Volume 2008, Article ID 454181,23pages doi:10.1155/2008/454181
Research Article
Strong Convergence of a Modified Iterative
Algorithm for Mixed-Equilibrium Problems in
Hilbert Spaces
Xueliang Gao and Yunrui Guo
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China Correspondence should be addressed to Xueliang Gao,[email protected]
Received 8 July 2008; Accepted 1 August 2008
Recommended by Ram U. Verma
The purpose of this paper is to study the strong convergence of a modified iterative scheme to find a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a relaxed cocoercive mapping, as well as the set of solutions of a mixed-equilibrium problem. Our results extend recent results of Takahashi and Takahashi2007, Marino and Xu2006, Combettes and Hirstoaga2005, Iiduka and Takahashi 2005, and many others.
Copyrightq2008 X. Gao and Y. Guo. 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.
1. Introduction and preliminaries
LetH be a real Hilbert space whose inner product and norm are denoted by·,·and·, respectively. LetCbe a nonempty closed convex subset ofHand letA:C→Hbe a nonlinear map.PCbe the projection ofHonto the convex subsetC. The classical variational inequality
problem, denoted by VIC, A, is to findu∈Csuch that
Au, v−u ≥0 ∀v∈C. 1.1
For a givenz∈H, u∈Csatisfies the inequality
u−z, v−u ≥0 ∀v∈C, 1.2
if and only ifu PCz. It is known that the projection operatorPCis nonexpansive. It is also
known thatPC satisfies
x−y, PCx−PCy ≥ PCx−PCy2 1.3
forx, y∈H. Moreover,PCxis characterized by the propertiesPCx∈Candx−PCx, PCx−y ≥
One can see that the variational inequality problem1.1is equivalent to some fixed-point problems.
The elementu∈Cis a solution of the variational inequality problem1.1if and only ifu ∈ Csatisfies the relationu PCu−λAu, whereλ > 0 is a constant. The alternative
equivalent formulation has played a significant role in the studies of the the variational inequalities and related optimization problems.
Recall the following definitions.
1Bis calledv-strongly monotone if for eachx, y∈C, we have
Bx−By, x−y ≥vx−y2 1.4
for a constantv >0. This implies that
Bx−By ≥vx−y, 1.5
that is,Bisv-expansive and whenv1, it is expansive.
2Bis calledv-cocoercive1,2if for eachx, y∈C, we have
Bx−By, x−y ≥vBx−By2 1.6
for a constantv >0. Clearly, everyv-cocoercive mapBis 1/v-Lipschitz continuous.
3Bis called relaxedu-cocoercive if there exists a constantu >0 such that
Bx−By, x−y ≥−uBx−By2 ∀x, y∈C. 1.7
4Bis called relaxedu, v-cocoercive if there exist two constantsu, v >0 such that
Bx−By, x−y ≥−uBx−By2vx−y2 ∀x, y∈C 1.8
for u 0, B is v-strongly monotone. This class of maps is more general than the class of strongly monotone maps. It is easy to see that we have the following implication:v-strongly monotonicity⇒relaxedu, v-cocoercivity.
5A mapping T : C → Cis called nonexpansive ifTx−Ty ≤ x−y ∀x, y ∈ C. Next, we denote byFTthe set of fixed points ofT.
6A mappingf :H →His said to be a contraction if there exists a coefficientα0 < α <1such that
fx−fy ≤αx−y ∀x, y∈H. 1.9
7An operatorAis strongly positive if there exists a constantγ >0 with the property
8A set-valued mappingT :H →2His called monotone if for allx, y ∈H, one has thatf ∈Txandg∈Tyimplyx−y, f−g ≥0. A monotone mappingT :H→2His maximal if the graphGTofTis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingTis maximal if and only if forx, f∈H×H,x−y, f−g ≥0 implies thatf ∈Txfor everyy, g∈GT. LetBbe a monotone map ofCintoHand letNCvbe the normal cone toCatv∈C,
that is,
NCv{ω∈H:v−u, ω ≥0 ∀u∈C} 1.11
and define
Tv
BvNCv, v∈C,
∅, v /∈C. 1.12
Then,T is maximal monotone and 0∈Tvif and only ifv∈VIC, B see3.
LetFbe an equilibrium bifunction ofC×CintoR, whereRis the set of real numbers. The equilibrium problem forF:C×C→Ris to findx∈Csuch that
Fx, y≥0 ∀y∈C. 1.13
The set of solutions of 1.13 is denoted by EPF. Given a mapping T : C → H, let Fx, y Tx, y−xforx, y ∈C. Then,z ∈EPFif and only ifTz, y−z ≥ 0 fory ∈C. A number of problems in physics, optimization, and economics can be reduced to finding a solution of1.13. Equilibrium problems have been studied extensivelysee, e.g.,4,5. Recently, Combettes and Hirstoaga4introduced an iterative scheme for finding the best approximation to the initial data when EPFis nonempty and proved a strong convergence theorem.
Very recently, S. Takahashi and W. Takahashi6introduced an new iterative:
Fyn, u r1
nu−yn, yn−xn ≥0 ∀u∈C,
xn1αnfxn 1−αnTyn ∀n≥1
1.14
for approximating a common element of the set of fixed points of a non-self nonexpansive mapping and the set of solutions of the equilibrium problem and obtained a strong convergence theorem in a real Hilbert space.
Iterative methods for nonexpansive mapping have recently been applied to solve convex minimization problems see, e.g., 7–16 and the references therein. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH:
min
x∈C
1
2Ax, x − x, b, 1.15
whereAis a linear-bounded operator,Cis the fixed-point set of a nonexpansive mappingS, andbis a given point inH. In10,11, it is proved that the sequence{xn}defined by the
iterative method below, with the initial guessx0∈Hchosen arbitrarily,
converges strongly to the unique solution of the minimization problem1.15provided that the sequenceαnsatisfies certain conditions. Recently, Marino and Xu8introduced a new
iterative scheme by the viscosity approximation
xn1 I−αnASxnαnγfxn, n≥0. 1.17
They proved that the sequence {xn} generated by the above iterative scheme converges
strongly to the unique solution of the variational inequality
A−γfx∗, x−x∗ ≥0, x∈C, 1.18
which is the optimality condition for the minimization problem
min
x∈C
1
2Ax, x −hx, 1.19
whereCis the fixed-point set of a nonexpansive mappingS, his a potential function forγf
i.e.,hx γfxforx∈H.
