An Iteration Method for Common Solution of a System of Equilibrium Problems in Hilbert Spaces

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Volume 2011, Article ID 780764,15pages doi:10.1155/2011/780764

Research Article

An Iteration Method for Common Solution of

a System of Equilibrium Problems in Hilbert Spaces

Jong Kyu Kim

1

_{and Nguyen Buong}

2

1_{Department of Mathematics Education, Kyungnam University, Masan Kyunganm 631-701,} Republic of Korea

2_{Department of Mathematics, Vietnamse Academy of Science and Technology,}

Institute of Information Technology, 18, Hoang Quoc Viet, q. Cau Giay, Hanoi 122100, Vietnam

Correspondence should be addressed to Jong Kyu Kim,[email protected]

Received 11 December 2010; Revised 3 March 2011; Accepted 4 March 2011

Academic Editor: Qamrul Hasan Ansari

Copyrightq2011 J. K. Kim and N. Buong. 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.

The strong convergence theorem is proved for finding a common solution for a system of equilibrium problems: findu∗ ∈ S : ∩N_i₁EPFi,EPFi : {z ∈ C : Fiz, v ≥ 0∀v ∈ C}, i 1, . . . , N,whereCis a closed convex subset of a Hilbert spaceHandFiareNbifunctions from C×CintoRgiven exactly or approximatively. As an application, finding a common solution for a system of variational inequality problems is given.

1. Introduction

LetHbe a real Hilbert space with the scalar product and the norm denoted by the symbols

·,· and · , respectively. Let C be a nonempty closed convex subset of H, and let

Fii1, . . . , NbeNbifunctions fromC×CintoR. In this paper, we consider the system of

equilibrium problems:

findu∗∈S:∩_iN₁EPFi,

EPFi:{z∈C:Fiz, v≥0∀v∈C}, i1, . . . , N.

1.1

We assume thatS /∅and the bifunctionsFisatisfy the following conditions.

Condition 1. The bifunctionFsatisfies the following conditions:

A1Fu, u 0 for allu∈C.

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A3For everyu∈C,Fu,·:C →Ris lower semicontinuous and convex.

A4lim_t→0F1−tutz, v≤Fu, vfor allu, z, v∈C×C×C.

Definition 1.1. A mappingAofCintoHis called monotone if

Ax−Ay, x−y≥0, 1.2

for allx, y∈C.

Now, we consider the variational inequality problem: findu∗∈Csuch that

Au∗_{, x}₋_u∗_≥₀_, _1.3

for allx∈C. We denote VIC, Athe set of solutions of the variational inequality problem.

Definition 1.2. A mapping T of C into H is called k-strictly pseudocontractive in the terminology of Browder and Petryshyn1, if there exists a numberk∈0,1such that

Tx−Ty 2_≤_x₋_y 2_k _I₋_T_x₋_I₋_T_y 2_, _1.4

whereIis the identity operator inH.

The above inequality is equivalent to

Ax−Ay, x−y≥λ Ax−Ay 2_, _1.5

where the operatorA:I−Tisλ 1−k/2-inverse strongly monotonehence monotone and Lipschitz continuous with the Lipschitz constant 2/1−k. Clearly, whenk 0, T is nonexpansive, that is,

Tx−Ty ≤ x−y 1.6

for allx, y ∈ DT, the domain ofT. It means that the class of k-strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings. Denote byFTthe set of fixed points of the operatorTinC, that is,

FT {x∈C:xTx}. 1.7

If N 1, then 1.1 is a single equilibrium problem 2, 3 to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, Nash equilibria in noncooperative games, vector equilibrium problems, as well as certain fixed point problems.

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If N > 1, then 1.1 is a problem of finding a common solution for a system of equilibrium problems which is studied firstly in 5 cf. 16 under the condition that

Fi i 1, . . . , N are bounded, Fr´echet diﬀerentiable with respect tov and ∇vFiu, uare

Lipschitz continuous, that is,

∇vFix, x− ∇vFiy, y ≤L x−y ∀x, y∈C, i1,2, . . . , N, 1.8

whereLis a positive constant. With the case that

Fiu, v I−Tiu, v−u, 1.9

and Ti i 2, . . . , Nare N−1 strictly pseudocontractive mappings, 1.1 is a problem of

finding a solution of an equilibrium problem which is also a common fixed point for a system of a finite family of strictly pseudocontractive mappings17–19.

