Volume 2010, Article ID 756492,22pages doi:10.1155/2010/756492
Research Article
Strong Convergence for Mixed Equilibrium
Problems of Infinitely Nonexpansive Mappings
Jintana Joomwong
Division of Mathematics, Faculty of Science, Maejo University, Chiang Mai 50290, Thailand Correspondence should be addressed to Jintana Joomwong,[email protected]
Received 29 March 2010; Accepted 24 May 2010
Academic Editor: Tomonari Suzuki
Copyrightq2010 Jintana Joomwong. 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.
We introduce a new iterative scheme for finding a common element of infinitely nonexpansive mappings, the set of solutions of a mixed equilibrium problems, and the set of solutions of the variational inequality for anα-inverse-strongly monotone mapping in a Hilbert Space. Then, the strong converge theorem is proved under some parameter controlling conditions. The results of this paper extend and improve the results of Jing Zhao and Songnian He2009and many others. Using this theorem, we obtain some interesting corollaries.
1. Introduction
LetHbe a real Hilbert space with norm·and inner product·,·. And letCbe a nonempty closed convex subset ofH. Letϕ :C → Rbe a real-valued function and letΘ:C×C → R be an equilibrium bifunction, that is,Θu, u 0 for eachu∈C. Ceng and Yao1considered the following mixed equilibrium problem.
Findx∗∈Csuch that
Θx∗, yϕy−ϕx∗≥0, ∀y∈C. 1.1
The set of solutions of1.1is denoted by MEPΘ, ϕ.It is easy to see thatx∗is the solution of problem 1.1and x∗ ∈ domϕ {x ∈ ϕx < ∞}. In particular, ifϕ ≡ 0, the mixed equilibrium problem1.1reduced to the equilibrium problem.
Findx∗∈Csuch that
The set of solutions of1.2 is denoted by EPΘ. Ifϕ ≡ 0 and Θx, y Ax, y−x for allx, y ∈C, whereAis a mapping fromCtoH, then the mixed equilibrium problem1.1 becomes the following variational inequality.
Findx∗∈Csuch that
Ax∗, y−x∗, ∀y∈C. 1.3
The set of solutions of1.3is denoted by VIA, C.
The variational inequality and the mixed equilibrium problems which include fixed point problems, optimization problems, variational inequality problems have been extensively studied in literature. See, for example,2–8.
In 1997, Combettes and Hirstoaga 9 introduced an iterative method for finding the best approximation to the initial data and proved a strong convergence theorem. Subsequently, Takahashi and Takahashi7introduced another iterative scheme for finding a common element of EPΘ and the set of fixed points of nonexpansive mappings. Furthermore,Yao et al.8,10introduced an iterative scheme for finding a common element of EPΘand the set of fixed points of finitelyinfinitelynonexpansive mappings.
Very recently, Ceng and Yao 1 considered a new iterative scheme for finding a common element of MEPΘ, ϕ and the set of common fixed points of finitely many nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem.
Now, we recall that a mappingA:C → His said to be
imonotone ifAu−Av, u−v ≥0,for allu, v∈C,
iiL-Lipschitz if there exists a constant L > 0 such that Au − Av ≤ Lu −
v,for allu, v∈C,
iiiα-inverse strongly monotone if there exists a positive real numberαsuch thatAu−
Av, u−v ≥αAu−Av2,for allu, v∈C.
It is obvious that anyα-inverse strongly monotone mappingAis monotone and Lipscitz. A mappingS:C → Cis called nonexpansive ifSu−Sv ≤ u−v,for allu, v∈C.We denote byFS:{x∈C:Sxx}the set of fixed point ofS.
In 2006, Yao and Yao11introduced the following iterative scheme.
LetCbe a closed convex subset of a real Hilbert space. LetAbe anα-inverse strongly monotone mapping ofCintoHand letSbe a nonexpansive mapping ofCinto itself such thatFS∩VIA, C/∅. Suppose thatx1u∈Cand{xn}and{yn}are given by
ynPCxn−λnAxn,
xn1 αnuβnxnγnSPCyn−λnAyn,
1.4
where{αn},{βn}, and{γn}are sequence in0,1and{λn}is a sequence in0,2λ. They proved
that the sequence{xn}defined by1.4converges strongly to a common element ofFS∩
VIA, Cunder some parameter controlling conditions.
problem for anα-inverse strongly monotone mapping in a real Hilbert space. Suppose that
x1u∈Cand{xn},{yn}, and{un}are given by
Θun, yr1
n
y−un, un−xn≥0, ∀y∈C,
ynPCun−λnAun,
xn1 αnuβnxnγnSPCyn−λnAyn,
1.5
where{αn},{βn}, and{γn}are sequence in 0,1,{λn}is a sequence in0,2λ, and {rn} ⊂ 0,∞. Under some parameter controlling conditions, they proved that the sequence {xn}
defined by1.5converges strongly toPFS∩VIA,C∩EPΘu.