For finding a common element of the set of fixed points of nonexpansive mappings and the set of solution of variational inequalities forα-cocoercive map, Takahashi and Toyoda
17introduced the following iterative process:
xn1αnxn 1−αnSPCxn−λnAxn 1.20
for everyn0,1,2, . . . ,whereAisα-cocoercive,x0x∈C, αnis a sequence in0,1, andλn
is a sequence in0,2α. They show that ifFS∩VIC, Ais nonempty, then the sequence
{xn}generated by1.20converges weakly to somez∈FS∩VIC, A. Recently, Iiduka and
Takahashi18studied similar scheme as follows:
xn1αnx 1−αnSPCxn−λnAxn 1.21
for everyn0,1,2, . . . ,wherex0 x∈C, αnis a sequence in0,1, andλn is a sequence in
0,2α. They proved that the sequence{xn}converges strongly toz∈FS∩VIC, A. Very
recently, Chen et al.19studied the following iterative process:
x1∈C, xn1αnfx 1−αnSPCxn−λnAxn, n≥1, 1.22
and also obtained a strong convergence theorem by the so-called viscosity approximation method20.
LetTi : C→ C, wherei 1,2, . . . , N be a a finite family of nonexpansive mappings,
let FTi denote the fixed-point set ofTi, that is, FTi : {x ∈ C : Tix x}. Finding an
optimal point in the intersection∩Ni1FTiof the fixed-point sets of a family of nonexpansive
mappings is a task that occurs frequently in various areas of mathematical sciences and engineering. For example, the well-known convex feasibility problem reduces to finding a point in the intersection of the fixed-point sets of a family of nonexpansive mappings
see, e.g., 21,22. The problem of finding an optimal point that minimizes a given cost function over∩Ni1FTi is of wide interdisciplinary interest and practical importancesee,
e.g.,12,16,23–25. A simple algorithmic solution to the problem of minimizing a quadratic function over∩Ni1FTiis of extreme value in many applications including set theoretic signal
We study the mappingWndefined by
Un0I,
Un1λn1T1Un0 1−λn1I,
Un2λn2T2Un1 1−λn2I,
.. .
Un,N−1λn,N−1TN−1Un,N−2 1−λn,N−1I,
Wn:UnNλn,NTNUn,N−1 1−λnNI,
1.23
where{λn1},{λn2}, . . . ,{λnN} ∈0,1. Such a mappingWnis called theW-mapping generated
byT1, T2, . . . , TNand{λn1},{λn2},. . .,{λnN}. Nonexpansivity ofTi yields the nonexpansivity
ofWn. Moreover, in27, Lemma 3.1, it is shown thatFWn ∩Ni1FTi. In28, Qin et al.
introduce a more general iterative process as follows:X1∈H
Fyn, u r1
nu−yn, yn−xn ≥0 ∀u∈C,
xn1αnγfWnxn 1−αnAWnPCI−snByn ∀n≥1,
1.24
whereWnis defined by1.23,Ais a linear-bounded operator, andBis relaxed cocoercive.
They prove that the sequence {xn} generated by the above iterative scheme converges
strongly to a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of the variational inequalities for relaxed cocoercive maps, and the set of solutions of the equilibrium problems1.13, which solves another variational inequality:
γfq−Aq, p−q ≤0 ∀p∈F, 1.25
where F ∩Ni1FixTi ∩ VIC, B ∩ EPF, and it is also the optimality condition for
the minimization problem minx∈F1/2Ax, x −hx, where his a potential function for
γfhx γfxforx∈H.
Recently, Ceng and Yao14introduce a mixed-equilibrium problemMEPas follows. LetHbe a real Hilbert space and letCbe a nonempty closed convex subset ofH. Letϕ:C→ Rbe a real-valued function andΘ:C×C→Rbe an equilibrium bifunction, that is,Θu, u 0 for eachu∈C, the MEP is given as follows, which is to findx∗∈Csuch that
MEP :Θx∗, y ϕy−ϕx∗≥0 ∀y∈C. 1.26
In particular, ifϕ≡0, this problem reduces to the equilibrium problemEP, which is to find x∗∈Csuch that
EP :Θx∗, y≥0 ∀y∈C. 1.27
Recall that a mappingf:C→Cis called contractive if there exists a constantα∈0,1 such that
fx−fy ≤αx−y ∀x, y∈C. 1.28
Recall also that a mappingS:C→His said to be nonexpansive if
Sx−Sy ≤ x−y ∀x, y∈C. 1.29
Denote the set of fixed points ofS by FixS. It is well known that ifCis bounded closed convex andS:C→Cis nonexpansive, then FixS/∅.
Inspired and motivated by the ongoing research in this field, we investigate the problem of finding a common element of the set of solution of1.26and the set of common fixed points of finite many nonexpansive mappings in a Hilbert space. First, we introduce a hybrid iterative scheme for finding a common element of the set of solutions of MEP and the set of common fixed points of finite many nonexpansive mapping. Furthermore, we prove that the sequences generated by the hybrid iterative scheme converge strongly to a common element of the set of solutions of MEP and the set of common fixed points of finite many nonexpansive mapping. Our results extend the recent ones announced by Chen et al.19, Combettes and Hirstoaga4, Iiduka and Takahashi18, Marino and Xu8, Qin et al.28, S. Takahashi and W. Takahashi6, Wittmann30, and many others.
