Volume 2009, Article ID 519065,32pages doi:10.1155/2009/519065
Research Article
A General Iterative Method for Variational
Inequality Problems, Mixed Equilibrium Problems,
and Fixed Point Problems of Strictly
Pseudocontractive Mappings in Hilbert Spaces
Rattanaporn Wangkeeree and Rabian Wangkeeree
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
Correspondence should be addressed to Rabian Wangkeeree,[email protected]
Received 23 April 2009; Accepted 22 June 2009
Recommended by Anthony To Ming Lau
We introduce an iterative scheme for finding a common element of the set of fixed points of a k-strictly pseudocontractive mapping, the set of solutions of the variational inequality for an inverse-strongly monotone mapping, and the set of solutions of the mixed equilibrium problem in a real Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above three sets are obtained. As applications, at the end of the paper we first apply our results to study the optimization problem and we next utilize our results to study the problem of finding a common element of the set of fixed points of two families of finitely k-strictly pseudocontractive mapping, the set of solutions of the variational inequality, and the set of solutions of the mixed equilibrium problem. The results presented in the paper improve some recent results of Kim and Xu2005, Yao et al.2008, Marino et al.2009, Liu2009, Plubtieng and Punpaeng2007, and many others.
Copyrightq2009 R. Wangkeeree and R. Wangkeeree. 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
Throughout this paper, we always assume thatHis a real Hilbert space with inner product
·,·and norm·, respectively,Cis a nonempty closed convex subset ofH. Letϕ:C → Rbe a real-valued function and letΘ:C×C → Rbe an equilibrium bifunction, that is,Θu, u 0 for eachu∈C. Ceng and Yao1considered the following mixed equilibrium problem:
Findx∗∈Csuch thatΘx∗, yϕy≥ϕx∗, ∀y∈C. 1.1
In particular, ifϕ ≡ 0, the mixed equilibrium problem1.1becomes the following equilibrium problem:
Findx∗∈Csuch thatΘx∗, y≥0, ∀y∈C. 1.2
The set of solutions of1.2is denoted by EPΘ.
Ifϕ≡0 andΘx, y Bx, y−x ≥0 for allx, y∈C, whereBis a mapping formCinto
H, then the mixed equilibrium problem1.1becomes the following variational inequality:
Find x∗∈Csuch thatBx∗, y−x∗≥0, ∀y∈C. 1.3
The set of solutions of 1.3 is denoted by VIB, C. The variational inequality has been extensively studied in literature. See, for example,2–13and the references therein.
The problem 1.1 is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games and others; see for instance,1,2,14,15.
First we recall some relevant important results as follows.
In 1997, Combettes and Hirstoaga 14 introduced an iterative method of finding the best approximation to the initial data when EPΘ is nonempty and proved a strong convergence theorem. Subsequently, S. Takahashi and W. Takahashi 16 introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of EPΘand the set of fixed point points of a nonexpansive mapping. Using the idea of S. Takahashi and W. Takahashi16, Plubtieng and Punpaeng17introduced an the general iterative method for finding a common element of the set of solutions of EPΘ and the set of fixed points of a nonexpansive mapping which is the optimality condition for the minimization problem in a Hilbert space. Furthermore, Yao et al.11introduced some new iterative schemes for finding a common element of the set of solutions of EPΘand the set of common fixed points of finitelyinfinitelynonexpansive mappings. Very recently, Ceng and Yao1considered a new iterative scheme for finding a common element of the set of solutions of MEPΘand the set of common fixed points of finitely many nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem which used the following condition:
EK : C → Risη-strongly convex and its derivativeKis sequentially continuous from the weak topology to the strong topology.
Their results extend and improve the corresponding results in6,11,14. We note that the conditionEfor the functionK :C → Ris a very strong condition. We also note that the condition Edoes not cover the case Kx x2/2 and ηx, y x−y. Motivated
We recall that a mappingB:C → His said to be:
imonotone ifBx−By, x−y ≥0, for allx, y∈C,
iiL-Lipschitz if there exists a constant L > 0 such that Bx − By ≤ Lx −
y, for allx, y∈C,
iiiα-inverse-strongly monotone19,20if there exists a positive real numberαsuch that
Bx−By, x−y ≥αBx−By2, ∀x, y∈C. 1.4
It is obvious that anyα-inverse-strongly monotone mapping Bis monotone and Lipschitz continuous. Recall that a mapping T : C → C is called a k-strictly pseudocontractive mapping if there exists a constant 0≤k <1 such that
Tx−Ty2≤x−y2kI−Tx−I−Ty2, ∀x, y∈C. 1.5
Note that the class of k-strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings which are mappingsT onCsuch that
Tx−Ty≤x−y, ∀x, y∈C. 1.6
That is, T is nonexpansive if and only if T is 0-strictly pseudocontractive. We denote by
FT:{x∈C:Txx}the set of fixed points ofT.
Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example,21–24and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points of nonexpansive mapping on a real Hilbert space:
θx min
x∈C
1
2Ax, x − x, b, 1.7
whereAis a linear bounded operator,Cis the fixed point set of a nonexpansive mappingT, andbis a given point inH. Recall that a linear bounded operatorAis strongly positive if there is a constantγ >0 with property
Ax, x ≥γx2 ∀x∈H. 1.8
Recently, Marino and Xu25introduced the following general iterative scheme based on the viscosity approximation method introduced by Moudafi26:
where A is a strongly positive bounded linear operator on H. They proved that if the sequence {αn} of parameters satisfies appropriate conditions, then the sequence {xn}
generated by1.9converges strongly to the unique solution of the variational inequality
A−γfx∗, x−x∗≥0, x∈C, 1.10
which is the optimality condition for the minimization problem
min
x∈C
1
2Ax, x −hx, 1.11
wherehis a potential function forγfi.e., hx γfxforx∈H.
Recall that the construction of fixed points of nonexpansive mappings via Manns algorithm 27 has extensively been investigated in literature; see, for example 27–32
and references therein. If T is a nonexpansive self-mapping of C, then Mann’s algorithm generates, initializing with an arbitraryx1∈C, a sequence according to the recursive manner
xn1αnxn 1−αnTxn, ∀n≥1, 1.12
where{αn}is a real control sequence in the interval0,1.
