Finding Common Solutions of a Variational Inequality, a General System of Variational Inequalities, and a Fixed-Point Problem via a Hybrid Extragradient Method

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Volume 2011, Article ID 626159,22pages doi:10.1155/2011/626159

Research Article

Finding Common Solutions of a Variational

Inequality, a General System of Variational

Inequalities, and a Fixed-Point

Problem via a Hybrid Extragradient Method

Lu-Chuan Ceng,

1

Sy-Ming Guu,

2

and Jen-Chih Yao

3

1_{Department of Mathematics, Shanghai Normal University, Scientific Computing Key Laboratory of}

Shanghai Universities, Shanghai 200234, China

2_{Department of Business Administration, College of Management, Yuan-Ze University, Taoyuan Hsien,}

Chung-Li City 330, Taiwan

3_{Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan}

Correspondence should be addressed to Sy-Ming Guu,[email protected]

Received 25 September 2010; Accepted 20 December 2010

Academic Editor: Jong Kim

Copyrightq2011 Lu-Chuan Ceng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We propose a hybrid extragradient method for finding a common element of the solution set of a variational inequality problem, the solution set of a general system of variational inequalities, and the fixed-point set of a strictly pseudocontractive mapping in a real Hilbert space. Our hybrid method is based on the well-known extragradient method, viscosity approximation method, and Mann-type iteration method. By constrasting with other methods, our hybrid approach drops the requirement of boundedness for the domain in which various mappings are defined. Furthermore, under mild conditions imposed on the parameters we show that our algorithm generates iterates which converge strongly to a common element of these three problems.

1. Introduction

LetHbe a real Hilbert space with inner product·,·and norm · . LetCbe a nonempty closed convex subset ofHandS:C → Cbe a self-mapping onC. We denote by FixSthe set of fixed points ofSand byPCthe metric projection ofHontoC. Moreover, we also denote byRthe set of all real numbers. For a given nonlinear operatorA:C → H, we consider the following variational inequality problem of findingx∗∈Csuch that

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The solution set of the variational inequality 1.1 is denoted by VIA, C. Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, equilibrium problems. See, for example,1–21and the references therein.

For finding an element of FixS ∩ VIA, C when C is closed and convex, S is nonexpansive andAisα-inverse strongly monotone, Takahashi and Toyoda22introduced the following Mann-type iterative algorithm:

xn1αnxn 1−αnSPCxn−λnAxn, ∀n≥0, 1.2

wherePC is the metric projection ofHontoC,x0 x∈C,{αn}is a sequence in0,1, and {λn} is a sequence in0,2α. They showed that, if FixS∩VIA, C/∅, then the sequence

{xn}converges weakly to somez ∈FixS∩VIA, C. Nadezhkina and Takahashi23and

Zeng and Yao24proposed extragradient methods motivated by Korpeleviˇc25for finding a common element of the fixed point set of a nonexpansive mapping and the solution set of a variational inequality problem. Further, these iterative methods were extended in26to develop a new iterative method for finding elements in FixS∩VIA, C.

LetB1, B2 : C → H be two mappings. Now we consider the following problem of

findingx∗, y∗∈C×Csuch that

μ1B1y∗x∗−y∗, x−x∗

≥0, ∀x∈C,

μ2B2x∗y∗−x∗, x−y∗

≥0, ∀x∈C, 1.3

which is called a general system of variational inequalities whereμ1 >0 andμ2 >0 are two

constants. The set of solutions of problem1.3is denoted by GSVIB1, B2, C. In particular, if

B1B2A, then problem1.3reduces to the problem of findingx∗, y∗∈C×Csuch that

μ1Ay∗x∗−y∗, x−x∗

≥0, ∀x∈C,

μ2Ax∗y∗−x∗, x−y∗

≥0, ∀x∈C, 1.4

which was defined by Verma27 see also28and is called the new system of variational inequalities. Further, if x∗ y∗ additionally, then problem 1.4 reduces to the classical variational inequality problem1.1.

Ceng et al.29studied the problem1.3by transforming it into a fixed-point problem. Precisely and for easy reference, we state their results in the following lemma and theorem.

Lemma CWYsee29. For givenx, y∈C,x, yis a solution of problem1.3if and only ifxis

a fixed point of the mappingG:C → Cdefined by

Gx PCPCx−μ2B2x

−μ1B1PCx−μ2B2x

, ∀x∈C, 1.5

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Throughout this paper, the fixed-point set of the mappingGis denoted byΓ. Utilizing Lemma CWY, they introduced and studied a relaxed extragradient method for solving problem1.3.

