An Extragradient Method for Fixed Point Problems and Variational Inequality Problems

Volume 2007, Article ID 38752,12pages doi:10.1155/2007/38752

Research Article

An Extragradient Method for Fixed Point Problems and Variational Inequality Problems

Yonghong Yao, Yeong-Cheng Liou, and Jen-Chih Yao Received 11 September 2006; Accepted 10 December 2006 Recommended by Yeol-Je Cho

We present an extragradient method for fixed point problems and variational inequal- ity problems. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for monotone mapping.

Copyright © 2007 Yonghong Yao 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.

1. Introduction

LetCbe a closed convex subset of a real Hilbert spaceH. Recall that a mappingAofC intoH is called monotone if

Au−Av,u−v ≥0, (1.1)

for allu,v∈C.Ais calledα-inverse strongly monotone if there exists a positive real num- berαsuch that

Au−Av,u−v ≥αAu−Av², (1.2) for allu,v∈C. It is well known that the variational inequality problem VI(A,C) is to find u∈Csuch that

Au,v−u ≥0, (1.3)

for allv∈C(see [1–3]). The set of solutions of the variational inequality problem is denoted byΩ. The variational inequality has been extensively studied in the literature, see, for example, [4–6] and the references therein. A mappingSofCinto itself is called nonexpansive if

Su−Sv ≤ u−v, (1.4)

for allu,v∈C. We denote byF(S) the set of fixed points ofS.

For finding an element ofF(S)∩Ωunder the assumption that a setC⊂H is closed and convex, a mappingSof Cinto itself is nonexpansive and a mappingA of Cinto H isα-inverse strongly monotone, Takahashi and Toyoda [7] introduced the following iterative scheme:

xn+1=αnxn+1−αn

SPC

xn−λnAxn

, (1.5)

for everyn=0, 1, 2,. . ., whereP_Cis the metric projection ofHontoC,x0=x∈C,{α_n} is a sequence in (0, 1), and{λ_n}is a sequence in (0, 2α). They showed that ifF(S)∩Ω is nonempty, then the sequence{xn}generated by (1.5) converges weakly to somez∈ F(S)_∩Ω. Recently, Nadezhkina and Takahashi [8] introduced a so-called extragradient method motivated by the idea of Korpeleviˇc [9] for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem. They obtained the following weak convergence theorem.

Theorem 1.1 (see Nadezhkina and Takahashi [8]). LetCbe a nonempty closed convex sub- set of a real Hilbert spaceH. LetA:C→Hbe a monotonek-Lipschitz continuous mapping, and letS:C→Cbe a nonexpansive mapping such thatF(S)∩Ω= ∅. Let the sequences {xn},{yn}be generated by

x0=x∈H, yn=PC

xn−λnAxn

, xn+1=αnxn+1−αn

SPC

xn−λnAyn

, ∀n≥0,

(1.6)

where{λ_n} ⊂[a,b] for somea,b∈(0, 1/k) and{α_n} ⊂[c,d] for somec,d∈(0, 1). Then the sequences{x_n},{y_n}converge weakly to the same pointP_F(S)∩Ω(x0).

Very recently, Zeng and Yao [10] introduced a new extragradient method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem. They obtained the following strong conver- gence theorem.

Theorem 1.2 (see Zeng and Yao [10]). LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Let A:C→H be a monotone k-Lipschitz continuous mapping, and let S:C→Cbe a nonexpansive mapping such thatF(S)∩Ω= ∅. Let the sequences{xn},{yn}

be generated by

x0=x∈H, yn=PC

xn−λnAxn

, xn+1=αnx0+1−αn

SPC

xn−λnAyn

, ∀n≥0,

(1.7)

where{λ_n}and{α_n}satisfy the following conditions:

(a){λnk} ⊂(0, 1−δ) for someδ∈(0, 1);

(b){αn} ⊂(0, 1),^∞_n=0αn= ∞, limn→∞αn=0.

Then the sequences{x_n}and{y_n}converge strongly to the same pointP_F(S)∩Ω(x0) pro- vided that

nlim→∞xn+1−xn=0. (1.8)

Remark 1.3. The iterative scheme (1.6) inTheorem 1.1has only weak convergence. The iterative scheme (1.7) inTheorem 1.2has strong convergence but imposed the assump- tion (1.8) on the sequence{x_n}.

In this paper, motivated by the iterative schemes (1.6) and (1.7), we introduced a new extragradient method for finding a common element of the set of fixed points of a nonex- pansive mapping and the set of solutions of the variational inequality problem for mono- tone mapping. We obtain a strong convergence theorem under some mild conditions.

