Hybrid Algorithm for Finding Common Elements of the Set of Generalized Equilibrium Problems and the Set of Fixed Point Problems of Strictly Pseudocontractive Mapping

Volume 2011, Article ID 274820,19pages doi:10.1155/2011/274820

Research Article

Hybrid Algorithm for Finding Common

Elements of the Set of Generalized Equilibrium

Problems and the Set of Fixed Point Problems of

Strictly Pseudocontractive Mapping

Atid Kangtunyakarn

Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand

Correspondence should be addressed to Atid Kangtunyakarn,[email protected]

Received 8 November 2010; Accepted 14 December 2010

Academic Editor: Qamrul Hasan Ansari

Copyrightq2011 Atid Kangtunyakarn. 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 paper is to prove the strong convergence theorem for finding a common element of the set of fixed point problems of strictly pseudocontractive mapping in Hilbert spaces and two sets of generalized equilibrium problems by using the hybrid method.

1. Introduction

Let C be a closed convex subset of a real Hilbert spaceH, and let F : C ×C → R be a bifunction. Recall that the equilibrium problem for a bifunctionFis to findx∈Csuch that

Fx, y≥0, ∀y∈C. 1.1

The set of solutions of1.1is denoted by EPF. Given a mappingT : C → H, letFx, y

Tx, y−x for allx, y ∈ C. Then,z ∈ EPFif and only ifTz, y−z ≥ 0 for all y ∈ C; that is,zis a solution of the variational inequality. LetA : C → Hbe a nonlinear mapping. The variational inequality problem is to find au∈Csuch that

for allv∈C. The set of solutions of the variational inequality is denoted by VIC, A. Now, we consider the following generalized equilibrium problem:

Find z∈Csuch thatFz, yAz, y−z≥0, ∀y∈C. 1.3

The set ofz∈Cis denoted by EPF, A, that is,

EPF, A z∈C:Fz, yAz, y−z≥0, ∀y∈C. 1.4

In the case of A ≡ 0, EPF, Ais denoted by EPF. In the case ofF ≡ 0, EPF, Ais also denoted by VIC, A. Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and economics are reduced to find a solution of1.3; see, for instance,1–3.

A mappingAofCintoHis called inverse strongly monotone mapping, see4, if there exists a positive real numberαsuch that

x−y, Ax−Ay ≥αAx−Ay2 1.5

for allx, y∈C. The following definition is well known.

Definition 1.1. A mappingT : C → Cis said to be aκ-strict pseudocontraction if there exists

κ∈0,1such that

_Tx−Ty2_≤

x−y2κI−Tx−I−Ty2, ∀x, y∈C. 1.6

A mappingTis called nonexpansiveif

_{Tx−Ty≤x−y 1.7}

for allx, y∈C.

We know thatκ-strict pseudocontraction includes a class of nonexpansive mappings. Ifκ 1,T is said to be a pseudocontractivemapping.T is strong pseudocontraction if there exists a positive constantλ ∈ 0,1such thatT λI is pseudocontraction. In a real Hilbert spaceH,1.6is equivalent to

Tx−Ty, x−y≤x−y2−1−κ

2 I−Tx−I−Ty

2

, ∀x, y∈DT. 1.8

Tis pseudocontraction if and only if

Tx−Ty, x−y≤x−y2, ∀x, y∈DT. 1.9

Then,Tis strong pseudocontraction if there exists positive constantλ∈0,1

The class ofκ-strict pseudocontractions falls into the one between classes of nonex-pansive mappings, and the pseudocontraction mappings, and the class of strong pseudocon-traction mappings is independent of the class ofκ-strict pseudocontraction.