Theorem CWYsee29, Theorem 3.1. LetCbe a nonempty closed convex subset of a real Hilbert

spaceH. Let the mappingBi :C → Hbeβi-inverse strongly monotone fori1,2. LetS:C → C

be a nonexpansive mapping withFixS∩Γ/∅. Supposex1u∈Cand{xn}is generated by

ynPCxn−μ2B2xn,

xn1αnuβnxnγnSPCyn−μ1B1yn,

1.6

whereμi∈0,2βifori1,2, and{αn},{βn},{γn}are three sequences in0,1such that

iαnβnγn1, for alln≥1;

iilimn→ ∞αn0,∞n0αn∞;

iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1.

Then{xn} converges strongly to x PFixS∩Γu and x, y is a solution of problem1.3, where

yPCx−μ2B2x.

It is clear that the above result unifies and extends some corresponding results in the literature.

Based on the relaxed extragradient method and viscosity approximation method, Yao et al.30proposed and analyzed an iterative algorithm for finding a common element of the solution set of the general system1.3of variational inequalities and the fixed-point set of a strictly pseudocontractive mapping in a real Hilbert spaceH.

Theorem YLKsee30, Theorem 3.2. LetCbe a nonempty bounded closed convex subset of a

real Hilbert spaceH. Let the mappingBi :C → Hbeμi-inverse strongly monotone fori1,2. Let

S:C → Cbe ak-strictly pseudocontractive mapping such thatFixS∩Γ/∅. LetQ:C → Cbe a

ρ-contraction withρ∈0,1/2. For givenx0∈Carbitrarily, let the sequences{xn},{yn}, and{zn}

be generated iteratively by

znPCxn−μ2B2xn,

ynαnQxn 1−αnPCzn−μ1B1zn

,

xn1βnxnγnPCzn−μ1B1zn

δnSyn, ∀n≥0,

1.7

whereμi∈0,2βifori1,2and{αn},{βn},{γn},{δn}are four sequences in0,1such that

iβnγnδn1andγnδnk≤γn<1−2ρδnfor alln≥0;

iilimn→ ∞αn0and∞n0αn∞;

iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1andlim infn→ ∞δn>0;

ivlimn→ ∞γn1/1−βn1−γn/1−βn 0.

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Motivated by the above work, in this paper, we introduce an iterative algorithm for finding a common element of the solution set of the variational inequality1.1, the solution set of the general system1.3and the fixed-point set of the strictly pseudocontractive map-pingS:C → Cvia a hybrid extragradient method based on the well-known extragradient method, viscosity approximation method, and Mann-type iteration method, that is,

zn PCxn−λnAxn,

ynαnQxn 1−αnPCPCzn−μ2B2zn−μ1B1PCzn−μ2B2zn,

xn1βnxnγnynδnSyn, ∀n≥0,

1.8

where{λn} ⊂ 0,∞,{αn},{βn},{γn},{δn} ⊂ 0,1such thatβnγnδn 1 for alln ≥ 0. Moreover, we prove that the studied iterative algorithm converges strongly to an element of FixS∩Γ∩VIA, Cunder some mild conditions imposed on algorithm parameters. Our method improves and extends Yao et al.30, Theorem 3.2in the following aspects:

ithe problem of finding an element of FixS∩Γin30, Theorem 3.2is extended to the the problem of finding an element of FixS∩Γ∩VIA, C;

iithe requirement of boundedness ofCin30, Theorem 3.2is removed;

iiithe condition γn δnk ≤ γn < 1−2ρδn, for alln ≥ 0 in 30, Theorem 3.2is

replaced by the oneγnδnk≤γn, for alln≥0;

ivthe argument of Step 5 in the proof of30, Theorem 3.2is simplified under the lack of the conditionγn<1−2ρδn, for alln≥0;

vour iterative algorithm is similar to but diﬀerent from the one of30, Theorem 3.2 because the problem of finding an element of FixS ∩ Γ ∩ VIA, C is more challenging than the problem of finding an element of FixS ∩ Γ in 30, Theorem 3.2.

2. Preliminaries

In this section, we collect some notations and lemmas. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. A mappingA:C → His called monotone if

Ax−Ay, x−y≥0, ∀x, y∈C. 2.1

A mappingA:C → His called Lipschitz continuous if there exists a real numberL >0 such that

Ax−Ay ≤L x−y , ∀x, y∈C. 2.2

Recall that a mappingA:C → His calledα-inverse strongly monotone if there exists a real numberα >0 such that

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It is clear that every inverse strongly monotone mapping is a monotone and Lipschitz continuous mapping. Also, recall that a mapping S : C → C is said to be k-strictly pseudocontractive 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. _2.4

For such a case, we also say thatSis ak-strict pseudo-contraction31. It is clear that, in a real Hilbert spaceH, inequality2.4is equivalent to the following:

Sx−Sy, x−y≤ x−y 2₋1−k

2 I−Sx−I−Sy

2_, _∀_{x, y}_∈_C. _2.5

This immediately implies that ifSis a k-strictly pseudocontractive mapping, then I−Sis

1−k/2-inverse strongly monotone; see32for more details. We use FixSto denote the set of fixed points ofS. It is well known that the class of strict pseudo-contractions strictly includes the class of nonexpansive mappings which are mappings S : C → Csuch that

Sx−Sy ≤ x−y, for allx, y ∈C. A mappingQ :C → Cis called a contraction if there exists a constantρ∈0,1such thatQx−Qy ≤ρx−yfor allx, y∈C.

