Hybrid Algorithms of Common Solutions of Generalized Mixed Equilibrium Problems and the Common Variational Inequality Problems with Applications

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Volume 2011, Article ID 971479,28pages doi:10.1155/2011/971479

Research Article

Hybrid Algorithms of Common Solutions of

Generalized Mixed Equilibrium Problems

and the Common Variational Inequality Problems

with Applications

Thanyarat Jitpeera, Uamporn Witthayarat, and Poom Kumam

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

Correspondence should be addressed to Poom Kumam,[email protected]

Received 5 January 2011; Accepted 20 February 2011 Academic Editor: Tomonari Suzuki

Copyrightq2011 Thanyarat Jitpeera 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.

We introduce new iterative algorithms by hybrid method for finding a common element of the set of solutions of fixed points of infinite family of nonexpansive mappings, the set of common solutions of generalized mixed equilibrium problems, and the set of common solutions of the variational inequality with inverse-strongly monotone mappings in a real Hilbert space. We prove the strong convergence of the proposed iterative method under some suitable conditions. Finally, we apply our results to complementarity problems and optimization problems. Our results improve and extend the results announced by many others.

1. Introduction

Throughout this paper, let H be a real Hilbert space with inner product ·,· and norm

· , and letCbe a nonempty closed convex subset ofH. A mapping T : C → Cis called

nonexpansiveifTx−Ty ≤ x−y, for allx, y ∈C. The set offixed pointsofT denoted by

FT; that is, FT {x ∈ C : Tx x}. IfC ⊂ H is bounded, closed, and convex andT

is a nonexpansive mapping ofCinto itself, thenFT/∅; see, for instance,1. LetF be a bifunction ofC×CintoÊ, whereÊ is the set of real numbers,A:C → Ha mapping, and

ϕ : C → Ê a real-valued function. Thegeneralized mixed equilibrium problem is for finding

x∈Csuch that

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The set of solutions of1.1is denoted by GMEPF, ϕ, A; that is,

GMEPF, ϕ, Ax∈C:Fx, yAx, y−xϕy−ϕx≥0, ∀y∈C. 1.2

The generalized mixed equilibrium problem include fixed point problems, optimization problems, variational inequalities problems, Nash equilibrium problems, noncooperative games, economics, and the equilibrium problems as special cases2–7.

In particular, ifA≡ 0, the problem1.1is reduced into themixed equilibrium problem

8for findingx∈Csuch that

Fx, yϕy−ϕx≥0, ∀y∈C. 1.3

The set of solutions of1.3is denoted by MEPF, ϕ.

Ifϕ ≡ 0,1.1is reduced into thegeneralized equilibrium problem9for findingx∈ C

such that

Fx, yAx, y−x≥0, ∀y∈C. 1.4

The set of solutions of1.4is denoted by GEPF, A, which this problem was studied by S. Takahashi and W. Takahashi10.

IfA≡0 andϕ≡0, then the generalized mixed equilibrium problem1.1becomes the followingequilibrium problemwhich is to findx∈Csuch that

Fx, y≥0, ∀y∈C. 1.5

The set of solutions of1.5is denoted by EPF. Many problems in applied sciences, such as numerous problems in physics, optimization, and economics reduce into finding a solution of 1.5. Some methods have been proposed to solve the generalized mixed equilibrium problems, equilibrium problems, and fixed point problems 2, 6, 11–29 and references therein. IfF ≡ 0 andϕ ≡0, then the generalized mixed equilibrium problem1.1becomes the followingvariational inequality problem,denoted by VIC, A, is to findx∈Csuch that

Ax, y−x≥0, ∀y∈C. 1.6

The variational inequality problem has been extensively studied in the literature. See, for example30,31and the references therein. A mappingAofCintoHis called monotoneif

Ax−Ay, x−y≥0, ∀x, y∈C. 1.7

Ais called anα-inverse-strongly monotoneif there exists a positive real numberα >0 such that

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In 2008, Takahashi et al.32introduced an iterative method for finding the set of fixed point by Hybrid method in Hilbert spaces. Starting withC1C,x1PC1x0, define sequence

{xn},{yn}as follows:

ynαnxn 1−αnTxn, n≥1,

Cn1

z∈Cn:yn−z≤ xn−z

, n≥1,

xn1PCn1x0, n≥1,

1.9

where PC is a metric projection ofH onto C andT is a nonexpansive mapping ofC into

itself. They proved that if the sequence{αn}of parameters satisfies appropriate conditions,

then{xn}generated by1.9converges strongly toPFTx0. In 2009, Kumam20introduced

an iterative method for finding a common element of the set of common fixed points of nonexpansive mapping, the set of solutions of a variational inequality problem, and the set of solutions of an equilibrium problem in Hilbert spaces. Starting with an arbitraryC1 C,

x1PC1x0, define sequence{xn},{zn}as follows:

Fzn, y

1

rn

y−zn, zn−xn

≥0, ∀y∈C,

ynαnxn 1−αnTPCzn−λnBzn, n≥1,

Cn1

z∈Cn:yn−z≤ xn−z

, n≥1,

xn1PCn1x0, n≥1,

1.10

whereT is a nonexpansive mapping ofCinto itself andBis aβ-inverse-strongly monotone mapping of CintoH. He proved that if the sequences{αn},{rn}, and{λn} of parameters

satisfies appropriate conditions, then {xn} generated by 1.10 converges strongly to

PFT∩EPF∩VIC,Bx0. In 2010, Kangtunyakarn33 introduced a new method for a common

of generalized equilibrium problems, common of variational inequality problems, and fixed point problems by usingS-mapping generated by a finite family of nonexpansive mappings and real numbers in Hilbert spaces. Starting with an arbitrary x1, u, v in C, define the

sequences{xn},{yn}as follows:

