Convergence Theorems on Generalized Equilibrium Problems and Fixed Point Problems with Applications

Volume 2010, Article ID 189036,14pages doi:10.1155/2010/189036

Research Article

Convergence Theorems on

Generalized Equilibrium Problems and

Fixed Point Problems with Applications

Yan Hao,

_{Sun Young Cho,}

_{and Xiaolong Qin}

1_{School of Mathematics, Physics and Information Science, Zhejiang Ocean University,}

Zhoushan 316004, China

2_{Department of Mathematics, Gyeongsang National University, Jinju 660-701, South Korea} 3_{Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China} Correspondence should be addressed to Yan Hao,[email protected]

Received 2 October 2009; Accepted 23 January 2010

Academic Editor: Jong Kim

Copyrightq2010 Yan Hao 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.

The purpose of this work is to introduce an iterative method for finding a common element of a solution set of a generalized equilibrium problem, of a solution set solutions of a variational inequality problem and of a fixed point set of a strict pseudocontraction. Strong convergence theorems are established in the framework of Hilbert spaces.

1. Introduction and Preliminaries

LetHbe a real Hilbert space,Ca nonempty closed and convex subset ofH,andB:C → H

a nonlinear mapping. Recall the following definitions.

aThe mappingBis said to be monotoneif

Bx−By, x−y≥0, ∀x, y∈C. 1.1

bBis said to beβ-strongly monotoneif there exists a constantβ >0 such that

cBis said to beβ-inverse-strongly monotoneif there exists a constantβ >0 such that

Bx−By, x−y≥βBx−By2, ∀x, y∈C. 1.3

The classical variational inequality problem is to findu∈Csuch that

Bu, v−u ≥0, ∀v∈C. 1.4

In this paper, we use VIC, Bto denote the solution set of the problem1.4. One can easily see that the variational inequality problem is equivalent to a fixed point problem.u∈Cis a solution to the problem1.4if and only ifuis a fixed point of the mappingPCI−λB, where

λ >0 is a constant andIis the identity mapping.

LetS:C → Cbe a nonlinear mapping. In this paper, we useFSto denote the fixed point set ofS. Recall the following definitions.

dThe mappingSis said to be nonexpansiveif

_{Sx−Sy≤x−y, ∀x, y∈C.} 1.5

eSis strictly pseudocontractivewith a constantk∈0,1if

_Sx−Sy2_≤

x−y2kI−Sx−I−Sy2, ∀x, y∈C. 1.6

For such a case,Sis called ak-strict pseudocontraction.

fSis said to be pseudocontractiveif

_Sx−Sy2_≤

x−y2I−Sx−I−Sy2, ∀x, y∈C. 1.7

Clearly, the class of strict pseudocontractions falls into the one between a class of nonexpansive mappings and a class of pseudocontractions.

Recently, many authors considered the problem of finding a common element of the solution set of the variational inequality1.4and of fixed point set of a nonexpansive mapping in Hilbert spaces; see, for examples,1–5and the references therein.

In 2005, Iiduka and Takahashi2obtained the following theorem in a real Hilbert space.

Theorem IT. LetC be a closed convex subset of a real Hilbert space H. Let B be anα -inverse-strongly monotone mapping ofCintoHand letSbe a nonexpansive mapping ofCinto itself such thatFS∩VIC, B/∅. Supposex1x∈Cand{xn}is given by

for everyn1,2, . . . ,where{αn}is a sequence in0,1and{λn}is a sequence ina, b. If{αn}and

{λn}are chosen so that{λn} ∈a, bfor somea, bwith0< a < b <2α,

lim n→ ∞αn0,

∞

n1

αn∞, ∞

n1

|αn1−αn|<∞, ∞

n1

|λn1−λn|<∞, 1.9

then{xn}converges strongly toPFS∩VIC,Bx.