IfT :C → Cis a nonexpansive mapping with a fixed point and if the control sequence
{αn} is chosen so that∞n1αn1−αn ∞, then the sequence {xn} generated by Manns
algorithm converges weakly to a fixed point ofT. Reich33showed that the conclusion also holds good in the setting of uniformly convex Banach spaces with a Fr´ehet differentiable norm. It is well known that Reich’s result is one of the fundamental convergence results. However, this scheme has only weak convergence even in a Hilbert space34. Therefore, many authors try to modify normal Mann’s iteration process to have strong convergence; see, for example,35–40and the references therein.
Kim and Xu36introduced the following iteration process:
yn βnxn1−βnTxn,
xn1 αnu 1−αnyn, n≥1,
1.13
whereT is a nonexpansive mapping ofCinto itself andu∈Cis a given point. They proved the sequence {xn}defined by1.13strongly converges to a fixed point ofT provided the control sequences{αn}and{βn}satisfy appropriate conditions.
In41, Yao et al. also modified iterative algorithm1.13to have strong convergence by using viscosity approximation method. To be more precisely, they considered the following iteration process:
yn βnxn1−βnTxn,
xn1 αnfxn 1−αnyn, n≥1,
whereT is a nonexpansive mapping ofCinto itself andfis anβ-contraction. They proved the sequence {xn}defined by1.14strongly converges to a fixed point ofT provided the
control sequences{αn}and{βn}satisfy appropriate conditions.
Very recently, motivated by Acedo and Xu35, Kim and Xu36, Marino and Xu42, and Yao et al.41, Marino et al.43introduced a composite iteration scheme as follows:
yn βnxn1−βnTxn,
xn1αnγfxn I−αnAyn, n≥1,
1.15
where T is a k-strictly pseudocontractive mapping on H, f is an β-contraction, and A is a linear bounded strongly positive operator. They proved that the iterative scheme {xn}
defined by1.15converges to a fixed point ofT, which is a unique solution of the variational inequality1.10and is also the optimality condition for the minimization problem provided
{αn}and{βn}are sequences in0,1satifies the following control conditions:
C1limn→ ∞αn0, ∞n1αn∞,
∞
n1|αn1−αn|<∞,
C20≤k≤βn< ε <1 for alln≥0 and∞n1|βn1−βn|<∞.
Moreover, for finding a common element of the set of fixed points of a k-strictly pseudocontractive nonself mapping and the set of solutions of an equilibrium problem in a real Hilbert space, Liu44introduced the following iterative scheme:
x1x∈Cchosen arbitrarily,
Θun, yrn1y−un, un−xn≥0, ∀y∈C,
ynβnun1−βnTun,
xn1αnγfxn I−αnAyn, n≥1,
1.16
whereT is ak-strictly pseudocontractive mapping onH, f is anα-contraction and, Ais a linear bounded strongly positive operator. They proved that the iterative scheme{xn}defined
by 1.16 converges to a common element ofFT∩EPΘ, which solves some variation inequality problems provided{αn},{βn},and{rn}are sequences in0,1satifies the control
conditionsC1and the following conditions:
C2 0≤k≤βn< ε <1 for alln≥1, limn→ ∞βnε, and∞n1|βn1−βn|<∞;
C3lim infn→ ∞rn>0, n∞1|rn1−rn|<0.
All of the above bring us the following conjectures?
Question 1. iCould we weaken or remove the control condition ∞n1|αn1−αn| < ∞on
parameter{αn}inC1?
ii Could we weaken or remove the control condition ∞n1|βn1 − βn| < ∞ on
parameter{βn}inC2andC2?
iiiCould we weaken or remove the control condition limn→ ∞βn εon the parameter
{βn}inC2?
vCould we construct an iterative algorithm to approximate a common element of
FT∩VIB, C∩MEPΘ, ϕ?
It is our purpose in this paper that we suggest and analyze an iterative scheme for finding a common element of the set of fixed points of a k-strictly pseudocontractive mapping, the set of solutions of a variational inequality and the set of solutions of a mixed equilibrium problem in the framework of a real Hilbert space. Then we modify our iterative scheme to finding a common element of the set of common fixed points of two finite families ofk-strictly pseudocontractive mappings, the set of solutions of a variational inequality and the set of solutions of a mixed equilibrium problem. Application to optimization problems which is one of the motivation in this paper is also given. The results in this paper generalize and improve some well-known results in17,36,41,43,44.
2. Preliminaries
LetHbe a real Hilbert space with norm · and inner product·,·and letCbe a closed convex subset ofH. We denote weak convergence and strong convergence by notations and →, respectively. It is well known that for anyλ∈0,1,
λx 1−λy2λx2 1−λy2−λ1−λx−y2, ∀x, y∈H. 2.1
For every pointx∈H, there exists a unique nearest point inC, denoted byPCx, such that
x−PCx ≤x−y ∀y∈C. 2.2
PC is called the metric projection of Honto C.It is well known thatPC is a nonexpansive mapping ofHontoCand satisfies
x−y, PCx−PCy≥PCx−PCy2 2.3
for everyx, y∈H.Moreover,PCxis characterized by the following properties:PCx∈Cand
x−PCx, y−PCx≤0, x−y2≥ x−P
Cx2y−PCx2,
2.4
for allx∈H, y∈C. It is easy to see that the following is true:
u∈VIB, C⇐⇒uPCu−λBu, λ >0. 2.5
A set-valued mappingS:H → 2His called monotone if for allx, y∈H,f ∈Sxand
g∈Syimplyx−y, f−g ≥0. A monotone mappingS:H → 2His maximal if the graph of
everyy, g∈GSimpliesf∈Sx. LetBbe a monotone map ofCintoHand letNCvbe the normal cone toCatv∈C, that is,NCv{w∈H:u−v, w ≥0, ∀u∈C}and define
Sv
⎧ ⎨ ⎩
BvNCv, v∈C,
∅, v /∈C. 2.6
ThenSis the maximal monotone and 0∈Svif and only ifv∈VIB, C; see45.
The following lemmas will be useful for proving the convergence result of this paper.
Lemma 2.146. Assume{an}is a sequence of nonnegative real numbers such that
an1≤1−αnanσn, n≥1, 2.7
where{αn}is a sequence in0,1and{σn}is a sequence inRsuch that
1∞n1αn∞
2lim supn→ ∞σn/αn ≤0or∞n1|σn|<∞.
Thenlimn→ ∞an0.
Lemma 2.247. Let{xn}and{ln}be bounded sequences in a Banach spaceEand let{βn}be a sequence in0,1with0<lim infn→ ∞βn≤lim supn→ ∞βn <1.Supposexn1 1−βnlnβnxn for all integersn≥1andlim supn→ ∞ln1−ln − xn1−xn≤0.Then,limn→ ∞ln−xn0.
