Volume 2010, Article ID 827082,16pages doi:10.1155/2010/827082
Research Article
Convergence of Iterative Sequences for
Generalized Equilibrium Problems Involving
Inverse-Strongly Monotone Mappings
Shin Min Kang,
1Sun Young Cho,
2and Zeqing Liu
31Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, South Korea
2Department of Mathematics, Gyeongsang National University, Jinju 660-701, South Korea 3Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China
Correspondence should be addressed to Sun Young Cho,[email protected]
Received 11 December 2009; Accepted 25 January 2010
Academic Editor: Yeol Je Cho
Copyrightq2010 Shin Min Kang 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 weak convergence of an iterative sequence for finding a common element in the set of solutions of generalized equilibrium problems, in the set of solutions of classical variational inequalities, and in the set of fixed points of nonexpansive mappings.
1. Introduction and Preliminaries
Throughout this paper, we always assume that H is a real Hilbert space with the inner product·,·and the norm·andCis a nonempty closed convex subset ofH. LetS:C → C
be a nonlinear mapping. In this paper, we useFSto denote the fixed point set ofS. Recall that the mappingSis said to be nonexpansive if
Sx−Sy ≤ x−y, ∀x, y∈C. 1.1
LetA:C → Hbe a mapping. Recall thatAis said to be monotone if
Ais said to be inverse-strongly monotone if there exists a constantα >0 such that
Ax−Ay, x−y≥αAx−Ay2, ∀x, y∈C. 1.3
A set-valued mappingR :H → 2His said to be monotone if for allx, y ∈ H,f ∈Rxand g ∈Ryimplyx−y, f−g>0. A monotone mappingR:H → 2His maximal if the graph GRofRis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingRis maximal if and only if, for anyx, f∈H×H,x−y, f−g ≥0 for ally, g∈GRimpliesf ∈Rx.
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. 1.4
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. 1.5
Next, we give two special cases of problem1.4.
IIfA≡0, then the generalized equilibrium problem1.4is reduced to the following equilibrium problem:
Find x∈Csuch thatFx, y≥0, ∀y∈C. 1.6
In this paper, the set of such anx∈Cis denoted by EPF, that is,
EPF x∈C:Fx, y≥0, ∀y∈C. 1.7
Numerous problems in physics, optimization, and economics reduce to finding a solution of the equilibrium problem.
To study problems 1.4 and 1.6, we may assume that F 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
IIIfF ≡0, then problem1.4is reduced to the classical variational inequality. Find
x∈Csuch that
Ax, y−x≥0, ∀y∈C. 1.9
It is known thatx∈ Cis a solution to1.9if and only ifxis a fixed point of the mapping
PCI−λA, whereλ >0 is a constant andIis the identity mapping.
In 2003, Takahashi and Toyoda 1 considered the variational inequality1.9 and proved the following theorem.
Theorem 1.1. Let Cbe a closed convex subset of a real Hilbert space H. Let A be an α -inverse-strongly monotone mapping ofCinto H, and letSbe a nonexpansive mapping ofCinto itself such thatFFS∩V IC, A/∅.Let{xn}be a sequence generated by
x0∈C, xn1αnxn 1−αnSPCxn−λnAxn, ∀n≥0, 1.10
whereλn ∈a, bfor somea, b∈0,2αandαn∈c, dfor somec, d∈0,1. Then,{xn}converges
weakly toz∈FS∩V IC, A, wherezlimn→ ∞PFxn.
In 2007, Tada and Takahashi2considered the equilibrium problem1.6and proved the following result.
Theorem 1.2. LetCbe a nonempty closed convex subset ofH. LetFbe a bifunction fromC×CtoR satisfying (A1)–(A4) and letSbe a nonexpansive mapping ofCintoHsuch thatFS∩EPF/∅.
