An Extragradient Method for Mixed Equilibrium Problems and Fixed Point Problems

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Volume 2009, Article ID 632819,15pages doi:10.1155/2009/632819

Research Article

An Extragradient Method for Mixed Equilibrium

Problems and Fixed Point Problems

Yonghong Yao,

1

_{Yeong-Cheng Liou,}

2

_{and Yuh-Jenn Wu}

3

1_{Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China}

2_{Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan}

3_{Department of Applied Mathematics, Chung Yuan Christian University, Chung Li 320, Taiwan}

Correspondence should be addressed to Yeong-Cheng Liou,simplex [email protected]

Received 2 November 2008; Revised 8 April 2009; Accepted 23 May 2009

Recommended by Nan-Jing Huang

The purpose of this paper is to investigate the problem of approximating a common element of the set of fixed points of a demicontractive mapping and the set of solutions of a mixed equilibrium problem. First, we propose an extragradient method for solving the mixed equilibrium problems and the fixed point problems. Subsequently, we prove the strong convergence of the proposed algorithm under some mild assumptions.

Copyrightq2009 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

LetH be a real Hilbert space and letCbe a nonempty closed convex subset ofH. Letϕ :

C → Rbe a real-valued function andΘ: C×C → Rbe an equilibrium bifunction, that is,

Θu, u 0 for eachu ∈ C. We consider the following mixed equilibrium problemMEP which is to findx∗∈Csuch that

Θx∗, yϕy−ϕx∗≥0, ∀y∈C. MEP

In particular, ifϕ≡0, this problem reduces to the equilibrium problemEP, which is to find

x∗∈Csuch that

Θx∗, y≥0, ∀y∈C. EP

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inequality problems, Nash equilibrium problems, and the equilibrium problems as special cases; see, for example, 1–5. Some methods have been proposed to solve the equilibrium problems, see, for example, 5–21.

In 2005, Combettes and Hirstoaga 6introduced an iterative algorithm of finding the best approximation to the initial data whenΓ/∅and proved a strong convergence theorem. Recently by using the viscosity approximation method S. Takahashi and W. Takahashi 8 introduced another iterative algorithm for finding a common element of the set of solutions ofEPand the set of fixed points of a nonexpansive mapping in a real Hilbert space. Let

S : C → H be a nonexpansive mapping and f : C → Cbe a contraction. Starting with arbitrary initialx1∈H, define the sequences{xn}and{un}recursively by

Θun, y

1

rn y−un, un−xn ≥

0, ∀y∈C,

xn1αnfxn 1−αnSun, ∀n≥0.

TT

S. Takahashi and W. Takahashi proved that the sequences {xn} and {un} defined byTT

converge strongly toz∈FixS∩Γwith the following restrictions on algorithm parameters

{αn}and{rn}:

ilimn→ ∞αn0 and∞n0αn∞;

iilim infn→ ∞rn>0;

iii A1:∞_n₀|αn1−αn|<∞; andR1: _∞

n0|rn1−rn|<∞.

Subsequently, some iterative algorithms for equilibrium problems and fixed point problems have further developed by some authors. In particular, Zeng and Yao 16 introduced a new hybrid iterative algorithm for mixed equilibrium problems and fixed point problems and Mainge and Moudafi 22introduced an iterative algorithm for equilibrium problems and fixed point problems.

On the other hand, for solving the equilibrium problemEP, Moudafi 23presented a new iterative algorithm and proved a weak convergence theorem. Ceng et al. 24introduced another iterative algorithm for finding an element of FixS∩Γ. LetS :C → Cbe ak-strict pseudocontraction for some 0 ≤ k < 1 such that FixS∩Γ/∅. For given x1 ∈ H, let the

sequences{xn}and{un}be generated iteratively by

Θun, y

1

rn y−un, un−xn ≥

0, ∀y∈C,

xn1αnun 1−αnSun, ∀n≥1,

CAY

where the parameters{αn}and{rn}satisfy the following conditions:

i{αn} ⊂ α, βfor someα, β∈k,1;

ii{rn} ⊂0,∞and lim infn→ ∞rn>0.