Lemma 2.342, Proposition 2.1. Assume thatCis a closed convex subset of Hilbert spaceH, and letT :C → Cbe a self-mapping ofC,
iifT is ak-strictly pseudocontractive mapping, thenTsatisfies the Lipscchitz condition
Tx−Ty≤ 1κ
1−κx−y ∀x, y∈C. 2.8
iiifT is ak-strictly pseudocontractive mapping, then the mappingI−Tis demiclosed(at0). That is, if{xn}is a sequence inCsuch thatxn xandI−Txn → 0, thenI−Tx0.
iiiifT is ak-strictly pseudocontractive mapping, then the fixed point setFTofT is closed and convex so that the projectionPFTis well defined.
Lemma 2.425. AssumeAis a strongly positive linear bounded operator on a Hilbert spaceH with coefficientγ >0and0< ρ≤ A−1.ThenI−ρA ≤1−ργ.
The following lemmas can be obtained from Acedo and Xu35, Proposition 2.6easily.
Lemma 2.5. LetHbe a Hilbert space, Cbe a closed convex subset ofH. For any integerN ≥ 1, assume that, for each1 ≤ i≤ N, Ti :C → His aki-strictly pseudocontractive mapping for some
Lemma 2.6. Let{Ti}Ni1 and {ξi}iN1 be as inLemma 2.5. Suppose that{Ti}Ni1 has a common fixed point inC. ThenFNi1ξiTi Ni1FTi.
For solving the mixed equilibrium problem, let us give the following assumptions for a bifunctionΘ, ϕand the setC:
A1 Θx, x 0 for allx∈C;
A2 Θis monotone, that is,Θx, y Θy, x≤0 for allx, y∈C;
A3for eachx, y, z∈C,limt→0Θtz 1−tx, y≤Θx, y;
A4for eachx∈C, y→Θx, yis convex and lower semicontinuous;
B1For eachx∈Handr >0, there exists a bounded subsetDx ⊂C,andyx ∈Csuch that for anyz∈C\Dx,
Θz, yϕyx1ryx−z, z−x< ϕz, 2.9
B2Cis a bounded set.
By similar argument as in48, proof of Lemma 2.3, we have the following result.
Lemma 2.7. LetCbe a nonempty closed convex subset ofH. LetΘ: C×C → Rbe a bifunction satifies (A1)–(A4) and letϕ:C → R∪ {∞}be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. Forr > 0 andx ∈ H, define a mappingTr : H → Cas follows:
Trx
z∈C:Θz, yϕy1
r
y−z, z−x≥ϕz, ∀y∈C
2.10
for allx∈H. Then, the following conditions hold:
ifor eachx∈H,Trx/∅;
iiTr is single- valued;
iiiTr is firmly nonexpansive, that is, for anyx, y∈H, Trx−Try2≤ Trx−Try, x−y;
ivFTr MEPΘ, ϕ;
vMEPΘ, ϕis closed and convex.
3. Main Results
In this section, we derive a strong convergence of an iterative algorithm which solves the problem of finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of ak-strictly pseudocontractive mapping ofCinto itself and the set of the variational inequality for anα-inverse-strongly monotone mapping ofCintoHin a Hilbert space.
convex function. LetTbe ak-strictly pseudocontractive mapping ofCinto itself. Letfbe a contraction ofCinto itself with coefficientβ ∈ 0,1,Banα-inverse-strongly monotone mapping ofCintoH such thatΩ : FT∩VIB, C∩MEPΘ, ϕ/∅. LetAbe a strongly bounded linear self-adjoint operator with coefficientγ > 0and0 < γ < γ/β. Assume that either (B1) or (B2) holds. Given the
sequences{αn}, {βn}, {δn}, {λn},and{rn}in0,1satisfyies the following conditions
D1limn→ ∞αn0, ∞n1αn∞;
D20<lim infn→ ∞δn≤lim supn→ ∞δn<1;
D30≤k≤βn< ε <1for alln≥0,andlimn→ ∞|βn1−βn|0;
D4{λn} ⊂a, bfor somea, bwith0< a < b <2α,andlimn→ ∞|λn1−λn|0;
D5lim infn→ ∞rn>0, limn→ ∞|rn1−rn|0.
Let{xn}, {un},and{yn}be sequences generated by
x1x∈Cchosen arbitrarily,
Θun, yϕy−ϕun r1
n
y−un, un−xn≥0, ∀y∈C,
ynβnun1−βnTun,
xn1αnγfxn δnxn 1−δnI−αnAPC
yn−λnByn, n≥1.
3.1
Then{xn}, {un},and{yn}converge strongly to a pointz ∈Ωwhich is the unique solution of the variational inequality
A−γfz, z−x≤0, ∀x∈Ω. 3.2
Equivalently, one haszPΩI−Aγfz.
Proof. Since limn→ ∞αn0, we may assume, without loss of generality, thatαn<A−1for all n. ByLemma 2.4, we haveI−αnA ≤1−αnγ. We will assume thatI−A ≤1−γ. Observe thatPΩI−Aγfis a contraction. Indeed, for allx, y∈C, we have
PΩI−Aγfx−PΩI−Aγfy≤I−Aγfx−I−Aγfy
≤ I−Ax−yγfx−fy
≤1−γx−yγβx−y
1−γ−γβx−y.
3.3
SinceHis complete, there exists a unique elementz∈Csuch thatzPΩI−Aγfz.On
the other hand, sinceAis a linear bounded self-adjoint operator, one has
Observing that
1−δnI−αnAx, x1−δn−αnAx, x
≥1−δn−αnA
≥0,
3.5
we obtain1−δnI−αnAis positive. It follows that
1−δnI−αnAsup{1−δnI−αnAx, x:x∈H,x1}
sup{1−δn−αnAx, x:x∈H,x1}
≤1−δn−αnγ.
3.6
Next, we divide the proof into six steps as follows.
Step 1. First we prove thatI−λnBis nonexpansive. For allx, y∈Candλn∈0,2α,
I−λnBx−I−λnBy2x−y−λnBx−B
y2
x−y2−2λ
nx−y, Bx−Byλ2nBx−By2
≤x−y2λnλn−
2αBx−By2,
3.7
which implies thatI−λnBis nonexpansive.