Let H be a real Hilbert space with inner product ·,· and norm ·. Let C be a nonempty closed convex subset ofH. Then, for anyx ∈ H, there exists a unique nearest pointu∈Csuch that
x−u ≤ x−y ∀y∈C. 1.30
We denoteubyPCx, wherePCis called the metric projection ofHontoC. It is well known
thatPC is nonexpansive. Furthermore, forx∈Handu∈C,
uPCx⇐⇒ x−u, u−y ≥0 ∀y∈C. 1.31
In this paper, for solving the MPE for an equilibrium bifunction,Θ:C×C→Rsatisfies the following conditions:
H1 Θis monotone, that is,Θx, y Θy, x≤0∀x, y∈C;
H2for each fixedy∈C, x→Θx, yis concave and upper semicontinuous;
H3for eachx∈C, y→Θx, yis convex.
LetF:C→Handη:C×C→Hbe two mapping. Then,Fis called
iη-monotone if
Fx−Fy, ηx, y ≥0 ∀x, y∈C; 1.32
iiη-strongly monotone if there exists a constantα >0 such that
iiiLipschitz continuous if there exists a constantβ >0 such that
Fx−Fy, ηx, y ≤βx−y ∀x, y∈C 1.34
whenηx, y x−y∀x, y∈Cand if there exists a constantλ >0 such that
ηx, y ≤λx−y ∀x, y∈C. 1.35
A differentiable functionK:C→Ron a convex setCis called
iη-convex14if
Ky−Kx≥ Kx, ηy, x ∀x, y∈C, 1.36
whereKxis the Frechet derivative ofKatx;
iiη-strongly convex15if there exists a constantμ >0 such that
Ky−Kx− Kx, ηy, x ≥ μ
2x−y
2 ∀x, y∈C. 1.37
LetCbe a nonempty closed convex subset of real Hilbert spaceH, ϕ : C → Rbe a real-valued function, andΘ : C×C → Rbe an equilibrium bifunction. Letr be a positive parameter. For a given point x ∈ C, consider the auxiliary problem for MEPMEPx, r which consists of findingy∈Csuch that
Θy, z ϕz−ϕy 1
rKy−Kx, ηz, y ≥0 ∀z∈C, 1.38
whereη:C×C→HandKxis the Frechet derivative of a functionalK:C→Ratx. Let Tr :C→Cbe the mapping such that for eachx∈C, Trxis the solution of MEPx, r, that
is,
Trx
y∈C:Θy, z ϕz−ϕy 1
rKy−Kx, ηz, y ≥0, ∀z∈C
. 1.39
Lemma 1.1see14. LetCbe a nonempty closed convex subset of a real Hilbert spaceHand let
ϕ:C→Rbe a lower semicontinuous and convex functional. LetΘ:C×C→Rbe an equilibrium bifunction satisfying conditions (H1)–(H3).
Assume that
iη:C×C→His Lipschitz continuous with constantλ >0such that aηx, y ηy, x 0 ∀x, y∈C,
bη·,·is affine in the first variable,
cfor each fixedy∈C, x→ηy, xis sequentially continuous from the weak topology to the weak topology;
iiifor each x ∈ C, there exist a bounded subset Dx ⊆ Cand zx ∈ Csuch that for any
y∈C\Dx,
Θy, zx ϕzx−ϕy 1rKy−Kx, ηzx, y<0. 1.40
Then, there existsy∈Csuch that
Θy, z ϕz−ϕy 1
rKy−Kx, ηz, y ≥0 ∀z∈C. 1.41
Lemma 1.2see14. Assume thatΘsatisfies the same assumptions asLemma 2.1forr >0and
x∈C, the mappingTr :C→Ccan be defined as follows:
Trx
y∈C:Θy, z ϕz−ϕy 1
rKy−Kx, ηz, y ≥0 ∀z∈C
1.42
for ally∈C. Then, the following hold:
iTr is single-valued;
ii aKx1−Kx2, ηu1, u2 ≥ Ku1−Ku2, ηu1, u2 ∀x1, x2 ∈ C×C;
where uiSrxi, i1,2;
bTris nonexpansive ifKis Lipschitz continuous with constantν >0such thatμ≥λν;
iiiFTr Ω;
iv Ωis closed and convex.
Lemma 1.3see24. Let{xn}and{yn}be bounded sequences in a Banach spaceX and let{βn}
be a sequence in [0, 1] with
0≤lim inf
n→∞ βn≤lim supn→∞ βn≤1. 1.43
Suppose
xn1 1−βnynβnxn 1.44
for all integern≥0and
lim sup
n→∞ yn1−yn − xn1−xn≤0. 1.45 Then,limn→∞yn−xn0.
Lemma 1.4see23. Assumeanis sequence of nonexpansive real number such that
an1≤1−γnanδn, 1.46
where{γn}is a sequence in (0,1) and{δn}is a sequence such that
1∞n1γn∞;
2lim supn→∞δn/γn≤0or∞n1|δn|<∞.
2. Iterative scheme and strong convergence
Now, we introduced the following hybrid iterative scheme. Letfbe a contraction ofHinto itself with coefficientα ∈0,1and letAbe a strongly positive bounded linear operator on Hwith coefficientγ >0 such that 0<γ < γ/α , whereγ >0 is some constant. Givenx0 ∈H,
suppose the sequences{xn}and{yn}are generated iterative by
Θyn, x ϕx−ϕyn 1rKyn−Kxn, ηx, yn ≥0 ∀x∈C,
xn1 αnγfWnxn βnxn 1−βnI−αnAWnPCI−snByn ∀n≥1,
2.1
whereWnis defined by1.23,Ais a linear bounded operator, andBis relaxed cocoercive,
we prove that the sequence{xn}generated by the above iterative scheme converges strongly
to a common element of the set of common fixed points of a finite family of nonexpansive mappings, the set of solutions of the variational inequalities for relaxed cocoercive maps, and the set of solutions of the equilibrium problems1.26, which solves another variational inequality
γfq−Aq, p−q ≤0 ∀p∈F, 2.2
whereF∩Ni1FixTi∩VIC, B∩Ωand is also the optimality condition for the minimization
problem minx∈F1/2Ax, x −hx, where h is a potential function for γfx i.e., h
γfxforc ∈ H. The results obtained in this paper improve and extend the recent ones announced by Chen et al. 19, Combettes and Hirstoaga 4, Iiduka and Takahashi 18, Marino and Xu8, Qin et al.28, S. Takahashi and W. Takahashi 6, Wittmann30, and many others.