For every pointx∈H, there exists a unique nearest point inC, denoted byPCxsuch that

x−PCx ≤ x−y , ∀y∈C. 2.6

The mapping PC is called the metric projection ofH ontoC. It is well known that PC is a nonexpansive mapping and satisfies

x−y, PCx−PCy≥ PCx−PCy 2_, _∀_{x, y}_∈_H. _2.7

It is known thatPCxis characterized by the following property:

x−PCx, y−PCx≤0, ∀x∈H, y∈C. 2.8

In order to prove the main result in this paper, we will need the following lemmas in the sequel.

Lemma 2.1see33. Let{xn}and{yn}be bounded sequences in a Banach spaceXand let{βn}be

a sequence in0,1with0<lim infn→ ∞βn≤lim supn→ ∞βn<1. Supposexn1 1−βnynβnxn

for all integersn≥0andlim sup_n_{→ ∞}yn1−yn − xn1−xn≤0. Then,limn→ ∞yn−xn0.

Lemma 2.2see34, Proposition 2.1. LetCbe a nonempty closed convex subset of a real Hilbert

spaceHandS:C → Cbe a self-mapping ofC.

iIfSis ak-strict pseudocontractive mapping, thenSsatisfies the Lipschitz condition

_Sx₋_Sy _≤ 1k

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iiIfSis ak-strict pseudocontractive mapping, then the mappingI−Sis demiclosed at 0, that is, if{xn}is a sequence inCsuch thatxn → xweakly andI−Sxn → 0strongly, then

I−Sx0.

iiiIfSisk-(quasi-)strict pseudo-contraction, then the fixed-point setFixSofSis closed and convex so that the projectionPFixSis well defined.

Lemma 2.3see9, Lemma 2.1. Let{sn}be a sequence of nonnegative real numbers satisfying

the condition

sn1≤1−αnsnαnβn, ∀n≥0, 2.10

where{αn},{βn}are sequences of real numbers such that

i{αn} ⊂0,1and∞_n₀αn∞, or equivalently,

∞

n0

1−αn:_nlim → ∞

n

k1

1−αk 0; 2.11

iilim sup_n_{→ ∞}βn≤0; or

ii∞_n₀αnβnis convergent.

Then,limn→ ∞sn0.

Lemma 2.4see30. LetC be a nonempty closed convex subset of a real Hilbert spaceH. Let

S:C → Cbe ak-strictly pseudocontractive mapping. Letγandδbe two nonnegative real numbers. Assumeγδk≤γ. Then

γx−yδSx−Sy ≤γδ x−y , ∀x, y∈C. 2.12

The following lemma is an immediate consequence of an inner product.

Lemma 2.5. In a real Hilbert spaceH, there holds the inequality

xy 2

≤ x2₂_{y, x}_y_, _∀_{x, y}_∈_H. _2.13

LetAbe a monotone mapping ofCintoH. In the context of the variational inequality problem the characterization of projection2.8implies that

u∈VIA, C⇐⇒uPCu−λAu, ∀λ >0. 2.14

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monotone and Lipschitz continuous mapping ofCintoH. LetNCvbe the normal cone toCatv∈C, that is,

NCv{w∈H:v−u, w ≥0, ∀ ∈C}. 2.15

Define

Tv

⎧ ⎨ ⎩

AvNCv ifv∈C,

∅ ifv /∈C.

2.16

It is known that in this case the mappingT is maximal monotone, and0 ∈ Tv if and only ifv ∈ VIA, C; see [35] for more details.

3. Main Results

The main idea for showing strong convergence of the sequence{xn}generated by1.8to an element of VIA, Cis first to transform the variational inequality problem1.1into the zero point problem of a maximal monotone mappingT and then to derive the strong convergence of{xn}to a zero ofT by using the technique in10. We are now in a position to state and prove the main result in this paper.

Theorem 3.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetA:C → H

beα-inverse strongly monotone andBi : C → Hbeβi-inverse strongly monotone fori 1,2. Let

S : C → C be ak-strictly pseudocontractive mapping such thatFixS∩Γ∩VIA, C/∅. Let

Q : C → Cbe aρ-contraction withρ ∈ 0,1/2. For givenx0 ∈ Carbitrarily, let the sequences {xn},{yn}and{zn}be generated iteratively by

ynαnQxn 1−αnPCPCzn−μ2B2zn

−μ1B1PC

zn−μ2B2zn

,

3.1

whereμi∈0,2βifori1,2,{λn} ⊂0,2αand{αn},{βn},{γn},{δn} ⊂0,1such that

iβnγnδn1andγnδnk≤γnfor alln≥0;

ivlimn→ ∞γn1/1−βn1−γn/1−βn 0;

v0<lim infn→ ∞λn≤lim supn→ ∞λn<2αandlimn→ ∞|λn1−λn|0.