2. Preliminaries

LetHbe a real Hilbert space with inner product·,·and norm · and letCbe a closed convex subset ofH. It is well known that for anyu∈H, there exists uniquey0∈Csuch that

u−y0=infu−y:y∈C. (2.1)

We denotey0byP_Cu, whereP_Cis called the metric projection ofH ontoC. The metric projectionPCofHontoChas the following basic properties:

(i)PCx−PCy ≤ x−y, for allx,y∈H,

(ii)x−y,P_Cx−P_Cy ≥ P_Cx−P_Cy², for everyx,y∈H, (iii)x−PCx,y−PCx ≤0, for allx∈H,y∈C,

(iv)x−y²≥ x−PCx²+y−PCx², for allx∈H,y∈C.

Such property ofP_Cwill be crucial in the proof of our main results. LetAbe a monotone mapping ofCintoH. In the context of the variational inequality problem, it is easy to see from (iv) that

u∈Ω⇐⇒u=P_C(u−λAu), ∀λ >0. (2.2)

A set-valued mappingT:H→2^H is called monotone if for allx,y∈H, f ∈Txand g∈T yimplyx−y,f −g ≥0. A monotone mappingT:H→2^His maximal if its graph G(T) is not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingT is maximal if and only if for (x,f)∈H×H,x−y,f − g ≥ 0 for every (y,g)∈G(T) implies that f ∈Tx. LetAbe a monotone mapping ofC intoH and letN_Cvbe the normal cone toCatv∈C, that is,

N_Cv=

w∈H:v−u,w ≥0,∀u∈C. (2.3) Define

Tv=

⎧⎨

⎩

Av+N_Cv ifv∈C,

∅ ifv /∈C. (2.4)

ThenTis maximal monotone and 0∈Tvif and only ifv∈VI(C,A) (see [11]).

Now, we introduce several lemmas for our main results in this paper.

Lemma 2.1 (see [12]). Let (E,·,·) be an inner product space. Then, for allx,y,z∈Eand α,β,γ_∈[0, 1] withα+β+γ₌1, one has

αx+βy+γz²=αx²+βy²+γz²−αβx−y²−αγx−z²−βγy−z². (2.5) Lemma 2.2 (see [13]). Let{x_n}and{y_n}be bounded sequences in a Banach spaceXand let{βn}be a sequence in [0, 1] with 0<lim infn→∞βn≤lim sup_n→∞βn<1. Supposexn+1= (1−βn)yn+βnxnfor all integersn≥0 and lim sup_n→∞(yn+1−yn − xn+1−xn)≤0.

Then, lim_n→∞y_n−x_n =0.

Lemma 2.3 (see [14]). Assume{a_n}is a sequence of nonnegative real numbers such that an+1≤

1−γn

an+δn, (2.6)

where{γn}is a sequence in (0, 1) and{δn}is a sequence such that (1)^∞_n=1γn= ∞;

(2) lim sup_n→∞δ_n/γ_n≤0 or^∞_n=1|δ_n|<∞. Then limn→∞an=0.

3. Main results

Theorem 3.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetAbe a monotoneL-Lipschitz continuous mapping ofCintoH, and letSbe a nonexpansive mapping ofCinto itself such thatF(S)∩Ω= ∅. For fixedu∈Hand givenx0∈H arbitrary, let the sequences{x_n},{y_n}be generated by

y_n=P_Cx_n−λ_nAx_n,

x_n+1=α_nu+β_nx_n+γ_nSP_Cx_n−λ_nAy_n, (3.1)

where{αn},{βn},{γn}are three sequences in [0, 1] satisfying the following conditions:

(C1)αn+βn+γn=1,

(C2) limn→∞α_n=0,^∞_n=0α_n= ∞,

(C3) 0<lim infn→∞βn≤lim sup_n→∞βn<1, (C4) limn→∞λn=0.

Then{x_n}converges strongly toP_F(S)∩Ωu.

Proof. Letx^∗∈F(S)∩Ω, thenx^∗=P_C(x^∗−λ_nAx^∗). Putt_n=P_C(x_n−λ_nAy_n). Substi- tutingxbyxn−λnAynandybyx^∗in (iv), we have

t_n−x^∗2≤x_n−λ_nAy_n−x^∗2−x_n−λ_nAy_n−t_n²

=xn−x^∗2−2λn

Ayn,xn−x^∗+λ²_nAyn²

−xn−tn²+ 2λn

Ayn,xn−tn

−λ²_nAyn²

=xn−x^∗2+ 2λn

Ayn,x^∗−tn

−xn−tn²

=xn−x^∗2−xn−tn²+ 2λn

Ayn−Ax^∗,x^∗−yn

+ 2λ_nAx^∗,x^∗₋y_n+ 2λ_nAy_n,y_n−t_n.