Fun, u Axn, u−un

1

rn

u−un, un−xn ≥0, ∀u∈C,

Gvn, v Bxn, v−vn 1

sn

v−vn, vn−xn ≥0, ∀v∈C,

ynδnPCun−λnAun 1−δnPC

vn−ηnBvn

, n≥1,

xn1αnfxn βnxnγnSnyn, ∀n≥1,

1.11

whereSnis theS-mapping andA,Bareα,β-inverse-strongly monotone mappings ofCinto

H, respectively. He proved that if the sequences{αn},{βn},{γn},{rn},{sn},{ηn}, and{λn}of

parameters satisfies appropriate conditions, then{xn}generated by1.11converges strongly

toP:∩∞

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Recently, Shehu 34 motivated Chantarangsi et al. 35 who studied the problem of approximating a common element of the set of fixed points of an infinite family of nonexpansive mapping, the set of common solutions of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in a real Hilbert spaces. In this paper, motivated by the above results, we present a new hybrid iterative scheme for finding a common element of the set of solutions of a common of generalized mixed equilibrium problems, the common solutions of the variational inequality for inverse-strongly monotone mapping, and the set of fixed points of infinite family of nonexpansive mappings in the set of Hilbert spaces. Then, we prove strong convergence theorems under some mild conditions. Finally, we give some applications of our results. The results presented in this paper generalize, extend, and improve the results of Takahashi et al.32, Kumam20, Kangtunyakarn33, and many authors.

2. Preliminaries

LetHbe a real Hilbert space with norm · and inner product·,·, and letCbe a closed convex subset ofH. When{xn}is a sequence inH,xn xmeans{xn}converges weakly to

x, andxn → xmeans{xn}converges strongly tox. In a real Hilbert spaceH, we have

_x₋_y2

x2−y2−2x−y, y,

λx 1−λy2λx2 1−λy2−λ1−λx−y2,

2.1

for allx, y∈Handλ∈0,1. For every pointx∈H, there exists a unique nearest point inC, denoted byPCx, such that

x−PCx ≤x−y, ∀y∈C. 2.2

PC is called the metric projectionof H onto C. It is well known thatPC is a nonexpansive

mapping ofHontoCand satisfies

x−y, PCx−PCy

≥PCx−PCy2, ∀x, y∈H. 2.3

Moreover,PCxis characterized by the following properties:PCx∈C,

x−PCx, y−PCx ≤0, 2.4

x−y2≥ x−PCx2y−PCx2, 2.5

for allx∈H,y ∈C. It is also known thatHsatisfies the Opial condition; for any sequence

{xn}withxn x, the inequality

lim inf

n→ ∞ xn−x<lim infn→ ∞ xn−y 2.6

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It is obvious that any α-inverse-strongly monotone mapping A is 1/α-Lipschitz monotone and continuous mapping. We also have that for allx, y∈Handλ >0,

_I₋_rAx₋_I₋_rAy2

x−y−rAx−Ay2

x−y2−2rx−y, Ax−Ayr2Ax−Ay2

≤x−y2rr−2αAx−Ay2.

2.7

So, ifr ≤2α, thenI−rAis a nonexpansive mapping ofCintoH.

For solving the generalized mixed equilibrium problem, let us assume that the bifunctionF : C×C → Ê, the nonlinear mapping A : C → H is continuous monotone,

ϕ:C → Êis convex, and lower semicontinuous satisfies the following conditions:

A1Fx, x 0 for allx∈C,

A2Fis monotone; that is,Fx, y Fy, x≤0 for anyx, y∈C,

A3Fis upper-hemicontinuous; that is, for eachu, x, y∈C,

lim sup

t→0 F

tu 1−tx, y≤Fx, y, _2.8

A4Fx,·is convex and lower semicontinuous for eachx∈C,

B1For eachx ∈ H andr > 0, there exists a bounded subsetDx ⊆ Candyx ∈ C ∩

domϕsuch that for anyu∈C\Dx,

Fu, yx

Au, yx−u

ϕyx

1

r

yx−u, u−x

< ϕu, 2.9

B2Cis a bounded set.

The following lemma appears implicitly in2. We need the following lemmas for proving our main result.

Lemma 2.1see2. LetCbe a nonempty closed convex subset ofH, and letFbe a bifunction of

C×CintoÊsatisfying (A1)–(A4). Letr >0andx∈H. Then, there existsu∈Csuch that

Fu, y 1 r

y−u, u−x≥0, ∀y∈C. 2.10

The following lemma was also given in36.

Lemma 2.2see36. Assume thatF :C×C → Ê satisfies (A1)–(A4). Forr > 0andx ∈H,

define a mappingKr :H → Cas follows:

Krx u∈C:F

u, y1 r

y−u, u−x≥0, ∀y∈C

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for allx∈H. Then, the following hold:

1Kris single-valued,

2Kris firmly nonexpansive, that is, for anyx, y∈H,Krx−Kry2≤ Krx−Kry, x−y,

3FKr EPF,

4EPFis closed and convex.

Lemma 2.3see37. LetC be a nonempty closed convex subset of a real Hilbert spaceH. Let

F : C×C → Ê be a bifunction mapping satisfies (A1)–(A4), and letϕ : C → Ê be convex and

lower semicontinuous such thatC∩domϕ /∅. Assume that either (B1) or (B2) holds. Forr >0and

x∈H, there existsu∈Csuch that

Fu, yϕy−ϕu 1 r

y−u, u−x. 2.12

Define a mappingKr :H → Cas follows:

TrF,ϕx u∈C:F

u, yϕy−ϕu 1 r

y−u, u−x≥0, ∀y∈C

, 2.13

for allx∈H. Then, the following hold:

1TrF,ϕis single-valued,

2TrF,ϕis firmly nonexpansive, that is, for anyx, y ∈H,TrF,ϕx−TrF,ϕy2 ≤ TrF,ϕx−

TrF,ϕy, x−y,

3FTrF,ϕ MEPF, ϕ,

4MEPF, ϕis closed and convex.