In addition, whenF1u, v A1u, v−u whereA1 is a monotone operator,1.1

is a problem of finding an element which is a solution of a variational inequality problem and a common fixed point for a finite family of strictly pseudocontractive mappings and investigated intensively in 20–32. If all Fi have the form 1.9, then 1.1is a problem of

finding a common fixed point for a finite family of strictly pseudocontractive mappingsTi

fromCintoH14,33–35.

In this paper, we present an iteration method for solving1.1, where the iteration sequence{xn}is defined by

x0x∈H,

ui

n∈C:Fi

ui n, v

ui

n−xn, v−uin ≥0, ∀v∈C, i1, . . . , N,

xn1xn−βn

_n

i1

xn−uin

αnxn

,

1.10

where{αn},{βn}are two sequences of positive numbers satisfying some conditions.

As an application, we find a common solution for a system ofNvariational inequality problems with monotone mappings.

2. Main Results

The strong and weak convergence of any sequence are denoted by → and, respectively. We formulate the following facts which are necessary in the proof of our main results.

Lemma 2.1see5. LetCbe a nonempty closed convex subset of a Hilbert spaceH, and letFbe a bifunction ofC×CintoRsatisfying the Condition1. Letr >0andx∈H. Then, there existsz∈C such that

Fz, v 1

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Lemma 2.2see5. Assume thatF :C×C →Rsatisfies the Condition1. Forr >0andx∈H, define a mappingTr :H → Cas follows:

Trx

z∈C:Fz, v 1

rz−x, v−z ≥0, ∀v∈C

. 2.2

Then, the following hold:

iTr is single-valued;

iiTr is firmly nonexpansive, that is, for anyx, y∈H,

Trx−Try 2 ≤Trx−Try, x−y; 2.3

iiiFTr EPF;

ivEPFis closed and convex.

Lemma 2.3. LetFhu, vbe a bifunction approximating the bifunctionFu, vin the sense

Fh_{u, v}₋_F_{u, v}_≤_hg _u _u₋_v_∀_{u, v}_∈_{C, h >}₀_, _2.4

wheregtis a real positive function. Then, for eachr >0andx∈H, we have

Th

rx−Trx ≤rhg Trx , 2.5

where

Th rx

z∈C:Fhz, v 1

rz−x, v−z ≥0∀v∈C

. 2.6

Proof. Letxbe an arbitrary element ofH. By replacingvbyzin2.2and byzin2.6, we obtain

Fz,z Fh_{z, z} _≥ 1

rx−z,z−zz−x,z−z. 2.7

Therefore, by virtue ofA2in Condition1, we can write

Fz,z−Fh_z,_z_≥ 1

r z−z 2. 2.8

Consequently,

z−z ≤rhg Trx . 2.9

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Lemma 2.4 see36. Let {an},{bn}, and{cn} be sequences of positive numbers satisfying the conditions:

ian1≤1−bnancn, bn<1, ii∞_n₀bn ∞, limn→∞cn/bn 0. Then,limn→∞an0.

Lemma 2.5see37. Assume thatT is a nonexpansive mapping of a closed convex subsetCof a Hilbert spaceH. ThenI−T is demiclosed at zero; that is whenever{xn}is a sequence inCweakly converging to some x ∈ Cand the sequence {I −Txn} strongly converges to zero, it follows I−Tx 0.

Lemma 2.6see17. LetAbe aλ-inverse strongly monotone mapping fromCintoHsuch that

SA/∅, whereSA{x∈C:Ax 0}. Then,SA VIC, A.

Now, consider the firmly nonexpansive mappingsTidefined by

Tix {z∈C:Fiz, v z−x, v−z ≥0, ∀v∈C}, i1, . . . , N. 2.10

By virtue ofLemma 2.2, we can seeTi is nonexpansive. Consequently,Ai : I−Tiis1/2

-inverse strongly monotone and Lipschitz continuous with the Lipschitz constantLi 2,i

1, . . . , N.

We construct a Tikhonov regularization solutionynfor1.1by solving the following

operator equation: findyn∈Hsuch that

N

i1

Aiynαnyn0, 2.11

where the positive regularization parameterαn → 0 asn → ∞. We have the following

result.

Theorem 2.7. iFor eachαn >0, problem2.11has a unique solutionyn. iilimn→∞ynu∗,u∗∈S, u∗ ≤ y , for ally∈S.

iii yn−ym ≤|αn−αm|/αn u∗ .