We denote byFTthe set of fixed points ofT. IfC⊂His bounded, closed, and convex, andTis a nonexpansive mapping ofCinto itself, thenFTis nonempty; for instance, see5. Browder and Petryshyn6show that if aκ-strict pseudocontractionThas a fixed point inC, then starting with an initialx0∈C, the sequence{xn}generated by the recursive formula:

xn1αxn 1−αTxn, 1.11

whereαis a constant such that 0< α <1, converges weakly to a fixed point ofT. Marino and Xu7have extended Browder and Petryshyns above-mentioned result by proving that the sequence{xn}generated by the following Manns algorithm8:

xn1αnxn 1−αnTxn 1.12

converges weakly to a fixed point ofT provided the control sequence{αn}∞n0 satisfies the conditions thatκ < αn<1 for allnand ∞n0αn−κ1−αn ∞. In 1974, S. Ishikawa proved the following strong convergence theorem of pseudocontractive mapping.

Theorem 1.2see9. LetCbe a convex compact subset of a Hilbert spaceH, and letT : C → C

be a Lipschitzian pseudocontractive mapping. For any x1 ∈ C, suppose that the sequence {xn} is

defined by

yn

1−βn

xnβnTxn,

xn1 1−αnxnαnTyn, ∀n∈N,

1.13

where{αn},{βn}are two real sequences in0,1satisfying

iαn≤βn, for alln∈N,

iilimn→ ∞βn0 ,

iii ∞_n1αnβn∞.

Then{xn}converges strongly to a fixed point ofT.

Theorem 1.3see7. LetCbe a closed convex subset of a Hilbert spaceH. LetT : C → Cbe aκ-strict pseudocontraction for some0 ≤ κ < 1, and assume that the fixed point setFTofT is nonempty. Let{xn}∞n1be the sequence generated by the followingCQalgorithm:

x1∈C,

ynαnxn 1−αnTxn,

Cn

z∈C:yn−z2≤1−αnκ−αnxn−Txn2

,

Qn{z∈C:xn−z, x1−xn},

xn1PCn∩Qnx1.

1.14

Assume that the control sequence {αn}∞n1 is chosen so that αn < 1 for all n ∈ N. Then {xn}

converges strongly to PFTx1. Very recently, in 2010, [10] established the hybrid algorithm for Lipschitz pseudocontractive mapping as follows:

ForC1C, x1PC1x1,

yn 1−αnxnαnTzn,

zn

1−βn

xnβnTxn,

Cn1

z∈Cn :αnI−Tyn2≤2αn

xn−z,I−Tyn

2αnβnLxn−Txnyn−xnαnI−Tyn,

xn1PCn1x1, ∀n∈N.

1.15

Under suitable conditions of{αn}and {βn}, they proved that the sequence {xn}defined by 1.15

converges strongly toPFTx1.

Many authors study the problem for finding a common element of the set of fixed point problem and the set of equilibrium problem in Hilbert spaces, for instance, [2,3,11–15]. The motivation of

1.14,1.15, and the research in this direction, we prove the strong convergence theorem for finding solution of the set of fixed points of strictly pseudocontractive mapping and two sets of generalized equilibrium problems by using the hybrid method.

2. Preliminaries

In order to prove our main results, we need the following lemmas. LetCbe closed convex subset of a real Hilbert spaceH, and letPC be the metric projection ofHontoC; that is, for

x∈H,PCxsatisfies the property

x−PCxmin y∈C

_{x−y. 2.1}

Lemma 2.1 see5. Given thatx ∈ Hand y ∈ C, thenPCx yif and only if the following

inequality holds:

x−y, y−z≥0, ∀z∈C. 2.2

The following lemma is well known.

Lemma 2.2. LetH be Hilbert space, and letCbe a nonempty closed convex subset ofH. LetT :

C → Cbeκ-strictly pseudocontractive, then the fixed point setFTofTis closed and convex so that the projectionPFTis well defined.

Lemma 2.3demiclosedness principle see16. IfTis aκ-strict pseudocontraction on closed

convex subsetCof a real Hilbert spaceH, thenI−T is demiclosed at any pointy∈H.

To solve the equilibrium problem for a bifunctionF : C×C → R, assume that F

satisfies the following conditions:

A1Fx, x 0 for allx∈C,

A2F is monotone,that is, Fx, y Fy, x≤0, for allx, y∈C,

A3for allx, y, z∈C,

lim t→0F

tz 1−tx, y≤Fx, y, 2.3

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

The following lemma appears implicitly in1.