Step 2. Next we prove that{xn}, {yn}, {un}, {Bxn}, {Byn}and{Bun}are bounded. Indeed, pick anyp∈Ω. From2.5, we havepPCp−λnBp.SettingvnPCyn−λnByn, we obtain
from the nonexpansivity ofI−λnBthat
vn−pPCyn−λnByn−PCp−λnBp
≤yn−λnByn−p−λnBp≤yn−p. 3.8
From2.1, we have
yn−p2βnun−p
1−βnTun−p2
≤βnun−p2−1−βnβnun−Tun21−βnTun−p2
3.9
so, by3.9and thek-strict pseudocontractivity ofT, it follows that yn−p2≤un−p2−1−βnβn−kun−Tun2
that is,
yn−p≤un−p. 3.11
Observe that
un−pTrnxn−Trnp≤xn−p. 3.12
From3.8,3.11and the last inequality, we have
vn−p≤xn−p. 3.13
It follows that
xn1−pαnγfxn δnxn 1−δnI−αnAvn−p
αnγfxn−Apδnxn−p 1−δnI−αnAvn−p
≤αnγfxn−Apδnxn−p1−δn−αnγvn−p
≤αnγfxn−fpαnγfp−Ap1−αnγxn−p
1−αnγ−γβxn−pαnγfp−Ap
1−αnγ−γβxn−pαnγ−γβ
γfp−Ap γ−γβ .
3.14
By simple induction, we have
xn−p≤max
x1−p,
Ap−γf
p γ−γβ
, 3.15
which gives that the sequence{xn}is bounded, so are{yn}, {un}, {Bxn}, {Byn},and{Bun}.
Step 3. Next we claim that
lim
n→ ∞xn1−xn0. 3.16
Notice that
vn−vn−1PCyn−λnByn−PCyn−1−λn−1Byn−1
≤yn−λnByn−yn−1−λn−1Byn−1
yn−λnByn−yn−1−λnByn−1
λn−1−λnByn−1
≤yn−λnByn−yn−1−λnByn−1|λn−1−λn|Byn−1
≤yn−yn−1|λn−1−λn|Byn−1.
Next, we define
Vn1−βnTβnI. 3.18
As shown in 19, from the k-strict pseudocontractivity of T and the conditions D4, it follows thatVnis a nonexpansive maping for whichFT FVn.
Observing that
ynVnun,
yn−1Vn−1un−1,
3.19
we have
yn−yn−1Vnun−Vn−1un−1
≤ Vnun−Vnun−1Vnun−1−Vn−1un−1
≤ un−un−1Vnun−1−Vn−1un−1
un−un−1βnun−1
1−βnTun−1
−βn−1un−1
1−βn−1
Tun−1
≤ un−un−1M1βn−βn−1,
3.20
whereM1is an appropriate constant such thatM1≥supn≥1{un,Tun}. Substituting3.20
into3.17, we obtain
vn−vn−1 ≤yn−yn−1|λn−1−λn|Byn−1
≤ un−un−1M1βn−βn−1|λn−1−λn|Byn−1.
3.21
On the other hand, fromunTrnxn∈domϕandun1Trn1xn1∈domϕ,we note that
Θun, yϕy−ϕun 1 rn
y−un, un−xn≥0 ∀y∈C, 3.22
Θun1, yϕy−ϕun1 rn1
1
y−un1, un1−xn1
≥0 ∀y∈C. 3.23
Puttingyun1in3.22andyunin3.23, we have
Θun, un1 ϕun1−ϕun
1
rnun1−un, un−xn ≥0,
Θun1, un ϕun−ϕun1 rn1
1
un−un1, un1−xn1 ≥0.
So, fromA2we have
un1−un, un−xn
rn −
un1−xn1 rn1
≥0, 3.25
and hence
un1−un, un−un1un1−xn−rnrn
1
un1−xn1
≥0. 3.26
Without loss of generality, let us assume that there exists a real numbercsuch thatrn> c >0 for alln∈N.Then, we have
un1−un2≤
un1−un, xn1−xn
1− rn
rn1
un1−xn1
≤ un1−un
xn1−xn1−rrn n1
un1−xn1
,
3.27
and hence
un1−un ≤ xn1−xnrn1
1|
rn1−rn|un1−xn1
≤ xn1−xn1c|rn1−rn|M2,
3.28
whereM2sup{un−xn:n∈N}. It follows from3.21and the last inequality that
vn−vn−1 ≤ xn1−xnM
1
c|rn1−rn|βn−βn−1
|λn−1−λn|Byn−1, 3.29
whereMmax{M1, M2}.
Define a sequence{ln}such that
Then, we have
ln1−ln xn2−δn1xn1
1−δn1 −
xn1−δnxn
1−δn
αn1γfxn1 1−δn1I−αn1Avn1
1−δn1
− αnγfxn 1−δnI−αnAvn
1−δn
αn1
1−δn1
γfxn1−Avn1
αn
1−δn
Avn−γfxn
vn1−vn.
3.31
It follows from3.29that
ln1−ln − xn−xn1 ≤ αn1
1−δn1
γfxn1−Avn1
αn
1−δn
Avn−γfxnvn1−vn − xn−xn1
≤ αn1
1−δn1
γfxn1−Avn1 αn
1−δnAvn−γfxn
M
1
c|rn1−rn|βn−βn−1
|λn−1−λn|Byn−1.
3.32
Observing the conditionsD1,D3,D4,D5, and taking the superior limit asn → ∞, we get
lim sup
n→ ∞ ln1−ln − xn−xn1≤0. 3.33
We can obtain limn→ ∞ln−xn0 easily byLemma 2.2. Observing that
xn1−xn 1−δnln−xn, 3.34
we obtain
lim
n→ ∞xn1−xn0. 3.35
Hence3.16is proved.
