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Volume 2009, Article ID 632819, 15 pagesdoi:10.1155/2009/632819 Research Article An Extragradient Method for Mixed Equilibrium Problems and Fixed Point Problems 1 Department of Mathemati

Volume 2009, Article ID 632819, 15 pages

doi:10.1155/2009/632819

Research Article

An Extragradient Method for Mixed Equilibrium Problems and Fixed Point Problems

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 liou@hotmail.com

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

Copyrightq 2009 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

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H Let ϕ :

C → R be a real-valued function and Θ : C × C → R be an equilibrium bifunction, that is,

Θu, u  0 for each u ∈ C We consider the following mixed equilibrium problem MEP which is to find x∗∈ C such that

Θx∗, y

 ϕy

− ϕx∗ ≥ 0, ∀y ∈ C. MEP

In particular, if ϕ ≡ 0, this problem reduces to the equilibrium problem EP, which is to find

x∗∈ C such that

Θx∗, y

≥ 0, ∀y ∈ C. EP Denote the set of solutions ofMEP by Ω and the set of solutions of EP by Γ The mixed equilibrium problems include fixed point problems, optimization problems, variational

inequality problems, Nash equilibrium problems, and the equilibrium problems as special cases; see, for example, 1 5

problems, see, for example, 5 21

In 2005, Combettes and Hirstoaga 6

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

ofEP and 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 → C be a contraction Starting with

arbitrary initial x1∈ H, define the sequences {x n } and {u n} recursively by

Θu n , y

 1

r n y − u n , u n − x n

x n1  α n f x n   1 − α n Su n , ∀n ≥ 0.

TT

S Takahashi and W Takahashi proved that the sequences {x n } and {u n} defined by TT

converge strongly to z ∈ FixS ∩ Γ with the following restrictions on algorithm parameters {α n } and {r n}:

i limn → ∞ α n 0 and∞n0 α n ∞;

ii lim infn → ∞ r n > 0;

iii A1:∞

n0 |α n1 − α n | < ∞; and R1:∞

n0 |r n1 − r n | < ∞.

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 22

problems and fixed point problems

On the other hand, for solving the equilibrium problemEP, Moudafi 23

new iterative algorithm and proved a weak convergence theorem Ceng et al 24

another iterative algorithm for finding an element of FixS ∩ Γ Let S : C → C be a k-strict pseudocontraction for some 0 ≤ k < 1 such that FixS ∩ Γ / ∅ For given x1 ∈ H, let the

sequences{x n } and {u n} be generated iteratively by

Θu n , y

 1

r n y − u n , u n − x n

x n1  α n u n  1 − α n Su n , ∀n ≥ 1,

CAY

where the parameters{α n } and {r n} satisfy the following conditions:

i {α n

ii {r n } ⊂ 0, ∞ and lim inf n → ∞ r n > 0.

Then, the sequences{x n } and {u n} generated by CAY converge weakly to an element of FixS ∩ Γ

At this point, we should point out that all of the above results are interesting and valuable At the same time, these results also bring us the following conjectures

1 Could we weaken or remove the control condition iii on algorithm parameters in

S Takahashi and W Takahashi 8

2 Could we construct an iterative algorithm for k-strict pseudocontractions such that

the strong convergence of the presented algorithm is guaranteed?

3 Could 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

closed convex subset of H.

Let T : C → C be a mapping We use FixT to denote the set of the fixed points of T.

Recall what follows

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

Tx − x∗2≤ x − x∗2 kx − Tx2 2.1

for all x ∈ C and x∗∈ FixT, which is equivalent to

x − Tx, x − x∗ 1− k

2 x − Tx2. 2.2

For such case, we also say that T is a k-demicontractive mapping.

ii T is called nonexpansive if

Tx − Ty ≤ x − y 2.3

for all x, y ∈ C.

iii T is called quasi-nonexpansive if

Tx − x∗ ≤ x − x∗ 2.4

for all x ∈ C and x∗∈ FixT.

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

Tx − Ty2≤ x − y2 kx − Tx − y − Ty2 2.5

for all x, y ∈ C.