Then, the sequences{xn}and {un}generated byCAYconverge weakly to an element of

FixS∩Γ.

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Questions

1Could we weaken or remove the control conditioniiion algorithm parameters in S. Takahashi and W. Takahashi 8?

2Could we construct an iterative algorithm fork-strict pseudocontractions such that the strong convergence of the presented algorithm is guaranteed?

3Could we give some proof methods which are diﬀerent from those in 8,12,16,24? It is our purpose in this paper that we introduce a general iterative algorithm for approximating a common element of the set of fixed points of a demicontractive mapping and the set of solutions of a mixed equilibrium problem. Subsequently, we prove the strong convergence of the proposed algorithm under some mild assumptions. Our results give positive answers to the above questions.

2. Preliminaries

LetHbe a real Hilbert space with inner product ·,·and norm · . LetCbe a nonempty closed convex subset ofH.

LetT :C → Cbe a mapping. We use FixTto denote the set of the fixed points ofT. Recall what follows.

iT is called demicontractive if there exists a constant 0≤k <1 such that

Tx−x∗2≤ x−x∗2kx−Tx2 2.1

for allx∈Candx∗∈FixT, which is equivalent to

x−Tx, x−x∗ ≥ 1−k

2 x−Tx

2_. _2.2

For such case, we also say thatTis ak-demicontractive mapping.

iiT is called nonexpansive if

Tx−Ty ≤ x−y 2.3

for allx, y∈C.

iiiT is called quasi-nonexpansive if

Tx−x∗ ≤ x−x∗ 2.4

for allx∈Candx∗∈FixT.

ivT is called strictly pseudocontractive if there exists a constant 0≤k <1 such that

Tx−Ty2≤ x−y2kx−Tx−y−Ty2 2.5

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It is worth noting that the class of demicontractive mappings includes the class of the nonexpansive mappings, the quasi-nonexpansive mappings and the strictly pseudo-contractive mappings as special cases.

Let us also recall thatTis called demiclosed if for any sequence{xn} ⊂Handx∈H,

we have

xn−→xweakly, I−Txn −→0 strongly⇒x∈FixT. 2.6

It is well-known that the nonexpansive mappings, strictly pseudo-contractive mappings are all demiclosed. See, for example, 25–27.

An operatorA : C → H is said to beδ-strongly monotone if there exists a positive constantδsuch that

Ax−Ay, x−y ≥δx−y2 2.7

for allx, y∈C.

Now we concern the following problem: findx∗∈FixT∩Ωsuch that

Ax∗, x−x∗ ≥0, ∀x∈FixT∩Ω. 2.8

In this paper, for solving problem2.8with an equilibrium bifunctionΘ:C×C → R, we assume thatΘsatisfies the following conditions:

H1 Θis monotone, that is,Θx, y Θy, x≤0 for allx, y∈C;

H2for each fixedy∈C,x→Θx, yis concave and upper semicontinuous;

H3for eachx∈C,y→Θx, yis convex.

A mappingη : C×C → His called Lipschitz continuous, if there exists a constant

λ >0 such that

ηx, y ≤λx−y, ∀x, y∈C. 2.9

A diﬀerentiable functionK:C → Ron a convex setCis called

iη-convex if

Ky−Kx≥ Kx, ηy, x, ∀x, y∈C, 2.10

whereKis the Frechet derivative ofKatx;

iiη-strongly convex if there exists a constantσ >0 such that

Ky−Kx− Kx, ηy, x ≥σ

2

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LetCbe a nonempty closed convex subset of a real Hilbert spaceH,ϕ : C → Rbe real-valued function andΘ : C×C → Rbe an equilibrium bifunction. Letr be a positive number. For a given pointx∈C, the auxiliary problem forMEPconsists of findingy∈C

such that

Θy, zϕz−ϕy 1

r K

_y₋_K_x_{, η}_z,_y_≥₀_, _∀_z_∈_C. _2.12

LetSr : C → Cbe the mapping such that for eachx ∈ C,Srxis the solution set of the

auxiliary problem, that is,∀x∈C,

Srx

y∈C:Θy, zϕz−ϕy1 r

Ky−Kx, ηz, y ≥0,∀z∈C

. 2.13

We need the following important and interesting result for proving our main results.