Lemma 2.4see38. Assume that{an}is a sequence of nonnegative real numbers such that

an1≤1−αnanδn, n≥0, 2.14

where{αn}is a sequence in0,1and{δn}is a sequence inÊsuch that

1∞_n₁αn∞,

2lim sup_n_{→ ∞}δn/αn≤0or

_∞

n1|δn|<∞.

Then,limn→ ∞an0.

3. Main Result

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the set of fixed points of infinite family of nonexpansive mappings in the set of Hilbert spaces.

Theorem 3.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetF1,F2 be a

bifunction ofC×C into real numbersÊ satisfying (A1)–(A4), and let ϕ1, ϕ2 : C → Ê∪ {∞}

be a proper lower semicontinuous and convex function. Let A, B, D, and E be α, β, δ, and η -inverse-strongly monotone mapping ofCintoH, respectively. Let{Ti}∞i1be an infinite nonexpansive

mapping such thatΘ:∞_i₁FTi∩GMEPF1, ϕ1, A∩GMEPF2, ϕ2, B∩VIC, D∩VIC, E/∅.

Assume that either (B1) or (B2) holds. Let {xn} be a sequence generated by x0 ∈ C, C1,i C,

C1∩∞_i₁C1,i, x1PC1x0and

tnTrnF1,ϕ1xn−rnAxn,

unTsFn2,ϕ2xn−snBxn,

wnξnPCun−λnDun 1−ξnPC

tn−μnEtn

,

yn,iαn,ix0 1−αn,iTiwn,

Cn1,i

z∈Cn,i:yn,i−z2≤ xn−z2αn,i

x022wn−x0, z

,

Cn1

∞

i1

Cn1,i,

xn1PCn1x0,

3.1

for everyn ≥ 0, where{rn},{sn} ⊂ 0,∞,λn ∈ 0,2δandμn ∈0,2ηsatisfying the following

conditions:

i0< a≤rn≤b <2α,

ii0< c≤sn≤d <2β,

iiilimn→ ∞αn,i0,

ivlimn→ ∞ξnξ∈0,1,

v0< e≤λn≤f <2δ,

vi0< g≤μn≤j <2η.

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Proof. Letp ∈ Θ, thenp TF1,ϕ1

rn p−rnAp,p T

F2,ϕ2

sn p−snBp,p PCp−λnDp, and

pPCp−μnEp. By nonexpansiveness ofPC, TrnF1,ϕ1, andT

F2,ϕ2

sn , we have

wn−p2

ξnPCun−λnDun 1−ξnPC

tn−μnEtn

−ξnPC

p−λnDp

−1−ξnPC

p−μnEp2

ξn

PCun−λnDun−PC

p−λnDp

1−ξn

PC

tn−μnEtn

−PC

p−μnEp2

≤ξnun−λnDun−

p−λnDp2 1−ξntn−μnEtn

−p−μnEp2

ξnun−p

−λn

Dun−Dp2 1−ξntn−p

−μn

Etn−Ep2

ξn

un−p2−λn2δ−λnDun−Dp2

1−ξn

tn−p2−μn

2η−μnEtn−Ep2

≤ξn TsFn2,ϕ2xn−snBxn−T

F2,ϕ2

sn

p−snBp

2

−λn2δ−λnDun−Dp2

1−ξn TrnF1,ϕ1xn−rnAxn−T

F1,ϕ1

rn

p−rnAp

2

−μn

≤ξn

xn−snBxn−

p−snBp2

1−ξn

xn−rnAxn−

p−rnAp2

≤ξnxn−p2 1−ξnxn−p2

≤xn−p2.

3.2

Since bothI−rnAandI−snBare nonexpansive for eachn≥1 and2.7, we have

un−p2TsFn2,ϕ2I−snBxn−T

F2,ϕ2

sn I−snBp

2

≤I−snBxn−I−snBp2

≤xn−p2sn

sn−2βBxn−Bp2

≤xn−p2,

_t_n₋_p2

TF1,ϕ1

rn I−rnAxn−T

F1,ϕ1

rn I−rnAp

2

≤I−rnAxn−I−rnAp2

≤xn−p2rnrn−2αAxn−Ap2

≤xn−p2.

3.3

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Next, we will divide the proof into four steps.

Step 1. We show that{xn}is well defined. Letn 1, thenC1,i Cis closed and convex for

eachi≥1. Suppose thatCn,iis closed convex for somen >1. Then, by definition ofCn1,i, we

know thatCn1,iis closed convex forn≥1. Hence,Cn,iis closed convex forn≥1 and for each

i≥ 1. This implies thatCnis closed convex forn≥ 1. Moreover, we show thatΘ ⊂Cn. For

n1,Θ⊂CC1,i. Forn≥2, letp∈Θ. Then,

_y_n,i₋_p2

αn,i

x0−p

2

1−αn,i

Tiwn−p

2

≤αn,ix0−p2 1−αn,iwn−p2

wn−p2αn,i

x0−p2−wn−p2

≤xn−p2αn,i

x022

wn−x0, p

,

3.4

which shows thatp ∈Cn,i, for alln≥2, for alli≥1. So,Θ ⊂Cn,i, for alln≥1, for alli≥ 1.

Therefore, it follows that∅/ Θ⊂Cn, for alln≥1. This implies that{xn}is well defined.

Step 2. We claim that limn→ ∞xn1−xn0 and limn→ ∞yn,i−xn0.

Fromxn PCnx0, we get

x0−xn, xn−y

≥0, 3.5

for eachy∈Cn. SinceΘ⊂Cn, we have

x0−xn, xn−p

≥0 for eachp∈Θ, n∈Æ. 3.6

Hence, forp∈Θ, we obtain

0≤x0−xn, xn−p

x0−xn, xn−x0x0−p

−x0−xn, x0−xn

x0−xn, x0−p

≤ −x0−xn2x0−xnx0−p.