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 that T is called demiclosed if for any sequence {x n } ⊂ H and x ∈ H,

we have

x n −→ x weakly, I − Tx n −→ 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 operator A : C → H is said to be δ-strongly monotone if there exists a positive constant δ such that

for all x, y ∈ C.

Now we concern the following problem: find x∗∈ FixT ∩ Ω such that

In this paper, for solving problem2.8 with an equilibrium bifunction Θ : C × C → R,

we assume thatΘ satisfies the following conditions:

H1 Θ is monotone, that is, Θx, y  Θy, x ≤ 0 for all x, y ∈ C;

H2 for each fixed y ∈ C, x → Θx, y is concave and upper semicontinuous;

H3 for each x ∈ C, y → Θx, y is convex.

A mapping η : C × C → H is called Lipschitz continuous, if there exists a constant

λ > 0 such that

ηx, y

 ≤ λx − y, ∀x, y ∈ C. 2.9

A differentiable function K : C → R on a convex set C is called

i η-convex if

y

− Kx ≥ Kx, ηy, x

2.10

where Kis the Frechet derivative of K at x;

ii η-strongly convex if there exists a constant σ > 0 such that

y

− Kx − Kx, ηy, x σ

2



x − y2, ∀x, y ∈ C. 2.11

Let C be a nonempty closed convex subset of a real Hilbert space H, ϕ : C → R be

real-valued function andΘ : C × C → R be an equilibrium bifunction Let r be a positive

number For a given point x ∈ C, the auxiliary problem for MEP consists of finding y ∈ C

such that

Θy, z

 ϕz − ϕy

 1

r K

y

− Kx, ηz, y

2.12

Let S r : C → C be the mapping such that for each x ∈ C, S r x is the solution set of the

auxiliary problem, that is,∀x ∈ C,

S r x 



y, z

 ϕz − ϕy

1

r



K

y

− 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

ϕ : C → R be a lower semicontinuous and convex functional Let Θ : C × C → R be an equilibrium

bifunction satisfying conditions (H1)–(H3) Assume what follows.

i η : C × C → H is Lipschitz continuous with constant λ > 0 such that

a ηx, y  ηy, x  0, ∀x, y ∈ C,

b η·, · is affine in the first variable,

c for each fixed y ∈ C, x → ηy, x is sequentially continuous from the weak topology

to the weak topology.

ii K : C → R is η-strongly convex with constant σ > 0 and its derivative Kis sequentially continuous from the weak topology to the strong topology.

iii For each x ∈ C, there exist a bounded subset D x ⊂ C and z x ∈ C such that for any

y ∈ C \ D x ,

Θy, z x



 ϕz x  − ϕy

1

r K

y

− Kx, ηz x , y

2.14

Then there hold the following:

i S r is single-valued;

ii S r is nonexpansive if Kis Lipschitz continuous with constant ν > 0 such that σ ≥ λν and



Kx1 − Kx2, ηu1, u2 ≥Ku1 − Ku2, ηu1, u2 , ∀x1, x2 ∈ C × C, 2.15

where u i  S r x i  for i  1, 2;

iii FixS r   Ω;

iv Ω is closed and convex.

3 Main Results

Let H be a real Hilbert space, ϕ : H → R be a lower semicontinuous and convex real-valued

function,Θ : H × H → R be an equilibrium bifunction Let A : H → H be a mapping

and T : H → H be a mapping In this section, we first introduce the following new iterative

algorithm

sequence in 0, 1 Define the sequences {x n }, {y n }, and {z n} by the following manner:

x0∈ C chosen arbitrarily, Θz n , x   ϕx − ϕz n 1

r Kz n  − Kx n , ηx, z n

y n  z n − λ n Az n ,

x n1  1 − α n y n  α n Ty n

3.1

Now we give a strong convergence result concerningAlgorithm 3.1as follows

Theorem 3.2 Let H be a real Hilbert space Let ϕ : H → R be a lower semicontinuous and convex

A : H → H be an L-Lipschitz continuous and δ-strongly monotone mapping and T : H → H be a

i η : H × H → H is Lipschitz continuous with constant λ > 0 such that

a ηx, y  ηy, x  0, ∀x, y ∈ H,

b η·, · is affine in the first variable,

c for each fixed y ∈ H, x → ηy, x is sequentially continuous from the weak topology

to the weak topology.

ii K : H → R is η-strongly convex with constant σ > 0 and its derivative Kis not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant ν > 0 such that σ ≥ λν.

iii For each x ∈ H; there exist a bounded subset D x ⊂ H and z x ∈ H such that, for any

y / ∈ D x ,

Θy, z x



 ϕz x  − ϕy

1

r



K

y

− Kx, ηz x , y

iv α n n → ∞ λ n  0 and∞n0 λ n  ∞.