Proof. i Since the mapping n_i₁Ai is a monotone and Lipschitz continuous mapping

defined onH, it is maximal monotone. Therefore,2.11has a unique solution for eachαn>0 38.

iiFor eachy ∈S, on the base ofLemma 2.2, we have thatAiy 0,i 1, . . . , N.

Thus, from2.11it follows that

N

i1

Aiyn−Aiy, yn−yαnyn, yn−y0. 2.12

Since everyAiis monotone, from the last equality, we obtain

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Hence,

yn ≤ y , ∀y∈S. 2.14

It means that the sequence{yn}is bounded. Let{ynk}be a subsequence of the sequence{yn}

such thatynk yask → ∞.

Again, letybe an arbitrary element ofS. From the1/2-inverse strongly monotone property ofAl, andAly 0,l1, . . . , N, it implies that

1

2 ynk−Tl

ynk 2≤Alynk, ynk−y

≤N

i1

Aiynk, ynk−y

≤ −αnkynk, ynk −y

−αnkynk−y, ynk −y−αnky, ynk−y

≤ −αnky, ynk−y

≤αnk2 y 2,

2.15

that is,

ynk −Tlynk ≤2 y αnk. 2.16

Therefore,

lim

k→ ∞ Al

ynk 0. 2.17

ByLemma 2.5,Aly 0, that is,y ∈ FTl,l 1, . . . , N. It means thaty ∈ S. BecauseSis

a closed convex subset in Hilbert space, it has a unique minimal elementu∗in norm. From

2.14and the weak convergence of{ynk}to y u∗, it also follows that ynk → u∗ , as

k → ∞. Moreover, the sequence{yn}converges strongly tou∗asn → ∞. iiiFrom2.11,2.14, and the monotone property ofAi, it follows

αnyn, yn−ym−αmym, yn−ym≤0 2.18

or

yn−ym ≤ |αn_α−αm|

n ym ≤

|αn−αm|

αn u

∗ _, _2.19

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Theorem 2.8. Suppose thatαn, βnsatisfy the following conditions:

αn, βn>0 αn≤1, _nlim

→ ∞αnn→ ∞lim

|αn−αn1|

α2

nβn

0,

∞

n0

αnβn∞, lim n→ ∞βn

2Nαn2

αn <1.

2.20

Then,

lim

n→ ∞xnu

∗_∈_S, _2.21

wherexnis defined by1.10.

Proof. Letynbe a solution of2.11. SetΔn xn−yn . Then,

Δn1 xn1−yn1 ≤ xn1−yn yn1−yn ,

xn1−yn xn−yn−βn

_N

i0

Aixn−Aiynαnxn−yn.

2.22

From the monotone and Lipschitz continuous properties ofAi,i1, . . . , N,2.11, anduin

Tixn, we can write

xn−yn−βn

_N

i1

Aixn−Aiynαnxn−yn

2

xn−yn 2β2n

_N

i1

Aixn−Aiynαnxn−yn

2

−2βn

_N

i1

Aixn−Aiynαnxn−yn, xn−yn

≤ xn−yn 2

1−2βnαnβ2n2Nαn2

.

2.23

Hence,

xn1−yn ≤Δn

1−2βnαnβ2n2Nαn2 1/2

. 2.24

Therefore,

Δn1≤Δn

|αn−_ααn1|

n u

∗

≤Δn1−αnβn1/2|αn−_ααn1|

n u

∗ _.

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Thus, applying the inequality

ab2_≤₁_ε

a2b2 ε

ε >0, ε αnβn

2 , 2.26

we obtain

0≤Δ2_n₁≤Δ2n1−αnβn1 1

2αnβn

_α

n−αn1 αn u

∗ 2 2

αnβn

11 2αnβn

≤a2

n

1−1 2αnβn−

1 2

αnβn2

αn−αn1 α2

nβn u

∗

2

2αnβn

1 1 2αnβn

.

2.27

Set

bnαnβn

1 2

1 2αnβn

,

cn

αn−αn1 α2

nβn

u∗

2

2αnβn

11 2αnβn

.

2.28

It is not diﬃcult to check thatbnandcn satisfy the conditions inLemma 2.4for suﬃciently

largen. Hence, limn→∞Δ2n0. Since limn→ ∞ynu∗, we have

lim

n→ ∞xnu

∗_∈_S.

