An Extragradient Method for Fixed Point Problems and Variational Inequality Problems

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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

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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,. . ., wherePCis the metric projection ofHontoC,x0=x∈C,{αn}

is a sequence in (0, 1), and{λn}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.

Theorem1.1 (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{λn} ⊂[a,b]for somea,b∈(0, 1/k)and{αn} ⊂[c,d]for somec,d∈(0, 1). Then the

sequences{xn},{yn}converge weakly to the same pointPF(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.

Theorem1.2 (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

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be generated by

x0=x∈H,

yn=PC

xn−λnAxn

,

xn+1=αnx0+

1−αn

SPC

xn−λnAyn

, ∀n≥0,

(1.7)

where{λn}and{αn}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{xn}and{yn}converge strongly to the same pointPF(S)∩Ω(x0)

pro-vided that

lim

n→∞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{xn}.

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₌_inf

u−y:y∈C. (2.1)

We denotey0byPCu, wherePCis 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,PCx−PCy ≥ PCx−PCy2, for everyx,y∈H,

(iii)x−PCx,y−PCx ≤0, for allx∈H,y∈C,

(iv)x−y2_≥_x₋_P

Cx2+y−PCx2, for allx∈H,y∈C.

Such property ofPCwill 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

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A set-valued mappingT:H→2H _{is called monotone if for all}_x,_y_∈_H, _f _∈_Tx_and

g∈T yimplyx−y,f −g ≥0. A monotone mappingT:H→2H_{is 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 letNCvbe the normal cone toCatv∈C, that is,

NCv=

w∈H:v−u,w ≥0,∀u∈C. (2.3)

Define

Tv= ⎧ ⎨ ⎩

Av+NCv 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.

Lemma2.1 (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₌_α_x2₊_β_y2₊_γ_z2₋_αβ_x₋_y2₋_αγ_x₋_z2₋_βγ_y₋_z2_. (2.5)

Lemma2.2 (see [13]). Let{xn}and{yn}be bounded sequences in a Banach spaceXand

let{βn}be a sequence in[0, 1]with0<lim infn→∞βn≤lim supn→∞βn<1. Supposexn+1=

(1−βn)yn+βnxnfor all integersn≥0andlim supn→∞(yn+1−yn − xn+1−xn)≤0.

Then,limn→∞yn−xn =0.

Lemma2.3 (see [14]). Assume{an}is a sequence of nonnegative real numbers such that

an+1≤

1−γnan+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_→∞δn/γn≤0or

_∞

n=1|δn|<∞.

Thenlimn→∞an=0.

3. Main results

Theorem3.1. 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{xn},{yn}be generated by

yn=PC

xn−λnAxn

, xn+1=αnu+βnxn+γnSPC

xn−λnAyn

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

(C1)αn+βn+γn=1,

(C2) limn→∞αn=0,

_∞

n=0αn= ∞,

(C3) 0<lim infn→∞βn≤lim supn→∞βn<1,

(C4) limn→∞λn=0.

Then{xn}converges strongly toPF(S)∩Ωu.

Proof. Letx∗∈F(S)∩Ω, thenx∗=PC(x∗−λnAx∗). Puttn=PC(xn−λnAyn).

Substi-tutingxbyxn−λnAynandybyx∗in (iv), we have

_t_n₋_x∗2

≤xn−λnAyn−x∗

2

−xn−λnAyn−tn

2

=xn−x∗2−2λnAyn,xn−x∗+λ2nAyn2

−xn−tn2+ 2λnAyn,xn−tn−λ2nAyn2

=xn−x∗2+ 2λn

Ayn,x∗−tn

−xn−tn2

=xn−x∗2−xn−tn2+ 2λn

Ayn−Ax∗,x∗−yn

+ 2λn

Ax∗,x∗−yn

+ 2λn

Ayn,yn−tn

.

(3.2)

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

Ayn−Ax∗,x∗−yn

≤0, Ax∗,x∗−yn

≤0. (3.3)

It follows from (3.2) and (3.3) that

_t_n₋_x∗2

≤xn−x∗

2

−xn−tn

2 + 2λn

Ayn,yn−tn

=xn−x∗

2

−xn−yn

+yn−tn

2 + 2λn

Ayn,yn−tn

=xn−x∗

2

−xn−yn

2

−2xn−yn,yn−tn

−yn−tn2+ 2λnAyn,yn−tn

=xn−x∗2−xn−yn2−yn−tn2+ 2

xn−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.