Step 4. Next we prove that
lim
aFirst we prove that limn→ ∞xn−vn0. Observing that
xn−vnxn−xn1xn1−vn
xn−xn1αnγfxn δnxn 1−δnI−αnAvn−vn
xn−xn1αnγfxn−Avnδnxn−vn,
3.37
we arrive at
1−δnxn−vn xn−xn1αn
γfxn−Avn, 3.38
which implies that
1−δnxn−vn ≤ xn−xn1αnγfxn−Avn. 3.39
Therefore, it follows from3.16,D1, andD2that
lim
n→ ∞xn−vn0. 3.40
bNext, we will show that limn→ ∞Byn−Bp0 for anyp∈Ω.Observe that
xn1−p21−δnI−αnAvn−p δnxn−p αnγfxn−Ap2
1−δnI−αnAvn−p δnxn−p2α2nγfxn−Ap2
2δnαnxn−p, γfxn−Ap2αn1−δnI−αnAvn−p, γfxn−Ap
≤1−δn−αnγvn−pδnxn−p2α2nγfxn−Ap2
2δnαnxn−p, γfxn−Ap2αn1−δnI−αnAvn−p, γfxn−Ap
1−δn−αnγ2vn−p2δ2nxn−p2
21−δn−αnγδnvn−pxn−pcn
≤1−δn−αnγ2vn−p2δ2nxn−p2
1−δn−αnγδnvn−p2xn−p2
cn
1−αnγ2−21−αnγδnδ2n
vn−p2δn2xn−p2
1−αnγδn−δ2nvn−p2xn−p2
1−αnγ2vn−p2−1−αnγδnvn−p21−αnγδnxn−p2cn
1−αnγ1−δn−αnγvn−p21−αnγδnxn−p2cn
≤1−αnγ1−δn−αnγyn−λnByn−p−λnBp2
1−αnγδnxn−p2cn
≤1−αnγ1−δn−αnγyn−p2λnλn−2αByn−Bp2
1−αnγδnxn−p2cn
≤xn−p2bb−
2αByn−Bp2cn,
3.41
where
cnα2nγfxn−Ap22δnαnxn−pγfxn−Ap
2αn1−δnI−αnAvn−pγfxn−Ap.
3.42
This implies that
−bb−2αByn−Bp2≤xn−p2−xn1−p2cn
≤ xn−xn1xn−pxn1−pcn.
3.43
It is easy to see that limn→ ∞cn0 and then from3.16, we obtain
lim
n→ ∞Byn−Bp0. 3.44
cNext we prove that limn→ ∞xn−un0. From2.3, we have
vn−p2PCyn−λnByn−PCp−λnBp2
≤yn−λnByn−p−λnBp, vn−p
1
2
yn−λnByn−p−λnBp2vn−p2
−yn−λnByn−p−λnBp−vn−p2
≤ 1
2
yn−p2vn−p2−yn−vn−λnByn−Bp2
1
2
yn−p2vn−p2−yn−vn2
2λnyn−vn, Byn−Bp−λ2nByn−Bp2,
so, we obtain
vn−p2≤yn−p2−yn−vn2
2λnyn−vn, Byn−Bp−λ2nByn−Bp2. 3.46
It follows that xn1−p2≤
1−αnγ1−δn−αnγvn−p21−αnγδnxn−p2cn
≤1−αnγ1−δn−αnγ
×yn−p2−yn−vn22λnyn−vn, Byn−Bp− λ2nByn−Bp2
1−αnγδnxn−p2cn
≤1−αnγxn−p2−1−αnγ1−δn−αnγyn−vn2
2λn1−αnγ1−δn−αnγyn−vnByn−Bp
−λ2 n
1−αnγ1−δn−αnγByn−Bp2cn,
3.47
which implies that
1−αnγ1−δn−αnγyn−vn2≤xn−p2−xn1−p2
2λn1−αnγ1−δn−αnγyn−vnByn−Bp
−λ2 n
1−αnγ1−δn−αnγByn−Bp2cn
≤ xn−xn1xn−pxn1−p
2λn1−αnγ1−δn−αnγyn−vnByn−Bp
−λ2
n1−αnγ1−δn−αnγByn−Bp2cn.
3.48
Applying3.16,3.44, lim supn→ ∞δn<1, and limn→ ∞cn0 to the last inequality, we obtain
that
lim
n→ ∞yn−vn0. 3.49
It follows from3.40and3.49that
xn−yn≤ xn−vnvn−yn−→0 asn−→ ∞. 3.50
Then it follows fromD1,3.49and3.50that
xn1−ynαnγfxn−Aynδnxn−yn 1−δnI−αnAvn−yn
For anyp∈Ω, we have fromLemma 2.7,
un−p2 Tr
nxn−Trnp
2≤Tr
nxn−Trnp, xn−p
un−p, xn−p 1
2
un−p2xn−p2− xn−un2
. 3.52
Hence
un−p2≤xn−p2
− xn−un2. 3.53
From3.41we observe that
xn1−p2≤
1−δn−αnγ2vn−p2δn2xn−p2
21−δn−αnγδnvn−pxn−pcn
≤1−δn−αnγ2un−p2δn2xn−p2
21−δn−αnγδnun−pxn−pcn
≤1−δn−αnγ2un−p2δn2xn−p2
1−δn−αnγδnun−p2xn−p2
cn
1−αnγ2−2δn1−αnγδn2un−p2δ2nxn−p2
1−αnγδnun−p2xn−p2
−δ2
nun−p2xn−p2
cn
1−αnγ2−2δn1−αnγδn21−αnγδn−δn2un−p2δ2nxn−p2
1−αnγδnxn−p2−δ2nxn−p2cn
1−αnγ2−δn1−αnγun−p21−αnγδnxn−p2cn
≤1−αnγ1−αnγ−δnxn−p2− xn−un2
1−αnγδnxn−p2cn
1−αnγ2xn−p2−1−αnγ1−αnγ−δnxn−un2cn
1−2αnγαnγ2xn−p2−1−αnγ1−αnγ−δnxn−un2cn
≤xn−p2αnγ2xn−p2−
1−αnγ1−αnγ−δnxn−un2cn.
Hence
1−αnγ1−αnγ−δnxn−un2≤xn−p2−xn1−p2
αnγ2xn−p2cn
xn−p−xn1−pxn−pxn1−p
αnγ2xn−p2cn
≤ xn−xn1xn−pxn1−p
αnγ2xn−p2cn.
3.55
UsingD1,D2and3.16, we obtain
lim
n→ ∞un−xn0. 3.56
dNext we prove that limn→ ∞xn−Txn0. UsingLemma 2.3i, we have
Txn−xn ≤ xn−xn1xn1−ynyn−Txn
≤ xn−xn1xn1−ynβnun−Txn
1−βnTun−Txn
≤ xn−xn1xn1−ynβnun−xnβnxn−Txn
1−βn1k
1−kun−xn,
3.57
which implies that
1−βnTxn−xn ≤ xn−xn1xn1−yn
1k 1−k βn
1− 1k 1−k
un−xn −→0 asn−→ ∞. 3.58
By3.16,3.51, and3.56, we have
lim
n→ ∞Txn−xn0. 3.59
Observing that
xn1−vn ≤αn
γfxn−Avnδnxn−vn
≤αnγfxn−Avnδnxn−vn −→0 asn−→ ∞. 3.60
Using3.40and the last inequality, we obtain that
FromLemma 2.3i,3.59, and3.61, we have
Tvn−vn ≤ Tvn−TxnTxn−xnxn−vn
≤
11k 1−k
vn−xnTxn−xn −→0 as n−→ ∞.
