Generalized Equilibrium Problems and Strictly Pseudocontractive Mappings

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Volume 2010, Article ID 312602,18pages doi:10.1155/2010/312602

Research Article

Hybrid Projection Algorithms for

Generalized Equilibrium Problems and Strictly Pseudocontractive Mappings

Jong Kyu Kim,

¹

Sun Young Cho,

²

and Xiaolong Qin

³

1Department of Mathematics Education, Kyungnam University, Masan 631-701, Republic of Korea

2Department of Mathematics, Gyeongsang National University, Chinju 660-701, Republic of Korea

3Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

Correspondence should be addressed to Jong Kyu Kim,[email protected] Received 12 October 2009; Accepted 19 July 2010

Academic Editor: Andr´as Ront ´o

Copyrightq2010 Jong Kyu Kim 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 paper is to consider the problem of finding a common element in the solution set of equilibrium problems and in the fixed point set of a strictly pseudocontractive mapping.

Strong convergence of the purposed hybrid projection algorithm is obtained in Hilbert spaces.

1. Introduction and Preliminaries

LetHbe a real Hilbert space with inner product·,·and norm · . LetCbe a nonempty closed convex subset ofHandS :C → Ca nonlinear mapping. In this paper, we useFS to denote the fixed point set ofS. Recall that the mappingSis said to be nonexpansive if

Sx−Sy≤x−y, ∀x, y∈C. 1.1 Sis said to bek-strictly pseudocontractive if there exists a constantk∈0,1such that

Sx−Sy²≤x−y²kx−Sx−

y−Sy², ∀x, y∈C. 1.2 Sis said to be pseudocontractive if

Sx−Sy²≤x−y²x−Sx−

y−Sy², ∀x, y∈C. 1.3

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The class of strictly pseudocontractive mappings was introduced by Browder and Petryshyn 1in 1967. It is easy to see that the class of strictly pseudocontractive mappings falls into the class of nonexpansive mappings and the class of pseudocontractions.

LetA:C → Hbe a mapping. Recall thatAis said to be monotone if Ax−Ay, x−y

≥0, ∀x, y∈C. 1.4

Ais said to be inverse-strongly monotone if there exists a constantα >0 such that Ax−Ay, x−y

≥αAx−Ay², ∀x, y∈C. 1.5 LetF be a bifunction ofC×CintoR, whereRdenotes the set of real numbers and A:C → Han inverse-strongly monotone mapping. In this paper, we consider the following generalized equilibrium problem.

Findx∈Csuch thatF x, y

Ax, y−x

≥0, ∀y∈C. 1.6

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

x∈C:F x, y

Ax, y−x

≥0, ∀y∈C

. 1.7

To study the generalized equilibrium problems1.6, we may assume thatF satisfies 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 F

tz 1−tx, y

≤F x, y

; 1.8

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

Next, we give two special cases of the problem1.6.

IIfA≡0, then the generalized equilibrium problem1.6is reduced to the following equilibrium problem:

Find x∈Csuch thatF x, y

≥0, ∀y∈C. 1.9

In this paper, the set of such anx∈Cis denoted by EPF, that is,

EPF

x∈C:F x, y

≥0, ∀y∈C

. 1.10

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IIIf F ≡ 0, then the problem1.6is reduced to the following classical variational inequality. Findx∈Csuch that

Ax, y−x

≥0, ∀y∈C. 1.11

It is known thatx∈Cis a solution to1.11if and only ifxis a fixed point of the mappingPCI−ρA, whereρ >0 is a constant andIis the identity mapping.

Recently, many authors studied the problems 1.6 and 1.9 based on iterative methods; see, for example,2–18.

In 2007, Tada and Takahashi 17 considered the problem 1.9 and proved the following result.

Theorem TT. LetCbe a nonempty closed convex subset ofH. LetFbe a bifunction fromC×CtoR satisfyingA1–A4and letSbe a nonexpansive mapping ofCintoHsuch thatFS∩EPF/∅.