3.7

It follows that

x0−xn ≤x0−p, ∀p∈Θ, n∈Æ. 3.8

FromxnPCnx0andxn1PCn1x0 ∈Cn1 ⊂Cn, we have

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Forn∈Æ, we compute

0≤ x0−xn, xn−xn1

x0−xn, xn−x0x0−xn1

−x0−xn, x0−xnx0−xn, x0−xn1

≤ −x0−xn2x0−xn, x0−xn1

≤ −x0−xn2x0−xnx0−xn1,

3.10

and then

x0−xn ≤ x0−xn1, ∀n∈Æ. 3.11

Thus, the sequence{xn−x0}is a bounded and nondecreasing sequence, so limn→ ∞xn−x0

exists. That is, there existsmsuch that

m lim

n→ ∞xn−x0. 3.12

Hence,{xn}is bounded and so are{Axn},{Bxn},{un},{Dun},{tn},{Etn},{wn},{Tiwn}, and

{yn,i}fori1,2, . . ., andn≥1. From3.9, we get

xn−xn12xn−x0x0−xn12

xn−x022xn−x0, x0−xn1x0−xn12

xn−x022xn−x0, x0−xnxn−xn1x0−xn12

xn−x02−2xn−x0, xn−x02xn−x0, xn−xn1x0−xn12

−xn−x022xn−x0, xn−xn1x0−xn12

≤ −xn−x02x0−xn12.

3.13

By3.12, we obtain

lim

n→ ∞xn−xn10. 3.14

Sincexn1 PCn1x0∈Cn1⊂Cn, we have

yn,i−xn12≤ xn−xn12αn,i

x022wn−x0, xn1

. 3.15

Byiiiand3.14, we get

lim

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It follows that

_y_n,i₋_x_n_≤_y_n,i₋_x_n₁_x_n₋_x_n₁_. _3.17

By3.14and3.16, we have

lim

n→ ∞yn,i−xn0, i1,2, . . . . 3.18

Step 3. We claim that the following statements hold:

S1limn→ ∞xn−un0,

S2limn→ ∞xn−tn0,

S3limn→ ∞wn−xn0.

For3.2, we note that

_y_n,i₋_p2_≤

αn,ix0−p2 1−αn,iTiwn−p2

αn,ix0−p2 1−αn,iwn−p2

≤αn,ix0−p2 1−αn,i

×ξnxn−snBxn−

p−snBp2 1−ξnxn−rnAxn−

p−rnAp2

≤αn,ix0−p2 1−αn,i

×ξn

xn−p2sn

sn−2βBxn−Bp2

1−ξn

xn−p2rnrn−2αAxn−Ap2

αn,ix0−p2 1−αn,i

×xn−p2ξnsn

sn−2βBxn−Bp2 1−ξnrnrn−2αAxn−Ap2

αn,ix0−p2xn−p2 1−αn,iξnsn

sn−2βBxn−Bp2

1−αn,i1−ξnrnrn−2αAxn−Ap2

αn,ix0−p2xn−p2 1−αn,iξnsn

sn−2βBxn−Bp2.

3.19

Since 0< c≤sn≤d≤2β, 0≤ki≤αn,i≤hi <1, we have

1−hiξc

2β−dBxn−Bp2 ≤αn,ix0−p2xn−p2−yn,i−p2

≤αn,ix0−p2yn,i−xnxn−pyn,i−p.

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By conditioniiiand3.18, limn→ ∞Bxn−Bp0 by using the same method with3.20.

Hence, from3.19, since 0< a≤rn≤b≤2α, 0≤ki≤αn,i≤hi<1, we have

1−hi1−ξa2α−bAxn−Ap2≤αn,ix0−p2xn−p2−yn,i−p2

≤αn,ix0−p2yn,i−xnxn−pyn,i−p.

3.21

By conditioniiiand3.18, then we have limn→ ∞Axn−Ap 0. On the other hand, we

compute

_u_n₋_p2

TF2,ϕ2

sn I−snBxn−T

F2,ϕ2

sn I−snBp

2

≤xn−snBxn−

p−snBp

, un−p

1

2

xn−snBxn−

p−snBp2un−p2

−xn−snBxn−

p−snBp

−un−p2

≤ 1

2

xn−p2un−p2−xn−snBxn−

p−snBp

−un−p2

1

2

xn−p2un−p2− un−xn22sn

xn−un, Bxn−Bp

−s2_nBxn−Bp2

,

3.22

and hence,

_u_n₋_p2_≤

xn−p2− un−xn22sn

xn−un, Bxn−Bp

−s2_nBxn−Bp2

≤xn−p2− un−xn22snxn−unBxn−Bp.

3.23

By using the same method as3.23, we also have

_t_n₋_p2_≤

xn−p2− tn−xn22rn

xn−tn, Axn−Ap

−r_n2Axn−Ap2

≤xn−p2− tn−xn22rnxn−tnAxn−Ap.

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Furthermore, we observe that

_y_n,i₋_p2

≤αn,ix0−p2 1−αn,iTiwn−p2

≤αn,ix0−p2 1−αn,i

ξnPCun−λnDun 1−ξnPC

tn−μnEtn

−p2

≤αn,ix0−p2 1−αn,i

×ξn

1−ξn

tn−p2−μn

≤αn,ix0−p2 1−αn,i

ξnun−p2 1−ξntn−p2

≤αn,ix0−p2 1−αn,i

×ξn

xn−p2− un−xn22snxn−unBxn−Bp

1−ξn

xn−p2− tn−xn22rnxn−tnAxn−Ap

≤αn,ix0−p2xn−p2−1−αn,iξnun−xn2

1−αn,iξn2snxn−unBxn−Bp−1−αn,i1−ξntn−xn2

1−αn,i1−ξn2rnxn−tnAxn−Ap

≤αn,ix0−p2xn−p2−1−αn,iξnun−xn2 1−αn,iξn2snxn−unBxn−Bp

1−αn,i1−ξn2rnxn−tnAxn−Ap.