3.62
Hence3.36is proved.
Step 5. We claim that
lim sup
n→ ∞
A−γfz, z−vn≤0. 3.63
We choose a subsequence{vni}of{vn}such that
lim
i→ ∞
A−γfz, z−vni
lim sup
n→ ∞
A−γfz, z−vn. 3.64
Since{vni}is bounded, there exists a subsequence{vnij}of{vni}which converges weakly to
q∈C.
Next, we show thatq∈Ω:FT∩VIB, C∩MEPΘ, ϕ.
aWe first show q ∈ FT. In fact, using Lemma 2.3ii and 3.36, we obtain that
q∈FT.
bNext, we prove q ∈ VIB, C. For this purpose, letS be the maximal monotone mapping defined by2.6:
Sv
⎧ ⎨ ⎩
BvNCv, v∈C;
∅, v /∈C. 3.65
For any givenv, w∈GS, hencew−Bv∈NCv. Sincevn∈C,we have
v−vn, w−Bv ≥0. 3.66
On the other hand, fromvnPCyn−λnByn, we have
v−vn, vn−yn−λnByn≥0 3.67
that is,
v−vn,vnλ−yn n Byn
Therefore, we obtian
v−vni, w ≥ v−vni, Bv ≥ v−vni, Bv −
v−vni,
vni−yni
λni
Byni
v−vni, Bv−Byni−
vni−yni
λni
v−vni, Bv−Bvni
v−vni, Bvni−Byni
−
v−vni,
vni −yni
λni
≥ v−vni, Bvni −
v−vni,
vni−yni
λni
Byni
v−vni, Bvni−Byni
−
v−vni,
vni−yni
λni
.
3.69
Noting thatvni−yni → 0 asi → ∞andBis Lipschitz continuous, hence from3.69, we
obtain
v−q, w≥0. 3.70
SinceSis maximal monotone, we haveq∈S−10, and henceq∈VIB, C.
cWe showq∈MEPΘ, ϕ. In fact, byunTrnxn∈domϕ, and we have,
Θun, yϕy−ϕun 1 rn
y−un, un−xn≥0, ∀y∈C. 3.71
FromA2, we also have
ϕy−ϕun rn1y−un, un−xn≥Θy, un, ∀y∈C, 3.72
and hence
ϕy−ϕun
y−uni,
uni−xni
rni
≥Θy, uni
Fromun−xn → 0, xn−Tvn → 0,andTvn−vn → 0,we getuni q. It follows from A4,uni −xni/rni → 0, and the lower semicontinuous ofϕthat
Θy, zϕq−ϕy≤0 ∀y∈C. 3.74
Fortwith 0< t≤1 andy∈C,letytty 1−tq.Sincey∈Candq∈C,we haveyt∈Cand henceΘyt, q ϕq−ϕyt≤0.So, fromA1andA4and the convexity ofϕ, we have
0 Θyt, ytϕyt−ϕyt
≤tΘyt, y 1−tΘyt, qtϕy 1−tϕq−ϕyt
≤tΘyt, yϕy−ϕyt.
3.75
Dividing byt, we have
Θyt, yϕy−ϕyt≥0, ∀y∈C. 3.76
Lettingt → 0, it follows from the weakly semicontinuity ofϕthat
Θq, yϕy−ϕq≥0, ∀y∈C. 3.77
Henceq ∈ MEPΘ, ϕ. Therefore, the conclusionq ∈ Ω : FT∩VIB, C∩MEPΘ, ϕis proved.
Consequently
lim sup
n→ ∞
A−γfz, z−vn lim
i→ ∞
A−γfz, z−vni
A−γfz, z−q≤0 3.78
as required. This together with3.40implies that
lim sup
n→ ∞
γfz−Az, xn−zlim sup
n→ ∞
γfz−Az,xn−vn vn−z
≤lim sup
n→ ∞
γfz−Az, vn−z
≤0.
Step 6. Finally, we show thatxn → z, yn → z, un → z. Indeed, we note that
xn1−z2αnγfxn δnxn 1−δnI−αnAvn−z2
1−δnI−αnAvn−z δnxn−z αnγfxn−Az2
1−δnI−αnAvn−z δnxn−z2α2nγfxn−Az2
2δnαnxn−z, γfxn−Az
2αn1−δnI−αnAvn−z, γfxn−Az
≤1−δn−αnγvn−zδnxn−z2α2nγfxn−Az2
2δnαnγxn−z, fxn−fz2δnαnxn−z, γfz−Az
21−δnγαnvn−z, fxn−fz21−δnαnvn−z, γfz−Az
−2α2nAvn−z, γfz−Az
≤1−δn−αnγxn−zδnxn−z2α2nγfxn−Az2
2δnαnγαxn−z22δnαnxn−z, γfq−Az
21−δnγαnαxn−z221−δnαnvn−z, γfz−Az
−2α2nAvn−z, γfq−Az
1−αnγ22δnαnγα21−δnγαnα
xn−z2αn2γfxn−Az2
2δnαnxn−z, γfz−Az21−δnαnvn−z, γfz−Az
−2α2
nAvn−z, γfz−Az
≤1−2γ−αnγαnxn−z2γ2α2nxn−z2α2nγfxn−Az2
2δnαnxn−z, γfz−Az21−δnαnvn−z, γfz−Az
2α2
nAvn−zγfz−Az
1−2γ−αnγαnxn−z2
αnαnγ2xn−z2γfxn−Az2
2Avn−zγfz−Az2δnxn−z, γfz−Az
21−δnvn−z, γfz−Az.
3.80
Since{xn},{fxn},and{vn}are bounded, we can take a constantK >0 such that
γ2xn−z2γfxn−Az2
for alln≥0. It then follows that
xn1−z2≤
1−2γ−αnγαnxn−z2αnσn, 3.82
where
σn2δnxn−z, γfz−Az21−δnvn−z, γfz−AzαnK 3.83
UsingD1, and 3.79, we get lim supn→ ∞δn ≤ 0. Now applyingLemma 2.1to3.82, we
conclude thatxn → z. Fromxn−yn → 0 andxn−un → 0, we obtainyn → z, un → z. The proof is now complete.