2.29

Now, letFn_iu, v:F_ihnu, vbe bifunctions approximating the bifunctionsFiu, vin

the sense2.4wherehn → 0, asn → ∞, andgtis a real positive and boundedthe image

of any bounded set is boundedfunction. Then, the sequence of iterations{xn}is defined by

x0x∈H,

ui

n∈C:Fin

ui n, v

ui

n−xn, v−uni ≥0 ∀v∈C, i1, . . . , N,

xn1xn−βn

_n

i1

xn−uin

αnxn

,

2.30

where{αn},{βn}are two sequences of positive numbers satisfying some conditions.

We have the following result.

Theorem 2.9. Suppose thatαn, βn, andhnsatisfy the conditions inTheorem 2.8and

lim

n→ ∞

hnhn1 α2

nβn

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Then, we have

lim

n→ ∞xnu

∗_∈_S,

2.32

wherexnis defined by2.30.

Proof. Letynbe a solution of the following equation:

N

i1 An

i

ynαnyn0, An_i I−T_in, 2.33

where eachT_inis defined by

Tn ix

z∈C:F_inz, v z−x, v−z ≥0, ∀v∈C, i1, . . . , N. 2.34

Since

xn−u∗ ≤ xn−yn yn−yn yn−u∗ , 2.35

and limn→ ∞ynu∗, in order to prove that limn→ ∞xnu∗, it is necessary to prove that

lim

n→ ∞ xn−yn nlim→ ∞ yn−yn 0. 2.36

For this purpose, first we estimate the value yn−yn . On the basis ofLemma 2.3, we have

Aix−Anix Tix−Tinx ≤hng Tix . 2.37

Therefore, from2.11,2.33, and the monotone property ofAn_i it implies that

yn−yn 2 _α1 n

N

i1

An i

yn−Aiyn, yn−yn

≤ 1 αn

N

i1

An i

yn−Aiyn, yn−yn.

2.38

Consequently, we have

yn−yn ≤ _α1 n

N

i1 An

i

yn−Aiyn

≤Nhn αng

Tiyn .

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On the other hand,

Tiyn Tiyn−Tiu∗ u∗

≤ yn−u∗ u∗

≤ yn 2 u∗

≤3 u∗ .

2.40

Therefore,

yn−yn ≤C0N hn

αn, 2.41

whereC0sup{gt: 0< t≤3 u∗ }. It means that limn→ ∞ynu∗because limn→ ∞hn/αn

0.

Secondly, to prove

lim

n→ ∞ xn−yn 0, 2.42

as in the proof ofTheorem 2.8, first we need to estimate the value yn1−yn . By the argument

as in the proof ofTheorem 2.7, we have

N

i1

An i

yn−Ani1

yn1

,yn−yn1 αnyn,yn−yn1

−αn1

yn1,yn−yn1

0. 2.43

Thus,

yn−yn1 2 αn−_ααn1 n

−yn1,yn−yn1

_α1

n N

i1

An1

i

yn1

−An i

yn,yn−yn1

≤ αn−αn1 αn

−yn,yn−yn1

1

αn N

i1

An1

i

yn1

−An i

yn,yn−yn1

≤ αn−αn1

αn yn yn−yn1

1

αn N

i1

An1

i

yn−Ani

yn,yn−yn1 .

2.44

Therefore,

yn−yn1 ≤ αn−_ααn1

n yn

1

αn N

i1 An1

i

yn−Aiyn Aiyn−Ani

yn

≤ αn−αn1

αn yn N

hnhn1 αn g

yn .

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Using2.14and2.41, we have

yn ≤ u∗ C0N hn

αn. 2.46

Consequently, there exists a positive constantCsuch that yn ≤Cforn≥0. Finally, we have

yn−yn1 ≤ αn−_ααn1

n CNC1

hnhn1

αn , 2.47

whereC1sup{gt: 0< t < C}. Now, setΔn xn−yn . It is not diﬃcult to verify that

xn1−yn ≤Δn

,

Δn1≤Δn1−αnβn1/2|αn−_ααn1|

n CNC1

hnhn1 αn .

2.48

Therefore, limn→ ∞Δn0. The proof is completed.

Remark. The sequencesαn 1n−p, 0< p <1/2, andβnγ0αnwith

0< γ0<

1

2Nα02

, 2.49

satisfy all the necessary conditions inTheorem 2.8.

3. Applications

Consider the following problem: find an elementu∗∈Csuch that

Aiu∗, v−u∗ ≥0, ∀v∈C, i1, . . . , N, 3.1

whereAi areN monotone hemicontinuous mappings from a closed convex subsetC of a

Hilbert spaceHintoH.