Lemma 2.4see1. LetCbe a nonempty closed convex subset ofH, and letFbe a bifunction of

C×CintoRsatisfyingA1–A4. Letr >0, andx∈H. Then, there existsz∈Csuch that

Fz, y1 r

y−z, z−x, 2.4

for allx∈C.

Lemma 2.5see11. Assume thatF :C×C → RsatisfiesA1–A4. Forr >0andx∈H,

define a mappingTr : H → Cas follows:

Trx

z∈C:Fz, y 1 r

y−z, z−x≥0, ∀y∈C

, 2.5

for allz∈H. Then, the following hold:

2Tr is firmly nonexpansive, that is,

_Trx−Try2_≤

Trx−Tr

y, x−y, ∀x, y∈H, 2.6

3FTr EPF;

4EPFis closed and convex.

Lemma 2.6see17. LetCbe a closed convex subset ofH. Let{xn}be a sequence inHandu∈H.

LetqPCu; if{xn}is such thatωxn⊂Cand satisfy the condition

xn−u ≤u−q, ∀n∈N, 2.7

thenxn → q, as n → ∞.

Lemma 2.7see7. For a real Hilbert space H, the following identities hold: if{xn}is a sequence

in H weak convergence toz, then

lim sup n→ ∞

_xn−y2

lim sup n→ ∞

xn−z2z−y2, 2.8

for ally∈H.

3. Main Result

Theorem 3.1. Let Cbe a nonempty closed convex subset of a Hilbert space H. LetF and G be

bifunctions fromC×CintoRsatisfyingA1–A4, respectively. LetA:C → Hbe anα-inverse strongly monotone mapping, and letB : C → Hbe a β-inverse strongly monotone mapping. Let T :C → Cbe aκ-strict pseudocontraction mapping withFFT∩EPF, A∩EPG, B/∅. Let {xn}be a sequence generated byx1∈CC1and

Fun, u Axn, u−un 1

rnu−un, un−xn ≥0, ∀u∈C,

Gvn, v Bxn, v−vn 1

snv−vn, vn−xn ≥

0, ∀v∈C,

znδnun 1−δnvn,

ynαnzn 1−αnTzn,

Cn1z∈Cn:yn−z≤ xn−z

,

xn1PCn1x1, ∀n≥1,

3.1

where {αn}∞n0 is sequence in 0,1,rn ∈ a, b ⊂ 0,2α, and sn ⊂ c, d ⊂ 0,2β satisfy the

ilimn→ ∞δn δ∈0,1,

ii0≤κ≤αn<1, for alln≥1.

Thenxnconverges strongly toPFx1.

Proof. First, we show thatI−rnAis nonexpansive. Letx, y∈C. SinceAisα-inverse strongly monotone mapping andrn<2α, we have

_{I−rnAx−I−rnAy}2

x−y−rn

Ax−Ay2

x−y2−2rn

x−y, Ax−Ayrn2Ax−Ay

2

≤x−y2−2αrnAx−Ay2rn2Ax−Ay

2

x−y2rnrn−2αAx−Ay2

≤x−y2.

3.2

ThusI−rnAis nonexpansive, so areI−snB,TrnI−rnA, andTsnI−snB. Since

Fun, u Axn, u−un 1

rnu−un, un−xn ≥

0, ∀u∈C, 3.3

then we have

Fun, u 1

rnu−un, un−I−rnAxn ≥

0. 3.4

By Lemma2.5, we haveun TrnI−rnAxn. By the same argument as above, we conclude

thatvnTsnI−snBxn.

Letz∈F. ThenFz, y y−z, Az ≥0 andGz, y y−z, Bz ≥0. Hence

Fz, y 1 rn

y−z, z−zrnAz

≥0,

Gz, y 1 sn

y−z, z−zsnBz

≥0.

3.5

Again by Lemma2.5, we havez Trnz−rnAz Tsnz−snBz. By nonexpansiveness of TrnI−rnAandTsnI−snB, we have

un−zTrnI−rnAxn−TrnI−rnAz

≤ xn−z,

vn−zTsnI−snAxn−TsnI−snAz

≤ xn−z.