By Theorem 3.1, we can obtain some new and interesting strong convergence theorems. Now we give some examples as follows.
Settingϕ0 inTheorem 3.1, we have the following result.
Corollary 3.2. Let C be a nonempty closed convex subset of a Hilbert space H. LetΘbe a bifunction from C ×C to R satifies (A1)–(A4). Let T be a k-strictly pseudocontractive mapping of C into itself. Let f be a contraction of C into itself with coefficient β ∈ 0,1,B an α-inverse-strongly monotone mapping ofCintoHsuch thatΩ : FT∩VIB, C∩EPΘ/∅. LetAbe a strongly bounded linear self-adjoint operator with coefficient γ > 0 and0 < γ < γ/β. Given the sequences
{αn}, {βn}, {δn}, {λn},and{rn}in0,1satisfies the following conditions
D1limn→ ∞αn0, ∞n1αn∞;
D20<lim infn→ ∞δn≤lim supn→ ∞δn<1;
D30≤k≤βn< ε <1for alln≥0,andlimn→ ∞|βn1−βn|0;
D4{λn} ⊂a, bfor somea, bwith0< a < b <2α,andlimn→ ∞|λn1−λn|0
D5lim infn→ ∞rn>0,limn→ ∞|rn1−rn|0. Let{xn}, {un},and{yn}be sequences generated by
x1x∈Cchosen arbitrarily,
Θun, yrn1y−un, un−xn≥0, ∀y∈C,
ynβnun1−βnTun,
xn1αnγfxn δnxn 1−δnI−αnAPCyn−λnByn, n≥1.
3.84
Then{xn},{un}and {yn} converge strongly to a pointz ∈ Ωwhich is the unique solution of the variational inequality
A−γfz, z−x≤0, ∀x∈Ω. 3.85
SettingΘ 0, rn 1 andϕ 0 inTheorem 3.1, we havexn un, then the following result is obtained.
Corollary 3.3. Let C be a nonempty closed convex subset of a Hilbert space H. LetT be ak-strictly pseudocontractive mapping ofCinto itself. Letfbe a contraction ofCinto itself with coefficientβ∈
0,1,Banα-inverse-strongly monotone mapping ofCintoHsuch thatΩ:FT∩VIB, C/∅. LetAbe a strongly bounded linear self-adjoint operator with coefficientγ >0and0< γ < γ/β. Given
the sequences{αn}, {βn}, {δn}and{λn}in0,1satifies the following conditions
D1limn→ ∞αn0, ∞n1αn∞;
D20<lim infn→ ∞δn≤lim supn→ ∞δn<1;
D30≤k≤βn< ε <1for alln≥0,andlimn→ ∞|βn1−βn|0;
D4{λn} ⊂a, bfor somea, bwith0< a < b <2αandlimn→ ∞|λn1−λn|0.
Let{xn}and{yn}be sequences generated by
x1x∈Cchosen arbitrarily,
yn βnxn1−βnTxn,
xn1αnγfxn δnxn 1−δnI−αnAPC
yn−λnByn, n≥1.
3.86
Then{xn}and{yn}converge strongly to a pointz∈Ωwhich is the unique solution of the variational inequality
A−γfz, z−x≤0, ∀x∈Ω. 3.87
Equivalently, one haszPΩI−Aγfz.
Remark 3.4. i Since the conditionsC1and C2have been weakened by the conditions
D1 and D3 respectively. Theorem 3.1 and Corollary 3.2 generalize and improve 44, Theorem 3.2.
iiWe can remove the control condition limn→ ∞βnεon the parameter{βn}inC2.
iiiSince the conditionsC1andC2have been weakened by the conditionsD1 andD3respectively.Theorem 3.1andCorollary 3.3generalize and improve43, Theorem 2.1.
Setting ϕ 0, βn 0, B 0 and T is nonexpansive in Theorem 3.1, we have the following result.
Corollary 3.5. Let C be a nonempty closed convex subset of a Hilbert space H. LetΘbe a bifunction fromC×CtoRsatifies (A1)–(A4). LetT be a nonexpansive mapping ofC into itself. Letf be a contraction ofCinto itself with coefficient β ∈ 0,1such thatΩ : FT∩EPΘ/∅. LetAbe a strongly bounded linear self-adjoint operator with coefficient γ > 0and 0 < γ < γ/β. Given the
sequences{αn},{δn},and{rn}in0,1satifies the following conditions
D1limn→ ∞αn0, ∞n1αn∞;
D20<lim infn→ ∞δn≤lim supn→ ∞δn<1;
Let{xn},{un},and{yn}be sequences generated by
x1x∈Cchosen arbitrarily,
Θun, yrn1y−un, un−xn≥0, ∀y∈C,
xn1 αnγfxn δnxn 1−δnI−αnATun, n≥1.
3.88
Then{xn}, {un}and{yn}converge strongly to a pointz ∈ Ωwhich is the unique solution of the variational inequality
A−γfz, z−x≤0, ∀x∈Ω. 3.89
Equivalently, one haszPΩI−Aγfz.
Remark 3.6. Since the conditions∞n1|αn1 −αn| < ∞ and
∞
n1|rn1 −rn| < ∞have been
weakened by the conditions limn→ ∞|αn1−αn| 0 and limn→ ∞|rn1−rn| 0, respectively.
HenceCorollary 3.5generalize, extend and improve17, Theorem 3.3.
4. Applications
First, we will utilize the results presented in this paper to study the following optimization problem:
min
y∈C ϕ
y, 4.1
whereCis a nonempty bounded closed convex subset of a Hilbert space andϕ :C → R∪
{∞}is a proper convex and lower semicontinuous function. We denote by Argminϕthe set of solutions in 4.1. Let Θx, y 0 for all x, y ∈ C,γ ≡ 1, A ≡ I, T I andf : x inTheorem 3.1, then MEPΘ, ϕ Argminϕ. It follows fromTheorem 3.1that the iterative sequence{xn}is defined by
x1x∈Cchosen arbitrarily,
unargmin y∈C
ϕy 1
2rny−xn
2,
xn1αnxδnxn 1−δn−αnPCun−λnBun, n≥1,
4.2
LetΘx, y 0 for allx, y∈C,T I,γ ≡1, A≡I, f :xandB≡0 inTheorem 3.1, then MEPΘ, ϕ Argminϕ. It follows fromTheorem 3.1that the iterative sequence{xn}
defined by
x1x∈Cchosen arbitrarily,
unargmin
y∈C
ϕy 1
2rn
y−xn2,
xn1αnxδnxn 1−δn−αnun, ∀n≥1,
4.3
where {αn},{δn} ⊆ 0,1, and {rn} ⊆ 0,∞ satisfy the conditions D1, D2 and D5, respectively in Theorem 3.1. Then the sequence {xn} converges strongly to a solutionz PArgminϕx.