Let{xn}and{un}be sequences generated byx1x∈Hand let
un∈Csuch thatFun, u
1
rn
u−un, un−xn ≥0, ∀u∈C,
xn1αnxn 1−αnSun, ∀n≥1,
1.11
where{αn} ⊂a, bfor somea, b∈0,1and{rn} ⊂0,∞satisfieslim infn→ ∞rn>0. Then,{xn}
converges weakly tow∈FS∩EPF, wherewlimn→ ∞PFS∩EPFxn.
Very recently, Moudafi3considered the following iterative process:
x0∈C,
F1un, u
1
rn
u−un, un−xn ≥0, ∀u∈C,
F2vn, v 1
rn
v−vn, vn−xn ≥0, ∀v∈C,
xn1 unvn
2 , ∀n≥1,
1.12
whereF1andF2are bifunctions and{rn}is a control sequence. A weak convergence theorem
Weak convergence of iterative sequences has been studied recently for the problems
1.4,1.6, and 1.9; see1–14and the references therein. In this paper, we consider the generalized equilibrium problem1.4 and a nonexpansive mapping based on an iterative process. We show that the sequence generated in the purposed iterative process converges weakly to a common element in the set of solutions of the variational inequality 1.9, in the fixed point sets of a nonexpansive mapping and in the solution sets of the generalized equilibrium problem 1.4. The results presented in this paper improve and extend the corresponding results announced by Takahashi and Toyoda1and Tada and Takahashi2.
In order to prove our main results, we also need the following lemmas.
Lemma 1.3see1. LetCbe a nonempty closed convex subset ofH. Let{xn}be a sequence inH. Suppose that, for ally∈C,
xn1−y ≤ xn−y, ∀n≥1, 1.13
then{PCxn}converges strongly to somez∈C.
The following lemma can be found in15.
Lemma 1.4. LetTbe a monotone mapping ofCintoHandNCvthe normal cone toCatv∈C, that is,
NCv{w∈H:v−u, w ≥0, ∀u∈C} 1.14
and define a mappingRonCby
Rv
⎧ ⎨ ⎩
TvNCv, v∈C,
∅, v /∈C.
1.15
ThenRis maximal monotone and0∈Rvif and only ifTv, u−v ≥0for allu∈C.
The following lemma can be found in16,17.
Lemma 1.5. 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. 1.16
Further, define
Trx
z∈C:Fz, y1 r
y−z, z−x≥0, ∀y∈C
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
; 1.18
cFTr EPF;
dEPFis closed and convex.
Lemma 1.6see18. LetHbe a Hilbert space and0< p≤tn ≤q < 1for alln≥1. Suppose that
{xn}and{yn}are sequences inHsuch that
lim sup
n→ ∞ xn ≤r, lim supn→ ∞ yn ≤r,
lim
n→ ∞tnxn 1−tnynr
1.19
hold for somer ≥0,thenlimn→ ∞xn−yn0.
Lemma 1.7see19. LetHbe a real Hilbert space,Ca nonempty closed convex subset ofH, and
S:C → Ca nonexpansive mapping. Then the mappingI−Sis demiclosed at zero, that is, if{xn}is
a sequence inCsuch thatxn xandxn−Sxn → 0, thenx∈FS.
2. Main Results
Theorem 2.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetF1 and F2
be two bifunctions from C×CtoRwhich satisfy (A1)–(A4). Let A : C → H be an α -inverse-strongly monotone mapping,B : C → H aβ-inverse-strongly monotone mapping,T : C → H
anλ-inverse-strongly monotone mapping, andS : C → Ca nonexpansive mapping. Assume that F : EPF1, A∩EPF2, B∩V IC, T∩FS/∅. Let{αn}and{βn}be sequences in0,1. Let
{an}be a sequence in0,2α,{bn}a sequence in0,2β,and{tn}a sequence in0,2λ. Let{xn}be a sequence generated in the following manner:
x1∈C,
F1un, u Axn, u−un
1
anu−un, un−xn ≥0, ∀u∈C,
F2vn, v Bxn, v−vn
1
bn
v−vn, vn−xn ≥0, ∀v∈C,
ynβnun
1−βn
vn,
xn1αnxn 1−αnSPC
yn−tnTyn, ∀n≥1.