We will need the following result concerning theW-mappingWn.
Lemma 2.1 see 4. Let C be a nonempty closed convex subset of a Banach space X. Let
T1, T2, . . . , TN be a finite family of nonexpansive mappings of Cinto itself such that∩Ni1FixTi is
nonempty, and letλn1, λn2, . . . , λnNbe real numbers such that0 < λni ≤ a < 1fori 1,2, . . . , N.
For any n ≥ 1, let Wn be the W-mapping of C into itself generated by TN, TN−1, . . . ,1 and
λnN, λn,N−1, . . . , λn1. IfXis strictly convex, then FixWn ∩Ni1FixTi.
Now, we study the strong convergence of the hybrid iterative method2.1.
Theorem 2.2. LetCbe a nonempty closed convex subset of a real Hilbert spaceH, and letϕ:C→R
be a lower semicontinuous and convex functional. LetΘ:C×C→Rbe an equilibrium bifunction satisfying conditions (H1)–(H3), letT1, T2, . . . , TNbe a finite family of nonexpansive mappings onC
intoH, and letBbe aμ-Lipschitzian, relaxedu, v-cocoercive map ofCintoHsuch that
N
i1
FixTi∩Ω∩VIC, B/∅. 2.3
Letλn1, λn2, . . . , λnNbe a real number such thatlimn→∞λn1,i−λn,i 0 ∀i1,2, . . . , N. Suppose
{αn},{βn}are three sequences in (0,1), andris a positive parameter. Letfbe a contraction ofHinto
coefficient γ > 0 such thatA ≤ 1. Assume that0 < γ < γ/α . Let{xn} and{yn}be sequences
generated byx1∈Hand suppose that the following conditions are satisfied:
iη:C×C→His Lipschitz with constantλ >0such that
aηx, y ηy, x 0, ∀x, y∈C;
bη·,·is affine in the first variable;
cfor each fixedy∈C, x→ηy, xis sequentially continuous from the weak topology to the weak topology;
iiK : C → Risη-strongly convex with constantμ > 0and its derivativeKis not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constantν >0, μ≥λν;
iiifor each x ∈ C,there exists a bounded subset Dx ⊆ C and zx ∈ C,such that, for any
y∈C\Dx,
Θy, zx ϕzx−ϕy 1rKy−Kx, ηzx, y<0; 2.4
ivlimn→∞αn 0and∞n1αn∞; 0<lim infn→∞βn ≤lim supn→∞βn <1; ∞n1|sn1−
sn|<∞; {sn} ⊂a, bfor somea, bwith0≤a≤b≤2v−uμ2/μ2.
Givenx0 ∈ Carbitrarily, then the sequences {xn}and {yn}generated iteratively by2.1
converge strongly toq ∈ FixTi∩Ω∩VIC, B provided thatTr is firmly nonexpansive, where
qPFixTi∩Ω∩VIC,BI−Aγfqis a unique solution of variational inequalities:
A−γfq, q−x ≤0, ∀p∈
N
i1
FixTi∩Ω∩VIC, B, 2.5
which is the optimality condition for the minimization problem
min
p∈F
1
2Ap, p −hp, 2.6
wherehis a potential function forγf.
Proof. Note that for the control conditioniv, we may assume, without loss of generality, that αn≤1−βnA−1.
SinceAis linear bounded self-adjoint operator onC, then
Asup{|Au, u|:u∈C,u1}. 2.7
Observe that
1−βnI−αnAu, u1−βn−αnAu, u
≥1−βn−αnA
≥0,
that is,1−βnI−αnAis positive. It follows that
1−βnI−αnAsup{1−βnI−αnAu, u:u∈C,u1}
sup{1−βn−αnAu, u:u∈C,u1}
≤1−βn−αnγ.
2.9
LetQ P∩N
i1FixTi∩Ω. Note thatf is a contraction with coefficientα ∈0,1. Then, we
have
QI−Aγfx−QI−Aγfy ≤ I−Aγfx−I−Aγfy ≤ I−Ax−yγfx−fy ≤1−γx−yγαx−y
1−γ−γαx−y, ∀x, y∈C.
2.10
Therefore, QI −Aγf is a contraction ofC into itself, which implies that there exists a unique elementq∈Csuch thatqQI−Aγfq P∩N
i1FixTi∩ΩI−Aγfq.
First, we show thatI−snBis nonexpansive. Indeed, from the relaxedu, v-cocoercive
andμ-Lipschitzian definition onBand conditioniv, we have
I−snBx−I−snBx2x−y−snBx−By2
x−y2−2s
nx−y, Bx−Bys2nBx−By2
≤ x−y2−2s
n−uBx−By2vx−y2 s2nBx−By2
≤ x−y22s
nμ2ux−y2−2snvx−y2μ2s2nx−y2
12snμ2u−2snvμ2s2nx−y2
≤ x−y2,
2.11
which implies that the mapping I −snB is nonexpansive. Now, we observe that {xn} is
bounded. Indeed, pickp∈F. SinceynTrxn, we have
yn−pTrxn−Trp ≤ xn−p. 2.12
PuttingρnPCI−snByn, we have
ρn−p ≤ I−snByn−p ≤ yn−p ≤ xn−p. 2.13
It follows that
xn1−pαnγfWnxn−Ap βnxn−p 1−βnI−αnAWnρn−p
≤1−βn−αnγρn−pβnxn−pαnγfWnxn−Ap
≤1−αnγxn−pαnγfWnxn−fpγfp−Ap
≤1−γ−γααnxn−pγfp−Ap,
which gives that
xn−p ≤max
x0−p,
γfp−Ap
γ−γα
, n≥0. 2.15
Therefore, we obtain that{xn}is bounded, so is{yn}{Wnxn},{Wnρn}and{fWnxn}are all
bounded. Now, we show that
lim
n→∞xn1−xn0. 2.16
Letp∈Ni1FixTi∩Ω∩VIC, B.From the definition ofTr, we note thatynTrxn
andyn1 Trxn1. It follows that
yn1−ynTrxn1−Trxn
≤ xn1−xn. 2.17
Note that
ρn1−ρnPCI−sn1Byn1−PCI−snByn
≤ I−sn1Byn1−I−snByn
I−sn1Byn1−I−sn1Byn sn−sn1Byn
≤ yn1−yn|sn−sn1|Byn.