3.25

By conditioni–iv,3.18, limn→ ∞Axn−Ap0 and limn→ ∞Bxn−Bp0, then we get

1−αn,iξnun−xn2≤αn,ix0−p2xn−p2−yn,i−p2

1−αn,iξn2snxn−unBxn−Bp

≤αn,ix0−p2xn−yn,ixn−pyn,i−p

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Therefore, we have

lim

n→ ∞xn−un0. 3.27

Similar to 3.26, from 3.25 by conditions i–iv, 3.18, limn→ ∞Axn −Ap 0, and

limn→ ∞Bxn−Bp0, we get

1−αn,i1−ξntn−xn2 ≤αn,ix0−p2xn−p2−yn,i−p2

≤αn,ix0−p2xn−yn,ixn−pyn,i−p

3.28

Therefore, we have

lim

n→ ∞xn−tn0. 3.29

From3.1,3.3, we have

wn−p2ξnPCun−λnDun 1−ξnPC

tn−μnEtn

−ξnPC

p−λnDp

−1−ξnPC

p−μnEp2

ξnPCun−λnDun−PC

p−λnDp2

1−ξnPC

tn−μnEtn

−PC

p−μnEp2

≤ξn

1−ξn

tn−p2−μn

≤ξn

xn−p2sn

sn−2βBxn−Bp2−λn2δ−λnDun−Dp2

1−ξn

xn−p2rnrn−2αAxn−Ap2−μn

≤xn−p2ξnsn

sn−2βBxn−Bp2−ξnλn2δ−λnDun−Dp2

1−ξnrnrn−2αAxn−Ap2−1−ξnμn

2η−μnEtn−Ep2.

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Furthermore, we observe that

yn,i−p2≤αn,ix0−p2 1−αn,iTiwn−p2

≤αn,ix0−p2 1−αn,i

×xn−p2ξnsn

sn−2βBxn−Bp2−ξnλn2δ−λnDun−Dp2

1−ξnrnrn−2αAxn−Ap2−1−ξnμn

≤αn,ix0−p2xn−p2 1−αn,iξnsn

sn−2βBxn−Bp2−1−αn,iξnλn

×2δ−λnDun−Dp2 1−αn,i1−ξnrnrn−2αAxn−Ap2

−1−αn,i1−ξnμn

≤αn,ix0−p2xn−p2 1−αn,iξnsn

sn−2βBxn−Bp2

−1−αn,iξnλn2δ−λnDun−Dp2

1−αn,i1−ξnrnrn−2αAxn−Ap2.

3.31

Since 0< e≤λn≤f <2δ, 0≤ki≤αn,i≤hi<1, we have

1−hiξe

2δ−fDun−Dp2≤αn,ix0−p2xn−p2−yn,i−p2

1−αn,iξnsn

sn−2βBxn−Bp2

≤αn,ix0−p2yn,i−xnxn−p−yn,i−p

1−αn,iξnsn

sn−2βBxn−Bp2

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By conditions i–v, 3.18, limn→ ∞Axn −Ap 0, and limn→ ∞Bxn −Bp 0, then

limn→ ∞Dun−Dp 0. By using the same method with3.32. Hence, from 3.31, and

since 0< g≤μn≤j ≤2η, 0≤ki≤αn,i≤hi≤1, we have

1−hi1−ξg

2η−jEtn−Ep2

≤αn,ix0−p2xn−p2−yn,i−p2

1−αn,iξnsn

sn−2βBxn−Bp2

≤αn,ix0−p2yn,i−xnxn−p−yn,i−p

1−αn,iξnsn

sn−2βBxn−Bp2

3.33

By conditionsi–iv,vi,3.18, limn→ ∞Axn−Ap0, and limn→ ∞Bxn−Bp0, then

limn→ ∞Etn−Ep0. From3.1, we have

_w_n₋_p2 _≤

ξn

PCun−λnDun−PC

p−λnDp

1−ξn

PC

tn−μnEtn

−PC

p−μnEp2

≤ξnun−p

2

1−ξntn−p

2

.

3.34

Assume thatu_nPCun−λnDunandtn PCtn−μnEtn. By nonexpansiveness ofI−λnD

andI−μnE, we also have

u_n−p2≤PCI−λnDun−PCI−λnDp2

≤un−λnDun−

p−λnDp

, u_n−p

1

2

un−λnDun−

p−λnDp2un−p

2

−un−λnDun−

p−λnDp

−u_n−p2

≤ 1

2

un−p2un−p

2₋_u

n−λnDun−

p−λnDp

−u_n−p2

1

2 T

F2,ϕ2

sn xn−snBxn−T

F2,ϕ2

sn

p−snBp

2

u_n−p2−un−un

2

2λn

un−un, Dun−Dp

−λ2_nDun−Dp2

≤ 1

2

xn−p2un−p

2₋

un−un

2

λnλn−2δDun−Dp2

.

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It follows that

u_n−p2≤xn−p2−un−un

2

λnλn−2δDun−Dp2. 3.36

Similar to3.36, we obtain

t_n−p2≤xn−p2−tn−tn

2

μn

μn−2ηEtn−Ep2. 3.37

Substituting3.36,3.37into3.34, we have

_w_n₋_p2_≤

ξnun−p

2

1−ξntn−p

2

≤ξn

xn−p2−un−un

2

λnλn−2δDun−Dp2

1−ξn

xn−p2−tn−tn

2

μn

μn−2ηEtn−Ep2

≤xn−p2−ξnun−un

2

ξnλnλn−2δDun−Dp2

−1−ξntn−tn

2

1−ξnμn

μn−2ηEtn−Ep2.