LetAbe an inverse-strongly monotone mapping, andFa bifunction ofC×CintoR, whereRis the set of real numbers. We consider the following equilibrium problem:

findz∈Csuch thatFz, yAz, y−z ≥0, ∀y∈C. 1.10

In this paper, the set of suchz∈Cis denoted by EPF, A, that is,

EPF, A z∈C:Fz, yAz, y−z≥0, ∀y∈C . 1.11

In the case ofA≡0, the zero mapping, the problem1.10is reduced to

Findz∈Csuch thatFz, y≥0, ∀y∈C. 1.12

In this paper, we use EPFto denote the solution set of the problem1.12, which was studied by many others; see, for examples,1,3,6–23and the reference therein. In the case ofF≡0, the problem1.10is reduced to the classical variational inequality1.4. The problem1.10is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others; see, for instances,15,24.

To study the problems1.10and1.12, we may assume that the bifunctionF : C× C → Rsatisfies the following conditions:

A1Fx, x 0 for allx∈C;

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

A3for eachx, y, z∈C,

lim sup t↓0

Ftz 1−tx, y≤Fx, y; _1.13

A4for eachx∈C,y→Fx, yis convex and weakly lower semicontinuous.

Theorem TT 1. LetCbe a nonempty closed convex subset ofH. LetFbe a bifunction fromC×Cto Rsatisfying (A1)–(A4), and letSbe a nonexpansive mapping ofCintoHsuch thatFS∩EPF/∅. Letfbe a contraction ofHinto itself, and let{xn}and{yn}be sequences generated byx1∈Hand

Fyn, u

1

rn

u−yn, yn−xn

≥0, ∀u∈C,

xn1αnfxn 1−αnSyn, n≥1,

1.14

where{αn} ∈0,1and{rn} ⊂0,∞satisfy

lim n→ ∞αn0,

∞

n1

αn∞, ∞

n1

|αn1−αn|<∞,

lim inf n→ ∞ rn>0,

∞

n1

|rn1−rn|<∞.

1.15

Then{xn}and{yn}converge strongly toz∈FS∩EPF,wherezPFS∩EPFfz.

Very recently, S. Takahashi and W. Takahashi 22 further considered the problem

1.10. Strong convergence theorems of common elements are established. More precisely, they obtained the following result.

Theorem TT 2. LetCbe a closed convex subset of a real Hilbert spaceH,and letF:C×C → Rbe a bifunction satisfying (A1), (A2), (A3) and (A4). LetAbe anα-inverse-strongly monotone mapping ofCintoHand letSbe a nonexpansive mapping ofCinto itself such thatFS∩EPF, A/∅. Let

u∈Candx1∈Cand let{zn} ⊂Cand{xn} ⊂Cbe sequences generated by

Fzn, y

Axn, y−zn

1

λn

y−zn, zn−xn

≥0, ∀y∈C,

xn1βnxn

1−βn

Sαnu 1−αnzn, ∀n≥1,

1.16

where{αn} ⊂0,1,{βn} ⊂0,1, and{λn} ⊂0,2αsatisfy

0< c≤βn≤d <1, 0< a≤λn≤b <2α,

lim

n→ ∞λn−λn1 0, nlim→ ∞αn0, ∞

n1

αn∞.

1.17

Then,{xn}converges strongly tozPFS∩EPF,Au.

In this paper, motivated by Theorem IT, Theorem TT1, and Theorem TT2, we introduce a general iterative method for the problem of finding a common element of a solution set of a generalized equilibrium problem1.10, of a solution set of a variational inequality problem

In order to prove our main results, we need the following lemmas. The following lemmas can be found in11,24.

Lemma 1.1. LetCbe a nonempty closed convex subset ofHand letF:C×C → Rbe a bifunction satisfying (A1)–(A4). Then, for anyr >0andx∈H, there existsz∈Csuch that

Fz, y1 r

y−z, z−x≥0, ∀y∈C. 1.18

Further, define a mappingTr by

Trx

z∈C:Fz, y1 r

y−z, z−x≥0, ∀y∈C

1.19

for allr >0andx∈H.Then, the following hold. 1Tr is single-valued;

2Tr is firmly nonexpansive, that is, for anyx, y∈H,

_Trx−Try2_≤

Trx−Try, x−y

; 1.20

3FTr EPF;

4EPFis closed and convex.