Let{xn}and{un}be sequences generated byx1x∈Hand let

F un, y

1 rn

y−un, un−xn

≥0, ∀y∈C,

wn 1−αnxnαnSun, Cn{z∈H:wn−z ≤ xn−z}, Dn{z∈H:xn−z, x−xn ≥0},

x_n1PCn∩Dnx,

1.12

for everyn≥1, where{αn} ⊂a,1for somea∈0,1and{rn} ⊂0,∞satisfies lim infn→ ∞rn >

0. Then,{xn}converges strongly toPFS∩EPFx.

In this paper, we consider the generalized equilibrium problem1.6 and a strictly pseudocontractive mapping based on the shrinking projection algorithm which was first introduced by Takahashi et al. 18. A strong convergence of common elements of the fixed point sets of the strictly pseudocontractive mapping and of the solution sets of the generalized equilibrium problem is established in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by Tada and Takahashi17.

In order to prove our main results, we also need the following definitions and lemmas.

Lemma 1.1see19. LetCbe a nonempty closed convex subset of a Hilbert spaceHandT :C → Cak-strict pseudocontraction. ThenTis1k/1−k-Lipschitz andI−Tis demiclosed, this is, if {xn}is a sequence inCwithxn xandxn−Txn → 0, thenx∈FT.

The following lemma can be found in2,3.

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Lemma 1.2. LetCbe a nonempty closed convex subset ofHand letF:C×C → Rbe a bifunction satisfyingA1–A4. Then, for anyr >0 andx∈H, there existsz∈Csuch that

F z, y

1 r

y−z, z−x

≥0, ∀y∈C. 1.13

Further, define

Trx z∈C:F z, y

1 r

y−z, z−x

≥0, ∀y∈C

1.14

for allr >0 andx∈H.Then, the following hold:

aTr is single-valued;

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

Trx−Try²≤

Trx−Try, x−y

; 1.15

cFTr EPF;

dEPFis closed and convex.

Lemma 1.3see1. LetCbe a nonempty closed convex subset of a real Hilbert spaceHandS : C → Cak-strict pseudocontraction with a fixed point. DefineS:C → CbySaxax 1−aSx for eachx∈C. Ifa∈k,1, thenSais nonexpansive withFSa FS.

2. Main Results

Theorem 2.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetF1 and F2

be two bifunctions fromC×CtoRwhich satisfiesA1–A4. LetA : C → H be anα-inverse- strongly monotone mapping,B:C → Haβ-inverse-strongly monotone mapping, andS:C → C

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ak-strict pseudocontraction. Let{rn} and {sn} be two positive real sequences. Assume thatF : EPF1, A∩FPF2, B∩FSis not empty. Let{xn}be a sequence generated in the following manner:

x1∈C, C1C, F1un, u Axn, u−un 1

rnu−un, un−xn ≥0, ∀u∈C, F2vn, v Bxn, v−vn 1

snv−vn, vn−xn ≥0, ∀v∈C, znγnun

1−γn

vn, ynαnxn 1−αn

βnzn 1−βn

Szn

, Cn1

w∈Cn:yn−w≤ xn−w , xn1PCn1x1, n≥1,

Υ

where{αn},{βn}, and{γn}are sequences in0,1. Assume that{αn},{βn},{γn},{rn}, and{sn} satisfy the following restrictions:

a0≤αn≤a <1;

b0≤k≤βn< b <1;

c0≤c≤γn≤d <1;

d0< e≤rn≤f <2αand 0< e≤sn≤f<2β.

Then the sequence{xn}generated inΥconverges strongly to some pointx, wherexP_Fx1. Proof. Note thatuncan be rewritten as

unTrnxn−rnAxn, ∀n≥1 2.1

andvncan be rewritten as

vnTsnxn−snBxn, ∀n≥1. 2.2

Fixp∈ F. It follows that

pSpTrn

p−rnAp Tsn

p−snBp

, ∀n≥1. 2.3

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Note thatI−rnAis nonexpansive for eachn≥1.Indeed, for anyx, y∈ C, we see from the restrictiondthat

I−rnAx−I−rnAy²x−y

−rn

Ax−Ay² x−y²−2rn

x−y, Ax−Ay

r_n²Ax−Ay²

≤x−y²−rn2α−rnAx−Ay²

≤x−y².