Lemma 2.1 16,28. LetCbe a nonempty closed convex subset of a real Hilbert spaceHand let

ϕ:C → Rbe a lower semicontinuous and convex functional. LetΘ:C×C → Rbe an equilibrium bifunction satisfying conditions (H1)–(H3). Assume what follows.

iη:C×C → His Lipschitz continuous with constantλ >0such that

aηx, y ηy, x 0,∀x, y∈C,

bη·,·is aﬃne in the first variable,

cfor each fixedy∈C, x→ηy, xis sequentially continuous from the weak topology to the weak topology.

iiK:C → Risη-strongly convex with constantσ >0and its derivativeKis sequentially continuous from the weak topology to the strong topology.

iiiFor each x ∈ C, there exist a bounded subset Dx ⊂ C and zx ∈ C such that for any y∈C\Dx,

Θy, zx

ϕzx−ϕ

y1 r K

_y₋_K_x_{, η}_z

x, y

<0. 2.14

Then there hold the following:

iSris single-valued;

iiSris nonexpansive ifKis Lipschitz continuous with constantν >0such thatσ≥λνand

Kx1−Kx2, ηu1, u2 ≥

Ku1−Ku2, ηu1, u2 , ∀x1, x2∈C×C, 2.15

whereuiSrxifori1,2;

iiiFixSr Ω;

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3. Main Results

LetHbe a real Hilbert space,ϕ:H → Rbe a lower semicontinuous and convex real-valued function,Θ : H×H → Rbe an equilibrium bifunction. LetA : H → H be a mapping andT :H → Hbe a mapping. In this section, we first introduce the following new iterative algorithm.

Algorithm 3.1. Letrbe a positive parameter. Let{λn}be a sequence in 0,∞and{αn}be a

sequence in 0,1. Define the sequences{xn},{yn},and{zn}by the following manner:

x0∈Cchosen arbitrarily,

Θzn, x ϕx−ϕzn

1

r K

_z

n−Kxn, ηx, zn ≥0, ∀x∈C,

ynzn−λnAzn,

xn1 1−αnynαnTyn.

3.1

Now we give a strong convergence result concerningAlgorithm 3.1as follows.

Theorem 3.2. LetHbe a real Hilbert space. Letϕ:H → Rbe a lower semicontinuous and convex functional. LetΘ:H×H → Rbe an equilibrium bifunction satisfying conditions (H1)–(H3). Let

A:H → Hbe anL-Lipschitz continuous andδ-strongly monotone mapping andT :H → Hbe a demiclosed andk-demicontractive mapping such thatFixT∩Ω/∅. Assume what follows.

iη:H×H → His Lipschitz continuous with constantλ >0such that

aηx, y ηy, x 0,∀x, y∈H,

cfor each fixedy∈H,x→ηy, xis sequentially continuous from the weak topology to the weak topology.

iiK :H → Risη-strongly convex with constantσ > 0and its derivativeKis not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constantν >0such thatσ≥λν.

iiiFor eachx ∈ H; there exist a bounded subsetDx ⊂ H andzx ∈ H such that, for any y /∈Dx,

Θy, zx

ϕzx−ϕ

y1 r

Ky−Kx, ηzx, y <0. 3.2

ivαn∈ γ,1−k/2for someγ >0,limn→ ∞λn0and∞n0λn ∞.