3.38

By3.38, we have

yn,i−p2

≤αn,ix0−p2 1−αn,iTiwn−p2

αn,ix0−p2 1−αn,i

×xn−p2−ξnun−un

2

ξnλnλn−2δDun−Dp2

−1−ξntn−tn

2

1−ξnμn

μn−2ηEtn−Ep2

αn,ix0−p2xn−p2−1−αn,iξnun−un

2

1−αn,iξnλnλn−2δDun−Dp2−1−αn,i1−ξntn−tn

2

1−αn,i1−ξnμn

μn−2ηEtn−Ep2

αn,ix0−p2xn−p2−1−αn,iξnun−un2

1−αn,iξnλnλn−2δDun−Dp2 1−αn,i1−ξnμn

μn−2ηEtn−Ep2.

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It follows that

1−αn,iξnun−un2≤αn,ix0−p2xn−p2−yn,i−p2 1−αn,iξnλn

×λn−2δDun−Dp2 1−αn,i1−ξnμn

μn−2ηEtn−Ep2

≤αn,ix0−p2xn−yn,ixn−pyn,i−p 1−αn,iξnλn

×λn−2δDun−Dp2 1−αn,i1−ξnμn

μn−2ηEtn−Ep2.

3.40

By conditionsiii–vi,3.18, limn→ ∞Dun−Dp0 and limn→ ∞Etn−Ep0, we get

lim

n→ ∞un−u

n0. 3.41

By using3.41, we can prove that

lim

n→ ∞tn−t

n0. 3.42

Applying3.27and3.41, we also have

lim

n→ ∞xn−u

n0. 3.43

From3.29and3.42, we obtain

lim

n→ ∞xn−t

n0. 3.44

Sinceu_nPCun−λnDunandtnPCtn−μnEtn, we have

wn−xn ξn

u_n−xn

1−ξn

t_n−xn

. 3.45

By3.43and3.44, we obtain

lim

n→ ∞wn−xn0. 3.46

By conditioniii, we haveyn,iαn,ix0 1−αn,iTiwn, which implies that

_y_n,i₋_T_i_w_n_α_n,i_x₀₋_T_i_w_n_−→₀_, _n_{−→ ∞}_, _∀_i_≥₁_. _3.47

From3.18and limn→ ∞yn,i−Tiwn0, we have

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Since

wn−Tiwn ≤ wn−xnxn−Tiwn. 3.49

By3.46and3.48, we have that limn→ ∞wn−Tiwn0, for alli1,2, . . ..

Step 4. We show thatz∈Θ: ∞_i₁FTi∩GMEPF1, ϕ1, A∩GMEPF2, ϕ2, B∩VIC, D∩

VIC, E.

First, we show thatz ∈ ∞_i₁FTi. Assume thatz /∈

_∞

i1FTi. Since limn→ ∞wn −

xn0 and limn→ ∞xn−z0, we have that limn→ ∞wn−z0. By limn→ ∞wn−z0

and limn→ ∞wn−Tiwn0,i1,2, . . ., from Opial’s condition, we have

lim inf

i→ ∞ wni−z<lim infi→ ∞ wni −Tiz

≤lim inf

i→ ∞ wni −TiwniTiwni−Tiz

≤lim inf

i→ ∞ wni −z,

3.50

which is a contradiction. Thus, we obtainz∈∞_i₁FTi.

Next, we show thatz∈GMEPF1, ϕ, A. SincetnTrnF1,ϕ1xn−rnAxn, n≥1, we have for anyy∈Cthat

F1

tn, y

ϕ1

y−ϕ1tn

Axn, y−tn

1

rn

y−tn, tn−xn

≥0, ∀y∈C. 3.51

FromA2, we also have

ϕ1

y−ϕ1tn

Axn, y−tn

1

rn

y−tn, tn−xn

≥F1

y, tn

, ∀y∈C. 3.52

Fortwith 0< t≤1 andy∈C, letytty 1−tz. Sincey∈Candz ∈C, we haveyt∈C.

Then, we have

yt−tni, Ayt

≥yt−tni, Ayt

−ϕ1

yt

ϕ1tni−

yt−tni, Axni

−

yt−tni,

tni−xni

rni

F1

yt, tni

yt−tni, Ayt−Atni

yt−tni, Atni−Axni

−ϕ1

yt

ϕ1tni

−

yt−tni,

tni−xni

rni

F1

yt, tni

.

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Since tni − xni → 0, we have Atni −Axni → 0. Further, from an inverse-strongly monotonicity of A, we haveyt−tni, Ayt−Atni ≥ 0. So, fromA4 and the weak lower semicontinuity ofϕ1,tni−xni/rni → 0 andtni z, we have at the limit

yt−z, Ayt

≥ −ϕ1

yt

ϕ1z F1

yt, z

, 3.54

asi → ∞. FromA1,A4, and3.54, we also get

0F1

yt, yt

ϕ1

yt

−ϕ1

yt

≤tF1

yt, y

1−tF1

yt, z

tϕ1

y−1−tϕ1z−ϕ

yt

tF1

yt, y

ϕ1

y−ϕ1

yt

1−tF1

yt, z

ϕ1z−ϕ1

yt

≤tF1

yt, y

ϕ1

y−ϕ1

yt

1−tyt−z, Ayt

tF1

yt, y

ϕ1

y−ϕ1

yt

1−tty−z, Ayt

,

0≤F1

yt, y

ϕ1

y−ϕ1

yt

1−ty−z, Ayt

.

3.55

Lettingt → 0, we have, for eachy∈C,

F1

z, yϕ1

y−ϕ1z

y−z, Az≥0. 3.56

This implies that z ∈ GMEPF1, ϕ1, A. By the same arguments, we can show that z ∈

GMEPF2, ϕ2, B.

Lastly, by the same proof of39, Theorem3.1, pages 346-347, we can show thatz ∈

VIC, Dandz∈VIC, E. Therefore,z∈∞_i₁FTi∩GMEPF1, ϕ1, A∩GMEPF2, ϕ2, B∩

VIC, D∩VIC, E; that is,z∈Θ.

