008.02.014

3. Ceng, LC, Al-Homidan, S, Ansari, QH, Yao, JC: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J Comput Appl Math. 223, 967974 (2009). doi:10.1016/j.cam.2008.03.032

4. Ceng, LC, Ansari, QH, Yao, JC: Relaxed extragradient iterative methods for variational inequalities. Appl Math Comput (2011). doi:10.1016/j.amc.2011.01.061

5. Ceng, LC, Guu, SM, Yao, JC: A general iterative method with strongly positive operators for general variational inequalities. Comput Math Appl. 59, 14411452 (2010). doi:10.1016/j.camwa.2009.11.007

6. Ceng, LC, Schaible, S, Yao, JC: Implicit iterative scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings. J Optim Theory Appl. 139, 403418 (2008). doi:10.1007/ s10957-008-9361-y

7. Ceng, LC, Wang, CY, Yao, JC: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math Method Oper Res. 67, 375390 (2008). doi:10.1007/s00186-007-0207-4

8. Ceng, LC, Xu, HK, Yao, JC: The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces. Nonlinear Anal. 69, 14021412 (2008). doi:10.1016/j.na.2007.06.040

9. Ceng, LC, Yao, JC: A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem. Nonlinear Anal. 72, 19221937 (2010). doi:10.1016/j.na.2009.09.033

10. Ceng, LC, Yao, JC: Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Appl Math Comput. 198, 729741 (2008). doi:10.1016/j.amc.2007.09.011

11. Ceng, LC, Yao, JC: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J Comput Appl Math. 214, 186201 (2008). doi:10.1016/j.cam.2007.02.022

12. Ceng, LC, Yao, JC: An extragradient-like approximation method for variational inequality problems and fixed point problems. Appl Math Comput. 190, 205215 (2007). doi:10.1016/j.amc.2007.01.021

13. Censor, Y, Zenios, SA: Parallel Optimization Theory, Algorithms, and Applications, Numerical Mathematics and Scientific Computation. Oxford University Press, New York (1997)

14. Cho, SY, Kang, SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math Sci Eng Ser (2011). to appear

15. Iiduka, H, Takahashi, W: Strong convergence theorem by a hybrid method for nonlinear mappings of nonexpansive and monotone type and applications. Adv Nonlinear Var Inequal. 9, 110 (2006)

16. Iiduka, H, Takahashi, W: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal. 61, 341350 (2005). doi:10.1016/j.na.2003.07.023

17. Marino, G, Xu, HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J Math Anal Appl. 329, 336346 (2007). doi:10.1016/j.jmaa.2006.06.055

18. Nadezhkina, N, Takahashi, W: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J Optim. 16(4):12301241 (2006). doi:10.1137/050624315

19. Nadezhkina, N, Takahashi, W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J Optim Theory Appl. 128, 191201 (2006). doi:10.1007/s10957-005-7564-z

20. Takahashi, W, Toyoda, M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J Optim Theory Appl. 118, 417428 (2003). doi:10.1023/A:1025407607560

Page 18 of 19

Ceng et al. Journal of Inequalities and Applications 2012, 2012:88 http://www.journalofinequalitiesandapplications.com/content/2012/1/88

21. Verma, RU: Iterative algorithms and a new system of nonlinear quasivariational inequalities. Adv Nonlinear Var Inequal. 4(1):117124 (2001)

22. Verma, RU: On a new system of nonlinear variational inequalities and associated iterative algorithms. Math Sci Res Hot-line. 3(8):6568 (1999)

23. Xu, HK: Viscosity approximation methods for nonexpansive mappings. J Math Anal Appl. 298, 279291 (2004). doi:10.1016/j.jmaa.2004.04.059

24. Xu, HK, Kim, TH: Convergence of hybrid steepest-descent methods for variational inequalities. J Optim Theory Appl. 119, 185201 (2003)

25. Yamada, I: The hybrid steepest-descent method for the variational inequality problem over the intersection of fixed-point sets of nonexpansive mappings. In: Butnariu, D, Censor, Y, Reich, S (eds.) (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. pp. 473504. Kluwer Academic Publishers, Dordrecht, Netherlands (2001)

26. Yao, Y, Liou, YC, Kang, SM: Approach to common elements of variational inequality problems and fixed point problems via a related extragradient method. Comput Math Appl. 59, 34723480 (2010). doi:10.1016/j.camwa.2010.03.036

27. Yao, Y, Yao, JC: On modified iterative method for nonexpansive mappings and monotone mappings. Appl Math Comput. 186, 15511558 (2007). doi:10.1016/j.amc.2006.08.062

28. Zeng, LC, Schaible, S, Yao, JC: Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities. J Optim Theory Appl. 124, 725738 (2005). doi:10.1007/s10957-004-1182-z

29. Zeng, LC, Wong, NC, Yao, JC: Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities. J Optim Theory Appl. 132, 5169 (2007). doi:10.1007/s10957-006-9068-x

30. Zeng, LC, Wong, NC, Yao, JC: Strong convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type. Taiwanese J Math. 10, 837849 (2006)

31. Zeng, LC, Yao, JC: Mixed projection methods for systems of variational inequalities. J Global Optim. 41, 465478 (2008). doi:10.1007/s10898-007-9258-6

32. Zeng, LC, Yao, JC: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese J Math. 10, 12931303 (2006)

33. Korpelevich, GM: An extragradient method for finding saddle points and for other problems. Ekon Mate Metody. 12, 747756 (1976)

34. Ceng, LC, Guu, SM, Yao, JC: Finding common solutions of a variational inequality, a general system of variational inequalities, and a fixed-point problem via a hybrid extragradient method. Fixed Point Theory Appl 2011, 22 (2011). Article ID 626159. doi:10.1155/2011/626159. doi:10.1186/1687-1812-2011-22

35. Browder, FE, Petryshyn, WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J Math Anal Appl. 20, 197228 (1967). doi:10.1016/0022-247X(67)90085-6

36. Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory. In Cambridge Stud Adv Math, vol. 28,Cambridge University Press, Cambridge (1990)

37. Suzuki, T: Strong convergence of Krasnoselskii and Manns type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J Math Anal Appl. 305, 227239 (2005). doi:10.1016/j.jmaa.2004.11.017

doi:10.1186/1029-242X-2012-88Cite this article as: Ceng et al.: Generalized extragradient iterative method for systems of variational inequalities. Journal of Inequalities and Applications 2012 2012:88.

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