Lemma 1.2see25. LetC be a nonempty closed convex subset of a real Hilbert spaceH and

S : C → Cak-strict pseudocontraction. DefineSα : C → CbySαx αx 1−αSxfor each

x∈C. Then, asα∈k,1,Sαis nonexpansive andFSα FS.

Lemma 1.3see26. Let{xn}and{yn}be bounded sequences in a Banach spaceX and let{βn}

be a sequence in0,1with

0<lim inf

n→ ∞ βn≤lim sup_{n→ ∞} βn<1. 1.21

Suppose thatxn1 1−βnynβnxnfor all integersn≥0and

lim sup n→ ∞

_{yn1−yn− xn1−xn}

≤0. _1.22

Thenlimn→ ∞yn−xn0.

The following lemma can be deduced from Bruck8.

Lemma 1.4. LetCbe a closed convex subset of a strictly convex Banach spaceE. LetS1,S2,andS3 be three nonexpansive mappings onC. Suppose3_n1FSnis nonempty. Leta1,a2,anda3be three constant in0,1.Then the mappingSonCdefined by

Sxa1S1xa2S2xa3S3x 1.23

Lemma 1.5see6. LetH be a real Hilbert space,Ca nonempty closed and convex subset ofE,

andS:C → Ca nonexpansive mapping. ThenI−Sis demiclosed at zero.

Lemma 1.6see14. LetHbe a real Hilbert space,Ca nonempty closed and convex subset ofE

andS:C → Cak-strict pseudocontraction. ThenFSis closed and convex.

Lemma 1.7see27. Assume that{αn}is a sequence of nonnegative real numbers such that

αn1≤

1−γn

αnδn, 1.24

where{γn}is a sequence in0,1and{δn}is a sequence such that

a∞_n1γn∞;

blim sup_{n→ ∞}δn/γn≤0or

_∞

n1|δn|<∞.

Thenlimn→ ∞αn0.

2. Main Results

Theorem 2.1. Let C be a nonempty closed and convex subset of a real Hilbert spaceH and F a bifunction fromC×CtoRsatisfying (A1)–(A4). LetAbe anα-inverse-strongly monotone mapping ofCintoHandBaβ-inverse-strongly monotone mapping ofCintoH. LetS:C → Cbe ak-strict pseudocontraction with a fixed point. Assume thatΩ:EPF, A∩FS∩VIC, B/∅. Let{xn}be

a sequence inCgenerated by

x1∈C, chosen arbitrarily,

Fun, y

Axn, y−un

1

r

y−un, un−xn

≥0, ∀y∈C,

vnPCxn−λBxn,

ynδnxn 1−δnSxn,

xn1αnuβnxnγn

μ1,nynμ2,nunμ3,nvn

, ∀n≥1,

2.1

whereuis a fixed element inC,{αn},{βn},{γn},{μ1,n},{μ2,n}and{μ3,n}are sequences in0,1,

{δn}is sequence ink,1,r∈0,2αandλ∈0,2β. Assume that the above control sequences satisfy

the following restrictions

R1αnβnγnμ1,nμ2,nμ3,n1,for alln≥1;

R2limn→ ∞αn0,∞n1αn∞;

R30<lim infn→ ∞βn≤lim sup_{n→ ∞}βn<1;

R4limn→ ∞δn δ∈k,1andlimn→ ∞μi,nμi∈0,1for all1≤i≤3.

Then the sequence{xn}defined by the iterative process2.1converges strongly toxPΩu.

Proof. The proof is divided into six steps.