We remark that the algorithms4.2and4.3are variants of the proximal method for optimization problems introduced and studied by Martinet49, Rockafellar45, Ferris50
and many others.
Next, we give the strong convergence theorem for finding a common element of the set of common fixed point of a finite family of strictly pseudocontractive mappings, the set of solutions of the variational inequality problem and the set of solutions of the mixed equilibrium problem in a Hilbert space.
Theorem 4.1. Let C be a nonempty closed convex subset of a Hilbert space H. LetΘbe a bifunction fromC×CtoR satifies (A1)–(A4) and ϕ : C → R∪ {∞}be a proper lower semicontinuous and convex function. For each i 1,2, . . . , N,letTi be aki-strictly pseudocontractive mapping of Cinto itself for some0 ≤ ki < 1. Letf be a contraction ofCinto itself with coefficientβ ∈ 0,1, Banα−inverse-strongly monotone mapping of CintoH such thatΩ : iN1FTi∩VIB, C∩ MEPΘ, ϕ/∅. LetAbe a strongly bounded linear self-adjoint operator with coefficient γ > 0and
0 < γ < γ/β. Assume that either (B1) or (B2) holds. Given the sequences{αn}, {βn}, {δn}, {λn}
and{rn}in0,1satifies the following conditions
D1limn→ ∞αn0, ∞n1αn∞;
D20<lim infn→ ∞δn≤lim supn→ ∞δn<1;
D30≤max{ki:i1,2, . . . , N} ≤βn< β <1for alln≥0,andlimn→ ∞|βn1−βn|0;
D4{λn} ⊂a, bfor somea, bwith0< a < b <2αandlimn→ ∞|λn1−λn|0;
D5lim infn→ ∞rn>0, limn→ ∞|rn1−rn|0.
Let{xn}, {un}and{yn}be sequences generated by
x0x∈Cchosen arbitrarily,
Θun, yϕy−ϕun 1 rn
y−un, un−xn≥0, ∀y∈C,
ynβnun1−βn
N
i1 ηiTiun,
xn1αnγfxn δnxn 1−δnI−αnAPC
yn−λnByn, n≥1,
where ηi is a positive constant such thatη1 η2· · ·ηN 1.Then both{xn},{un}and {yn} converge strongly to a pointz∈Ωwhich is the unique solution of the variational inequality
A−γfz, z−x≤0, x∈Ω. 4.5
Equivalently, one haszPΩI−Aγfz.
Proof. Let{ηi}Ni1⊂0,1such that
N
i1ηi1 and defineTx
N
i1ηiTix. By Lemmas2.5and 2.6, we conclude thatT :C → Cis ak-strictly pseudocontractive mapping withkmax{ki: 1≤i≤N}andFT FNi1ηiTi Ni1FTi. FromTheorem 3.1, we can obtain the desired conclusion easily.
Finally, we will apply the main results to the problem for finding a common element of the set of fixed points of two finite families ofk-strictly pseudocontractive mappings, the set of solutions of the variational inequality and the set of solutions of the mixed equilibrium problem.
LetSi : C → Hbe aki-strictly pseudocontractive mapping for some 0≤ ki < 1. We define a mappingB I−Ni1ξiSi :C → H where{ξi}Ni1 is a positive sequence such that N
i1ξi1, thenBis a1−k/2-inverse-strongly monotone mapping withk max{ki: 1 ≤ i≤N}. In fact, fromLemma 2.5, we have
N
i1
ξiSix−N
i1 ξiSiy
2
≤x−y2k
I−N
i1
ξiSi x−
I−N
i1
ξiSi y 2
, ∀x, y∈C.
4.6
That is
I−Bx−I−By2≤x−y2kBx−By2. 4.7
On the other hand
I−Bx−I−By2x−y2−
2x−y, Bx−ByBx−By2. 4.8
Hence we have
x−y, Bx−By≥ 1−k
2 Bx−By
2. 4.9
This shows thatBis1−k/2-inverse-strongly monotone.
operator with coefficientγ > 0and0 < γ < γ/β. Assume that either (B1) or (B2) holds. Given the
sequences{αn}, {βn}, {δn}, {λn}and{rn}in0,1satifies the following conditions
D1limn→ ∞αn0, ∞n1αn∞;
D20<lim infn→ ∞δn≤lim supn→ ∞δn<1;
D30 ≤ max1≤i≤NkTi ≤ βn < β < 1 and0 ≤ max1≤i≤NkiS ≤ βn < β < 1for alln ≥ 0,and
limn→ ∞|βn1−βn|0;
D4{λn} ⊂a, bfor somea, bwith0< a < b <2αandlimn→ ∞|λn1−λn|0;
D5lim infn→ ∞rn>0, limn→ ∞|rn1−rn|0. Let{xn},{un}and{yn}be sequences generated by
x1x∈Cchosen arbitrarily,
Θun, yϕy−ϕun 1 rn
y−un, un−xn≥0, ∀y∈C,
ynβnun1−βn N
i1 ηiTiun,
xn1 αnγfxn δnxn 1−δnI−αnAPC
1−λnyn−λn
N
i1
ξiSiyn , n≥1,
4.10
where ηi and ξi are positive constants such that Ni1ηi 1 and Ni1ξi 1, respectively. Then
{xn}, {un}and{yn}converge strongly to a pointz∈Ωwhich is the unique solution of the variational inequality
A−γfz, z−x≤0, x∈Ω. 4.11
Equivalently, we havezPΩI−Aγfz.
Proof. TakingB I−Ni1ξiSi : C → H inTheorem 4.1, we know thatB : C → Hisα
-inverse strongly monotone withα 1−k/2. Hence,Bis a monotoneL-Lipschitz continuous mapping with L 2/1 −kT. From Lemma 2.6, we know that Ni1ξiSi is a kT-strictly pseudocontractive mapping withkT max{kTi : 1≤i≤N}and thenFNi1ξiSi VIB, C
byLemma 2.6. Observe that
PCyn−λnBynPC
1−λnyn−λn
N
i1
ξiSiyn . 4.12
The conclusion can be obtained fromTheorem 4.1.