On the other hand, Yao et al. 8introduced an iterative scheme 1.7for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed point of infinitely many nonexpansive mappings inH. Let {Tn}∞n1 be a sequence of nonexpansive mappings ofCinto itself and let{tn}n∞1be a sequence of real number in0,1. For eachn≥1, define a mappingWnofCinto itself as follows:
Un,n1I,
Un,ntnTnUn,n1 1−tnI,
Un,n−1tn−1Tn−1Un,n 1−tn−1I,
.. .
Un,ktkTkUn,k1 1−tkI,
Un,k−1tk−1Tk−1Un,k 1−tk−1I,
.. .
Un,2t2T2Un,3 1−t2I,
WnUn,1t1T1Un,2 1−t1I.
1.6
Such a mappingWnis called theW-mapping generated byTn, Tn−1, . . . , T1andtn, tn−1, . . . , t1. In8, givenx0∈Harbitrarily, the sequences{xn}and{un}are generated by
Θun, x r1
nx−un, un−xn ≥0, ∀x∈C,
xn1 αnfxn βnxnγnWnun.
1.7
They proved that under some parameter controlling conditions, {xn} generated by 1.7
converges strongly toz∈ ∩∞n1FTn∩EPΘ, wherezP∩∞
Subsequently, Ceng and Yao 13 introduced an iterative scheme by the viscosity approximation method:
Θun, x r1
nx−un, un−xn ≥0, ∀x∈C,
yn1−γnxnγnWnun,
xn1
1−αn−βnxnαnfynβnWnyn,
1.8
where{αn},{βn}and{γn}are sequence in0,1such thatαnβn≤1. Under some parameter controlling conditions, they proved that the sequence {xn} defined by 1.8 converges
strongly toz∈ ∩∞n1FTn∩EPΘ, wherezP∩∞
n1FTn∩EPΘfz.
Recently, Zhao and He14introduced the following iterative process. Suppose thatx1u∈C,
Θun, yr1
n
y−un, un−xn≥0, ∀y∈C,
yn snPCun−λnAun 1−snxn,
xn1αnuβnxnγnWnPCyn−λnAyn,
1.9
where{sn},{αn},{βn}, and{γn} ∈ 0,1such thatαnβnγn 1. Under some parameter
controlling conditions, they proved that the sequence {xn} defined by 1.9 converges
strongly toz∈ ∩∞i1FTi∩VIA, C∩EPΘ, wherezP∩∞
i1FTi∩VIA,C∩EPΘu.
Motivated by the ongoing research in this field, in this paper we suggest and analyze an iterative scheme for finding a common element of the set of fixed point of infinitely nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational of inequality problem for an α-inverse strongly monotone mapping in a real Hilbert space. Under some appropriate conditions imposed on the parameters, we prove another strong convergence theorem and show that the approximate solution converges to a unique solution of some variational inequality which is the optimality condition for the minimization problem. The results of this paper extend and improve the results of Zhao and He14and many others. For some related works, we refer the readers to15–22and the references therein.
2. Preliminaries
LetHbe a real Hilbert space and letCbe a closed convex subset ofH. Then, for anyx∈H, there exists a unique nearest point inC, denoted byPCxsuch that
x−PCx ≤ x−y, ∀y∈C. 2.1
PC is called the metric projection of H onto C. It is well known that PC is nonexpansive
mapping and satisfies
Moreover,PCis characterized by the following properties:Pcx∈Cand
x−PCx, y−PCx ≤0,
x−y2
≥ x−PCx2y−PCx2, ∀x∈H, y∈C.
2.3
It is clear thatu∈VIA, C⇔uPCu−λAu, λ >0.
A spaceX is said to satisfy Opials condition if for each sequence {xn} inX which
converges weakly to a pointx∈X, we have
lim inf
n→ ∞ xn−x<lim infn→ ∞ xn−y, ∀y∈X, y /x. 2.4
The following lemmas will be useful for proving the convergence result of this paper.
Lemma 2.1see23. Let{xn}and{yn}be bounded sequences in a Banach spaceXand let{βn}be a sequence in0,1with0<lim infn→ ∞βnlim supn→ ∞βn<1.Suppose thatxn1 1−βnyn
βnxnfor all integern≥1andlim supn→ ∞yn1−yn−xn1−xn0.Thenlimn→ ∞yn−xn
0.
Lemma 2.2see24. LetHbe a real Hilbert space, letCbe a closed convex subset ofH, and let
T :C → Cbe a nonexpansive mapping withFT/∅.If{xn}is a sequence inCweakly converging toxand ifI−Txnconverge strongly toy, thenI−Txy.