(3.2)

Using the fact thatAis monotonic andx^∗is a solution of the variational inequality prob- lem VI(A,C), we have

Ay_n−Ax^∗,x^∗−y_n≤0, Ax^∗,x^∗−y_n≤0. (3.3)

It follows from (3.2) and (3.3) that

t_n−x^∗2≤x_n−x^∗2−x_n−t_n²+ 2λ_nAy_n,y_n−t_n

=x_n−x^∗2−x_n−y_n+y_n−t_n²+ 2λ_nAy_n,y_n−t_n

=x_n−x^∗2−x_n−y_n²−2x_n−y_n,y_n−t_n

−yn−tn²+ 2λn

Ayn,yn−tn

=xn−x^∗2−xn−yn²−yn−tn²+ 2xn−λnAyn−yn,tn−yn

. (3.4) Substitutingxbyxn−λnAxnandybytnin (iii), we have

xn−λnAxn−yn,tn−yn

≤0. (3.5)

It follows that

xn−λnAyn−yn,tn−yn

=

xn−λnAxn−yn,tn−yn

+λnAxn−λnAyn,tn−yn

≤

λnAxn−λnAyn,tn−yn

≤λnLxn−yntn−yn. (3.6)

By (3.4) and (3.6), we obtain

tn−x^∗2≤xn−x^∗2−xn−yn²−yn−tn²+ 2λnLxn−yntn−yn

≤xn−x^∗2−xn−yn²−yn−tn²+λnL²xn−yn²+yn−tn²

≤xn−x^∗2+λ²_nL²−1xn−yn²+λ²_nL²−1yn−tn².

(3.7) Sinceλn→0 asn→ ∞, there exists a positive integerN0such thatλ²_nL²−1≤ −1/2 when n≥N0. It follows from (3.7) that

t_n−x^∗≤x_n−x^∗. (3.8)

By (3.1), we have

xn+1−x^∗=αnu+βnxn+γnStn−x^∗≤αnu−x^∗+βnxn−x^∗+γntn−x^∗

≤αnu−x^∗+1−αnxn−x^∗≤maxu−x^∗,x0−x^∗.

(3.9) Therefore,{xn}is bounded. Hence{tn},{Stn},{Axn}, and{Ayn}are also bounded.

For allx,y∈C, we get I−λ_nAx−

I−λ_nAy²=(x−y)−λ_n(Ax−Ay)²=x−y²−2λ_nx−y,Ax−Ay +λ²_nAx−Ay²≤ x−y²+λ²_nAx−Ay²

≤ x−y²+λ²_nL²x−y²=

1 +L²λ²_nx−y²,

(3.10) which implies that

I−λnAx−

I−λnAy≤ 1 +Lλn

x−y. (3.11)

By (3.1) and (3.11), we have tn+1−tn=PC

xn+1−λn+1Ayn+1

−PC

xn−λnAyn

≤xn+1−λn+1Ayn+1

−

xn−λnAyn

=xn+1−λn+1Axn+1

−

xn−λn+1Axn

+λn+1

Axn+1−Ayn+1−Axn

+λnAyn

≤xn+1−λn+1Axn+1

−

xn−λn+1Axn

+λ_n+1Ax_n+1+Ay_n+1+Ax_n+λ_nAy_n

≤

1 +λ_n+1Lx_n+1−x_n

+λ_n+1Ax_n+1+Ay_n+1+Ax_n+λ_nAy_n.

(3.12)

Setxn+1=(1−βn)zn+βnxn. Then, we obtain zn+1−zn=α_n+1u+γ_n+1St_n+1

1−β_n+1 ⁻

α_nu+γ_nSt_n 1−β_n

=

α_n+1

1−βn+1− α_n 1−βn

u+ γn+1

1−βn+1

Stn+1−Stn +

γn+1

1−βn+1− γn

1−βn

Stn.

(3.13)

Combining (3.12) and (3.13), we have zn+1−zn−xn+1−xn

≤ αn+1

1−β_n+1⁻ αn

1−β_n

u+ γ_n+1 1−β_n+1

1 +λn+1Lxn+1−xn + γn+1

1−βn+1

λn+1Axn+1+Ayn+1+Axn+λnAyn

+ γn+1

1−βn+1− γn

1−βn

St_n−x_n+1−x_n

≤ αn+1

1−βn+1− αn

1−βn

u+Stn+ γ_n+1

1−βn+1λn+1Lxn+1−xn + γn+1

1−βn+1

λn+1Axn+1+Ayn+1+Axn+λnAyn,

(3.14)

this together with (C2) and (C4) imply that lim sup

n→∞

z_n+1−z_n−x_n+1−x_n≤0. (3.15)

Hence byLemma 2.2, we obtainzn−xn →0 asn→ ∞. Consequently,

nlim→∞xn+1−xn=lim

n→∞

1−βnzn−xn=0. (3.16)

From (C4) and (3.12), we also havetn+1−tn →0 asn→ ∞.