Then the sequence{xn}generated by3.1converges strongly toxPFixS∩Γ∩VIA,C·Qxandx, y is a solution of the general system1.3of variational inequalities, whereyPCx−μ2B2x.

Proof. We divide the proof into several steps.

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Indeed, takex∗∈FixS∩Γ∩VIA, Carbitrarily. ThenSx∗x∗,x∗PCx∗−λnAx∗ and

x∗_P

CPCx∗−μ2B2x∗

−μ1B1PCx∗−μ2B2x∗

. 3.2

SinceA:C → Hbeα-inverse strongly monotone and 0< λn≤2α, we have for alln≥0,

zn−x∗2PCxn−λnAxn−PCx∗−λnAx∗2

≤ xn−λnAxn−x∗₋_λnAx∗2

xn−x∗₋_λn_Axn₋_Ax∗2

≤ xn−x∗2−λn2α−λnAxn−Ax∗2

≤ xn−x∗2.

3.3

For simplicity, we writey∗ PCx∗−μ2B2x∗andun PCzn−μ2B2znfor alln ≥0. Since

Bi:C → Hbeβi-inverse strongly monotone fori1,2 and 0< μi<2βifori1,2, we know that for alln≥0,

_PC

PCzn−μ2B2zn

−μ1B1PC

zn−μ2B2zn

−x∗ 2

PCPCzn−μ2B2zn−μ1B1PCzn−μ2B2zn

−PCPCx∗₋_μ

2B2x∗

−μ1B1PC

x∗₋_μ

2B2x∗ 2

≤ PCzn−μ2B2zn−μ1B1PCzn−μ2B2zn

−PCx∗₋_μ

2B2x∗

−μ1B1PC

x∗₋_μ

2B2x∗ 2

PCzn−μ2B2zn−PCx∗−μ2B2x∗

−μ1

B1PC

zn−μ2B2zn

−B1PC

x∗₋_μ

2B2x∗ 2

≤ PCzn−μ2B2zn−PCx∗−μ2B2x∗ 2

−μ1

2β1−μ1 B1PC

zn−μ2B2zn

−B1PC

x∗₋_μ

2B2x∗ 2

≤ zn−μ2B2zn−x∗−μ2B2x∗ 2−μ1

2β1−μ1 B1un−B1y∗ 2

zn−x∗−μ2B2zn−B2x∗ 2−μ1

2β1−μ1 B1un−B1y∗ 2

≤ zn−x∗2₋_μ 2

2β2−μ2

B2zn−B2x∗2−μ1

2β1−μ1 B1un−B1y∗ 2

≤ xn−x∗2−λn2α−λnAxn−Ax∗2

−μ2

2β2−μ2

2β1−μ1 B1un−B1y∗ 2

≤ xn−x∗2_.

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Hence we get

yn−x∗ _αn_Qxn₋_x∗₁₋_αn_PC_PC_zn₋_μ

2B2zn

−μ1B1PC

zn−μ2B2zn

−x∗

≤αnQxn−x∗ ₁₋_αn _PC_PC_zn₋_μ

2B2zn

−μ1B1PC

zn−μ2B2zn

−x∗

≤αnρxn−x∗Qx∗−x∗ 1−αnxn−x∗

1−1−ραnxn−x∗1−ραnQx

∗₋_x∗

1−ρ

≤max

xn−x∗_,Qx∗−x∗

1−ρ

.

3.5

Sinceγnδnk≤γnfor alln≥0, utilizingLemma 2.4we obtain from3.5

xn1−x∗ βnxn−x∗ γnyn−x∗δnSyn−x∗

≤βnxn−x∗ γnyn−x∗δnSyn−x∗

≤βnxn−x∗γnδn yn−x∗

≤βnxn−x∗_γn_δn_max_xn₋_x∗_,Qx∗−x∗

1−ρ

≤max

xn−x∗,Qx ∗₋_x∗

1−ρ

.

3.6

By induction, we obtain that for alln≥0

xn−x∗ ≤max

x0−x∗,

Qx∗₋_x∗

1−ρ

. 3.7

Hence,{xn}is bounded. Consequently, we deduce immediately that{zn},{yn},{Syn}, and

{un}are bounded, whereun PCzn−μ2B2znfor alln≥0.

Now, put

tn:PCPCzn−μ2B2zn−μ1B1PCzn−μ2B2zn, ∀n≥0. 3.8

Then it is easy to see that{tn}is bounded becausePC, B1, andB2 are Lipschitz continuous

and{zn}is bounded.