Lemma 2.3see25. Assume that{an}is a sequence of nonnegative real numbers such that
an1≤1−αnanδn, n≥0, 2.5
where{αn}is a sequence in0,1and{δn}is a sequence inRsuch that
1limn→ ∞αn0and∞n1αn∞.
2lim supn→ ∞δn/αn≤0or ∞n1|δn|<∞.
Thenlimn→ ∞an0.
In this paper, 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 anyx, y∈C;
A3 Θis upper-hemicontinuous, that is, for eachx, y, z∈C,
lim
t→0supΘ
tz 1−tx, y≤Θx, y; 2.6
B1for eachx ∈Handr >0, there exists a bounded subsetDx ⊂Candyx ∈ Csuch
that for anyz∈C\Dx,
Θz, yϕyxr1
n
yx−z, z−x< ϕz, 2.7
B2Cis a bounded set.
By a similar argument as in the proof of Lemma2.3in26, we have the following result.
Lemma 2.4. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetΘbe a bifunction fromC×C → Rthat satisfies (A1)–(A4) and letϕ:C → R∪{∞}be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. Forr >0andx∈H, define a mapping
Tr :H → Cas follows:
Trx
z∈C:Θz, yϕy1
r
y−z, z−x≥ϕz,∀y∈C 2.8
for allx∈H.Then, the following conditions hold:
1for eachx∈H, Trx/∅; 2Tr is single-valued;
3Tr is firmly nonexpansive, that is, for anyx, y∈H,Trx−Try2 ≤ Trx−Try, x−y;
4FTr MEPΘ, ϕ;
5MEPΘ, ϕis closed and convex.
Let {Tn}∞n1 be a sequence of nonexpansive mappings of Cinto itself, where C is a nonempty closed convex subset of a real Hilbert spaceH. Given a sequence{tn}n∞1in0,1, we define a sequence{Wn}∞n1 of self-mappings onCby1.6. Then We have the following result.
Lemma 2.5see27. LetC be a nonempty closed convex subset of a real Hilbert spaceH. Let
{Tn}∞n1be a sequence of nonexpansive self-mappings onCsuch that∩∞n1FTn/∅and let{tn}be a
sequence in0, bfor someb∈0,1. Then, for everyx∈Candk≥1,limn→ ∞Un,kxexists.
Remark 2.6see8. It can be shown from Lemma2.5that ifDis a nonempty bounded subset ofC, then for > 0, there existsn0 ≥ ksuch that for alln > n0, supx∈DUn,kx−Ukx ≤ ,
whereUkxlimn→ ∞Un,kx.
Remark 2.7see8. Using Lemma2.5, we define a mappingW :C → Cas follows:Wx limn→ ∞Wnxlimn→ ∞Un,1x, for allx∈C.Wis called theW-mapping generated byT1, T2, . . . andt1, t2, . . . .
SinceWnis nonexpansive,W:C → Cis also nonexpansive.
Indeed, for allx, y∈C,Wx−Wylimn→ ∞Wnx−Wny ≤ x−y.
If{xn}is a bounded sequence inC, then we putD {xn :n ≥ 0}. Hence it is clear
from Remark 2.6that for any arbitrary > 0, there exists n0 ≥ 1 such that for all n > n0,
This implies that limn→ ∞Wnxn−Wxn0.
Lemma 2.8see27. LetC be a nonempty closed convex subset of a real Hilbert spaceH. Let
{Tn}∞n1be a sequence of nonexpansive self-mappings onCsuch that∩∞n1FTn/∅and let{tn}be a
sequence in0, bfor someb∈0,1. ThenFW ∩∞n1FTn.
3. Main Results
Theorem 3.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Letϕ : C →
R∪ {∞}be a lower semicontinuous and convex function. LetΘbe a bifunction fromC×C → R satisfying (A1)–(A4), letAbe anα-inverse-strongly monotone mapping ofCintoH, and let{Tn}∞n1 be a sequence of nonexpansive self-mapping onCsuch that∩∞n1FTn∩VIA, C∩MEPΘ, ϕ/∅. Suppose that{sn},{αn},{βn}, and{γn}are sequences in0,1,{λn}is a sequence in0,2αsuch that
λn ∈a, bfor somea, bwith0< a < b <2α, and{rn} ⊂0,∞is a real sequence. Suppose that the following conditions are satisfied:
iαnβnγn1,
iilimn→ ∞αn0and∞n1αn∞,
iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1,
iv0<lim infn→ ∞sn≤lim supn→ ∞sn<1/2andlimn→ ∞|sn1−sn|0,
vlimn→ ∞|λn1−λn|0,
vilim infn→ ∞rn>0andlimn→ ∞|rn1−rn|0.
Letfbe a contraction ofCinto itself with coefficientβ∈0,1. Assume that either (B1) or (B2) holds.