2.4

This shows thatI−rnAis nonexpansive for eachn≥1. In a similar way, we can obtain that I−snBis nonexpansive for eachn≥1. It follows that

un−p≤xn−p, un−p≤xn−p. 2.5 This implies that

zn−p≤γnun−p

1−γnvn−p≤xn−p. 2.6 Now, we are in a position to show thatCnis closed and convex for eachn≥1.From the assumption, we see thatC1 Cis closed and convex. Suppose thatCmis closed and convex for somem ≥ 1. We show thatCm1 is closed and convex for the samem. Indeed, for any w∈Cm, we see that

ym−w≤ xm−w 2.7 is equivalent to

ym²− xm²−2

w, ym−xm

≥0. 2.8

ThusCm1is closed and convex. This shows thatCnis closed and convex for eachn≥1.

Next, we show thatF ⊂Cnfor eachn≥1.From the assumption, we see thatF ⊂C C1. Suppose thatF ⊂Cmfor somem≥1.Putting

SnβnI 1−βn

S, ∀n≥1, 2.9

we see fromLemma 1.3thatSnis a nonexpansive mapping for eachn≥1. For anyw ∈ F ⊂ Cm, we see from2.6that

ym−wαmxm 1−αmSmzm−w

≤αmxm−w 1−αmzm−w

≤ xm−w.

2.10

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This shows thatw∈Cm1. This proves thatF ⊂Cnfor eachn≥1. NotexnPCnx1. For each w∈ F ⊂Cn, we have

x1−xn ≤ x1−w. 2.11

In particular, we have

x1−xn ≤ x1−P_Fx1. 2.12

This implies that{xn}is bounded. SincexnPCnx1andx_n1PCn1x1∈C_n1⊂Cn, we have 0≤ x1−xn, xn−x_n1

x1−xn, xn−x1x1−xn1

≤ −x1−xn²x1−xnx1−xn1.

2.13

It follows that

xn−x1 ≤ x_n1−x1. 2.14

This proves that lim_n_{→ ∞}xn−x1exists. Notice that xn−x_n1²xn−x1x1−x_n1²

xn−x1²2xn−x1, x1−x_n1x1−x_n1²

xn−x1²2xn−x1, x1−xnxn−xn1x1−xn1² xn−x1²−2xn−x1²2xn−x1, xn−xn1x1−xn1²

≤ x1−xn1²− xn−x1².

2.15

It follows that

nlim→ ∞xn−x_n10. 2.16

Sincexn1PCn1x1∈Cn1, we see that

yn−xn1≤ xn−xn1. 2.17 This implies that

yn−xn≤yn−xn1xn−xn1 ≤2xn−xn1. 2.18

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From2.16, we obtain that

nlim→ ∞xn−yn0. 2.19

On the other hand, we have

xn−ynxn−αnxn−1−αnSnzn 1−αnxn−Snzn. 2.20

From the assumption 0≤αn≤a <1 and2.19, we have

nlim→ ∞xn−Snzn0. 2.21

For anyp∈ F, we have

un−p²TrnI−rnAxn−TrnI−rnAp² xn−p

−rn

Axn−Ap² xn−p²−2rn

xn−p, Axn−Ap

r_n²Axn−Ap²

≤xn−p²−rn2α−rnAxn−Ap².

2.22

In a similar way, we also have

vn−p²≤xn−p²−sn

2β−snBxn−Bp². 2.23

Note that

yn−p²αnxn 1−αnSnzn−p²

≤αnxn−p² 1−αnSnzn−p²

≤αnxn−p² 1−αnzn−p²

≤αnxn−p² 1−αnγnun−p² 1−αn

1−γnvn−p².

2.24

Substituting2.22and2.23into2.24, we arrive at

yn−p² ≤xn−p²−1−αnγnrn2α−rnAxn−Ap²

−1−αn 1−γn

sn

2β−snBxn−Bp².

2.25

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It follows that

1−αnγnrn2α−rnAxn−Ap²≤xn−p²−yn−p²

≤xn−pyn−pxn−yn. 2.26

In view of the restrictionsa–dand2.19, we obtain that

nlim→ ∞Axn−Ap0. 2.27

It also follows from2.25that 1−αn

1−γn

sn

2β−snBxn−Bp²≤xn−p²−yn−p²

≤xn−pyn−pxn−yn. 2.28

By virtue of the restrictionsa–dand2.19, we get that

nlim→ ∞Bxn−Bp0. 2.29

On the other hand, we have fromLemma 1.1that un−p²TrnI−rnAxn−TrnI−rnAp²

≤

I−rnAxn−I−rnAp, un−p 1

2

I−rnAxn−I−rnAp²un−p²−I−rnAxn−I−rnAp−

un−p²

≤ 1 2

xn−p²un−p²−xn−un−rnAxn−Ap² 1

2

xn−p²un−p²−

xn−un²−2rn

xn−un, Axn−Ap

r_n²Axn−Ap² . 2.30

This implies that

un−p²≤xn−p²− xn−un²2rnxn−unAxn−Ap. 2.31 In a similar way, we can also obtain that