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Indeed, definexn1βnxn 1−βnwnfor alln≥0. It follows that

wn1−wn xn2₁−₋βn_βn1xn1

1 −

xn1−βnxn

1−βn

γn1yn1δn1Syn1

1−βn1 −

γnynδnSyn

1−βn

γn1

yn1−ynδn1Syn1−Syn

1−βn1

_γ

n1

1−βn1 −

γn

1−βn

yn

_δn

1

1−βn1 −

δn

1−βn

Syn.

3.9

Sinceγnδnk≤γnfor alln≥0, utilizingLemma 2.4we have

γn1yn1−ynδn1Syn1−Syn ≤γn1δn1 yn1−yn . 3.10

Next, we estimateyn1−yn. Observe that

zn1−znPCxn1−λn1Axn1−PCxn−λnAxn ≤ xn1−λn1Axn1−xn−λnAxn

xn1−xn−λn1Axn1−Axn λn−λn1Axn

≤ xn1−xn−λn1Axn1−Axn|λn1−λn|Axn

≤ xn1−xn|λn1−λn|Axn,

3.11

tn1−tn2 PCPCzn1−μ2B2zn1

−μ1B1PC

zn1−μ2B2zn1

−PCPCzn−μ2B2zn

−μ1B1PC

zn−μ2B2zn 2 ≤ PCzn1−μ2B2zn1−μ1B1PCzn1−μ2B2zn1

−PCzn−μ2B2zn

−μ1B1PC

zn−μ2B2zn 2

PCzn1−μ2B2zn1−PCzn−μ2B2zn

−μ1

B1PCzn1−μ2B2zn1−B1PCzn−μ2B2zn 2

≤ PCzn1−μ2B2zn1

−PCzn−μ2B2zn 2

−μ1

2β1−μ1 B1PC

zn1−μ2B2zn1

−B1PC

zn−μ2B2zn 2

≤ PCzn1−μ2B2zn1

−PCzn−μ2B2zn 2

≤ zn1−μ2B2zn1

−zn−μ2B2zn 2

zn1−zn−μ2B2zn1−B2zn 2 ≤ zn1−zn2−μ2

2β2−μ2

B2zn1−B2zn2

≤ zn1−zn2.

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Combining3.11with3.12, we get

tn1−tn PCPCzn1−μ2B2zn1

−μ1B1PC

zn1−μ2B2zn1

−PCPCzn−μ2B2zn−μ1B1PCzn−μ2B2zn

≤ xn1−xn|λn1−λn|Axn.

3.13

This together with3.13implies that

yn1−yn tn1αn1Qxn1−tn1−tn−αnQxn−tn

≤ tn1−tnαn1Qxn1−tn1αnQxn−tn

≤ xn1−xn|λn1−λn|Axnαn1Qxn1−tn1αnQxn−tn.

3.14

Hence it follows from3.9,3.10, and3.14that

wn1−wn ≤

γn1yn1−ynδn1Syn1−Syn

1−βn1

γn1

1−βn1 −

γn

1−βn

yn δn1

1−βn1 −

δn

1−βn

Syn

≤ γn1δn1

1−βn1

yn1−yn ₁₋γn_β1

n1 −

γn

1−βn

yn Syn

yn1−yn ₁₋γn_βn1 1 −

γn

1−βn

yn Syn

≤ xn1−xn|λn1−λn|Axnαn1Qxn1−tn1αnQxn−tn

γn1

1−βn1 −

γn

1−βn

yn Syn .

3.15

Since{xn},{yn}, and{tn}are bounded, it follows from conditionsii,iv,vthat

lim sup

n→ ∞ wn1−wn − xn1−xn

≤lim sup

n→ ∞

|λn1−λn|Axnαn1Qxn1−tn1αnQxn−tn

γn1

1−βn1 −

γn

1−βn

yn Syn

0.

3.16

Hence byLemma 2.1we get limn→ ∞wn−xn0. Thus,

lim

n→ ∞xn1−xnnlim→ ∞

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Step 3. limn→ ∞B2zn−B2x∗ 0, limn→ ∞B1un−B1y∗ 0 and limn→ ∞Axn−Ax∗ 0,

wherey∗PCx∗−μ2B2x∗.

Indeed, utilizingLemma 2.4and the convexity of · 2, we get from3.1and3.4

xn1−x∗2 βnxn−x∗ γnyn−x∗δnSyn−x∗ 2

≤βnxn−x∗2_γn_δn 1

γnδn

γnyn−x∗_δn_Syn₋_x∗ 2

≤βnxn−x∗2γnδn yn−x∗ 2

≤βnxn−x∗2γnδnαnQxn−x∗2 1−αntn−x∗2

≤βnxn−x∗2_αn_Qxn₋_x∗2_γn_δn_tn₋_x∗2

≤βnxn−x∗2αnQxn−x∗2γnδn

×xn−x∗2−λn2α−λnAxn−Ax∗2−μ2

2β2−μ2

B2zn−B2x∗2

−μ1

2β1−μ1 B1un−B1y∗ 2

xn−x∗2αnQxn−x∗2−γnδn

×λn2α−λnAxn−Ax∗2

−μ2

2β2−μ2

2β1−μ1 B1un−B1y∗ 2

.