Noting that sincexnPCnx0, by2.4, we have

x0−xn, y−xn

≤0, ∀y∈Cn. 3.57

SinceΘ ⊂ Cnand by the continuity of inner product, we obtain from the above inequality

that

x0−z, y−z

≤0, ∀y∈C. 3.58

By2.4, again, we conclude thatzPΘx0. This completes the proof.

Using Theorem3.1, we obtain the following corollaries.

Corollary 3.2. LetCbe a nonempty closed convex subset of a real Hilbert SpaceH. LetF1,F2be a

bifunction ofC×Cinto real numbersÊ satisfying (A1)–(A4), and letϕ1, ϕ2 :C → Ê∪ {∞}be

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Θ: FT∩GMEPF1, ϕ1, A∩GMEPF2, ϕ2, B∩VIC, D∩VIC, E/∅. Assume that either

(B1) or (B2) holds. Let{xn}be a sequence generated byx0 ∈C,x1 PC1x0and

tn−μnEtn

,

ynαnx0 1−αnTwn,

Cn1

z∈Cn:yn−z2≤ xn−z2αn

x022wn−x0, z

,

xn1PCn1x0,

3.59

for every n ≥ 0, where{rn},{sn} ⊂ 0,∞,λn ∈ 0,2δ, and μn ∈ 0,2ηsatisfy the following

conditions:

i0< a≤rn≤b <2α,

ii0< c≤sn≤d <2β,

iiilimn→ ∞αn0,

v0< e≤λn≤f <2δ,

vi0< g≤μn≤j <2η.

Then,{xn}converges strongly toPΘx0.

Proof. TakingTi T fori 1,2, . . ., to be nonexpansive mappings, in Theorem3.1, we can

conclude the desired conclusion easily. This completes the proof.

A mappingT : C → Cis said to be aκ-strict pseudocontraction40if there exists a constant 0≤κ <1 such that

Tx−Ty2≤x−y2κI−Tx−I−Ty2, ∀x, y∈C, 3.60

whereIdenotes the identity operator onC.

Lemma 3.3see41. LetCbe a nonempty closed convex subset of a real Hilbert spaceH, and let

T :C → Cbe aκ-strict pseudocontraction. DefineSx:C → CbySxαx 1−αTxfor each

x∈C. Then, asα∈κ,1)Sis nonexpansive such thatFS FT.

Using Theorem3.1, we obtain the following result.

Theorem 3.4. LetCbe a nonempty closed convex subset of a real Hilbert SpaceH. LetF1,F2 be a

a proper lower semicontinuous and convex function. LetA,B,D, andEbeα,β,δ, andη -inverse-strongly monotone mapping of C intoH, respectively. Let T1, T2, . . . , TN be a finite family of κi

-psuedocontractions such thatΘ:N_i₁FTi∩GMEPF1, ϕ1, A∩GMEPF2, ϕ2, B∩VIC, D∩

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Snbe theS-mappings generated byTκ1, Tκ2, . . . , TκNandα1n,i, α2n,i, . . . , αNn,i. Assume that either (B1) or

(B2) holds. Let{xn}be a sequence generated byx0∈C,C1,iC,C1∞i1C1,i,x1PC1x0and

tn−μnEtn

,

Cn1,i

x022wn−x0, z

,

Cn1

∞

i1

Cn1,i,

xn1 PCn1x0,

3.61

conditions:

i0< a≤rn≤b <2α,

ii0< c≤sn≤d <2β,

v0< e≤λn≤f <2δ,

vi0< g≤μn≤j <2η.

Proof. From Theorem 3.1, {Ti}Ni1 is a finite family of κi-strict pseudocontraction. By

Lemma3.3, we have thatTκi is nonexpansive mappings. The conclusion of Theorem 3.4 can be obtained from Theorem3.1immediately.

4. Some Applications

4.1. Complementarity Problem

LetCbe a nonempty closed and convex cone inH, and letEbe an operator ofCintoH. We define thepolarofCinHto be the set

K∗:y∗∈H:x, y∗≥0, ∀x∈C. 4.1

Then, the elementu∈Cis called a solution of thecomplementarity problemif

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The set of solution of the complementarity problem is denoted byCC, D,CC, E. We will assume thatD,Esatisfies the following conditions:

E1D,Eis aδ,η-inverse-strongly monotone mapping, respectively,

E2CC, D, CC, E/∅.

B1For eachx∈Handr >0, there exist a bounded subsetDx⊆Candyx∈C∩domϕ

such that for anyz∈C\Dx,

Fz, yx

ϕyx

1

r

yx−z, z−x

< ϕz, 4.3

B2Cis a bounded set.

Corollary 4.1. LetCbe a nonempty closed convex subset of a real Hilbert SpaceH. LetF1,F2be a

a proper lower semicontinuous and convex function. LetA,B,D, andEbeα,β,δ, andη -inverse-strongly monotone mapping ofCintoH, respectively. LetT1, T2, . . .be infinite nonexpansive mapping

such thatΘ : ∞_i₁FTi ∩ GMEPF1, ϕ1, A ∩ GMEPF2, ϕ2, B ∩ CC, D ∩ CC, E/∅.

Assume that either (B1) or (B2) holds. Let{xn}be a sequence generated byx0 ∈C,C1,i C,C1

_∞

i1C1,i,x1PC1x0and

tn−μnEtn

,

Cn1,i

x022wn−x0, z

,

Cn1

∞

i1

Cn1,i,

xn1PCn1x0,

4.4

conditions:

i0< a≤rn≤b <2α,

ii0< c≤sn≤d <2β,

v0< e≤λn≤f <2δ,

vi0< g≤μn≤j <2η.