Then the sequences {x n }, {y n }, and {z n } generated by 3.1 converge strongly to x∗which solves the

Proof First, we prove that {x n }, {y n }, and {z n} are all bounded Without loss of generality,

we may assume that 0 < δ < L Given μ ∈ 0, 2δ/L2 and x, y ∈ H, we have

μA − Ix − μA − Iy2 μ2Ax − Ay2 x − y2

≤ μ2L2x − y2 x − y2− 2μδx − y2

1− 2μδ  μ2L2

x − y2

,

3.3

that is,

μA − I

y ≤

1− 2μδ  μ2L2x − y. 3.4

Take x∗∈ FixT ∩ Ω From 3.1, we have

y n1 − x∗− λ n1 Ax∗  z n1 − λ n1 Az n1  − x∗− λ n1 Ax∗



1−λ n1

μ



z n1 − x∗ −λ n1

μ



z n1−μA − I

x∗ 

≤ 1−λ n1

μ



z n1 − x∗ λ n1

μ μA − I

z n1−μA − I

x∗.

3.5 Therefore,

y n1 − x∗− λ n1 Ax∗ ≤ 1−λ n1 ω

μ



z n1 − x∗, 3.6

where ω  1 −

1− 2μδ  μ2L2∈ 0, 1.

Note that z n1  S r x n1 and S r are firmly nonexpansive Hence, we have

z n1 − x∗2  S r x n1 − S r x∗2

≤ S r x n1 − S r x∗, x n1 − x∗

 z n1 − x∗, x n1 − x∗

 1 2



z n1 − x∗2 x n1 − x∗2− x n1 − z n12

,

3.7

which implies that

z n1 − x∗2≤ x n1 − x∗2− x n1 − z n12. 3.8

From2.2 and 3.1, we have

x n1 − x∗2  1 − α n y n  α n Ty n − x∗2

 y n − x∗ − α n y n − Ty n2

 y n − x∗2− 2α n y n − Ty n , y n − x∗ 2

n y n − Ty n2

≤ y n − x∗2− 2α n

1− k

2 y n − Ty n2 α2

n y n − Ty n2

 y n − x∗2− α n 1 − k − α n y n − Ty n2

≤ y n − x∗2.

3.9

From3.6–3.9, we have

y n1 − x∗ ≤ y n1 − x∗− λ n1 Ax∗  λ n1 Ax∗

≤ 1− λ n1 ω

μ



z n1 − x∗  λ n1 Ax∗

≤ 1− λ n1 ω

μ



x n1 − x∗  λ n1 Ax∗

≤ 1− λ n1 ω

μ



y n − x∗  λ n1 Ax∗

 1− λ n1 ω

μ



y n − x∗ λ n1 ω

μ

 μ

ω Ax∗

≤ max



y n − x∗, μAx∗

ω

≤ · · ·

≤ max



y0− x∗, μAx∗

3.10

This implies that{y n } is bounded, so are {x n } and {z n}

From3.1, we can write yn − Ty n  1/α n y n − x n1 Thus, from 3.9, we have

x n1 − x∗2≤ y n − x∗2− α n 1 − k − α n y n − Ty n2

≤ y n − x∗2−1− k − α n

α n y n − x n12

.

3.11

Since α n n /α n≥ 1 Therefore, from 3.8 and 3.11, we obtain

x n1 − x∗2≤ y n − x∗2− y n − x n12

 z n − x∗− λ n Az n2− z n − x n1 − λ n Az n2

 z n − x∗2− 2λ n Az n , z n − x∗ 2

n Az n2

− z n − x n12 2λ n Az n , z n − x n1 2n Az n2

 z n − x∗2− 2λ n x n1 − x∗, Az n n1 − z n2

≤ x n − x∗2− x n − z n2− 2λ n x n1 − x∗, Az n n1 − z n2.