FromLemma 1.6, we see thatFSis closed and convex. On the other hand, we see that the mappingI−rA, wherer ∈ 0,2α, is nonexpansive. Indeed, for anyx, y ∈ C, we have that

_{I−rAx−I−rAy}2

x−y−rAx−Ay2

x−y2−2rx−y, Ax−Ayr2Ax−Ay2

≤x−y2−2rαAx−Ay2r2Ax−Ay2

x−y2−r2α−rAx−Ay2

≤x−y2.

2.2

This shows thatI−rAis nonexpansive mapping. Similarly, we can prove thatI−λB, where

λ ∈ 0,2βis nonexpansive. It follows that VIC, B FPCI−λB, for allλ > 0 is closed and convex. From Lemma 1.1, we see that EPF, A FTrI −rA. Since TrI −rA is nonexpansive, we obtain that EPF, Ais closed and convex. This shows that PΩuis well defined.

Step 2. Show that{xn}is bounded.

PutTnδnI 1−δnSfor eachn≥1.In view ofLemma 1.2andR4, we obtain that

Tnis nonexpansive andFTn FS. Lettingx∗∈Ω, we obtain that

x∗TrI−rAx∗Tnx∗Sx∗PCI−λBx∗, ∀n≥1. 2.3

Note thatynTnxnfor eachn≥1.It follows thatyn−x∗ ≤ xn−x∗.Putting

enμ1,nynμ2,nunμ3,nvn, ∀n≥1, 2.4

we have that

en−x∗ ≤μ1,nyn−x∗μ2,nun−x∗μ3,nvn−x∗

μ1,nyn−x∗μ2,nTrI−rAxn−TrI−rAx∗

μ3,nPCI−λAxn−PCI−λBx∗

≤μ1,nxn−x∗μ2,nxn−x∗μ3,nxn−x∗

xn−x∗.

It follows that

xn1−x∗αnuβnxnγnen−x∗

≤αnu−x∗βnxn−x∗γnen−x∗

≤αnu−x∗βnxn−x∗γnxn−x∗

αnu−x∗ 1−αnxn−x∗

≤max{u−x∗,x1−x∗}.

2.6

This shows that the sequence{xn}is bounded. Note that

un−x∗TrI−rAxn−x∗ ≤ xn−x∗,

vn−x∗PCxn−λBxn−x∗ ≤ xn−x∗.

2.7

This proves that the sequences{un}and{vn}are bounded, too.

Step 3. Show thatxn1−xn → 0 asn → ∞. Note that

_{yn1−ynTn1xn1−Tnxn}

≤ Tn1xn1−Tn1xnTn1xn−Tnxn

≤ xn1−xn|δn1−δn|M1,

2.8

whereM1is an appropriate constant such thatM1supn≥1{xn−Sxn}. On the other hand, we have

vn1−vnPCxn1−λBxn1−PCxn−λBxn ≤ xn1−xn,

un1−unTrI−rAxn1−TrI−rAxn ≤ xn1−xn.

2.9

It follows from2.1and2.9that

en1−enμ1,n1yn1μ2,n1un1μ3,n1vn1

− μ1,nynμ2,nunμ3,nvn

≤μ1,n1yn1−ynynμ1,n1−μ1,n

μ2,n1un1−ununμ2,n1−μ2,n

μ3,n1vn1−vnvnμ3,n1−μ3,n≤ xn1−xn

M2μ1,n1−μ1,nμ2,n1−μ2,nμ3,n1−μ3,n|δn1−δn|

,

whereM2is an appropriate constant such that

M2≥max

sup n≥1

_yn,sup n≥1

un,sup n≥1

vn, M1

. 2.11

Putln xn1−βnxn/1−βn, for eachn≥1, that is,

xn1

1−βn

lnβnxn, ∀n≥1. 2.12

Now, we computeln1−ln.Notice that

ln1−ln αn1uγn1en1 1−βn1 −

αnuγnen 1−βn

αn1

1−βn1u

1−βn1−αn1

1−βn1 en1−

αn 1−βnu−

1−βn−αn 1−βn en

αn1

1−βn1u−en1

αn

1−βnen−u en1−en.