2.18
Substituting2.17into2.18, we have
ρn1−ρn ≤ xn1−xn|sn−sn1|Byn. 2.19
Setxn1βnxn 1−βnzn ∀n≥0. Observe that from the definition ofzn, we obtain
zn1−zn xn1−βn1xn1
1−βn1 −
xn1−βnxn
1−βn
αn1γfWn1xn1 1−βn1I−αn1AWn1ρn1
1−βn1
−αnγfWnxn 1−βnI−αnAWnρn
1−βn
αn1
1−βn1
γfWn1xn1−
αn
1−βnγfWnxn Wn1ρn1
−Wnρn αn
1−βnAWnρn−
αn1
1−βn1
AWn1ρn1
αn1
1−βn1
γfxn1−AWn1ρn1
αn
1−βnAWnρn−γfWnxn
Wn1ρn1−Wn1ρnWn1ρn−Wnρn.
It follows that
zn1−zn − xn1−xn αn1
1−βn1
γfWn1xn1AWn1ρn1
αn
1−βnAWnρnγfWnxn
Wn1ρn1−Wn1ρnWn1ρn−Wnρn − xn1−xn
≤ αn1
1−βn1
γfWn1xn1AWn1ρn1
αn
1−βnAWnρnγfWnxn
Wn1ρn−Wnρnρn1−ρn − xn1−xn.
2.21
From1.23, sinceTiandUn,i ∀i1,2, . . . , Nare nonexpansive,
Wn1ρn−Wnρn
λn1,NTNUn1,N−1un 1−λn1,Nρn−λn,NTNUn,N−1ρn−1−λn,Nρn
≤ |λn1,N−λn,N|unλn1,NTNUn1,N−1ρn−λn,NTNUn,N−1ρn
≤ |λn1,N−λn,N|ρnλn1,NTNUn1,N−1ρn−TNUn,N−1ρn|λn1,N−λn,N|TNUn,N−1ρn
≤2M|λn1,N−λn,N|λn1,NUn1,N−1ρn−Un,N−1ρn.
2.22
Again, from1.23,
Un1,N−1ρn−Un,N−1ρn
λn1,NTN−1Un1,N−2ρn 1−λn1,N−1ρn−λn,N−1TN−1Un,N−2ρn−1−λn,N−1ρn
≤ |λn1,N−1−λn,N−1|ρnλn1,N−1TN−1Un1,N−2ρn−λn,N−1TN−1Un,N−2ρn
≤ |λn1,N−1−λn,N−1|ρnλn1,N−1TN−1Un1,N−2ρn−TN−1Un1,N−2ρn|λn1,N−1−λn,N−1|M
≤2M|λn1,N−1−λn,N−1|λn1,N−1Un1,N−2ρn−Un,N−2ρn
≤2M|λn1,N−1−λn,N−1|Un1,N−2ρn−Un,N−2ρn.
2.23
Therefore, we have
Un1,N−1ρn−Un,N−1ρn
≤2M|λn1,N−1−λn,N−1|2M|λn1,N−2−λn,N−2|Un1,N−3ρn−Un,N−3ρn
≤2M
N−1
i2
|λn1,i−λn,i|Un1,1un−Un,1ρn
λn1,1T1un 1−λn1,1ρn−λn,1T1un−1−λn,1ρn2M N−1
i2
|λn1,i−λn,i|,
and then
Un1,N−1ρn−Un,N−1ρn ≤ |λn1,1−λn,1|ρnλn1,1T1ρn−λn,1T1ρn
2M
N−1
i2
|λn1,i−λn,i| ≤2M N−1
i1
|λn1,i−λn,i|.
2.25
Substituting2.25into2.22, we have
Wn1ρn−Wnρn ≤2M|λn1,N−λn,N|2λn1,NM N−1
i1
|λn1,i−λn,i|
≤2M
N
i1
|λn1,i−λn,i|.
2.26
Using2.19and2.26in2.21, we get
zn1−zn − xn1−xn ≤ αn1
1−βn1
γfWn1xn1AWn1xn1
αn
1−βnAWnxnγfWnxn 2M N
i1
|λn1,i−λn,i|,
2.27
which implies that
lim sup
n→∞ zn1−zn − xn1−xn≤0. 2.28
Hence, byLemma 1.3, we have
lim
n→∞zn−xn0. 2.29
Consequently,
lim
n→∞xn1−xnnlim→∞1−βnzn−xn0. 2.30
From2.18,2.19,2.30, and conditioniv, we have
lim
n→∞ρn1−ρnnlim→∞yn1−yn0. 2.31
Sincexn1αnγfxn βnxn 1−βnI−αnAWnρn, we have
xn−Wnρn ≤ xn1−xnxn1−Wnρn
≤ xn1−xnαnγfWnxn−AWnρnβnxn−Wnρn,
that is,
xn−Wnρn ≤ 1
1−βnxn1−xn
αn
1−βnγfWnxn−AWnρn. 2.33
It follows that
lim
n→∞xn−Wnρn0. 2.34
Forp∈ ∩Ni1FixTi∩Ω∩VIC, B, note thatSr is firmly nonexpansive, then we have
yn−p2 ≤ Trxn−Trxn2
≤ Trxn−Trp, xn−p
yn−p, xn−p
1
2yn−p
2x
n−p2− xn−yn2,
2.35
and hence
yn−p2≤ xn−p2− xn−yn2. 2.36
Therefore, we have
xn1−p2αnγfWnxn−Ap βnxn−Wnρn I−αnAWnρn−p2
≤ I−αnAWnρn−p βnxn−Wnρn22αnγfWnxn−Ap, xn1−p
≤I−αnAWnρn−pβnxn−Wnρn22αnγfWnxn−Apxn1−p
≤1−αnγρn−pβnxn−Wnρn22αnγfWnxn−Apxn1−p
1−αnγ2ρn−p2β2nxn−Wnρn221−αnγβnρn−pxn−Wnρn
2αnγfWnxn−Apxn1−p
≤1−αnγ2{xn−p2− xn−yn2}βnxn−Wnρn2
21−αnγβnρn−pxn−Wnρn
2αnγfWnxn−Apxn1−p
1−2αnγ αnγ2xn−p2−1−αnγ2xn−yn2
βnxn−Wnρn221−αnγβnρn−pxn−Wnρn
2αnγfWnxn−Apxn1−p
≤ xn−p2 αnγ2xn−p2−1−αnγ2xn−yn2
βnxn−Wnρn221−αnγβnρn−pxn−Wnρn
2αnγfWnxn−Apxn1−p.