3.18

Therefore,

γnδnλn2α−λnAxn−Ax∗2μ2

2β2−μ2

B2zn−B2x∗2

μ1

2β1−μ1 B1un−B1y∗ 2

≤ xn−x∗2₋_xn

1−x∗2αnQxn−x∗2

≤xn−x∗xn1−x∗xn−xn1αnQxn−x∗2.

3.19

Sinceαn → 0,xn−xn1 → 0, lim infn→ ∞γnδn>0 and 0<lim infn→ ∞λn≤lim supn→ ∞λn<

2α, we have

lim

n→ ∞Axn−Ax

∗₀_, _lim

n→ ∞ B1un−B1y

∗ ₀_, _lim

n→ ∞B2zn−B2x

∗₀_. _3.20

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Indeed, noticing the firm nonexpansivity ofPCwe have

zn−x∗2PCxn−λnAxn−PCx∗−λnAx∗2

≤ xn−λnAxn−x∗₋_λnAx∗_{, zn}₋_x∗

1

2

xn−x∗₋_λn_Axn₋_Ax∗2_zn₋_x∗2

−xn−x∗₋_λn_Axn₋_Ax∗₋_zn₋_x∗2

≤ 1

2

xn−x∗2_zn₋_x∗2₋_xn₋_zn₋_λn_Axn₋_Ax∗2

1

2

xn−x∗2zn−x∗2− xn−zn2

2λnxn−zn, Axn−Ax∗ −λ2_nAxn−Ax∗2

≤ 1

2

xn−x∗2_zn₋_x∗2₋_xn₋_zn2₂_λn_xn₋_zn_Axn₋_Ax∗_,

3.21

that is,

zn−x∗2_≤_xn₋_x∗2₋_xn₋_zn2₂_λn_xn₋_zn_Axn₋_Ax∗_. _3.22

Similarly to the above argument, we obtain

un−y∗ 2 _PC_zn₋_μ 2B2zn

−PCx∗₋_μ

2B2x∗ 2

≤zn−μ2B2zn

−x∗₋_μ

2B2x∗

, un−y∗

1

2 zn−x

∗₋_μ

2B2zn−B2x∗ 2 un−y∗ 2

− zn−x∗−μ2B2zn−B2x∗−

un−y∗ 2

≤ 1

2

zn−x∗2 _un₋_y∗ 2₋ _zn₋_un₋_μ

2B2zn−B2x∗−

x∗₋_y∗ 2

1

2

zn−x∗2 _un₋_y∗ 2₋ _zn₋_un₋_x_∗₋_y_∗ 2

2μ2

zn−un−x∗−y∗, B2zn−B2x∗

−μ2

2B2zn−B2x∗2

,

3.23

that is,

_un₋_y∗ 2

≤ zn−x∗2₋ _zn₋_un₋_x∗₋_y∗ 2

2μ2 zn−un−

x∗₋_y∗ _B

2zn−B2x∗.

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Substituting3.22in3.24, we have

_un₋_y∗ 2

≤ xn−x∗2₋_xn₋_zn2₂_λn_xn₋_zn_Axn₋_Ax∗

− zn−un−x∗₋_y∗ 2

2μ2 zn−un−

x∗₋_y∗ _B

2zn−B2x∗.

3.25

Further, similarly to the above argument, we derive

tn−x∗2 PCun−μ1B1un−PCy∗−μ1B1y∗ 2

≤un−μ1B1un−y∗−μ1B1y∗

, tn−x∗

1

2 un−y

∗₋_μ

1

B1un−B1y∗ 2tn−x∗2

− un−y∗−μ1

B1un−B1y∗

−tn−x∗ 2

≤ 1

2 un−y

∗ 2_tn₋_x_∗2₋ _un₋_tn₋_μ 1

B1un−B1y∗

x∗₋_y∗ 2

1

2 un−y

∗ 2_tn₋_x_∗2₋ _un₋_tn_x∗₋_y∗ 2

2μ1

un−tnx∗−y∗, B1un−B1y∗

−μ2

1 B1un−B1y∗ 2

,

3.26

that is,

tn−x∗2≤ un−y∗ 2− un−tnx∗−y∗ 22μ1 un−tnx∗−y∗ B1un−B1y∗ .

3.27

Substituting3.25in3.27, we have

tn−x∗2_≤_xn₋_x∗2₋_xn₋_zn2₂_λn_xn₋_zn_Axn₋_Ax∗

− zn−un−x∗₋_y∗ 2

2μ2 zn−un−

x∗₋_y∗ _B

2zn−B2x∗

− un−tnx∗₋_y∗ 2

2μ1 un−tn

x∗₋_y∗ _B

1un−B1y∗ .