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By (3.4) and (3.6), we obtain

_t_n₋_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+

λ2nL2−1xn−yn2+

λ2nL2−1yn−tn2.

(3.7)

Sinceλn→0 asn→ ∞, there exists a positive integerN0such thatλ2nL2−1≤ −1/2 when

n≥N0. It follows from (3.7) that

_t_n₋_x∗_≤_x

n−x∗. (3.8)

By (3.1), we have

_x_n₊₁₋_x∗₌_α

nu+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_n_A

x−I−λnA

y2=(x−y)−λn(Ax−Ay)

2

=x−y2₋_2λ

nx−y,Ax−Ay

+λ2nAx−Ay2≤ x−y2+λ2nAx−Ay2

≤ x−y2₊_λ2

nL2x−y2=

1 +L2λ2n

x−y2_,

(3.10)

which implies that

_I₋_λ_n_A

x−I−λnA

y≤1 +Lλn

x−y. (3.11)

By (3.1) and (3.11), we have

_t_n₊₁₋_t_n₌_P_C

xn+1−λn+1Ayn+1

−PCxn−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

+λn+1Axn+1+Ayn+1+Axn+λnAyn

≤1 +λn+1Lxn+1−xn

+λn+1Axn+1+Ayn+1+Axn+λnAyn.

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Setxn+1=(1−βn)zn+βnxn. Then, we obtain

zn+1−zn=

αn+1u+γn+1Stn+1 1−βn+1 −

αnu+γnStn

1−βn

=

_α

n+1 1−βn+1−

αn

1−βn

u+ γn+1 1−βn+1

Stn+1−Stn

+

_γ

n+1 1−βn+1−

γn

1−βn

Stn.

(3.13)

Combining (3.12) and (3.13), we have

_z_n₊₁₋_z_n₋_x_n₊₁₋_x_n

≤ αn+1

1−βn+1− αn

1−βn

u+ γn+1 1−βn+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

Stn−xn+1−xn

≤ αn+1 1−βn+1−

αn

1−βn

u+Stn+

γn+1 1−βn+1λn+1L

_x_n₊₁₋_x_n

+ 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,

lim

n→∞xn+1−xn=nlim→∞

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

_x_n₊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

≤αnu−x∗

2

+βnxn−x∗

2

+γnxn−x∗

2 +λ2

nL2−1xn−yn

2 +λ2

nL2−1yn−tn

2

≤αnu−x∗2+xn−x∗2−1

2xn−yn 2

,

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which implies that

1

2xn−yn 2

≤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_C

xn−λnAxn

−PC

xn−λnAyn

≤λnAxn−Ayn≤λnLxn−yn−→0 asn−→ ∞,

_t_n₋_x_n_≤_t_n₋_y_n₊_y_n₋_x_n_−→₀ _as_n_{−→ ∞}_,

_Sy_n₋_x_n₊₁_≤_Sy_n₋_St_n₊_St_n₋_x_n₊₁_≤_y_n₋_t_n₊_α_n_St_n₋_u₊_β_n_St_n₋_x_n ≤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 +βnxn−tn+ 2tn−yn+αnStn−u+βnSxn−xn+xn+1−xn,

(3.20)

which implies that

_Sx_n₋_x_n_−→₀ _as_n_{−→ ∞}_. _(3.21)

Also we have

_St_n₋_t_n_≤_St_n₋_Sx_n₊_Sx_n₋_x_n₊_x_n₋_t_n ≤2tn−xn+Sxn−xn−→ ∞ asn−→ ∞.

(3.22)

Next we show that

lim sup

n→∞

u−z0,xn−z0

≤0, (3.23)

wherez0=PF(S)∩Ωu.

To show it, we choose a subsequence{tni}of{tn}such that

lim sup

n→∞

u−z0,Stn−z0

=lim

i→∞

u−z0,Stni−z0

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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

Stnizasi→ ∞. 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,tni−xni

λni +Ayni

=

v−tni,Av−Ayni−

tni−xni

λni

=v−tni,Av−Atni

+v−tni,Atni−Ayni

−

v−tni,

tni−xni

λni

≥v−tni,Atni

−v−tni,tni_λ−xni ni +

Ayni

.

