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Volume 2009, Article ID 362191, 21 pagesdoi:10.1155/2009/362191 Research Article Strong Convergence of an Iterative Method for Equilibrium Problems and Variational Inequality Problems 1

Volume 2009, Article ID 362191, 21 pages

doi:10.1155/2009/362191

Research Article

Strong Convergence of an Iterative

Method for Equilibrium Problems and

Variational Inequality Problems

1 Department of Mathematics, TianJin Polytechnic University, TianJin 300160, China

2 Department of Mathematics, Agricultural University of Hebei, BaoDing 071001, China

Correspondence should be addressed to HongYu Li,lhy x1976@eyou.com

Received 26 August 2008; Revised 11 November 2008; Accepted 9 January 2009

Recommended by Massimo Furi

We introduce an iterative method for finding a common element of the set of solutions ofequilibrium problems, the set of solutions of variational inequality problems, and the set of fixedpoints of finite many nonexpansive mappings We prove strong convergence of the iterativesequence generated by the proposed iterative algorithm to the unique solution of a variationalinequality, which is the optimality condition for the minimization problem

Copyrightq 2009 H Li and H Li This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

1 Introduction

R is the set of real number The equilibrium problem is to find a x ∈ C such that

The set of such solutions is denoted by EPf Numerous problems in physics, optimization,and economics reduce to find a solution of equilibrium problem Some methods have beenproposed to solve the equilibrium problems in Hilbert space, see, for instance, Blum andOettli1, Combettes and Hirstoaga 2, and Moudafi 3

A mappingA : C → H is called monotone if Au − Av, u − v ≥ 0 A is called relaxed

u, v-cocoercive, if there exist constants u > 0 and v > 0 such that

Ax − Ay, x − y ≥ −uAx − Ay2 vx − y2, ∀x, y ∈ C, 1.2

whenu  0, A is called v-strong monotone; when v  0, A is called relaxed u-cocoercive.

u ∈ C, such that

Au, v − u ≥ 0, ∀v ∈ C. 1.3

The set of solutions of variational inequality problem is denoted by VIC, A The variationalinequality problem has been extensively studied in literatures, see, for example,4,5 andreferences therein

there exists a constantγ > 0 such that Bx, x ≥ γx2, for all x ∈ H A typical problem is to

minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on

a real Hilbert spaceH:

min

where T is a nonexpansive mapping on H and b is a point on H.

A mappingT from C into itself is called nonexpansive, if Tx−Ty ≤ x−y, ∀x, y ∈ C.

The set of fixed points ofT is denoted by FT Let {T i}N

i1be a finite family of nonexpansivemappings andF N i1 FTi  / ∅, define the mappings

i1 ⊂ 0, 1 for all n ≥ 1 Such a mapping W nis calledW-mapping generated by

coefficient α ∈ 0, 1 Marino and Xu 7 considered the following general iterative scheme:

xn1  α nγfxn1− α nBSxn. 1.6They proved that{x n } converges strongly to z  P FS I −B γfz, where P FSis the metricprojection fromH onto FS.

By combining equilibrium problems and1.6, Plutbieng and Pumpaeng 8 proposedthe following algorithm:

On the other hand, letA : C → C be a α-cocoercive mapping, for finding common

element of the solution of variational inequality problems and the set of fixed point ofnonexpansive mappings, Takahashi and Toyoda12 introduced iterative scheme

They proved that{x n } converges weakly to z ∈ FS ∩ VIC, A Inspired by 1.10 and 13,

Y Yao and J.-C Yao14 given the following iterative process:

y n  P CI − λ n Ax n ,

and proved that {x n } converges strongly to z ∈ FS ∩ VIC, A By combining viscosity

approximation method and1.10, Chen et al 15 introduced the process

and studied the strong convergence of sequence{x n} generated by 1.12 Motivated by 1.6,

1.11, and 1.12, Qin et al 16 introduced the following general iterative process

and established a strong convergence theorem of{x n} to an element ofN i1 FTi  ∩ VIC, A.

The purpose of this paper is to introduce the iterative process:x1∈ H and

whereWnis defined by1.5, A is u, v-cocoercive, and B is a bounded linear operator We

should show that the sequences{x n} converge strongly to an element ofN

i1 FT i ∩VIC, A∩

EPF Our result extends the corresponding results of Qin et al 16 and Colao et al 9, andmany others

2 Preliminaries

convergence of{x n } to x by x n → x and weak convergence by x n  x Let PC : C → H

is a mapping such that for every pointx ∈ H, there exists a unique P C x ∈ C satisfying

x − P Cx ≤ x − y, for all y ∈ C PCis called the metric projection ofH onto C It is known

thatPC is a nonexpansive mapping fromH onto C It is also known that PC x ∈ C and





Lemma 2.1 see 17 Assume {a n } is a sequence of nonegative real number such that

Lemma 2.2 see 18 Let {x n }, {u n } be bounded sequences in Banach space E satisfying x n1 

τ n x n  1 − τ n u n for all n ≥ 0 and lim inf n → ∞ u n1 − u n  − x n1 − x n  ≤ 0 Let τ n be a sequence

in 0, 1 with 0 < lim inf n → ∞τn≤ lim supn → ∞ τn < 1 Then, limn → ∞ x n − u n   0.

Lemma 2.3 For all x, y ∈ H, there holds the inequality

x  y ≤ x2 2y, x  y. 2.4

Lemma 2.4 see 7 Assume that A is a strong positive linear bounded operator on a Hilbert space

For solving the equilibrium problem for a bifunctionF : C × C → R, we assume that

F satisfies the following conditions:

A1 Fx, x  0 for all x ∈ C;

A2 F is monotone: Fx, y  Fy, x ≤ 0 for all x, y ∈ C;

A3 for all x, y, z ∈ C, lim sup t↓0 Ftz  1 − tx, y ≤ Fx, y;

A4 for all x ∈ C, Fx, ·  is convex and lower semicontinuous.