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Thus from3.1and3.28, it follows that

xn1−x∗2 βnxn−x∗ γnyn−x∗δnSyn−x∗ 2

≤βnxn−x∗2γnδn yn−x∗ 2

βnxn−x∗2₁₋_βn _yn₋_x∗ 2

≤βnxn−x∗2₁₋_βn_αn_Qxn₋_x∗2 ₁₋_αn_tn₋_x∗2

≤βnxn−x∗2αnQxn−x∗21−βntn−x∗2

≤βnxn−x∗2_αn_Qxn₋_x∗2₁₋_βn

×xn−x∗2₋_xn₋_zn2₂_λn_xn₋_zn_Axn₋_Ax∗

− zn−un−x∗−y∗ 22μ2 zn−un−x∗−y∗ B2zn−B2x∗

− un−tnx∗−y∗ 22μ1 un−tnx∗−y∗ B1un−B1y∗

xn−x∗2_αn_Qxn₋_x∗2₁₋_βn

×2λnxn−znAxn−Ax∗2μ2 zn−un−

x∗₋_y∗ _B

2zn−B2x∗

2μ1 un−tn

x∗₋_y∗ _B

1un−B1y∗

−1−βnxn−zn2 zn−un−x∗−y∗ 2 un−tnx∗−y∗ 2,

3.29

which hence implies that

1−βnxn−zn2 zn−un−x∗−y∗ 2 un−tnx∗−y∗ 2

≤ xn−x∗2₋_xn

1−x∗2αnQxn−x∗21−βn

x∗₋_y∗ _B

2zn−B2x∗

2μ1 un−tn

x∗₋_y∗ _B

1un−B1y∗

≤xn−x∗_xn

1−x∗xn−xn1αnQxn−x∗21−βn

x∗₋_y∗ _B

2zn−B2x∗

2μ1 un−tn

x∗₋_y∗ _B

1un−B1y∗ .

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Since lim sup_n_{→ ∞}βn < 1, 0 < λn ≤ 2α,αn → 0, Axn −Ax∗ → 0, B2zn −B2x∗ → 0, B1un−B1y∗ → 0 andxn1−xn → 0, it follows from the boundedness of{xn},{zn},{un},

and{tn}that

lim

n→ ∞xn−zn0, nlim→ ∞ zn−un−

x∗₋_y∗ ₀_, _lim

n→ ∞ un−tn

x∗₋_y∗ ₀_.

3.31

Consequently, it immediately follows that

lim

n→ ∞zn−tn0, nlim→ ∞xn−tn0. 3.32

This together withyn−tn ≤αnQxn−tn → 0 implies that

lim

n→ ∞ xn−yn 0. 3.33

Since

δnSyn−xn ≤ xn1−xnγn yn−xn , 3.34

it follows that

lim

n→ ∞ Syn−xn 0, nlim→ ∞ Syn−yn 0. 3.35

Step 5. lim sup_n_{→ ∞}Qx−x, xn−x ≤0, wherexPFixS∩Γ∩VIA,C·Qx.

Indeed, since{xn}is bounded, there exists a subsequence{xni}of{xn}such that

lim sup

n→ ∞ Qx−x, xn−xilim→ ∞Qx−x, xni−x. 3.36

Also, sinceH is reflexive and{yn}is bounded, without loss of generality we may assume thatyni → pweakly for somep ∈C. First, it is clear fromLemma 2.2thatp ∈FixS. Now

let us show thatp∈Γ. We note that

yn−Gyn

≤αn Qxn−Gyn 1−αn PCPCzn−μ2B2zn−μ1B1PCzn−μ2B2zn−Gyn

αn Qxn−Gyn 1−αn Gzn−Gyn

≤αn Qxn−Gyn 1−αn xn−yn

−→0.

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According toLemma 2.2we obtainp∈Γ. Further, let us show thatp∈VIA, C. As a matter of fact, sincexn−zn → 0 andxn−yn → 0, we deduce thatxni → pweakly andzni → p

weakly. Let

Tv

⎧ ⎨ ⎩

AvNCv ifv∈C,

∅ ifv /∈C, 3.38

whereNCvis the normal cone toCatv∈C. In this case, the mappingTis maximal monotone, and 0∈Tvif and only ifv∈VIA, C; see10for more details. Let GphTbe the graph of

Tand letv, w∈GphT. Then, we havew∈TvAvNCvand hencew−Av∈NCv. So,

we havev−t, w−Av ≥0 for allt∈C. On the other hand, fromznPCxn−λnAxnand

v∈Cwe have

xn−λnAxn−zn, zn−v ≥0 3.39

and hence

v−zn,zn−xn λn Axn

≥0. 3.40

Fromv−t, w−Av ≥0 for allt∈Candzni ∈C, we have

v−zni, w ≥ v−zni, Av

≥ v−zni, Av −

v−zni,

zni−xni λni

Axni

v−zni, Av−Azniv−zni, Azni −Axni −

v−zni,

≥ v−zni, Azni−Axni −

v−zni,

,

3.41

Hence, we obtainv−p, w ≥0 asi → ∞. SinceTis maximal monotone, we havep ∈T−1₀

and hencep∈VIA, C. Therefore,p∈FixS∩Γ∩VIA, C. Hence it follows from2.8and

3.36that

lim sup

n→ ∞ Qx−x, xn−xilim→ ∞Qx−x, xni−x

Qx−x, p−x

≤0.