Then the sequences{xn},{yn}, and{zn}generated by3.1converge strongly tox∗which solves the

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Proof. First, we prove that{xn},{yn}, and{zn}are all bounded. Without loss of generality,

we may assume that 0< δ < L. Givenμ∈0,2δ/L2_and_{x, y}_∈_H_{, we have}

μA−Ix−μA−Iy2μ2Ax−Ay2x−y2−2μ Ax−Ay, x−y

≤μ2L2x−y2x−y2−2μδx−y2

1−2μδμ2L2x−y2,

3.3

that is,

μA−Ix−μA−Iy ≤

1−2μδμ2_L2_x₋_y_. _3.4

Takex∗∈FixT∩Ω. From3.1, we have

yn1−x∗−λn1Ax∗zn1−λn1Azn1−x∗−λn1Ax∗

1−λn1

μ

zn1−x∗−λn1

μ

μA−Izn1−

μA−Ix∗

≤

1−λn1

μ

zn1−x∗ λn1

μ

μA−Izn1−

μA−Ix∗.

3.5

Therefore,

yn1−x∗−λn1Ax∗ ≤

1−λn1ω

μ

zn1−x∗, 3.6

whereω1−

1−2μδμ2_L2_∈₀_,₁_.

Note thatzn1 Srxn1andSr are firmly nonexpansive. Hence, we have

zn1−x∗2 Srxn1−Srx∗2

≤ Srxn1−Srx∗, xn1−x∗

zn1−x∗, xn1−x∗

1

2

zn1−x∗2xn1−x∗2− xn1−zn12

,

3.7

which implies that

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From2.2and3.1, we have

xn1−x∗2 1−αnynαnTyn−x∗2

yn−x∗−αnyn−Tyn2

yn−x∗2−2αn yn−Tyn, yn−x∗αn2yn−Tyn2

≤ yn−x∗2−2αn

1−k

2 yn−Tyn

2_α2

nyn−Tyn2

yn−x∗2−αn1−k−αnyn−Tyn2

≤ yn−x∗2.

3.9

From3.6–3.9, we have

yn1−x∗ ≤ yn1−x∗−λn1Ax∗λn1Ax∗

≤

1− λn1ω

μ

zn1−x∗λn1Ax∗

≤

1− λn1ω

μ

xn1−x∗λn1Ax∗

≤

1− λn1ω

μ

yn−x∗λn1Ax∗

1− λn1ω

μ

yn−x∗λn1ω μ

_μ

ωAx

∗

≤max

yn−x∗,

μAx∗ ω

≤ · · ·

≤max

y0−x∗,μAx

∗

ω

.

3.10

This implies that{yn}is bounded, so are{xn}and{zn}.

From3.1, we can writeyn−Tyn 1/αnyn−xn1. Thus, from3.9, we have

xn1−x∗2≤ yn−x∗2−αn1−k−αnyn−Tyn2

≤ yn−x∗2−1−k−αn

αn yn−xn1

2

.

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Sinceαn∈0,1−k/2,1−k−αn/αn≥1. Therefore, from3.8and3.11, we obtain

xn1−x∗2≤ yn−x∗2− yn−xn12

zn−x∗−λnAzn2− zn−xn1−λnAzn2

zn−x∗2−2λn Azn, zn−x∗λ2nAzn2

− zn−xn122λn Azn, zn−xn1 −λ2nAzn2

zn−x∗2−2λn xn1−x∗, Azn − xn1−zn2

≤ xn−x∗2− xn−zn2−2λn xn1−x∗, Azn − xn1−zn2.

3.12

We note that{xn}and{zn}are bounded. So there exists a constantM≥0 such that

| xn1−x∗, Azn| ≤M ∀n≥0. 3.13

Consequently, we get

xn1−x∗2− xn−x∗2xn1−zn2xn−zn2≤2Mλn. 3.14

Now we divide two cases to prove that{xn}converges strongly tox∗.