2.13

It follows that

ln1−ln ≤ αn1

1−βn1u−en1

αn

1−βnen−uen1−en. 2.14

Substituting2.10into2.14, we arrive at

ln1−ln − xn1−xn

≤ αn1

1−βn1u−en1

αn

1−βnen−u

M2μ1,n1−μ1,nμ2,n1−μ2,nμ3,n1−μ3,n|δn1−δn|

.

2.15

It follows from the restrictionsR2–R4that

lim sup

n→ ∞ ln1−ln − xn1−xn1<0. 2.16

FromLemma 1.3, we obtain that

lim

n→ ∞ln−xn0. 2.17

From2.12, we see that

xn1−xn

1−βn

In view of2.17, we get that

lim

n→ ∞xn1−xn0. 2.19

Step 4. Show thatxn−en → 0 asn → ∞. From the iterative process2.1, we have

γnen−xn xn1−xnαnxn−u. 2.20

This implies that

γnen−xn ≤ xn1−xnαnxn−u. 2.21

It follows from the restrictionsR2andR3that we arrive at

lim

n→ ∞en−xn0. 2.22

Step 5. Show that lim sup_{n→ ∞}u−x, xn−x ≤0,wherexPΩu. To show that, we can choose a sequence{xni}of{xn}such that

lim sup

n→ ∞ u−x, xn−xilim→ ∞u−x, xni−x. 2.23

Since{xni}is bounded, we see that there exists a subsequence{xnij}of{xni}which converges

weakly toξ. Without loss of generality, we may assume thatxni ξ.

Next, we show that ξ ∈ Ω FS∩VIC, B∩EPF, A. In fact, define a mapping

Q:C → Cby

Qxμ1Txμ2TrI−rAxμ3PCI−λB, ∀x∈C, 2.24

whereT δI 1−δS. Note thatT is nonexpansive andFT FS. FromLemma 1.4, we see thatQis a nonexpansive mapping such that

FQ FT∩FTrI−rA∩FPCI−λB FS∩EPF, A∩VIC, B. 2.25

On the other hand, we have

en−Qxnμ1,nynμ2,nTrI−rAxnμ3,nPCI−λBxn

−μ1Txnμ2TrI−rAxnμ3PCI−λBxn

≤μ1,nδn−μ1δxnμ1,n1−δn−μ11−δSxn

TrI−rAxnμ2,n−μ2μ3,n−μ3PCI−λBxn.

It follows from the conditionR4thaten−Qxn → 0.Note that

Qxn−xnQxn−enen−xn. 2.27

From2.22, we arrive at

lim

n→ ∞Qxn−xn0. 2.28

It follows fromLemma 1.5that

ξ∈FQ Ω. 2.29

Thanks to2.23, we arrive at

lim sup

n→ ∞ u−x, xn−xu−x, ξ−x ≤0. 2.30

Step 6. Show thatxn → xasn → ∞. Notice that

xn1−x2αnuβnxnγnen−x2

αnu−x βnxn−x γnen−x, xn1−x

αnu−x, xn1−xβnxn−x, xn1−xγnen−x, xn1−x

≤αnu−x, xn1−xβnxn−xxn1−xγnen−xxn1−x

≤αnu−x, xn1−xβnxn−xxn1−xγnxn−xxn1−x

1−αnxn−xxn1−xαnu−x, xn1−x

≤ 1−αn 2

xn−x2xn1−x2

αnu−x, xn1−x,

2.31

which yields that

xn1−x2≤1−αnxn−x22αnu−x, xn1−x.

2.32

In view of the restrictions R2and 2.30, we from Lemma 1.7 can conclude the desired conclusion easily. This completes the proof.