3.42

Step 6. limn→ ∞xnx.

Indeed, sinceG:C → Cis nonexpansive, we have

(18)

Note that

Qxn−x, yn−xQxn−x, xn−xQxn−x, yn−xn

Qxn−Qx, xn−xQx−x, xn−xQxn−x, yn−xn

≤ρxn−x2_Qx₋_{x, xn}₋_x_Qxn₋_x _yn₋_xn _.

3.44

Utilizing Lemmas2.4and2.5, we obtain from3.4and the convexity of · 2

xn1−x2 βnxn−x γnyn−xδnSyn−x 2

≤βnxn−x2_γn_δn 1

γnδn

γnyn−xδnSyn−x 2

≤βnxn−x2γnδn yn−x 2

≤βnxn−x2_γn_δn₁₋_αn2_tn₋_x2₂_αn_Qxn₋_{x, yn}₋_x

≤βnxn−x2_γn_δn₁₋_αn_xn₋_x2₂_αn_Qxn₋_{x, yn}₋_x

1−γnδnαnxn−x2γnδn2αnQxn−x, yn−x

≤1−γnδnαnxn−x2

γnδn2αnρxn−x2Qx−x, xn−xQxn−x yn−xn

≤1−1−2ργnδnαnxn−x2

γnδn2αnQx−x, xn−xQxn−x yn−xn

1−1−2ργnδnαnxn−x2

1−2ργnδnαn2

Qx−x, xn−xQxn−x yn−xn

1−2ρ .

3.45

Note that lim infn→ ∞1−2ργnδn> 0. It follows thatn∞01−2ργnδnαn ∞. It is

clear that

lim sup

n→ ∞

2Qx−x, xn−xQxn−x yn−xn

1−2ρ ≤0 3.46

because lim sup_n_{→ ∞}Qx−x, xn−x ≤0 and limn→ ∞xn−yn0. Therefore, all conditions of

Lemma 2.3are satisfied. Consequently, we immediately deduce thatxn → x. This completes

the proof.

Corollary 3.2. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetA:C → H

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S:C → Cbe ak-strictly pseudocontractive mapping such thatFixS∩Γ∩VIA, C/∅. For fixed

u∈Cand givenx0∈Carbitrarily, let the sequences{xn},{yn}, and{zn}be generated iteratively by

ynαnu 1−αnPCPCzn−μ2B2zn−μ1B1PCzn−μ2B2zn,

3.47

iβnγnδn1andγnδnk≤γnfor alln≥0;

Then the sequence{xn}converges strongly tox PFixS∩Γ∩VIA,C·Qxandx, yis a solution of

the general system1.3of variational inequalities, whereyPCx−μ2B2x.

beα-inverse strongly monotone andBi :C → Hbeβi-inverse strongly monotone fori 1,2. Let

S :C → Cbe a nonexpansive mapping such thatFixS∩Γ∩VIA, C/∅. LetQ :C → Cbe a

ρ-contraction withρ∈0,1/2. For givenx0 ∈Carbitrarily, let the sequences{xn},{yn}and{zn}

be generated iteratively by

ynαnQxn 1−αnPCPCzn−μ2B2zn

−μ1B1PC

zn−μ2B2zn

,

3.48

iβnγnδn1for alln≥0;

Then the sequence{xn}converges strongly tox PFixS∩Γ∩VIA,C·Qxandx, yis a solution of the general system1.3of variational inequalities, whereyPCx−μ2B2x.

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S :C → Cbe a nonexpansive mapping such thatFixS∩Γ∩VIA, C/∅. For fixedu∈ Cand givenx0∈Carbitrarily, let the sequences{xn},{yn}and{zn}be generated iteratively by

ynαnu 1−αnPCPCzn−μ2B2zn

−μ1B1PC

zn−μ2B2zn

,

3.49

iβnγnδn1for alln≥0;

Then the sequence{xn}converges strongly tox PFixS∩Γ∩VIA,Cuandx, yis a solution of the

general system1.3of variational inequalities, whereyPCx−μ2B2x.

Acknowledgments

This research was partially supported by the National Science Foundation of China

10771141, 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. This research was partially supported by the Grant NSC 99-2115-M-110-004-MY3.

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