3.12

We note that{x n } and {z n } are bounded So there exists a constant M ≥ 0 such that

| x n1 − x∗, Az n 3.13 Consequently, we get

x n1 − x∗2− x n − x∗2 x n1 − z n2 x n − z n2≤ 2Mλ n 3.14 Now we divide two cases to prove that{x n } converges strongly to x∗

convergent Setting limn → ∞ x n − x∗  d.

i If d  0, then the desired conclusion is obtained.

ii Assume that d > 0 Clearly, we have

x n1 − x∗2− x n − x∗2−→ 0, 3.15

this together with λ n → 0 and 3.14 implies that

x n1 − z n2 x n − z n2−→ 0, 3.16 that is to say

x n1 − z n  −→ 0, x n − z n  −→ 0. 3.17

Let z ∈ H be a weak limit point of {z nk } Then there exists a subsequence of {z nk}, still denoted by{z nk } which weakly converges to z Noting that λ n → 0, we also have

y nk  z nk − λ nk Az nk −→ z weakly. 3.18

Combining3.1 and 3.17, we have

Ty nk − y nk  1

α nk y nk − x nk1

 1

α nk x nk1− z nk  λ nk Az nk

≤ x nk1− z nk   λ nk Az nk

−→ 0.

3.19

Since T is demiclosed, then we obtain z ∈ FixT.

Next we show that z ∈ Ω Since z n  S r x n, we derive

Θz n , x   ϕx − ϕz n  1

r Kz n  − Kx n , ηx, z n 3.20 From the monotonicity ofΘ, we have

1

r Kz n  − Kx n , ηx, z n n  ≥ −Θz n , x  ≥ Θx, z n , 3.21 and hence



Kz nk  − Kx nk

r , η x, z nk



 ϕx − ϕz nk  ≥ Θx, z nk . 3.22

SinceKz nk  − Kx nk /r → 0 and z nk → z weakly, from the weak lower semicontinuity

of ϕ and Θx, y in the second variable y, we have

Θx, z  ϕz − ϕx ≤ 0, 3.23

for all x ∈ C For 0 < t ≤ 1 and x ∈ C, let x t  tx  1 − tz Since x ∈ C and z ∈ C, we have

x t ∈ C and hence Θx t , z  ϕz − ϕx t ≤ 0 From the convexity of equilibrium bifunction

Θx, y in the second variable y, we have

0 Θx t , x t   ϕx t  − ϕx t

≤ tΘx t , x   1 − tΘx t , z   tϕx  1 − tϕz − ϕx t

≤ tΘx t , x   ϕx − ϕx t,

3.24

and henceΘx t , x  ϕx − ϕx t ≥ 0 Then, we have

Θz, x  ϕx − ϕz ≥ 0 3.25

for all x ∈ C and hence z ∈ Ω.

Therefore, we have

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

lim inf

Suppose that there exists another subsequence{z ni } which weakly converges to z It is easily

checked that z∈ FixT ∩ Ω and

lim inf

i → ∞ z ni − x∗, Ax∗ 

z− x∗, Ax∗

≥ 0. 3.28 Therefor, we have

lim inf

Since A is δ-strongly monotone, we have

x n1 − x∗, Az n n − x∗2 z n − x∗, Ax∗ n1 − z n , Az n 3.30

By3.17–3.30, we get

lim inf

From3.12, for 0 < < δd2, we deduce that there exists a positive integer number n0 large

enough, when n ≥ n0,

x n1 − x∗2− x n − x∗2≤ −2λ n



δd2− . 3.32 This implies that

x n1 − x∗2− x n0− x∗2≤ −2δd2− n

kn0

λ k 3.33

Since∞

n0 λ n  ∞ and {x n} is bounded, hence the last inequality is a contraction Therefore,

d  0, that is to say, x n → x∗

τ n  max{k ∈ N : k ≤ n, Γ k≤ Γk1 }. 3.34

Since α n n /α n≥ Therefore, from 3.8 and 3.11, we... n12. 3.8

From2.2 and 3.1, we have

x n1... nk −→ z weakly. 3.18

Combining3.1 and 3.17, we have

Ty nk