Then, we have
1−αnγ2xn−yn2≤ xn−p2− xn1−p2βn2xn−Wnρn2
αnγ2xn−p221−αnγβnρn−pxn−Wnρn
2αnγfWnxn−Apxn1−p
≤xn−pxn1−p× xn1−xn
αnγ2xn−p2βn2xn−Wnρn2
21−αnγβnρn−pxn−Wnρn2αnγfWnxn−Apxn1−p.
2.38 So, from2.30,2.34, and conditioniv, we have
lim
n→∞xn−yn0. 2.39
Forp∈F, we have
ρn−pPCI−snAyn−PCI−sAp2
≤ yn−p−snAyn−Ap2
yn−p2−2snyn−p, Ayn−Aps2nAyn−Ap2
≤ xn−p2−2sn−uAyn−Ap2vyn−p2 s2nAyn−Ap2
≤ xn−p22snuAyn−Ap2−2snvyn−p2s2nAyn−Ap2
≤ xn−p2
2sns2n−2sμn2v
Ayn−Ap2.
2.40
Observe that
xn1−p2αnγfWnxn−Ap βnxn−Wnρn I−αnAWnρn−p2
≤ I−αnAWnρn−p βnxn−Wnρn22αnγfWnxn−Ap, xn1−p
≤I−αnAWnρn−pβnxn−Wnρn22αnγfWnxn−Apxn1−p
≤1−αnγρn−pβnxn−Wnρn22αnγfWnxn−Apxn1−p
1−αnγ2ρn−p2β2nxn−Wnρn221−αnγβnρn−pxn−Wnρn
2αnγfWnxn−Apxn1−p
ρn−p2 2αnγ αnγ2ρn−p2β2nxn−Wnρn2
21−αnγβnρn−pxn−Wnρn2αnγfWnxn−Apxn1−p.
2.41 Substituting2.40into2.41, we have
xn1−p2xn−p2
2sns2n−2sμn2v
Ayn−Ap2
2αnγ αnγ2ρn−p2β2nxn−Wnρn2
21−αnγβnρn−pxn−Wnρn2αnγfWnxn−Apxn1−p.
It follows from the conditionivthat
2av
μ2 −2bu−b 2
Ayn−Ap2≤ xn−p2− xn1−p2
2αnγ αnγ2ρn−p2β2nxn−Wnρn2
21−αnγβnρn−pxn−Wnρn
2αnγfWnxn−Apxn1−p
≤xn−pxn1−pxn1−xn
2αnγ αnγ2ρn−p2β2nxn−Wnρn2
21−αnγβnρn−pxn−Wnρn
2αnγfWnxn−Apxn1−p.
2.43
From conditioniv,2.30, and2.34, we have
lim
n→∞Ayn−Ap0. 2.44
On the other hand, we have
ρn−p2PCI−snAyn−PCI−snAp2
≤ I−snAyn−I−snAp, ρn−p
1
2{I−snAyn−I−snAp
2ρ
n−p2−I−snAyn−I−snAp−ρn−p2}
≤ 1
2{yn−p
2ρ
n−p2− yn−ρn−snAyn−Ap2}
1
2{yn−p
2ρ
n−p2−yn−ρn2−s2nAyn−Ap22snyn−ρn, Ayn−Ap},
2.45 which yields that
ρn−p2≤ xn−p2− yn−ρn22snyn−ρnAyn−Ap. 2.46
Substituting2.44into2.40yields that
xn1−p2≤ xn−p2− yn−ρn22snyn−ρnAyn−Ap
2αnγ αnγ2ρn−p2β2nxn−Wnρn2
21−αnγβnρn−pxn−Wnρn2αnγfWnxn−Apxn1−p.
2.47
It follows that
yn−ρn2≤ xn−p2− xn1−p22snyn−ρnAyn−Ap
2αnγ αnγ2ρn−p2β2nxn−Wnρn2
21−αnγβnρn−pxn−Wnρn2αnγfWnxn−Apxn1−p
≤xn−p − xn1−pxn−xn12snyn−ρnAyn−Ap
2αnγ αnγ2ρn−p2β2nxn−Wnρn2
21−αnγβnρn−pxn−Wnρn2αnγfWnxn−Apxn1−p.
From conditioniv,2.30,2.34, and2.41, we have
lim
n→∞yn−ρn0. 2.49
Observe that
yn−Wnyn ≤ Wnyn−WnρnWnρn−xnxn−ynyn−ρn
≤2yn−ρnWnρn−xnxn−yn.
2.50
From2.34,2.39, and2.49, we have
lim
n→∞yn−Wnyn0. 2.51
Observe thatPFγf I−Ais a contraction. Indeed, for allx, y∈H, we have
PFγf I−Ax−PFγf I−Ay ≤ γf I−Ax−γf I−Ay
≤γfx−fyI−Ax−y ≤γαx−y 1−γx−y
γα1−γx−y.
2.52
The Banach contraction mapping principle guarantees thatPFγfI−Ahas a unique fixed
point, sayq∈H. That is,qPFγf I−Aq. Next, we show that
lim sup
n→∞ γfq−Aq, xn−q ≤0. 2.53
To see this, we choose a subsequence{xni}of{xn}such that
lim sup
n→∞ γfq−Aq, xn−qlim supn→∞ γfq−Aq, xni−q. 2.54
Correspondingly, there exists a subsequence{yni}of{yn}. Since{yni}is bounded, there exists
a subsequence{yni}of{yn}which converges weakly toω. Without loss of generality, we can
assume thatyni ω.