Let the sequences{xn},{un}, and{yn}be generated by,x1∈Cand
Θun, yϕy−ϕun r1
n
y−un, un−xn≥0, ∀y∈C,
yn snPCun−λnAun 1−snxn,
xn1αnfxn βnxnγnWnPCyn−λnAyn,
3.1
for alln∈N, whereWnis defined by1.6and{tn}is a sequence in0, b, for someb∈0,1. Then the sequence{xn}converges strongly to a point x∗ ∈ ∩∞n1FTn∩VIA, C∩MEPΘ, ϕ, where
x∗P ∩∞
n1FTn∩VIA,C∩MEPΘ,ϕfx
∗.
Proof. For anyx, y∈Candλn ∈a, b⊂0,2α, we note that
I−λnAx−I−λnAy2x−y−λ
nAx−Ay2
x−y2−
2λnx−y, Ax−Ayλ2nAx−Ay2
≤x−y2λnλ
n−2αAx−Ay2
≤x−y2,
3.2
Let{Trn}be a sequence of mappping defined as in Lemma2.4and letx∗∈ ∩∞n1FTn∩ VIA, C∩MEPΘ, ϕ. Thenx∗ Wnx∗andx∗ PCx∗−λnAx∗ Trnx∗. Putvn PCyn−
λnAyn. From3.2we have
vn−x∗PCyn−λnAyn−PCx∗−λnAx∗
≤ yn−λnAyn−x∗−λnAx∗
≤ yn−x∗
snPCun−λnAun 1−snxn−snPCx∗−λnAx∗−1−snx∗
≤snPCun−λnAun−PCx∗−λnAx∗ 1−snxn−x∗
≤snun−x∗ 1−snxn−x∗
snTrnxn−Trnx∗ 1−snxn−x∗
≤snxn−x∗ 1−snxn−x∗
xn−x∗.
3.3
Hence, we obtain that
xn1−x∗αnfxn−βnxn−γnWnvn−x∗
≤αnfxn−x∗βnxn−x∗γnWnvn−x∗
≤αnfxn−fx∗αnfx∗−x∗βnxn−x∗γnvn−x∗
≤αnβxn−x∗αnfx∗−x∗βnxn−x∗γnxn−x∗
1−βαnfx
∗−x∗
1−β
1−1−βαnxn−x∗
≤max
xn−x∗,fx
∗−x∗
1−β
≤max
x0−x∗,fx
∗−x∗
1−β .
3.4
Therefore {xn} is bounded. Consequently, {fxn},{un},{yn},{vn},{Wnvn},{Aun}, and
{Ayn}are also bounded.
Indeed, settingxn1βnxn 1−βnzn,for alln≥1,it follows that
zn1−zn αn1fxn11 −βγn1Wn1vn1
n1 −
αnfxn γnWnvn
1−βn
αn1fxn1 γn1Wn1vn1
1−βn1 −
γn1Wn1vn
1−βn1
γn1Wn1vn
1−βn1 −
αnfxn γnWnvn
1−βn
αn1fxn1
1−βn1 −
αnfxn
1−βn
γn1
1−βn1Wn1vn1−Wn1vn
1−βn1−αn1
1−βn1 Wn1vn−
1−βn−αn
1−βn Wnvn
αn1fxn1
1−βn1 −
αnfxn
1−βn
γn1
1−βn1Wn1vn1−Wn1vn
wn1vn−wnvn 1−αnβ
nWnvn−
αn1
1−βn1Wn1vn.
3.5
Now, we estimateWn1vn−WnvnandWn1vn1−Wn1vn.
From the definition of{Wn},1.6, and sinceTi,Un,iare nonexpansive, we deduce that,
for eachn≥1,
Wn1vn−Wnvnt1T1Un1,2vn−t1T1Un,2vn
≤t1Un1,2vn−Un,2vn
t1t2T2Un1,3vn−t2T2Un,3vn
≤t1t2Un1,3vn−Un,3vn
.. .
≤
n
i1
ti
Un1,n1vn−Un,n1vn
≤Mn
i1
ti,
3.6
we note that
Wn1vn1−Wn1vn ≤ vn1−vn
PCyn1−λn1Ayn1
−PCyn−λnAyn
≤ yn1−λn1Ayn1
−yn−λnAyn
≤ I−λn1Ayn1−I−λn1Ayn|λn−λn1|Ayn
≤ yn1−yn|λn−λn1|Ayn,
3.7
yn1−ynsn1PCun1−λn1Aun1 1−sn1xn1
−snPCun−λnAun−1−snxn
sn1PCun1−λn1Aun1−sn1PCun−λnAun
sn1−snPCun−λnAun 1−sn1xn1
−1−sn1sn1−snxn
≤sn1un1−λn1Aun1−un−λnAun
|sn1−sn|un−λnAun 1−sn1xn1−xn|sn1−sn|xn
≤sn1{un1−λn1Aun1−un−λnAun
|λn−λn1|Aun}|sn1−sn|unλnAunxn
1−sn1xn1−xn
≤sn1un1−unsn1|λn−λn1|Aun
|sn1−sn|Q 1−sn1xn1−xn,
3.8
whereQsup{un, λnAun,xn:n≥1}.