Case 1. Assume that the sequence{xn−x∗}is a monotone sequence. Then{xn−x∗}is

convergent. Setting limn→ ∞xn−x∗d.

iIfd0, then the desired conclusion is obtained.

iiAssume thatd >0. Clearly, we have

xn1−x∗2− xn−x∗2−→0, 3.15

this together withλn → 0 and3.14implies that

xn1−zn2xn−zn2−→0, 3.16

that is to say

xn1−zn −→0, xn−zn −→0. 3.17

Let z ∈ H be a weak limit point of{znk}. Then there exists a subsequence of{znk}, still denoted by{znk}which weakly converges toz. Noting thatλn → 0, we also have

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Combining3.1and3.17, we have

Tynk−ynk 1

αnk

ynk−xnk1

1

αnk

xnk1−znk λnkAznk

≤ xnk1−znkλnkAznk

−→0.

3.19

SinceT is demiclosed, then we obtainz∈FixT.

Next we show thatz∈Ω. Sincezn Srxn, we derive

Θzn, x ϕx−ϕzn 1

r K

_z

n−Kxn, ηx, zn ≥0, ∀x∈C. 3.20

From the monotonicity ofΘ, we have

1

r K

_z

n−Kxn, ηx, znϕx−ϕzn≥ −Θzn, x≥Θx, zn, 3.21

and hence

Kznk−Kxnk

r , ηx, znk

ϕx−ϕznk≥Θx, znk. 3.22

SinceKznk−Kxnk/r → 0 andznk → zweakly, from the weak lower semicontinuity ofϕandΘx, yin the second variabley, we have

Θx, z ϕz−ϕx≤0, 3.23

for allx∈C. For 0 < t≤1 andx∈C, letxttx 1−tz. Sincex∈Candz∈C, we have xt ∈Cand henceΘxt, z ϕz−ϕxt ≤ 0. From the convexity of equilibrium bifunction

Θx, yin the second variabley, we have

0 Θxt, xt ϕxt−ϕxt

≤tΘxt, x 1−tΘxt, z tϕx 1−tϕz−ϕxt

≤tΘxt, x ϕx−ϕxt

,

3.24

and henceΘxt, x ϕx−ϕxt≥0. Then, we have

Θz, x ϕx−ϕz≥0 3.25

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Therefore, we have

z∈FixT∩Ω. 3.26

Thus, ifx∗is a solution of problem2.8, we have

lim inf

k→ ∞ znk−x

∗_{, Ax}∗ _z₋_x∗_{, Ax}∗_≥₀_. _3.27

Suppose that there exists another subsequence{zni}which weakly converges toz. It is easily checked thatz∈FixT∩Ωand

lim inf

i→ ∞ zni−x∗, Ax∗

z−x∗, Ax∗ ≥0. 3.28

Therefor, we have

lim inf

n→ ∞ zn−x

∗_{, Ax}∗_≥₀_. _3.29

SinceAisδ-strongly monotone, we have

xn1−x∗, Azn ≥δzn−x∗2 zn−x∗, Ax∗ xn1−zn, Azn. 3.30

By3.17–3.30, we get

lim inf

n→ ∞ xn1−x ∗_{, Az}

n ≥δd2. 3.31

From3.12, for 0< < δd2_{, we deduce that there exists a positive integer number}_n 0 large

enough, whenn≥n0,

xn1−x∗2− xn−x∗2≤ −2λn

δd2−. 3.32

This implies that

xn1−x∗2− xn0−x∗

2_{≤ −}₂

δd2− n

kn0

λk. 3.33

Since∞_n₀λn ∞and{xn}is bounded, hence the last inequality is a contraction. Therefore, d0, that is to say,xn → x∗.