The following result is in Blum and Oettli1

Lemma 2.5 see 1 Let C be a nonempty closed convex subset of a Hilbert space E, let F be a

such that

We also know the following lemmas

Lemma 2.6 see 19 Let C be a nonempty closed convex subset of Hilbert space H, let F be a

A monotone operatorT : H → 2 His said to be maximal monotone if its graphGT 

{u, v : v ∈ Tu} is not properly contained in the graph of any other monotone operators Let

v ∈ C, that is

N C v x ∈ H : v − y, x ≥ 0, ∀y ∈ C. 2.8Define

It is known that in this caseT is maximal monotone, and 0 ∈ Tv if and only if v ∈ VIC, A.

3 Strong Convergence Theorem

Theorem 3.1 Let H be a real Hilbert space and C be a nonempty closed convex subset of H {T i}N i1

α-contraction with 0 ≤ α < 1 and B a strong positive linear bounded operator with coefficient γ > 0,

i1 FT i  ∩ VIC, A ∩ EPF / ∅ and

i limn → ∞ α n  0, ∞n1 α n  ∞;

ii limn → ∞ |r n1 − r n |  0, lim inf n → ∞ r n > 0;

iii {s n }, {t n } ∈ a, b for some a, b with 0 ≤ a ≤ b ≤ 2v −uμ2/μ2and lim n → ∞ |s n1 −s n| limn → ∞ |t n1 − t n |  0;

iv limn → ∞ |λ n1 − λ n|  limn → ∞ |b n1 − b n |  0.

I − s nAx −I − snAy  ≤ x − y,

I − t nAx −I − tnAy  ≤ x − y. 3.3

We divide the proof into several steps.

we have

u n − p  T r n xn − T r n p  ≤ x n − p. 3.4Sincep  W n P C p − s n Ap, we have

Putvn  P C y n − t nAyn, we have

for someM1> 0 Submitting 3.28 into 3.30, we have

v n − p2≤x n − p2−y n − v n2− t2

n Ay n − Ap2

 2t n y n − v n  · Ay n − Ap. 3.36

Submitting3.36 into 3.30, we have

x n1 − p2≤ 1 − βx n − p2−y n − v n2− t2

n Ay n − Ap2 2t n y n − v n  · Ay n − Ap

 βx n − p2 2α n M1.

3.37This implies that

which implies

u n − p2≤x n − p2−u n − x n2, 3.49which together with3.30 gives

x n − W n v n ≤ 11− β x n − x n1   α n

1− β



γ fW n x n  BWn v n , 3.54hence

lim

Note that

W n v n − v n  ≤ W n v n − x n   x n − u n   u n − v n , 3.56thus from3.47–3.55, we have

lim

inequalityγfx∗− Bx∗, x − x∗ ≤ 0, for all x ∈ C0.

Take a subsequence{x n j } of {x n1}, such that

Since{x n } is bounded, without loss of generality, we assume {x n j} itself converges weakly to

a pointp We should prove p ∈ C0N

i1 FT i  ∩ VIC, A ∩ EPF.

with the same argument as used in14, we can derive p ∈ T−10, sinceT is maximal monotone,

we knowp ∈ VIC, A.

Next, fromA2, for all y ∈ C we have

Divide by t in both side yields Fty  1 − tp, y ≥ 0, let t → 0, by A3 we conclude

Fp, y ≥ 0, for all y ∈ C Therefore, p ∈ EPF.

Finally, fromx n − v n  → 0 we know that v n j  p j → ∞ Assume p /∈N i1 FTi,that is,p / Wnp, for all n ∈ N Since Hilbert space satisfies Opial’s condition, we have

From the definition of{x n} and Lemmas2.3, and2.4, we have

which implies that

so, byLemma 2.1, we concludex n → x∗ This completes the proof

PuttingF ≡ 0 and bn  β  0 for all n ≥ 1 in Theorem 3.1, we obtain the followingcorollary

Corollary 3.2 Let H be a real Hilbert space and C be a nonempty closed convex subset of H {T i}N

i1 a

in 0, 1 and β be a constant in 0, 1 Assume C0N i1 FTi  ∩ VIC, A / ∅ and

xn1  α nγfWnxn βx n1 − βI − α nBWnPCI − tnAyn, 3.69

is,



≤ 0, ∀x ∈ C0. 3.70PuttingPC I − s nA  PC I − t nA  I and N  1, T1  S, β  b n  0 inTheorem 3.1,

we obtain the following corollary

Corollary 3.3 Let H be a real Hilbert space and C be a nonempty closed convex subset of H S a

and {r n } in 0, ∞ Assume C0 FS ∩ EPF / ∅ and

i limn → ∞ α n  0, ∞n1 α n  ∞;

ii limn → ∞ |r n1 − r n |  0, lim inf n → ∞rn > 0.

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variational inequality problems and fixed point problems in Hilbert spaces,” Applied Mathematics and< /i>... J Shang, Y F Su, and X L Qin, “A general iterative method for equilibrium problems and fixed

point problems in Hilbert spaces,” Fixed Point Theorey and Applications, vol 2007, Article. .. Yao, ? ?Convergence theorem for equilibrium problems and fixed point

problems of infinite family of nonexpansive mappings,” Fixed Point Theory and Applications, vol 2007,

Article