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Proof. Using Lemma 7.1.1 of42, we have that VIC, D CC, Dand VIC, E CC, E. Hence, by Corollary4.1, we can conclude the desired conclusion easily. This completes the proof.

4.2. Optimization Problem

In this section, we study a kind of multiobjective optimization problem by using the result of this paper. We will give an iterative algorithm of solution for the followingoptimization problemwith nonempty set of solutions:

minh1x,

minh2x,

x∈C,

4.5

wherehxis a convex and lower semicontinuous functional, and defineCas a closed convex subset of a real Hilbert spaceH. We denote the set of solutions of4.5byMh1andMh2.

Let Fi : C×C → Ê be a bifunction defined byFix, y hiy−hix. We consider the

equilibrium problem, it is obvious that EPFi Mhi,i1,2. Therefore, from Theorem3.1,

we obtained the following corollary.

Corollary 4.2. LetCbe a nonempty closed convex subset of a real Hilbert SpaceH. LetF1, F2be a

bifunction ofC×Cinto real numbersÊsatisfying (A1)–(A4), and letϕ1, ϕ2:C → Ê∪ {∞}be a

proper lower semicontinuous and convex function. LetA,B,D, andE beα,β,δ, andη -inverse-strongly monotone mapping of C into H, respectively. Let T1, T2, . . . be an infinite nonexpansive

mapping such thatΘ:∞_i₁FTi∩MEPF1, ϕ1∩MEPF2, ϕ2∩VIC, D∩VIC, E/∅. Assume

that either (B1) or (B2) holds. Let{xn}be a sequence generated byx0 ∈C,C1,iC,C1

_∞

i1C1,i,

x1PC1x0, and

h1t−h1tn

1

rnt−tn, tn−xn ≥0, ∀t∈C,

h2u−h2un

1

sn

u−un, un−tn ≥0, ∀u∈C,

tn−μnEtn

,

Cn1,i

x022wn−x0, z

,

Cn1

∞

i1

Cn1,i,

xn1PCn1x0,

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conditions:

ilimn→ ∞αn,i0,

iilimn→ ∞ξnξ∈0,1,

iii0< e≤λn≤f <2δ,

iv0< g≤μn≤j <2η.

Proof. From Theorem 3.1, put F1tn, t h1t− h1tn, F2un, u h2u −h2un, and

A, B, ϕ1, ϕ2 ≡ 0. The conclusion of Corollary 4.2 can be obtained from Theorem 3.1

immediately.

4.3. Minimization Problem

In this section, we study the problem for finding a minimizer of a continuously Fr`echet diﬀerentiable convex functional in a Hilbert space.

First, we use the following lemma in our result.

Lemma 4.3see43. LetEbe a Banach space, letfbe a continuously Fr`echet diﬀerentiable convex functional onE, and let∇fbe the gradient off. If∇fis1/α-Lipschitz continuous, then∇fis an

α-inverse-strongly monotone.

Letf1,f2be functionals onHwhich satisfies the following conditions:

C1let,f1,f2be a continuously Fr`echet diﬀerentiable convex functional onH, and let,

∇f1,∇f2be1/δ,1/η-Lipschitz continuous, respectively,

C2 ∇f1−10{z1∈H:f1z1 miny1∈Hf1y1}/∅and∇f2

−1₀_{_z

2∈H:f2z2

miny2∈Hf2y2}/∅.

Corollary 4.4. LetHbe a real Hilbert Space. LetF1,F2be a bifunction ofH×Hinto real numbersÊ

satisfying (A1)–(A4), and letϕ1, ϕ2:C → Ê∪ {∞}be a proper lower semicontinuous and convex

function. LetA,Bbeα,β-inverse-strongly monotone mapping ofHintoH, respectively. LetT1, T2, . . .

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(C1) and (C2). Suppose thatΘ : ∞_i₁FTi ∩ GMEPF1, ϕ1 ∩ GMEPF2, ϕ2 ∩ ∇f1−10 ∩

∇f2−10/∅. Assume that either (B1) or (B2) holds. Let{xn}be a sequence generated byx0 ∈ C,

C1,iC,C1

_∞

i1C1,i,x1PC1x0, and

wnξn

un−λn∇f1un

1−ξn

tn−μn∇f2tn

,

Cn1,i

x022wn−x0, z

,

Cn1

∞

i1

Cn1,i,

xn1PCn1x0,

4.7

for everyn≥ 0, where{rn},{sn} ⊂0,∞,λn ∈0,2δ, andμn ∈0,2ηsatisfying the following

conditions:

i0< a≤rn≤b <2α,

ii0< c≤sn≤d <2β,

v0< e≤λn≤f <2δ,

vi0< g≤μn≤j <2η.

Proof. We know form conditionC1and Lemma4.3that∇f1,∇f2is aδ,η-inverse-strongly

monotone operator fromHin to itself, respectively. The conclusion of Corollary4.4can be obtained from Theorem3.1immediately.

Acknowledgments

The authors are grateful to the anonymous referees for their helpful comments which improved the presentation of the original version of this work. The authors would like to thank The National Research University Project of Thailand’s Oﬃce of the Higher Education Commission for financial supportunder the NRU-CSEC Project no. 54000267. Furthermore, P. Kumam’s research supported by the Commission on Higher Education and the Thailand Research Fund under Grant no. MRG5380044.

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18 P. Kumam, “A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping,” Nonlinear Analysis. Hybrid Systems, vol. 2, no. 4, pp. 1245–1255, 2008.

19 P. Kumam, “Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space,”Turkish Journal of Mathematics, vol. 33, no. 1, pp. 85–98, 2009.

20 P. Kumam, “A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping,”Journal of Applied Mathematics and Computing, vol. 29, no. 1-2, pp. 263–280, 2009.

21 P. Kumam and C. Jaiboon, “A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems,”Nonlinear Analysis. Hybrid Systems, vol. 3, no. 4, pp. 510–530, 2009.