Combining3.7and3.8, we obtain
vn1−vn ≤sn1un1−unsn1|λn−λn1|Aun|sn1−sn|Q
1−sn1xn1−xn|λn−λn1|Ayn.
3.9
On the other hand, fromunTrnxnandun1Trn1xn1, we note that
Θun, yϕy−ϕun r1
n
y−un, un−xn≥0, ∀y∈C, 3.10
Θun1, y
ϕy−ϕun1 r1
n1
y−un1, un1−xn1
Puttingyun1in3.10andyunin3.11, we have
Θun, un1 ϕun1−ϕun r1
nun1−un, un−xn ≥0,
Θun1, un ϕun−ϕun1
1
rn1un−un1, un1−xn1 ≥ 0.
3.12
So, fromA2we getun1−un,un−xn/rn−un1−xn1/rn1 ≥0. Henceun1−un, un−un1un1−xn−rn/rn1un1−xn1 ≥0.
Without loss of generality, we may assume that there exists a real numbercsuch that
rn> c >0, for alln≥1. Then we get
un1−un2≤
un1−un, xn1−xn
1− rn
rn1
un1−xn1
≤ un1−un
xn1−xn1−rrn n1
un1−xn1 ,
3.13
and hence
un1−un ≤ xn1−xnr1
n1|rn1−rn|un1−xn1
≤ xn1−xn 1
c|rn1−rn|L,
3.14
whereLsup{un−xn:n≥1}. Hence from3.9and3.14, we have
Wn1vn1−Wn1vn ≤ xn1−xnsn1
L
c|rn1−rn||λn−λn1|Aun
|sn1−sn|Q|λn−λn1|Ayn.
3.15
Combining3.5,3.6, and3.15, we get
zn1−zn − xn1−xn ≤ 1−αnβ1 n1
fxn1Wn1vn1−αnβ n
fxnWnvn
γn1
1−βn1
xn1−xnsn1
L
c|rn1−rn||λn−λn1|Aun
|sn1−sn|Q|λn−λn1|Ayn
Mn
i1
≤ αn1 1−βn1
fxn1Wn1vn
αn
1−βn
fxnWnvn
γn1
1−βn1
sn1
L
c|rn1−rn||λn−λn1|Aun
|sn1−sn|Q|λn−λn1|Ayn M
n
i1
ti.
3.16
It follows from3.16and conditionsi–viand 0< ti≤b <1,for alli≥1 that
lim sup
n→ ∞ zn1−zn − xn1−xn≤0. 3.17
By Lemma2.1, we have limn→ ∞zn−xn0.Consequently,
lim
n→ ∞xn1−xnnlim→ ∞
1−βnzn−xn0. 3.18
From conditionsiv–vi,3.7,3.8,3.14, and3.18, we also get
lim
n→ ∞un1−un0, nlim→ ∞yn1−yn0, nlim→ ∞vn1−vn0. 3.19
Sinceαnβnγn1 and from the definition of{xn}, we havexn1−xnαnfxn−
xn γnWnvn−xn. Then we have
Wnvn−xn ≤ γ1 n
xn1−xnαnfxn−xn−→0, asn−→ ∞. 3.20
Forx∗∈ ∩∞n1FTn∩VIA, C∩MEPΘ, ϕ, we have
un−x∗2 Trnxn−Trnx∗2
≤ Trnxn−Trnx∗, xn−x∗
un−x∗, xn−x∗
1
2
un−x∗2xn−x∗2− xn−un2
,
3.21
and henceun−x∗2≤ x
From3.3, we have
xn1−x∗2αnfxn−βnxn−γnWnvn−x∗2
≤αnfxn−x∗2βnxn−x∗2γnWnvn−x∗2
≤αnfxn−x∗2βnxn−x∗2γnvn−x∗2
≤αnfxn−x∗2βnxn−x∗2
γn
snxn−x∗2−snxn−un2 1−snxn−x∗2
≤αnfxn−x∗2βnγnxn−x∗2−γnsnxn−un2
≤αnfxn−x∗2xn−x∗2−γnsnxn−un2.
3.22
That is,
xn−un2≤ γ1 nsn
αnfxn−x∗2xn−x∗2− xn1−x∗2
≤ 1
γnsn
αnfxn−x∗2xn1−xnxn−x∗xn1−x∗
.