vn−p²≤xn−p²− xn−vn²2snxn−vnBxn−Bp. 2.32

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Substituting2.31and2.32into2.24, we obtain that

yn−p²≤xn−p²−1−αnγnxn−un²2rn1−αnγnxn−unAxn−Ap

−1−αn 1−γn

xn−vn²2sn1−αn 1−γn

xn−vnBxn−Bp

≤xn−p²−1−αnγnxn−un²2rnxn−unAxn−Ap

−1−αn 1−γn

xn−vn²2snxn−vnBxn−Bp.

2.33

It follows that

1−αnγnxn−un²≤xn−p²−yn−p²2rnxn−unAxn−Ap 2snxn−vnBxn−Bp

≤xn−pyn−pxn−yn2rnxn−unAxn−Ap 2snxn−vnBxn−Bp.

2.34

In view of the restrictionsaandc, we obtain from2.27and2.29that

nlim→ ∞xn−un0. 2.35

It also follows from2.33that 1−αn

1−γn

xn−vn²≤xn−p²−yn−p²2rnxn−unAxn−Ap 2snxn−vnBxn−Bp

≤xn−pyn−pxn−yn2rnxn−unAxn−Ap 2snxn−vnBxn−Bp.

2.36

Thanks to the restrictionsaandc, we obtain from2.27and2.29that

nlim→ ∞xn−vn0. 2.37

Note that

zn−xn ≤γnun−xn 1−γn

vn−xn. 2.38

From2.35and2.37, we see that

n→ ∞limxn−zn0. 2.39

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On the other hand, we see from2.21that βnzn−xn

1−βn

Szn−xn−→0 2.40

asn → ∞.In view of2.39and the restrictionb, we obtain that

nlim→ ∞xn−Szn0. 2.41

Note that

Sxn−xn ≤ Sxn−SznSzn−xn ≤ 1k

1−kxn−znSzn−xn. 2.42 It follows from2.39and2.41that

n→ ∞limxn−Sxn0. 2.43

Since{xn}is bounded, we assume that a subsequence{xni}of{xn}converges weakly toξ.

Next, we show thatξ∈FS∩EPF1, A∩EPF2, B.First, we prove thatξ∈EPF1, A.

SinceunTrnxn−rnAxnfor anyu∈C, we have

F1un, u Axn, u−un 1

rnu−un, un−xn ≥0. 2.44 From the conditionA2, we see that

Axn, u−un 1

rnu−un, un−xn ≥F1u, un. 2.45 Replacingnbyni, we arrive at

Axni, u−uni

u−uni,uni−xni

rni

≥F1u, uni. 2.46

For anytwith 0< t≤1 andu∈C,letuttu 1−tξ. Sinceu∈Candξ∈C, we haveut∈C.

It follows from2.46that

ut−uni, Aut ≥ ut−uni, Aut − Axni, ut−uni −

ut−uni,uni−xni

rni

F1ut, uni ut−uni, Aut−Auniut−uni, Auni−Axni−

ut−uni,uni−xni

rni

F1ut, uni. 2.47

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SinceAis Lipschitz continuous, we obtain from2.35thatAuni −Axni → 0 asi → ∞. On the other hand, we get from the monotonicity ofAthat

ut−uni, Aut−Auni ≥0. 2.48

It follows fromA4and2.47that

ut−ξ, Aut ≥F1ut, ξ. 2.49

FromA1,A4, and2.49, we see that

0F1ut, ut≤tF1ut, u 1−tF1ut, ξ

≤tF1ut, u 1−tut−ξ, Aut tF1ut, u 1−ttu−ξ, Aut,

2.50

which yields that

F1ut, u 1−tu−ξ, Aut ≥0. 2.51

Lettingt → 0 in the above inequality, we arrive at

F1ξ, u u−ξ, Aξ ≥0. 2.52

This shows thatξ∈EPF1, A.In a similar way, we can obtain thatξ∈EPF2, B.