Case 2. Assume that{xn −x∗} is not a monotone sequence. SetΓn xn −x∗2 and let τ :N → Nbe a mapping for alln≥n0by

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Clearly,τis a nondecreasing sequence such thatτn → ∞asn → ∞andΓτn≤Γτn1for

n≥n0. From3.14, we have

xτn1−zτn2xτn−zτn2≤2Mλτn−→0, 3.35

thus

xτn1−zτn −→0, xτn−zτn −→0. 3.36

Therefore,

xτn1−xτn −→0. 3.37

SinceΓτn≤Γτn1, for alln≥n0, from3.12, we get

0≤ xτn1−x∗2− xτn−x∗2xτn1−zτn2xτn−zτn2

≤ −2λτnxτn1−x∗, Azτn ,

3.38

which implies that

xτn1−x∗, Azτn ≤0 ∀n≥n0. 3.39

Since{zτn}is bounded, there exists a subsequence of{zτn}, still denoted by{zτn}which

converges weakly toq∈H. It is easily checked thatq∈FixT∩Ω. Furthermore, we observe that

δzτn−x∗2≤

zτn−x∗, Azτn−Ax∗

xτn1−x∗, Azτn zτn−xτn1, Azτn −zτn−x∗, Ax∗ .

3.40

Hence, for alln≥n0,

δzτn−x∗2≤zτn−xτn1, Azτn −zτn−x∗, Ax∗ . 3.41

Therefore

lim sup

n→ ∞ zτn−x

∗2_{≤ −}1

δ

q−x∗, Ax∗ ≤0, 3.42

which implies that

lim

n→ ∞zτn−x

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Thus,

lim

n→ ∞xτn−x

∗₀_. _3.44

It is immediate that

lim

n→ ∞Γτnnlim→ ∞Γτn10. 3.45

Furthermore, for n ≥ n0, it is easily observed that Γn ≤ Γτn1 if n /τn i.e.,τn < n,

becauseΓj>Γj1forτn 1≤j≤n. As a consequence, we obtain for alln≥n0,

0≤Γn≤max

Γτn,Γτn1

Γτn1. 3.46

Hence limn→ ∞Γn 0, that is,{xn}converges strongly tox∗. Consequently, it easy to prove

that{yn}and{zn}converge strongly tox∗. This completes the proof.

Remark 3.3. The advantages of these results in this paper are that less restrictions on the parameters{λn}are imposed.

As direct consequence ofTheorem 3.2, we obtain the following.

Corollary 3.4. LetHbe a real Hilbert space. Letϕ:H → Rbe a lower semicontinuous and convex functional. LetΘ:H×H → Rbe an equilibrium bifunction satisfying conditions (H1)–(H3). Let

A:H → Hbe anL-Lipschitz continuous andδ-strongly monotone mapping andT :H → Hbe a nonexpansive mapping such thatFixT∩Ω/∅. Assume what follows.

iη:H×H → His Lipschitz continuous with constantλ >0such that;

aηx, y ηy, x 0,∀x, y∈H,

Θy, zx

ϕzx−ϕ

y1 r

Ky−Kx, ηzx, y <0. 3.47

ivαn∈ γ,1−k/2for someγ >0,limn→ ∞λn 0and _∞

n0λn∞.

Then the sequences{xn},{yn}, and{zn}generated by3.1converge strongly tox∗which solves the

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Corollary 3.5. LetHbe a real Hilbert space. Letϕ:H → Rbe a lower semicontinuous and convex functional. LetΘ:H×H → Rbe an equilibrium bifunction satisfying conditions (H1)–(H3). Let

A:H → Hbe anL-Lipschitz continuous andδ-strongly monotone mapping andT :H → Hbe a strictly pseudo-contractive mapping such thatFixT∩Ω/∅. Assume what follows.

iη:H×H → His Lipschitz continuous with constantλ >0such that

aηx, y ηy, x 0,∀x, y∈H,

Θy, zx

ϕzx−ϕ

y1 r

Ky−Kx, ηzx, y <0. 3.48

ivαn∈ γ,1−k/2for someγ >0,limn→ ∞λn 0and _∞

n0λn∞.

Then the sequences{xn},{yn}and{zn}generated by3.1converge strongly tox∗which solves the

problem2.8providedSr is firmly nonexpansive.

Acknowledgments

The authors are extremely grateful to the anonymous referee for his/her useful comments and suggestions. The first author was partially supposed by National Natural Science Foundation of China Grant 10771050. The second author was partially supposed by the Grant NSC 97-2221-E-230-017.

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