Corollary 2.2. Let Cbe a nonempty closed and convex subset of a real Hilbert space H and F a bifunction fromC×CtoRsatisfying (A1)–(A4). LetBbe aβ-inverse-strongly monotone mapping ofCintoH. LetS : C → Cbe ak-strict pseudocontraction with a fixed point. Assume thatΩ : EPF∩FS∩VIC, B/∅. Let{xn}be a sequence inCgenerated by

x1∈C, chosen arbitrarily,

Fun, y

1

r

y−un, un−xn

≥0, ∀y∈C,

vnPCxn−λBxn,

ynδnxn 1−δnSxn,

xn1αnuβnxnγn

μ1,nynμ2,nunμ3,nvn

, ∀n≥1,

2.33

whereuis a fixed element inC,{αn},{βn},{γn},{μ1,n},{μ2,n},and{μ3,n}are sequences in0,1,

{δn}is sequence ink,1,r∈0,∞,andλ∈0,2β. Assume that the above control sequences satisfy

the following restrictions:

R1αnβnγnμ1,nμ2,nμ3,n1,for all n≥1;

R2limn→ ∞αn0,

_∞

n1αn∞;

R30<lim infn→ ∞βn≤lim supn→ ∞βn<1;

R4limn→ ∞δn δ∈k,1andlimn→ ∞μi,nμi∈0,1for all1≤i≤3.

Then the sequence{xn}defined by the iterative process2.1converges strongly toxPΩu.

Proof. InTheorem 2.1, putA 0, the zero mapping. Then for anyα >0, we see that the the following inequality holds.

x−y, Ax−Ay≥αAx−Ay2. 2.34

Then, we can obtain the desired conclusion easily from Theorem 2.1. This completes the proof.

Corollary 2.3. Let Cbe a nonempty closed and convex subset of a real Hilbert space H and F a bifunction fromC×CtoRsatisfying (A1)–(A4). LetSA be akα-strict pseudocontraction ofCinto

HandKBakβ-strict pseudocontraction ofCintoH. LetS:C → Cbe ak-strict pseudocontraction

with a fixed point. Assume thatΩ:EPF, I−SA∩FS∩FSB/∅. Let{xn}be a sequence inC

generated by

x1∈C, chosen arbitrarily,

Fun, y

xn−SAxn, y−un

1

ry−un, un−xn ≥0, ∀y∈C, vn 1−λxnλSBxn,

ynδnxn 1−δnSxn,

xn1αnuβnxnγn

μ1,nynμ2,nunμ3,nvn

, ∀n≥1,

where uis a fixed element in C, {αn}, {βn},{γn},{μ1,n},{μ2,n}, and{μ3,n} are sequences in

0,1,{δn}is sequence ink,1,r ∈0,1−kα,andλ ∈0,1−kβ. Assume that the above control

sequences satisfy the following restrictions:

R1αnβnγnμ1,nμ2,nμ3,n1,for alln≥1;

R2limn→ ∞αn0,∞n1αn∞;

R30<lim infn→ ∞βn≤lim supn→ ∞βn<1;

R4limn→ ∞δn δ∈k,1andlimn→ ∞μi,nμi∈0,1for all1≤i≤3.

Then the sequence{xn}defined by the above iterative process converges strongly toxPΩu.

Proof. PutA I−SA andB I−SB. Then, we see thatAis1−kα/2-inverse-strongly monotone andBis1−kβ/2-inverse-strongly monotone; see7. We haveFSB VIC, B and

PCxn−λnBxn 1−λnxnλnSBxn. 2.36

It is easy to obtain the desired conclusion fromTheorem 2.1.

Remark 2.4. If f : C → C is a contractive mapping and we replace u by fxn in the recursion formula2.1, we can obtain the so-called viscosity iteration method. We note that all theorems and corollaries of this paper carry over trivially to the so-called viscosity iteration method; see28for more details.

Acknowledgment

This project is supported by the National Natural Science Foundation of China no. 10901140.

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