3.23
Fromiiand3.18, we obtain
xn−un −→0, asn−→ ∞. 3.24
From3.2-3.3, we get
xn1−x∗2≤αnfxn−x∗2βnxn−x∗2γnWnvn−x∗2
≤αnfxn−x∗2βnxn−x∗2γnvn−x∗2
≤αnfxn−x∗2βnxn−x∗2
γnyn−λnAyn−x∗−λnAx∗2
≤αnfxn−x∗2βnxn−x∗2
γnyn−x∗2λnλn−2αAyn−Ax∗2
≤αnfxn−x∗2βnxn−x∗2γnxn−x∗2
γnλnλn−2αAyn−Ax∗2
≤αnfxn−x∗2xn−x∗2γnab−2αAyn−Ax∗2.
Then we get,
−γnab−2αAyn−Ax∗2≤αnfxn−x∗2xn−x∗2− xn1−x∗2
≤αnfxn−x∗2 xn−x∗xn1−x∗xn−xn1.
3.26
Sinceαn → 0 andxn−xn1 → 0, we obtain
Ayn−Ax∗ −→0, asn−→ ∞. 3.27
We note that
vn−x∗2PCyn−λnAyn−PCx∗−λnAx∗2
≤yn−λnAyn−x∗−λnAx∗, vn−x∗
1
2
yn−λnAyn−x∗−λnAx∗2vn−x∗2
−yn−λnAyn−x∗−λnAx∗−vn−x∗2
≤ 1
2
yn−x∗2vn−x∗2−yn−vn−λnAyn−Ax∗2
1
2
yn−x∗2vn−x∗2−yn−vn2
2λnyn−vn, Ayn−Ax∗−λ2nAyn−Ax∗2.
3.28
Then we derive
vn−x∗2≤yn−x∗2−yn−vn22λnyn−vn, Ayn−Ax∗−λ2nAyn−Ax∗2
≤ xn−x∗2−yn−vn22λnyn−vn, Ayn−Ax∗.
3.29
Hence
xn1−x∗2≤αnfxn−x∗2βnxn−x∗2γnWnvn−x∗2
≤αnfxn−x∗2βnxn−x∗2γnvn−x∗2
≤αnfxn−x∗2βnxn−x∗2
γn
xn−x∗2−yn−vn22λnyn−vn, Ayn−Ax∗
≤αnfxn−x∗2xn−x∗2−γnyn−vn2
2γnλnyn−vnAyn−Ax∗,
which imply that
γnyn−vn2≤αnfxn−x∗2xn−x∗2− xn1−x∗2
2γnλnyn−vnAyn−Ax∗
≤αnfxn−x∗22γnλnyn−vnAyn−Ax∗
xn−x∗xn1−x∗xn−xn1.
3.31
From conditionii,3.18, and3.27, we get
lim
n→ ∞yn−vn0. 3.32
Since
un−yn ≤ un−xnxn−yn
un−xnsnPCun−λnAun−snxn
≤ un−xnsnun−ynsnvn−WnvnsnWnvn−xn,
3.33
we have
un−yn ≤ 1−1s
nun−xn
sn
1−snvn−Wnvn
sn
1−snWnvn−xn, 3.34
and then we obtain
Wnvn−vn ≤ Wnvn−xnxn−unun−ynyn−vn
≤ Wnvn−xnxn−un1−1s
nun−xn
1
1−snvn−Wnvn
sn
1−snWnvn−xnyn−vn.
3.35
So we get
1−2sn
1−sn Wnvn−vn ≤ 1
1−snWnvn−xn 2−sn
1−snun−xnyn−vn. 3.36
From conditionivand3.20,3.24, and3.32, we have limn→ ∞Wnvn−vn0.Moreover,
from Remark2.7we get limn→ ∞Wvn−vn0.
Next, we show that
lim sup
n→ ∞ fx
∗−x∗, x
where x∗ P∩∞
n1FTn∩VIA,C∩MEPΘ,ϕfx
∗. Indeed, we choose a subsequence {v
ni} of {vn}
such that
lim sup
n→ ∞
fx∗−x∗, Wv
n−x∗ilim→ ∞fx∗−x∗, Wvni−x∗
. 3.38
Since{vni}is bounded, there exists a subsequence{vnij}of{vni}which converges weakly to z. Without loss of generality, we can assume thatvni z.
FromWvn−vn → 0, we obtainWvni z.
Next, we show thatz∈ ∩∞n1FTn∩VIA, C∩MEPΘ, ϕ.