(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−1_{0, 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 infi→∞ tni−Sz=lim infi→∞ tni−Stni+Stni−Sz ≤lim inf

i→∞ tni−Stni

₊_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

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Therefore,

_x_n₊₁₋_z₀2

=αnu+βnxn+γnStn−z0,xn+1−z0

=αnu−z0,xn+1−z0+βnxn−z0,xn+1−z0+γnStn−z0,xn+1−z0

≤1

2βnxn−z0 2

+xn+1−z02

+αn

u−z0,xn+1−z0

+1

2γntn−z0 2

+xn+1−z0 2

≤1

2

1−αnxn−z0 2

+xn+1−z0 2

+αn

u−z0,xn+1−z0

,

(3.31)

which implies that

_x_n₊₁₋_z₀2

≤1−αnxn−z0 2

+ 2αn

u−z0,xn+1−z0

, (3.32)

this together with (3.30) andLemma 2.3, we can obtain the conclusion. This completes

the proof.

We observe that some strong convergence theorems for the iterative scheme (3.1) were established under the assumption that the mappingAisα-inverse strongly monotone in [15].

Corollary3.2. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Let

Abe a monotoneL-Lipschitz continuous mapping ofCintoH such thatΩ= ∅. For fixed

u∈Hand givenx0∈Harbitrary, let the sequences{xn},{yn}be generated by

yn=PC

xn−λnAxn

,

xn+1=αnu+βnxn+γnPC

xn−λnAyn

, (3.33)

(C2) limn→∞αn=0,

_∞

n=0αn= ∞,

Then{xn}converges strongly toPΩu.

4. Applications

A mappingT:C→Cis called strictly pseudocontractive if there existskwith 0≤k <1 such that

Tx−T y2_≤_x₋_y2₊_k_(I₋_T)x₋_(I₋_T_)y2_, _(4.1)

for allx,y∈C. PutA=I−T, then we have

_(I₋_A)x₋_(I₋_A)y2

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On the other hand,

_(I₋_A)x₋_(I₋_A)y2

= x−y2₊_Ax₋_Ay2₋₂_x₋_y,_Ax₋_Ay_. _(4.3)

Hence, we have

x−y,Ax−Ay ≥1−k

2 Ax−Ay

2_≥_0. _(4.4)

Theorem 4.1. LetC be a closed convex subset of a real Hilbert spaceH. LetT be ak

-strictly pseudocontractive mapping ofCinto itself, and letSbe a nonexpansive mapping of

Cinto itself such thatF(T)∩F(S)= ∅. For fixedu∈Hand givenx0∈Harbitrary, let the

sequences{xn},{yn}be generated by

yn=

1−λn

xn+λnTxn,

xn+1=αnu+βnxn+γnS

1−λn

yn+λnT yn

, (4.5)

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

(C2) limn→∞αn=0,∞n=0αn= ∞,

Then{xn}converges strongly toPF(T)∩F(S)u.

Proof. PutA=I−T. ThenA is monotone. We haveF(T)=ΩandPC(xn−λnAxn)=

(1−λn)xn+λnTxn. So, byTheorem 3.1, we can obtain the desired result. This completes

the proof.

Theorem4.2. LetHbe a real Hilbert space. LetAbe a monotone mapping ofHinto itself

and letSbe a nonexpansive mapping ofHinto itself such thatA−1₀_∩_F(S)_{= ∅}_{. For fixed}

u∈Hand givenx0∈Harbitrary, let the sequences{xn},{yn}be generated by

yn=xn−λnAxn,

xn+1=αnu+βnxn+γnS

yn−λnAyn

, (4.6)

(C2) limn→∞αn=0,∞n=0αn= ∞,

Then{xn}converges strongly toPA−1₀_∩_F₍_S₎u.

Proof. SinceA−1₀₌_Ω_{, putting}_P

H=I, byTheorem 3.1, we can obtain the conclusion.

This completes the proof.

Acknowledgment

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Yonghong Yao: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China

Email address:[email protected]

Yeong-Cheng Liou: Department of Information Management, Cheng Shiu University, Niaosong Township, Kaohsiung 833, Taiwan

Email address:[email protected]

Jen-Chih Yao: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan