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,
1Sun Young Cho,
2and Xiaolong Qin
31Department 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≤
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 thatFx, yAx, 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:Fx, yAx, 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
Ftz 1−tx, y≤Fx, 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 thatFx, y≥0, ∀y∈C. 1.9
In this paper, the set of such anx∈Cis denoted by EPF, that is,
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
Fun, 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},
xn1PCn∩Dnx,
1.12
for everyn≥1, where{αn} ⊂a,1for somea∈0,1and{rn} ⊂0,∞satisfieslim 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.
Lemma 1.2. LetCbe a nonempty closed convex subset ofHand letF:C×C → Rbe a bifunction satisfyingA1–A4. Then, for anyr >0andx∈H, there existsz∈Csuch that
Fz, y1 r
y−z, z−x≥0, ∀y∈C. 1.13
Further, define
Trx z∈C:F
z, y1 r
y−z, z−x≥0, ∀y∈C
1.14
for allr >0andx∈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
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 <2αand0< e≤sn≤f<2β.
Then the sequence{xn}generated inΥconverges strongly to some pointx, wherexPFx1.
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
Note thatI−rnAis nonexpansive for eachn≥1.Indeed, for anyx, y∈ C, we see from the restrictiondthat
I−rnAx−I−rnAy2
x−y−rn
Ax−Ay2
x−y2−2rn
x−y, Ax−Ayrn2Ax−Ay 2
≤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.
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−PFx1. 2.12
This implies that{xn}is bounded. SincexnPCnx1andxn1PCn1x1∈Cn1⊂Cn, we have
0≤ x1−xn, xn−xn1
x1−xn, xn−x1x1−xn1
≤ −x1−xn2x1−xnx1−xn1.
2.13
It follows that
xn−x1 ≤ xn1−x1. 2.14
This proves that limn→ ∞xn−x1exists. Notice that
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.
2.15
It follows that
lim
n→ ∞xn−xn10. 2.16
Sincexn1PCn1x1∈Cn1, we see that
yn−xn1≤ xn−xn1. 2.17
This implies that
From2.16, we obtain that
lim
n→ ∞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
lim
n→ ∞xn−Snzn0. 2.21
For anyp∈ F, we have
un−p2
TrnI−rnAxn−TrnI−rnAp2
xn−p
−rn
Axn−Ap2
xn−p2−2rn
xn−p, Axn−Ap
rn2Axn−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−p2
αnxn 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.
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
lim
n→ ∞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
lim
n→ ∞Bxn−Bp0. 2.29
On the other hand, we have fromLemma 1.1that
un−p2
TrnI−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
rn2Axn−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≤
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
lim
n→ ∞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
lim
n→ ∞xn−vn0. 2.37
Note that
zn−xn ≤γnun−xn
1−γn
vn−xn. 2.38
From2.35and2.37, we see that
lim
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
lim
n→ ∞xn−Szn0. 2.41
Note that
Sxn−xn ≤ Sxn−SznSzn−xn ≤
1k
1−kxn−znSzn−xn. 2.42
It follows from2.39and2.41that
lim
n→ ∞xn−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.
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−xn1 ≤ 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.
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
w∈Cn:yn−w≤ xn−w
,
xn1PCn1x1, n≥1,
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 <2αand0< e≤sn≤f<2β.
Then the sequence{xn}converges strongly to some pointx, wherexPFx1.
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
w∈Cn:yn−w≤ xn−w
,
xn1PCn1x1, n≥1,
2.57
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 <2αand0< 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
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,
Cn1
w∈Cn:yn−w≤ xn−w
,
xn1PCn1x1, 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 <2αand0< e≤sn≤f<2β.
Then the sequence{xn}converges strongly to some pointx, wherexPFx1.
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
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,
2.61
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;
c0≤c≤γn≤d <1;
d0< e≤rn≤f <∞and0< 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
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,
2.62
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 <1−kαand0< e≤sn ≤f<1−kβ.
Acknowledgment
This work was supported by a National Research Foundation of Korea Grant funded by the Korean Government2009-0076898.
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