Assume that the sequences{αn},{βn},{an},{bn},and{tn}satisfy the following restrictions:
a0< a≤αn≤a <1,0< b≤βn≤c <1;
b0< d≤an≤e <2α,0< f ≤bn≤g <2β,0< h≤tn≤j <2λ
for somea, a, b, c, d, e, f, g, h, j ∈ R,then the sequence{xn}generated inΔconverges weakly to some pointx∈ F, wherexlimn→ ∞PFxn.
Proof. Fixp∈ F. It follows that
pSpTanI−anApTbnI−bnBpPCI−tnTp, ∀n≥1. 2.1
Note thatI−tnTis nonexpansive for eachn≥1.Indeed, for anyx, y∈C, we see that
I−tnTx−I−tnTy2x−y
−tn
Tx−Ty2
x−y2−2tnx−y, Tx−Tyt2nTx−Ty2
≤x−y2−tn2λ−tnTx−Ty2
≤x−y2.
2.2
In a similar way, we can obtain thatI−anAandI−bnBare nonexpansive for eachn≥1.Note that
un−p ≤ TanI−anAxn−p ≤ xn−p,
vn−p ≤ TbnI−bnBxn−p ≤ xn−p.
2.3
PutznPCyn−tnTynfor eachn≥1.It follows from2.3that
xn1−p ≤αnxn−p 1−αnSzn−p
≤αnxn−p 1−αnzn−p
≤αnxn−p 1−αnyn−p
≤αnxn−p 1−αnβnun−p1−βnvn−p
≤ xn−p.
2.4
This implies that limn→ ∞xn−pexists. This shows that{xn}is bounded, so are{yn},{un},
and{vn}.
On the other hand, we have
un−p2TanI−anAxn−p
2≤x
n−p2−an2α−anAxn−Ap2, 2.5
vn−p2
TbnI−bnBxn−p
2≤
Combining2.5with2.6yields that
xn1−p2≤αnxn−p2 1−αnSzn−p2
≤αnxn−p2 1−αnzn−p2
≤αnxn−p2 1−αnyn−p2
≤αnxn−p2 1−αn
βnun−p2
1−βnvn−p2
≤αnxn−p2 1−αn
βn
xn−p2−an2α−anAxn−Ap2
1−βnxn−p2−bn
2β−anBxn−Bp2
≤xn−p2−1−αnβnan2α−anAxn−Ap2
−1−αn1−βnbn2β−anBxn−Bp2.
2.7
This implies that
1−αnβnan2α−anAxn−Ap2≤xn−p2−xn1−p2. 2.8
In view of the restrictionsaandb, we obtain that
lim
n→ ∞Axn−Ap0. 2.9
It also follows from2.7that
1−αn1−βnbn2β−anBxn−Bp2≤xn−p2−xn1−p2. 2.10
In view of the restrictionsaandb, we obtain that
lim
On the other hand, we see fromLemma 1.5that
un−p2
TanI−anAxn−TanI−anAp
2
≤I−anAxn−I−anAp, un−p
1
2
I−anAxn−I−anAp2un−p2
−I−anAxn−I−anAp−un−p2
≤ 1
2
xn−p2un−p2−xn−un−an
Axn−Ap2
1
2
xn−p2un−p2
−xn−un2−2an
xn−un, Axn−Ap
a2nAxn−Ap2
.