First, we show thatz∈MEPΘ, ϕ. In fact byunTrnxn∈domϕ, we have
Θun, yϕy−ϕun r1
n
y−un, un−xn≥0, ∀y∈C. 3.39
FromA2, we also have
ϕy−ϕun r1
ny−un, un−xn ≥Θ
y, un, ∀y∈C, 3.40
and hence
ϕy−ϕun
y−uni,
uni−xni rni
≥Θy, uni
, ∀y∈C. 3.41
Fromun−xn → 0,xn−Wvn → 0 andWvn−vn → 0, we getuni → z. It follows from
A4thatuni−xni/rni → 0 and from the lower semicontinuity ofϕthat
Θy, zϕz−ϕy≤0, ∀y∈C. 3.42
Fortwith 0< t≤1 andy ∈C, letytty 1−tz.Sincey∈Candz∈C, we haveyt∈C
and henceΘyt, z ϕz−ϕyt≤0. So, fromA1andA4, we have
0 Θyt, ytϕyt−ϕyt
≤tΘyt, y 1−tΘyt, ztϕy 1−tϕz−ϕyt
≤tΘyt, yϕy−ϕyt.
3.43
Dividing byt, we have
Lettingt → 0, it follows from the weakly semicontinuity ofϕthat
Θz, yϕy−ϕz≥0, ∀y∈C. 3.45
Hencez∈MEPΘ, ϕ.
Second, we show thatz ∈ FW ∩∞n1FTn. Assumez /∈FW. Sinceuni zand z /Wz, by Opial’s condition, we have
lim inf
i→ ∞ uni −z<lim infi→ ∞ uni−Wz
≤lim inf
i→ ∞ uni−WuniWuni −Wz
≤lim inf
i→ ∞ uni−z,
3.46
which derives a contradiction. Thus we havez∈FT.
Finally, by the same argument in the proof of 28, Theorem 3.1, we can show that
z∈VIA, C.
Hencez∈ ∩∞n1FTn∩VIA, C∩MEPΘ, ϕ. Sincex∗P∩∞
n1FTn∩VIA,C∩MEPΘ,ϕfx
∗andxn−Wvn → 0, we have
lim sup
n→ ∞
fx∗−x∗, x
n−x∗lim sup
n→ ∞ fx
∗−x∗, Wv
n−x∗
lim
i→ ∞fx
∗−x∗, Wv
ni−x∗
fx∗−x∗, z−x∗ ≤0.
3.47
Therefore,3.37holds.
Finally, we show thatxn → x∗. From definition of{xn}, we get
xn1−x∗2αnfxn βnxnγnWnvn−x∗2
αnfxn βnxnγnWnvn−x∗, xn1−x∗
αnfxn−x∗, xn1−x∗βnxn−x∗, xn1−x∗
γnWnvn−x∗, xn1−x∗
≤αnfxn−x∗, xn1−x∗
1
2βn
xn−x∗2xn1−x∗2
1
2γn
vn−x∗2xn1−x∗2
≤αnfxn−x∗, xn1−x∗
1
2βn
xn−x∗2xn1−x∗2
1
2γn
xn−x∗2xn1−x∗2
αnfxn−x∗, xn1−x∗ 1
21−αn
xn−x∗2xn1−x∗2
≤αnfxn−x∗, xn1−x∗
1
21−αnxn−x
∗21
2xn1−x
∗2,
3.48
which implies that
xn1−x∗2≤1−αnxn−x∗22αnfxn−x∗, xn1−x∗
. 3.49
By3.47and Lemma2.3, we get that{xn}converges strongly tox∗.
This completes the proof.
Settingfxn≡uandϕ0 in Theorem3.1., we have the following result.
Corollary 3.2see14, Theorem 2.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetΘbe a bifunction fromC×C → Rsatisfying (A1)–(A4), letAbe anα-inverse-strongly monotone mapping ofCintoH, and let{Tn}∞n1 be a sequence of nonexpansive self-mapping onC such that ∩∞n1FTn∩VIA, C∩EPΘ/∅. Suppose thatx1 u ∈ C, {sn},{αn}, {βn}, and
{γn}are sequences in 0,1,{λn}is a sequence in 0,2αsuch thatλn ∈ a, bfor somea, bwith
0 < a < b < 2αand {rn} ⊂ 0,∞ is a real sequence. Suppose that the following conditions are satisfied:
iαnβnγn1,
iilimn→ ∞αn0and∞n1αn∞,
iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1,
iv0<lim infn→ ∞sn≤lim supn→ ∞sn<1/2andlimn→ ∞|sn1−sn|0,
vlimn→ ∞|λn1−λn|0,
vilim infn→ ∞rn>0andlimn→ ∞|rn1−rn|0.
Let the sequence{xn}be generated by,
Θun, yr1
n
y−un, un−xn≥0, ∀y∈C,
yn snPCun−λnAun 1−snxn,
xn1αnuβnxnγnWn
PCyn−λnAyn,
3.50
for alln ∈ N, whereWn is defined by1.6and {tn} is a sequence in0, b, for someb ∈ 0,1.