Next, we show thatξ∈FS. We can conclude fromLemma 1.1the desired conclusion easily. This proves thatξ ∈ F.Putx PFx1.Sincex PFx1 ⊂ Cn1 andxn1 PCn1x1, we have

x1−x_n1 ≤ x1−x. 2.53

On the other hand, we have

x1−x ≤ x1−ξ

≤lim inf

i→ ∞ x1−xni

≤lim sup

i→ ∞ x1−xni

≤ x1−x.

2.54

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We, therefore, obtain that

x1−ξ lim

i→ ∞x1−xnix1−x. 2.55

This implies xni → ξ x.Since{xni}is an arbitrary subsequence of {xn}, we obtain that xn → xasn → ∞.This completes the proof.

IfSis nonexpansive, then we have fromTheorem 2.1the following result immediately.

Corollary 2.2. Let C be a nonempty closed convex subset of a real Hilbert space H. LetF1 and F2 be two bifunctions from C×C toR which satisfies A1–A4. Let A : C → H be an α- inverse-strongly monotone mapping, B : C → H a β-inverse-strongly monotone mapping, and S:C → Ca nonexpansive mapping. Let{rn}and{sn}be two positive real sequences. Assume that F:EPF1, A∩FPF2, B∩FSis not empty. Let{xn}be a sequence generated in the following manner:

x1∈C, C1C, F1un, u Axn, u−un 1

rnu−un, un−xn ≥0, ∀u∈C, F2vn, v Bxn, v−vn 1

snv−vn, vn−xn ≥0, ∀v∈C, ynαnxn 1−αnS

γnun 1−γn

vn

,

Cn1

2.56

where {αn} and {γn}are sequences in 0,1. Assume that {αn},{γn}, {rn}, and {sn} satisfy the following restrictions:

a0≤αn≤a <1;

b0≤c≤γn≤d <1;

c0< e≤rn≤f <2αand 0< e≤sn≤f<2β.

Then the sequence{xn}converges strongly to some pointx, wherexPFx1.

As applications ofTheorem 2.1, we consider the problems1.9and1.11.

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Theorem 2.3. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetA:C → Hbe anα-inverse-strongly monotone mapping,B:C → Haβ-inverse-strongly monotone mapping, and S :C → Cak-strict pseudocontraction. Let{rn}and{sn}be two positive real sequences. Assume thatF:V IC, A∩V IC, B∩FSis not empty. Let{xn}be a sequence generated in the following manner:

x1∈C, C1C, znγnPCI−rnAxn

1−γn

PCI−snBxn, ynαnxn 1−αn

βnzn 1−βn

Szn

, Cn1

2.57

a0≤αn≤a <1;

b0≤k≤βn< b <1;

c0≤c≤γn≤d <1;

d0< e≤rn≤f <2αand 0< e≤sn≤f<2β.

Then the sequence{xn}converges strongly to some pointx, wherexPFx1. Proof. PuttingF1F2≡0, we see that

Axn, u−un 1

rnu−un, un−xn ≥0, ∀u∈C, 2.58

is equivalent to

xn−rnAxn−un, un−u ≥0, ∀u∈C. 2.59

This implies thatunPCxn−rnAxn.We also havevnPCxn−snBxn.We can obtain from Theorem 2.1the desired results immediately.

Corollary 2.4. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetA:C → H be anα-inverse-strongly monotone mapping,B:C → Haβ-inverse-strongly monotone mapping, andS :C → Ca nonexpansive mapping. Let{rn}and{sn}be two positive real sequences. Assume

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thatF:V IC, A∩V IC, B∩FSis not empty. Let{xn}be a sequence generated in the following manner:

x1∈C, C1C, znγnPCI−rnAxn

1−γn

PCI−snBxn, yn αnxn 1−αnSzn,

C_n1

w∈Cn:yn−w≤ xn−w , xn1PC_n1x1, n≥1,

2.60

where {αn} and {γn}are sequences in 0,1. Assume that {αn},{γn}, {rn}, and {sn} satisfy the following restrictions:

a0≤αn≤a <1;

b0≤c≤γn≤d <1;

c0< e≤rn≤f <2αand 0< e≤sn≤f<2β.