By3.6, we have

zn−z ≤ xn−z. 3.7

Next, we show thatCn is closed and convex for everyn∈ N. It is obvious thatCnis closed. In fact, we know that, forz∈Cn,

_{yn−z≤ xn−z} is equivalent toyn−xn22

yn−xn, xn−z

≤0. 3.8

So, we have that for allz1, z2∈Cnandt∈0,1, it follows that

_yn−xn2

2yn−xn, xn−tz1 1−tz2

t2yn−xn, xn−z1

yn−xn2

1−t2yn−xn, xn−z2

yn−xn2

≤0.

3.9

Then, we have thatCn is convex. By Lemmas2.5and2.2, we conclude thatFis closed and convex. This implies thatPFis well defined. Next, we show thatF⊂Cnfor everyn∈N.

Takingp∈F, we have

_yn−p2

αn

zn−p

1−αn

Tzn−p2

αnzn−p2 1−αnTzn−p2−αn1−αnzn−Tzn2

≤αnzn−p2 1−αn

zn−p2κI−Tzn−I−Tp2

−αn1−αnzn−Tzn2

αnzn−p2 1−αnzn−p2κ1−αnzn−Tzn2

−αn1−αnzn−Tzn2

zn−p2 κ−αn1−αnzn−Tzn2

≤zn−p2

≤xn−p2.

3.10

It follows thatp∈Cn. Then, we haveF⊂Cn, for alln∈N. SincexnPCnx1, for everyw∈Cn,

we have

In particular, we have

xn−x1 ≤ PFx1−x1. 3.12

By3.11, we have that{xn}is bounded, so are{un},{vn},{zn},{yn}. Sincexn1 PCn1x1 ∈

Cn1⊂CnandxnPCnx1, we have

0≤ x1−xn, xn−xn1

x1−xn, xn−x1x1−xn1

≤ −xn−x12xn−x1x1−xn1.

3.13

It is implied that

xn−x1 ≤ xn1−x1. 3.14

Hence, we have that limn→ ∞xn−x1exists. Since

xn−xn12xn−x1x1−xn12

xn−x122xn−x1, x1−xn1x1−xn12

xn−x122xn−x1, x1−xnxn−xn1x1−xn12

xn−x12−2xn−x122xn−x1, xn−xn1x1−xn12

≤ x1−xn12− xn−x12,

3.15

it is implied that

lim

n→ ∞xn−xn10. 3.16

Sincexn1PCn1x1∈Cn1, we have

_{yn−xn1≤ xn−xn1, 3.17}

And by3.16, we have

lim

n→ ∞yn−xn10. 3.18

Since

by3.16and3.18, we have

lim

n→ ∞yn−xn0. 3.20

Next, we show that

lim

n→ ∞un−xn0, nlim→ ∞vn−xn0. 3.21

Letp∈F, by3.10and3.7, we have

_yn−p2

αn

zn−p

1−αn

Tzn−p2

αnzn−p2 1−αnTzn−p2−αn1−αnzn−Tzn2

≤αnzn−p2 1−αn

zn−p2κI−Tzn−I−Tp2

−αn1−αnzn−Tzn2

αnzn−p2 1−αnzn−p2κ1−αnzn−Tzn2

−αn1−αnzn−Tzn2

αnzn−p2 1−αnzn−p2 κ−αn1−αnzn−Tzn2

≤αnxn−p2 1−αnzn−p2

≤αnxn−p2 1−αn

δnun−p2 1−δnvn−p2

.

3.22

SinceunTrnI−rnAxn,pTrnI−rnAp, we have

_un−p2

TrnI−rnAxn−TrnI−rnAp

2

≤ I−rnAxn−I−rnAp2

xn−rnAxn−prnAp2

xn−p−rn

Axn−Ap2

xn−p2rn2Axn−Ap2−2rn

xn−p, Axn−Ap

≤xn−p2rn2Axn−Ap2−2rnαAxn−Ap2

xn−p2rnrn−2αAxn−Ap2.