2.12
This implies that
un−p2≤xn−p2− xn−un22anxn−unAxn−Ap. 2.13
In a similar way, we can obtain that
vn−p2 ≤xn−p2− xn−vn22bnxn−vnBxn−Bp. 2.14
It follows from2.13and2.14that
xn1−p2≤
αnxn−p2 1−αnSzn−p2
≤αnxn−p2 1−αnzn−p2
≤αnxn−p2 1−αnyn−p2
≤αnxn−p2 1−αn
βnun−p2
1−βnvn−p2
≤αnxn−p2 1−αn
×βnxn−p2− xn−un22anxn−unAxn−Ap
1−βnxn−p2− xn−vn22bnxn−vnBxn−Bp
≤xn−p2−1−αnβnxn−un22anxn−unAxn−Ap
−1−αn1−βnxn−vn22bnxn−vnBxn−Bp.
This shows that
1−αnβnxn−un2 ≤xn−p2−xn1−p22anxn−unAxn−Ap
2bnxn−vnBxn−Bp.
2.16
In view of the restrictiona, we obtain from2.9and2.11that
lim
n→ ∞xn−un0. 2.17
From2.15, we also have
1−αn1−βnxn−vn2≤xn−p2−xn1−p22anxn−unAxn−Ap
2bnxn−vnBxn−Bp.
2.18
In view of the restrictiona, we obtain from2.9and2.11that
lim
n→ ∞xn−vn0. 2.19
Since{xn}is bounded, we see that there exits a subsequence{xni}of{xn}which converges
weakly tox. It follows from2.17thatuni converges weakly tox. Note that
F1un, u Axn, u−un
1
an
u−un, un−xn ≥0, ∀u∈C. 2.20
FromA2, we see that
Axn, u−un 1
anu−un, un−xn ≥F1u, un, ∀u∈C. 2.21
Replacingnbyni, we arrive at
Axni, u−uni
1
ani
u−uni, uni −xni ≥F1u, uni, ∀u∈C. 2.22
Fortwith 0< t≤1 andu∈C,letuttu 1−tx. Sinceu∈Candx∈C, we haveut∈C. It
follows from2.22that
ut−uni, Aut ≥ ut−uni, Aut − Axni, ut−uni −
ut−uni,
uni−xni
ani
F1ut, uni
ut−uni, Aut−Auniut−uni, Auni−Axni
−
ut−uni,
uni−xni
ani
F1ut, uni.
From2.17, we haveAuni −Axni → 0 asi → ∞. On the other hand, we obtain from the
monotonicity ofAthatut−uni, Aut−Auni ≥0.It follows fromA4that
ut−x, Aut ≥F1ut, x. 2.24
FromA1,A4, and2.24, we obtain that
0F1ut, ut≤tF1ut, u 1−tF1ut, x
≤tF1ut, u 1−tut−x, Aut
tF1ut, u 1−ttu−x, Aut,
2.25
which yields that
F1ut, u 1−tu−x, Aut ≥0. 2.26
Lettingt → 0 in the above inequality, we arrive at
F1x, u u−x, Ax ≥0. 2.27
This shows thatx∈EPF1, A. In a similar way, we can obtain thatx∈EPF2, B.
Next, we claim thatx∈V IC, T
xn1−p2≤αnxn−p2 1−αnSPCyn−tnTyn−p2
≤αnxn−p2 1−αnyn−tnTyn−p−tnTp2
≤αnxn−p2 1−αn
yn−p2−tn2λ−tnTyn−Tp2
≤xn−p2−1−αntn2λ−tnTyn−Tp
2
.
2.28
It follows that
1−αntn2λ−tnTyn−Tp2≤xn−p2−xn1−p2. 2.29
This implies from conditionsaandbthat
lim
SincePCis firmly nonexpansive, we have
zn−p2
PCI−tnTyn−PCI−tnTp2
≤I−tnTyn−I−tnTp, zn−p
1
2
I−tnTyn−I−tnTp2zn−p2
−I−tnTyn−I−tnTp−zn−p2
≤ 1
2
yn−p2zn−p2−yn−zn−tn
Tyn−Tp2
1
2
yn−p2zn−p2−
yn−zn2−2tn
yn−zn, Tyn−Tp
t2
nTyn−Tp2
.