Then the sequence{xn}converges strongly to some pointx, wherexP_Fx1.

Theorem 2.5. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F1 and F2 be two bifunctions from C× C to R which satisfies A1–A4. Let S : C → C be a k- strict pseudocontraction. Let {rn} and {sn} be two positive real sequences. Assume that F : EPF1∩FPF2∩FSis not empty. Let{xn}be a sequence generated in the following manner:

x1∈C, C1C, F1un, u 1

rnu−un, un−xn ≥0, ∀u∈C, F2vn, v 1

1−γn

vn, ynαnxn 1−αn

βnzn 1−βn

Szn

, Cn1

w∈Cn:yn−w≤ xn−w , x_n1PCn1x1, n≥1,

2.61

a0≤αn≤a <1;

b0≤k≤βn< b <1;

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c0≤c≤γn≤d <1;

d0< e≤rn≤f <∞and 0< e≤sn≤f<∞.

Then the sequence{xn}converges strongly to some pointx, wherexPFx1.

Proof. Putting A B 0, we can obtain from Theorem 2.1 the desired conclusion immediately.

Remark 2.6. Theorem 2.5is generalization of Theorem TT. To be more precise, we consider a pair of bifunctions and a strictly pseudocontractive mapping.

Let T : C → Cbe ak-strict pseudocontraction. It is known thatI−T is a1−k/2- inverse-strongly monotone mapping. The following results are not hard to derive.

Theorem 2.7. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F1 and F2 be two bifunctions from C× C to R which satisfies A1–A4. Let TA : C → C be a kα-strict pseudocontraction, B : C → C a kβ-strict pseudocontraction, and S : C → C a k-strict pseudocontraction. Let {rn} and {sn} be two positive real sequences. Assume that F : EPF1, I−TA∩FPF2, I−TB∩FSis not empty. Let{xn}be a sequence generated in the following manner:

x1∈C, C1C, F1un, u I−TAxn, u−un 1

rnu−un, un−xn ≥0, ∀u∈C, F2vn, v I−TBxn, v−vn 1

1−γn

vn, ynαnxn 1−αn

βnzn 1−βn

Szn

, Cn1

w∈Cn:yn−w≤ xn−w , x_n1PCn1x1, n≥1,

2.62

a0≤αn≤a <1;

b0≤k≤βn< b <1;

c0≤c≤γn≤d <1;

d0< e≤rn≤f <1−kαand 0< e≤sn ≤f<1−kβ.

Then the sequence{xn}converges strongly to some pointx, wherexP_Fx1.

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Acknowledgment

This work was supported by a National Research Foundation of Korea Grant funded by the Korean Government2009-0076898.

References

1 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.

2 E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.

3 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.

4 L.-C. Ceng, S. Al-Homidan, Q. H. Ansari, and J.-C. Yao, “An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 967–974, 2009.

5 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.

6 S.-S. Chang, H. W. Joseph Lee, and C. K. Chan, “A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization,” Nonlinear Analysis: Theory, Methods & Applications.and Methods, vol. 70, no. 9, pp. 3307–3319, 2009.

7 Y. J. Cho, X. Qin, and J. I. Kang, “Convergence theorems based on hybrid methods for generalized equilibrium problems and fixed point problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 9, pp. 4203–4214, 2009.

8 C. Jaiboon, W. Chantarangsi, and P. Kumam, “A convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 1, pp. 199–215, 2010.

9 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.

10 A. Moudafi, “Weak convergence theorems for nonexpansive mappings and equilibrium problems,”

Journal of Nonlinear and Convex Analysis, vol. 9, no. 1, pp. 37–43, 2008.

11 S. Plubtieng and R. Punpaeng, “A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 548–558, 2008.

12 X. Qin, M. Shang, and Y. Su, “Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems,” Mathematical and Computer Modelling, vol. 48, no. 7-8, pp. 1033–1046, 2008.

13 X. Qin, S. M. Kang, and Y. J. Cho, “Convergence theorems on generalized equilibrium problems and fixed point problems with applications,” Proceedings of the Estonian Academy of Sciences, vol. 58, no. 3, pp. 170–183, 2009.

14 X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20–30, 2009.

15 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.

16 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.

17 A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,” Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370, 2007.

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18 W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 276–286, 2008.

19 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.