SincevnTsnI−snBxn,pTsnI−snBp, we have

_vn−p2

TsnI−snBxn−TsnI−snBp

2

≤I−snBxn−I−snBp2

xn−snBxn−psnBp2

xn−p−sn

Bxn−Bp2

xn−p2sn2Bxn−Bp2−2sn

xn−p, Bxn−Bp

≤xn−p2s2nBxn−Bp2−2snβBxn−Bp2

xn−p2sn

sn−2βBxn−Bp2.

3.24

Substituting3.23and3.24into3.22,

_yn−p2_≤

αnxn−p2 1−αn

δnun−p2 1−δnvn−p2

≤αnxn−p2

1−αn

δn

xn−p2rnrn−2αAxn−Ap2

1−δn

xn−p2sn

sn−2βBxn−Bp2

αnxn−p2

1−αn

δnxn−p2δnrnrn−2αAxn−Ap2

1−δnxn−p2sn1−δn

sn−2βBxn−Bp2

αnxn−p2 1−αn

×xn−p2δnrnrn−2αAxn−Ap2sn1−δn

sn−2βBxn−Bp2

αnxn−p2 1−αnxn−p2 1−αnδnrnrn−2αAxn−Ap2

sn1−αn1−δn

sn−2βBxn−Bp2

xn−p2 1−αnδnrnrn−2αAxn−Ap2

sn1−αn1−δn

sn−2βBxn−Bp2.

It is implied that

1−αnδnrn2α−rnAxn−Ap2≤xn−p2−yn−p2

sn1−αn1−δn

sn−2βBxn−Bp2

≤xn−pyn−pxn−yn.

3.26

By3.20and conditioni, we have

lim

n→ ∞Axn−Ap0. 3.27

By using the same method as3.27, we have

lim

n→ ∞Bxn−Bp0. 3.28

By Lemma2.5and firm nonexpansiveness ofTrn, we have

_un−p2

TrnI−rnAxn−TrnI−rnAp

2

≤I−rnAxn−I−rnAp, un−p

1

2

I−rnAxn−I−rnAp2un−p2

−I−rnAxn−I−rnAp−

un−p2

1

2

I−rnAxn−I−rnAp2un−p2

−xn−un−rn

Axn−Ap2

1

2

I−rnAxn−I−rnAp2un−p2

−xn−un2rn2Axn−Ap2−2rn

xn−un, Axn−Ap

≤ 1

2

xn−p2un−p2− xn−un2−rn2Axn−Ap2

2rn

xn−un, Axn−Ap

.

By3.29, it is implied that

_un−p2 _≤

xn−p2− xn−un2−rn2Axn−Ap2

2rn

xn−un, Axn−Ap

≤xn−p2− xn−un2−rn2Axn−Ap2

2rnxn−unAxn−Ap.

3.30

Again, by Lemma2.5and firm nonexpansiveness ofTsn, we have

_vn−p2

TsnI−snBxn−TsnI−snAp

2

≤I−snBxn−I−snBp, vn−p

1

2

I−snBxn−I−snBp2vn−p2

−I−snBxn−I−snBp−

vn−p2

1

2

I−snBxn−I−snBp2vn−p2−xn−vn−sn

Bxn−Bp2

1

2

I−snBxn−I−snBp2vn−p2

−xn−vn2s2nBxn−Bp2−2sn

xn−vn, Bxn−Bp

≤ 1

2

xn−p2vn−p2− xn−vn2−s2nBxn−Bp2

2sn

xn−vn, Bxn−Bp

.

3.31

By3.31, it is implied that

_vn−p2_≤

xn−p2− xn−vn2−s2nBxn−Bp22sn

xn−vn, Bxn−Bp

≤xn−p2− xn−vn2−s2nBxn−Bp22snxn−vn Bxn−Bp.