2.31
So, we obtain that
zn−p2≤xn−p2−yn−zn22tnyn−znTyn−Tp. 2.32
It follows that
xn1−p2≤αnxn−p2 1−αnSzn−p2
≤αnxn−p2 1−αnzn−p2
≤αnxn−p2 1−αn
xn−p2−yn−zn22tnyn−znTyn−Tp
.
2.33
Therefore we have
1−αnyn−zn2≤xn−p2−xn1−p22tnyn−znTyn−Tp. 2.34
From the restrictionaand2.30, we get that
lim
n→ ∞yn−zn0. 2.35
Note that
xn−zn ≤ xn−ynyn−zn
≤βnxn−un
1−βn
xn−vnyn−zn.
2.36
From2.17,2.19, and2.35, we obtain that
lim
Define a mappingRby
Rv
⎧ ⎨ ⎩
TvNCv, v∈C,
∅, v /∈C. 2.38
Letv, w∈GR.Sincew−Tv∈NCvandzn∈C, we obtain that
v−zn, w−Tv ≥0. 2.39
AsznPCyn−tnTynandv∈C,we get that
v−zn, zn−yn
tn Tyn
≥0. 2.40
From2.39and2.40, we obtain that
v−zni, w ≥ v−zni, Tv
≥ v−zni, Tv −
v−zni,
zni−yni
tni
Tyni
v−zni, Tv−Tyni −
zni−yni
tni
v−zni, Tv−Tzni
v−zni, Tzni−Tyni
−
v−zni,
zni−yni
tni
≥ v−zni, Tzni−Tyni −
v−zni,
zni −yni
tni
.
2.41
Note thatT is Lipschitz. On the other hand, we see from2.37thatzni x.Hence, we get
that
v−x, w ≥0. 2.42
SinceRis maximal monotone, we obtain thatx∈R−10.FromLemma 1.4, we get thatx ∈
V IC, T.
Finally, we show thatx∈FS.Note that
xn1−p ≤αnxn−p 1−αnSzn−p,
Since limn→ ∞xn −pexists, we may assume that limn→ ∞xn −p r for some positive
constantr. Then we see that
lim sup
n→ ∞
xn1−p ≤r, lim sup
n→ ∞
xn−p ≤r,
lim sup
n→ ∞ Szn−p ≤r.
2.44
In view ofLemma 1.6, we get that
lim
n→ ∞xn−Szn0. 2.45
Furthermore, we know that
Sxn−xn ≤ Sxn−SznSzn−xn
≤ xn−znSzn−xn.
2.46
From2.37and2.45, we obtain that
lim
n→ ∞Sxn−xn0. 2.47
Note thatxni xandSxni −xni → 0 asi → ∞.FromLemma 1.7, we arrive atx ∈ FS.
Assume that there exists another subsequence{xnj}of{xn},converges tox, wherex/x.In
view of the Opial’s condition, we see that
lim
n→ ∞xn−xlim infi→ ∞ xni−x<lim infi→ ∞ xni−x
lim
n→ ∞xn−x
lim inf
j→ ∞ xnj−x
< lim
j→ ∞xnj−xnlim→ ∞xn−x.
2.48
This is a contradiction. So, we havexx.
LethnPFxn. Sincex∈ F,we have
xn−hn, hn−x ≥0. 2.49
From2.4andLemma 1.3, we get that{hn}converges strongly to someυ ∈ F. Since{xn}
converges weakly tox,we have
Hence we obtain that
xυ lim
n→ ∞PFxn. 2.51
This completes the proof.