Forx^∗∈F(S)∩Ω, fromLemma 2.1, (3.1), and (3.7), we obtain whenn≥N0that xn+1−x^∗2=αnu+βnxn+γnStn−x^∗2

≤αnu−x^∗2+βnxn−x^∗2+γnStn−x^∗2

≤αnu−x^∗2+βnxn−x^∗2+γntn−x^∗2

≤α_nu−x^∗2+β_nx_n−x^∗2

+γ_nx_n−x^∗2+λ²_nL²−1x_n−y_n²+λ²_nL²−1y_n−t_n²

≤αnu−x^∗2+xn−x^∗2−1

2xn−yn²,

(3.17)

which implies that 1

2xn−yn²≤αnu−x^∗2+xn−x^∗2−xn+1−x^∗2

=αnu−x^∗2+xn−x^∗−xn+1−x^∗

×

xn−x^∗+xn+1−x^∗

≤αnu−x^∗2+xn−x^∗+xn+1−x^∗xn−xn+1.

(3.18)

Sinceαn→0 andxn−xn+1 →0, from (3.18), we havexn−yn →0 asn→ ∞. Noting that

y_n−t_n=P_Cx_n−λ_nAx_n−P_Cx_n−λ_nAy_n

≤λ_nAx_n−Ay_n≤λ_nLx_n−y_n−→0 asn−→ ∞, t_n−x_n≤t_n−y_n+y_n−x_n−→0 asn−→ ∞,

Syn−xn+1≤Syn−Stn+Stn−xn+1≤yn−tn+αnStn−u+βnStn−xn

≤yn−tn+αnStn−u+βnStn−Sxn+βnSxn−xn

≤yn−tn+αnStn−u+βntn−xn+βnSxn−xn.

(3.19) Consequently, from (3.19), we can infer that

Sx_n−x_n≤Sx_n−St_n+St_n−Sy_n+Sy_n−x_n+1+x_n+1−x_n

≤

1 +β_nx_n−t_n+ 2t_n−y_n+α_nSt_n−u+β_nSx_n−x_n+x_n+1−x_n, (3.20) which implies that

Sxn−xn−→0 asn−→ ∞. (3.21)

Also we have

St_n−t_n≤St_n−Sx_n+Sx_n−x_n+x_n−t_n

≤2t_n−x_n+Sx_n−x_n−→ ∞ asn−→ ∞. (3.22) Next we show that

lim sup

n→∞

u−z0,x_n−z0

≤0, (3.23)

wherez0=P_F(S)∩Ωu.

To show it, we choose a subsequence{t_ni}of{t_n}such that lim sup

n→∞

u−z0,St_n−z0

=lim

i→∞

u−z0,St_ni−z0

. (3.24)

As{tni}is bounded, we have that a subsequence{tni j} of{tni}converges weakly to z.

We may assume without loss of generality thattniz. SinceStn−tn →0, we obtain St_nizasi→ ∞. Then we can obtainz∈F(S)∩Ω. In fact, let us first show thatz∈Ω.

Let

Uv=

⎧⎨

⎩

Av+NCv, v∈C,

∅, v /∈C. (3.25)

ThenU is maximal monotone. Let (v,w)∈G(U). Since w−Av∈NCvandtn∈C, we havev−tn,w−Av ≥0. On the other hand, fromtn=PC(xn−λnAyn), we have

v−tn,tn−

xn−λnAyn

≥0, (3.26)

that is,

v−tn,tn−yn

λn +Ayn

≥0. (3.27)

Therefore, we have v−tni,w≥

v−tni,Av≥

v−tni,Av−

v−tni,t_ni−x_ni λni

+Ayni

=

v−t_ni,Av−Ay_ni−t_ni−x_ni λni

=

v−t_ni,Av−At_ni+v−t_ni,At_ni−Ay_ni−

v−t_ni,t_ni−x_ni λ_ni

≥

v−t_ni,At_ni−

v−t_ni,tni−xni

λ_ni +Ay_ni

.

(3.28)

Noting thattni−yni →0 asi→ ∞andAis Lipschitz continuous, hence from (3.28), we obtainv−z,w ≥0 asi→ ∞. SinceUis maximal monotone, we havez∈U⁻¹0, and hencez∈Ω.

Let us show thatz∈F(S). Assume thatz /∈F(S). From Opial’s condition, we have lim inf

i→∞ tni−z<lim inf

i→∞ tni−Sz=lim inf

i→∞ tni−Stni+Stni−Sz

≤lim inf

i→∞ t_ni−St_ni+St_ni−Sz=lim inf

i→∞

St_ni−Sz

≤lim inf

i→∞ tni−z.

(3.29)

This is a contradiction. Thus, we obtainz∈F(S).

Hence, from (iii), we have lim sup

n→∞

u−z0,xn−z0

=lim sup

n→∞

u−z0,Stn−z0

=lim

i→∞

u−z0,Stni−z0

=

u−z0,z−z0

≤0. (3.30)