Theorem 3.1. Letx0xbe an arbitrary element inH. If{αn},{βn}are chosen as inTheorem 2.8, and the iteration sequence{xn}is defined as follows:

ui n∈C,

Ai

ui n

, v−ui

n

ui

n−xn, v−uin ≥0, ∀v∈C, i1, . . . , N,

xn1xn−βn

_N

i1

xn−uin

αnxn

,

3.2

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IfC≡H, then we have a problem of finding a common zero for a system of monotone hemicontinous mappingsAi,i1, . . . , N. In this case, variational inequality in3.2has the

formAiuin uinxn. Therefore, we have the following result.

Theorem 3.2. LetAi, i1, . . . , NbeNhemicontinuous monotone mappings defined onH. Letx0 xbe an arbitrary element inH, let{αn}and{βn}be the sequences that are chosen as inTheorem 2.8, and, the iteration sequence{xn}be defined as follows:

ui n:Ai

ui n

ui

nxn,

xn1xn−βn

_N

i1

xn−uin

αnxn

.

3.3

Then the sequence{xn}converges strongly to an elementu∗such that

Aiu∗ 0, i1, . . . , N. 3.4

Without the strongly or uniformly monotone property forAi, each problem of3.1,

in general, is ill-posed. Some methods for finding a solution of each variational inequality in

3.1are presented in39.

Here we show an iterative regularization method for finding a common solution of these problems. Suppose that instead ofAi, we have their monotone approximationsAn_i such

thatDAn_i Cand

An

ix−Aix ≤hng x , i1, . . . , N, 3.5

where the positive parameterhn → 0 as n → ∞, andgtis a real positive and bounded

function. Obviously, the bifunctions

Fn_{u, v}_:_An

iu, v−u

, i1, . . . , N, 3.6

satisfy the Condition1and2.4. Therefore, we have the following theorem.

Theorem 3.3. Letx0xbe an arbitrary element inH. If{αn},{βn}are chosen as inTheorem 2.9, and the iteration sequence{xn}is defined as follows:

ui n∈C:

Ai

ui n

, v−ui

n

ui

n−xn, v−uni ≥0 ∀v∈C, i1, . . . , N,

xn1xn−βn

_N

i1

xn−uin

αnxn

,

3.7

then the sequence{xn}converges strongly to a common solution for3.1.

IfC ≡ H, then a common zero for a system of monotone hemicontinuous mappings

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Theorem 3.4. LetAi,i1, . . . , NbeNhemicontinuous monotone mappings defined onH. Letx0 xbe an arbitrary element inH, let{αn}and{βn}be the sequences that are chosen as inTheorem 2.9, and the iteration sequence{xn}be defined as follows:

ui n:Ai

ui n

ui

nxn,

xn1xn−βn

_N

i1

xn−uin

αnxn

.

3.8

Then the sequence{xn}converges strongly to an elementu∗such that

Aiu∗ 0, i1, . . . , N. 3.9

Acknowledgment

This work was supported by the Kyungnam University Research Fund, 2010.

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29 J. K. Kim and K. S. Kim, “A new system of generalized nonlinear mixed quasivariational inequalities and iterative algorithms in Hilbert spaces,”Journal of the Korean Mathematical Society, vol. 44, no. 4, pp. 823–834, 2007.

30 J. K. Kim and H. G. Li, “A hybrid proximal point algorithm and stability for set-valued mixed variational inclusions involvingA, n-accretive mappings,”East Asian Mathematical Journal, vol. 26, pp. 703–714, 2010.

31 Y. Yao, Y.-C. Liou, and J.-C. Yao, “An extragradient method for fixed point problems and variational inequality problems,”Journal of Inequalities and Applications, vol. 2007, Article ID 38752, 12 pages, 2007.

32 L.-C. Zeng and J.-C. Yao, “Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems,”Taiwanese Journal of Mathematics, vol. 10, no. 5, pp. 1293–1303, 2006.

33 Ng. Buong and Ph. van Son, “Regularization extragradient method for common fixed point of a finite family of strictly pseudocontractive mappings in Hilbert spaces,”International Journal of Mathematical Analysis, vol. 1, no. 25–28, pp. 1217–1226, 2007.

34 G. Wang, J. Peng, and H.-W. J. Lee, “Implicit iteration process with mean errors for common fixed points of a finite family of strictly pseudocontrative maps,” International Journal of Mathematical Analysis, vol. 1, no. 1–4, pp. 89–99, 2007.

35 L.-C. Zeng and J.-C. Yao, “Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings,”Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2507–2515, 2006.

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