Corollary 2.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction fromC×CtoRwhich satisfies (A1)–(A4). LetA : C → H be anα-inverse-strongly monotone mapping, T : C → H anλ-inverse-strongly monotone mapping, and S : C → Ca nonexpansive mapping. Assume thatF:EPF, A∩V IC, T∩FS/∅. Let{αn}be a sequence in0,1. Let{an}be a sequence in0,2α, and{tn} a sequence in0,2λ. Let{xn} be a sequence generated in the following manner:
x1∈C,
Fun, u Axn, u−un
1
anu−un, un−xn ≥0, ∀u∈C, xn1αnxn 1−αnSPCun−tnTun, ∀n≥1.
2.52
Assume that the sequences{αn},{an}, and{tn}satisfy the following restrictions:
a0< a≤αn≤a <1;
b0< d≤an≤e <2α,0< h≤tn≤j <2λ
for somea, a, d, e, h, j ∈ R,then the sequence{xn}converges weakly to some pointx ∈ F, where xlimn→ ∞PFxn.
Proof. PuttingF1 F2F, AB,andan bninTheorem 2.1, we see thatynun.From the
proof ofTheorem 2.1, we can conclude the desired conclusion immediately.
Corollary 2.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction fromC×CtoRwhich satisfies (A1)–(A4). LetA : C → H be anα-inverse-strongly monotone mapping andS:C → Ca nonexpansive mapping. Assume thatF:EPF, A∩FS/∅.
Let{αn}be a sequence in0,1. Let{an}be a sequence in0,2α. Let{xn}be a sequence generated in the following manner:
x1∈C,
Fun, u Axn, u−un 1 an
u−un, un−xn ≥0, ∀u∈C,
xn1αnxn 1−αnSun, ∀n≥1.
2.53
Assume that the sequences{αn}and{an}satisfy the following restrictions:
a0< a≤αn≤a <1;
for somea, a, d, e ∈ R,then the sequence{xn}converges weakly to some pointx ∈ F, wherex
limn→ ∞PFxn.
Proof. PuttingT 0 in Corollary 2.2, we can conclude the desired conclusion immediately.
Remark 2.4. Corollary 2.3 is a generalization of Theorem 1.2 in Section 1. More precisely,
Corollary 2.3is reduced toTheorem 1.2ifA0.
Corollary 2.5. 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,
T : C → Hanλ-inverse-strongly monotone mapping, andS : C → Ca nonexpansive mapping. Assume thatF:V IC, A∩V IC, B∩V IC, T∩FS/∅. Let{αn}and{βn}be sequences in
0,1. Let{an}be a sequence in0,2α,{bn}a sequence in0,2β,and{tn}a sequence in0,2λ.
Let{xn}be a sequence generated in the following manner:
x1∈C,
ynβnPCxn−anAxn 1−βnPCxn−bnBxn, xn1αnxn 1−αnSPC
yn−tnTyn
, ∀n≥1.
2.54
Assume that the sequences{αn},{βn},{an},{bn},and{tn}satisfy the following restrictions:
a0< a≤αn≤a <1,0< b≤βn≤c <1;
b0< d≤an≤e <2α,0< f ≤bn≤g <2β,0< h≤tn≤j <2λ
for somea, a, b, c, d, e, f, g, h, j ∈R,then the sequence{xn}converges weakly to some pointx∈ F,
wherexlimn→ ∞PFxn.
Proof. PuttingF1F20,we see that
Axn, u−un
1
an
u−un, un−xn ≥0 2.55
is equivalent to
unPCxn−anAxn, ∀n≥1. 2.56
In the same way, we can obtain that
vnPCxn−bnBxn, ∀n≥1. 2.57
From the proof ofTheorem 2.1, we can conclude the desired conclusion immediately.
Remark 2.6. Corollary 2.5 is a generalization of Theorem 1.1 in Section 1. More precisely,
Acknowledgments
This work was supported by the Korea Research Foundation Grant funded by the Korean Government KRF-2008-313-C00050 and the Science Research Foundation of Educational Department of Liaoning Province2009A419.
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