Substituting3.30and3.32into3.22, we have

_yn−p2_≤

αnxn−p2 1−αn

δnun−p2 1−δnvn−p2

≤αnxn−p2 1−αn

×δn

xn−p2− xn−un2−rn2Axn−Ap22rnxn−unAxn−Ap

1−δn

xn−p2− xn−vn2−s2nBxn−Bp2

2snxn−vnBxn−Bp

≤αnxn−p2

1−αn

δn

xn−p2− xn−un22rnxn−unAxn−Ap

1−δn

xn−p2− xn−vn22snxn−vnBxn−Bp

αnxn−p2

1−αn

δnxn−p2−δnxn−un22δnrnxn−unAxn−Ap 1−δn

×xn−p2−1−δnxn−vn221−δnsnxn−vnBxn−Bp

αnxn−p2 1−αn

×xn−p2−δnxn−un22δnrnxn−unAxn−Ap−1−δnxn−vn2

21−δnsnxn−vnBxn−Bp

≤xn−p2−1−αnδnxn−un22δnrnxn−unAxn−Ap

−1−αn1−δnxn−vn221−δnsnxn−vnBxn−Bp,

3.33

which implies that

1−αnδnxn−un2≤xn−p2−yn−p22δnrnxn−unAxn−Ap

−1−αn1−δnxn−vn221−δnsnxn−vnBxn−Bp

≤xn−ynxn−pyn−p2δnrnxn−unAxn−Ap

−1−αn1−δnxn−vn221−δnsnxn−vnBxn−Bp

≤xn−ynxn−pyn−p2δnrnxn−unAxn−Ap

21−δnsnxn−vnBxn−Bp,

and by3.27,3.28,3.20, and conditionsi,ii, we have

lim

n→ ∞xn−un0. 3.35

By using the same method as3.35, we have

lim

n→ ∞xn−vn0. 3.36

Since

zn−xn ≤δnun−xn 1−δnvn−xn, 3.37

from3.35,3.36, and conditioni, we have

lim

n→ ∞zn−xn0. 3.38

By3.20and3.38, we have

lim

n→ ∞yn−zn0. 3.39

Since

yn−zn 1−αnTzn−zn, 3.40

from3.39and conditionii, we have

lim

n→ ∞Tzn−zn0. 3.41

Letωxnbe the set of all weaksω-limit of{xn}. We will show thatωxn ⊂ F. Since{xn} is bounded, then ωxn/∅. Letting q ∈ ωxn, there exists a subsequence {xni} of {xn}

converging toq. By3.35, we haveuni qasi → ∞. Since un TrnI −rnAxn, for any y∈C, we have

Fun, y

Axn, y−un

1

rn

y−un, un−xn

≥0. 3.42

FromA2, we have

Axn, y−un

1

rn

y−un, un−xn

≥Fy, un

This implies that

Axni, y−uni

1

rni

y−uni, uni−xni

≥Fy, uni

. 3.44

Putzt ty 1−tqfor allt∈ 0,1andy ∈C. Then, we havezt ∈C. So, from3.44, we have

zt−uni, Azt ≥ zt−uni, Azt − zt−uni, Axni −

zt−uni,

uni−xni rni

Fzt, uni

zt−uni, Azt−Aunizt−uni, Auni−Axni −

zt−uni,

uni−xni rni

Fzt, uni.

3.45

Sinceuni−xni → 0, we haveAuni−Axni → 0. Further, from monotonicity ofA, we have

zt−uni, Azt−Auni ≥0. So, we have

zt−q, Azt

≥Fzt, q

asi−→ ∞. 3.46

FromA1,A4, and3.46, we also have

0Fzt, zt≤tF

zt, y

1−tFzt, q

≤tFzt, y

1−tzt−q, Azt

tFzt, y

1−tty−q, Azt

.

3.47

Thus

0≤Fzt, y

1−ty−q, Azt

. 3.48

Lettingt → 0, we have, for eachy∈C,

0≤Fq, yy−q, Aq. 3.49

This implies that

q∈EPF, A. 3.50

From3.36, we havevni q. SincevnTsnI−snBxn, for anyy∈C, we have

Gvn, y

Bxn, y−vn

1

sn

y−vn, vn−xn

By using the same method as3.50, we have

q∈EPG, B. 3.52

Sincexni qasi → ∞and 3.38, we havezni qasi → ∞. By Lemma2.3,I −T is

demiclosed at zero, and by3.41, we have

q∈FT. 3.53

From3.50,3.52, and 3.53, we haveq ∈ F. Henceωxn ⊂ F. Therefore, by3.12and Lemma2.6, we have that{xn}converges strongly toPFx1. The proof is completed.