Next, we showω ∈ ∩Ni1FixTi∩Ω∩VIC, B. First, we proveω ∈ Ω. Since{yni}is
bounded, there exists a subsequence{yij}of{yni}which converges weakly toω. Without loss
of generality, we can assume thatyni ω. FromWnyn−yn → 0, we obtainWnyni ω.
Now, we show thatω∈Ω. SinceynTrxn, we derive
Θyn, x ϕx−ϕyn 1rKyn−Kxn, ηx, yn ≥0, ∀x∈C. 2.55
From the monotonicity ofΘ, we have
1
and hence
Ky
ni−Kxni
r , ηx, yni
ϕx−ϕyni≥Θx, yni. 2.57
SinceKyni−Kxni/r →0, andyni → ωweakly, from the weak lower semicontinuity
ofϕandΘx, yin the second variabley, we have
Θx, ω ϕω−ϕx≤0, ∀x∈C 2.58
for 0< t≤1 andx∈C, letxttx 1−tω. Sincex∈Candω∈C, we havext∈Cand hence
Θxt, ω ϕω−ϕxt≤0. From the convexity of equilibriumΘx, yin the second variable
y, we have
0 Θxt, xt ϕxt−ϕxt
≤tΘxt, x 1−tΘxt, ω tϕx 1−tϕω−ϕxt
≤tΘxt, x ϕx−ϕxt,
2.59
and henceΘxt, x ϕx−ϕxt≥0. Then, we have
Θω, x ϕx−ϕω≥0, ∀x∈C, 2.60
and henceω∈Ω.
We will showω∈FixWn. Assumeω /∈FixWn. Sinceynj→ωweakly andω /Wnω,
fromOpialcondition, we have
lim inf
j→∞ ynj−ω<lim infj→∞ ynj−Wnω ≤lim inf
j→∞ ynj−WnynjWnynj−Wnω ≤lim inf
j→∞ ynj−ω.
2.61
This is a contradiction, so, we getω∈FixWn ∩Ni1FixTi. Therefore,ω∈FixWn∩Ω.
Next, let us first show thatω∈ ∩VIC, A, put
TW1
BW1NCW1, W1∈C,
∅, W1/∈C.
2.62
SinceBis relaxedu, v-cocoercive and conditioniv, we have
Bx−By, x−y ≥−μBx−By2vx−y2≥v−uμ2x−y2≥0, 2.63
which yieldsBis monotone. Thus,T is maximal monotone. Letω1, ω2∈GT.Sinceω2−
ω1∈NCω1andρn∈C, we have
ω1−ρn, ω2−Bω1 ≥0. 2.64
On the other hand, fromρnPCI−snByn,we have
and hence
ω1−ρn,ρns−yn n Byn
≥0. 2.66
It follows that
ω1−ρn, ω2 ≥ ω1−ρni, Bω1
≥ ω1−ρni, Bω1 −
ω1−ρni,
ρni−yni sn Byni
ω1−ρni, Bω1−
ρni−yni sni
−Byni
ω1−ρni, Bω1−Bρniω1−ρni, Bρni−Byni −
ω1−ρni,
ρni −yni sni
≥ ω1−ρni, Bρni−Byni −
ω1−ρni,
ρni−yni sni
,
2.67
which implies thatω1−ω, ω2 ≥ 0. We haveω ∈ T−10 and henceω ∈ VIC, A.That is,
ω∈F. SinceqPFγf I−Aq, we have
lim sup
n→∞ γfq−Aq, xn−qlim supn→∞ γfq, xni−q γfq−Aq, ω−q ≤0.
2.68
Finally, we prove that{xn}and{yn}converge strongly toq. From2.1, we have
xn1−q2αnγfWnxn−Aq βnxn−q1−βnI−αnAWnρn−q2
≤ βnxn−q 1−βnI−αnAWnρn−q22αnγfWnxn−Aq, xn1−q
≤1−βnI−αnAWnρn−qβnxn−q2
2αnγfWnxn−fq, xn1−q2αnγfq−Aq, xn1−q
≤1−βn−αnγρn−qβnxn−q2
2αnγαxn−qxn1−q2αnγfq−Aq, xn1−q
≤1−βn−αnγyn−qβnxn−q2
2αnγαxn−pxn1−q2αnγfp−Aq, xn1−q
≤1−αnγ2xn−q2αnγα{xn−q2xn1−q2}2αnγfq−Aq, xn1−q.
This implies that
xn1−q2
1−2αnγ αnγ2αnγα
1−αnγα xn−q
2 2αn
1−αnγαγfq−Aq, xn1−α
1−2γ−γααn 1−αnγα
xn−q2 αnγ
2
1−αnγαxn−p
2
2αn
1−αnγαγfp−Aq, xn1−q
≤
1−2γ−γααn 1−αnγα
xn−q22γ−γααn
1−αnγα
×
αnγ2M 1
2γ−γα 1
γ−γαγfq−Aq, xn1−q
1−δnxn−q2δnσn,
2.70
whereM1 sup{xn−q2 :n≥1}, δn 2γ−γααn/1−αnγα,andβn αnγ2M1/2γ−
γα 1/γ−γαγfq−Aq, xn1 −q. It is easy to see thatδn → 0, ∞n1δn ∞, and
lim supn→∞βn/δn ≤ 0. Hence, by Lemma 1.4, the sequence {xn} converges strongly to q.
Consequently, we can obtain that {yn} also converges strongly to q. This completes the
proof.
Corollary 2.3. LetCbe a nonempty closed convex subset of a real Hilbert spaceHand letϕ:C→R
be a lower semicontinuous and convex functional. LetΘ:C×C→Rbe an equilibrium bifunction satisfying conditions (H1)–(H3) such thatΩ/∅. Suppose{αn}and{βn}are two sequences in (0,1)
andris a positive parameter. Suppose that the following conditions are satisfied:
iη:C×C→His Lipschitz with constantλ >0such that aηx, y ηy, x 0 ∀x·y∈C;
bη·,·is affine in the first variable;
cfor each fixedy∈C, x→ηy, xis sequentially continuous from the weak topology to the weak topology;
iiK : C → Risη-strongly convex with constantμ > 0and its derivativeKis not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constantν >0, μ≥λν;
iiifor each x ∈ C, there exist a bounded subsetDx ⊆ C and zx ∈ H, such that for any
y∈C\Dx,
Θy, zx ϕzx−ϕy 1rKy−Kx, ηzx, y ≤0; 2.71
ivlimn→∞αn0and∞n1αn∞; 0<lim infn→∞βn≤lim supn→∞βn<1.
Givenx0∈Carbitrarily, then the sequences{xn}and{yn}generated iteratively by
Θyn, x ϕx−ϕyn 1rKyn−Kxn, ηx, yn ≥0, ∀x∈C,
xn1αnγfWnxn βnxn 1−βnI−αnAWnyn, ∀n≥1.
Then,{xn}and{yn}converge strongly toq∈FixTi∩Ωprovided thatTr is firmly nonexpansive,
whereqPFixTi∩ΩI−Aγfqis a unique solution of variational inequalities
A−γfq, q−x ≤0, ∀x∈
N
i1
FixTi∩Ω, −11
which is the optimality condition for the minimization problem
min
p∈FixTi
1
2Ap, p −hp, 2.72
wherehis a potential function forγf.
Proof. TakeTix x∀i 1,2, . . . , N and for allx ∈ Cin2.1. Then,Wnx x∀x ∈ C. The
conclusion follows immediately fromTheorem 2.2. This completes the proof.
Corollary 2.4. LetCbe a nonempty closed convex subset of a real Hilbert space. Let{Ti}Ni1be a finite
family of nonexpansive mappings ofCinto itself such that∩Ni1FixTi/∅. Letλn1, λn2, . . . , λnN be
real number such thatlimn→∞λn1,i−λn,i 0 ∀i 1,2, . . . , N. Suppose that{αn}and{βn}are
two sequences in0,1andris a positive parameter. Assume that the following hold:
ilimn→∞αn0and∞n1αn∞;
ii0<lim infn→∞βn≤lim supn→∞βn<1.
Givenx0∈Harbitrarily, then the sequences{xn}are generated iteratively by
xn1 αnγfWnxn βnxn 1−βnI−αnAWnxn, ∀n≥1. 2.73
Proof. Setϕx 0 andΘx, y 0∀x, y∈Cand putr 1. TakeKx x2/2 andηy, x y−x∀x, y∈C, we getynxninTheorem 2.2. Therefore, the conclusion follows.
References
1 R. U. Verma, “Generalized system for relaxed cocoercive variational inequalities and projection methods,”Journal of Optimization Theory and Applications, vol. 121, no. 1, pp. 203–210, 2004.
2 R. U. Verma, “General convergence analysis for two-step projection methods and applications to variational problems,”Applied Mathematics Letters, vol. 18, no. 11, pp. 1286–1292, 2005.
3 R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,”Transactions of the American Mathematical Society, vol. 149, no. 1, pp. 75–88, 1970.
4 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.
5 S. D. Fl˚am and A. S. Antipin, “Equilibrium programming using proximal-like algorithms,” Mathematical Programming, vol. 78, no. 1, pp. 29–41, 1997.
6 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.
7 F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,”Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 33–56, 1998.
8 G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43–52, 2006.
10 H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659–678, 2003.
11 I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8 ofStudies in Computational Mathematics, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001.
12 D. C. Youla, “Mathematical theory of image restoration by the method of convex projections,” in Image Recovery: Theory and Applications, H. Stark, Ed., pp. 29–77, Academic Press, Orlando, Fla, USA, 1987.
13 L.-C. Zeng, S.-Y. Wu, and J.-C. Yao, “Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems,”Taiwanese Journal of Mathematics, vol. 10, no. 6, pp. 1497–1514, 2006.
14 L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,”Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008. 15 Y. Yao, “A general iterative method for a finite family of nonexpansive mappings,”Nonlinear Analysis:
Theory, Methods & Applications, vol. 66, no. 12, pp. 2676–2687, 2007.
16 Y.-C. Liou, Y. Yao, and K. Kimura, “Strong convergence to common fixed points of a finite family of nonexpansive mappings,”Journal of Inequalities and Applications, vol. 2007, Article ID 37513, 10 pages, 2007.
17 W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,”Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003.
18 H. Iiduka and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings,”Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 3, pp. 341–350, 2005.
19 J. Chen, L. Zhang, and T. Fan, “Viscosity approximation methods for nonexpansive mappings and monotone mappings,”Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 1450–1461, 2007.
20 A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
21 H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,”SIAM Review, vol. 38, no. 3, pp. 367–426, 1996.
22 P. L. Combettes, “The foundations of set theoretic estimation,”Proceedings of the IEEE, vol. 81, no. 2, pp. 182–208, 1993.
23 H. H. Bauschke, “The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space,”Journal of Mathematical Analysis and Applications, vol. 202, no. 1, pp. 150–159, 1996. 24 P. L. Combettes, “Constrained image recovery in a product space,” in Proceedings of the IEEE
International Conference on Image Processing (ICIP ’95), vol. 2, pp. 25–28, IEEE Computer Society Press, Washington, DC, USA, October 1995.
25 F. Deutsch and H. Hundal, “The rate of convergence of Dykstra’s cyclic projections algorithm: the polyhedral case,”Numerical Functional Analysis and Optimization, vol. 15, no. 5-6, pp. 537–565, 1994. 26 A. N. Iusem and A. R. De Pierro, “On the convergence of Han’s method for convex programming
with quadratic objective,”Mathematical Programming, vol. 52, no. 2, pp. 265–284, 1991.
27 S. Atsushiba and W. Takahashi, “Strong convergence theorems for a finite family of nonexpansive mappings and applications,”Indian Journal of Mathematics, vol. 41, no. 3, pp. 435–453, 1999.
28 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.
29 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.