4. Applications

By using our main result, we have the following results in Hilbert spaces.

Theorem 4.1. Let Cbe a nonempty closed convex subset of a Hilbert space H. LetF and G be

bifunctions fromC×C intoRsatisfyingA1–A4, respectively. LetT : C → Cbe aκ-strict pseudocontraction mapping withFFT∩EPF∩EPG/∅. Let{xn}be a sequence generated

byx1∈CC1and

Fun, u 1

rnu−un, un−xn ≥0, ∀u∈C,

Gvn, v 1

snv−vn, vn−xn ≥

0, ∀v∈C,

znδnun 1−δnvn,

ynαnzn 1−αnTzn,

Cn1z∈Cn:yn−z≤ xn−z

,

xn1PCn1x1, ∀n≥1,

4.1

where{αn}∞n0is sequence in0,1,rn∈a, b, andsn⊂c, dsatisfy the following conditions:

ilimn→ ∞δn δ∈0,1,

ii0≤κ≤αn<1, for alln≥1.

Thenxnconverges strongly toPFx1.

Theorem 4.2. LetCbe a nonempty closed convex subset of a Hilbert spaceH. LetFbe bifunctions fromC×CintoRsatisfyingA1–A4, respectively. LetA : C → Hbe anα-inverse strongly monotone mapping, and let{xn}be a sequence generated byx1∈CC1and

Fun, u Axn, u−un 1

rnu−un, un−xn ≥

0, ∀u∈C,

ynαnun 1−αnTun,

Cn1z∈Cn:yn−z≤ xn−z

,

xn1PCn1x1, ∀n≥1,

4.2

where{αn}∞n0is sequence in0,1,rn∈a, b⊂0,2α, and0≤κ≤αn<1, for alln≥1. Thenxn

converges strongly toPFx1.

Proof. PuttingG ≡F,A≡B, andun vn, for alln≥1, in Theorem3.1, we have the desired conclusions.

References

1 E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,”

Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.

2 A. Moudafi and M. Th´era, “Proximal and dynamical approaches to equilibrium problems,” in

Ill-Posed Variational Problems and Regularization Techniques (Trier, 1998), vol. 477 ofLecture Notes in Economics and Mathematical Systems, pp. 187–201, Springer, Berlin, Germany, 1999.

3 S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 3, pp. 1025–1033, 2008.

4 H. Iiduka and W. Takahashi, “Weak convergence theorems by Ces´aro means for nonexpansive mappings and inverse-strongly-monotone mappings,”Journal of Nonlinear and Convex Analysis, vol. 7, no. 1, pp. 105–113, 2006.

5 W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000.

6 F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,”Journal of Mathematical Analysis and Applications, vol. 20, pp. 197–228, 1967.

7 G. Marino and H.-K. Xu, “Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces,”Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 336–346, 2007. 8 W. R. Mann, “Mean value methods in iteration,”Proceedings of the American Mathematical Society, vol.

4, pp. 506–510, 1953.

9 S. Ishikawa, “Fixed points by a new iteration method,”Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974.

10 Y.-C. Tanga, J.-G. Penga, and L.-W. Liu, “Strong convergence theorem for pseudo-contractive mappings in Hilbert spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 74, pp. 380–385, 2011.

11 P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.

12 A. Kangtunyakarn and S. Suantai, “A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings,”Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4448–4460, 2009.

14 S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,”Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506–515, 2007.

15 V. Colao, G. Marino, and H.-K. Xu, “An iterative method for finding common solutions of equilibrium and fixed point problems,”Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 340– 352, 2008.

16 H. Zhou, “Convergence theorems of fixed points for -strict pseudo-contractions in Hilbert spaces,”

Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 2, pp. 456–462, 2008.