Then the sequence{xn}converges strongly to a pointx∗ ∈ ∩∞n1FTn∩VIA, C∩EPΘ, where
x∗P∩∞
n1FTn∩VIA,C∩EPΘu.
Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert spaceH. Let Θbe a bifunction fromC×C → Rsatisfying (A1)–(A4), letAbe anα-inverse-strongly monotone mapping ofCintoH, and let{Tn}∞n1be a sequence of nonexpansive self-mapping onCsuch that∩∞n1FTn∩ VIA, C∩EPΘ/∅. Suppose that{sn}, {αn},{βn}, and{γn} are sequences in 0,1,{λn} is a sequence in0,2αsuch thatλn ∈a, bfor somea, bwith0 < a < b <2α, and{rn} ⊂0,∞is a real sequence. Suppose that the following conditions are satisfied:
iαnβnγn1,
iilimn→ ∞αn0and∞n1αn∞,
iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1,
iv0<lim infn→ ∞sn≤lim supn→ ∞sn<1/2andlimn→ ∞|sn1−sn|0,
vlimn→ ∞|λn1−λn|0,
vilim infn→ ∞rn>0andlimn→ ∞|rn1−rn|0.
Letfbe a contraction ofCinto itself with coefficientβ∈0,1and let the sequence{xn}be generated byx1∈Cand
Θun, yr1
n
y−un, un−xn≥0, ∀y∈C,
yn snPCun−λnAun 1−snxn,
xn1αnfxn βnxnγnWn
PCyn−λnAyn,
3.51
for alln ∈ N, whereWn is defined by1.6and {tn} is a sequence in0, b, for someb ∈ 0,1.
Then the sequence{xn}converges strongly to a pointx∗ ∈ ∩∞n1FTn∩VIA, C∩EPΘ, where
x∗P ∩∞
n1FTn∩VIA,C∩EPΘfx
∗.
By Theorem3.1, we obtain some interesting strong convergence theorems.
SettingTnx xthen we haveWnx xin Theorem3.1, and we have the following
result.
Corollary 3.4. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Letϕ : C →
R∪ {∞}be a lower semicontinuous and convex function. LetΘbe a bifunction fromC×C → R satisfying (A1)–(A4), and letAbe anα-inverse-strongly monotone mapping ofCintoHsuch that
VIA, C∩MEPΘ, ϕ/∅. Suppose that{sn},{αn},{βn}, and{γn}are sequences in0,1,{λn}is a sequence in0,2αsuch thatλn ∈a, bfor somea, bwith0< a < b <2αand{rn} ⊂0,∞is a real sequence. Suppose that the following conditions are satisfied:
iαnβnγn1,
iilimn→ ∞αn0and∞n1αn∞,
iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1,
iv0<lim infn→ ∞sn≤lim supn→ ∞sn<1/2andlimn→ ∞|sn1−sn|0,
vlimn→ ∞|λn1−λn|0,
Letfbe a contraction ofCinto itself with coefficientβ∈0,1. Assume that either (B1) or (B2) holds.
Then the sequences{xn},{un}, and{yn}generated by,x1∈Cand
Θun, yϕy−ϕun r1
n
y−un, un−xn≥0, ∀y∈C,
yn snPCun−λnAun 1−snxn,
xn1αnfxn βnxnγnPCyn−λnAyn
3.52
converge strongly to a pointx∗∈VIA, C∩MEPΘ, ϕ, wherex∗PVIA,C∩MEPΘ,ϕfx∗.
SettingΘ 0, ϕ0 andrn 1 then we haveun PCxn xnin Theorem3.1, and we have the following result.
Corollary 3.5. Let Cbe a nonempty closed convex subset of a real Hilbert space H. Let A be an
α-inverse-strongly monotone mapping ofC intoH and let{Tn}∞n1 be a sequence of nonexpansive self-mapping onCsuch that∩∞n1FTn∩VIA, C/∅. Suppose that{sn},{αn},{βn}, and{γn}are sequences in0,1,{λn}is a sequence in0,2αsuch thatλn∈a, bfor somea, bwith0< a < b <
2α. Suppose that the following conditions are satisfied:
iαnβnγn1,
iilimn→ ∞αn0and∞n1αn∞,
iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1,
iv0<lim infn→ ∞sn≤lim supn→ ∞sn<1/2andlimn→ ∞|sn1−sn|0,
vlimn→ ∞|λn1−λn|0.
Letf be a contraction ofCinto itself with coefficientβ∈0,1. Let the sequences{xn}and{yn}be generated byx1∈Cand
ynsnPCxn−λnAxn 1−snxn,
xn1αnfxn βnxnγnWnPCyn−λnAyn,
3.53
for all n ∈ N, where Wn defined by 1.6 and {tn} is a sequence in 0, b, for some b ∈ 0,1.
Then the sequences{xn}and{yn}converge strongly to a pointx∗ ∈ ∩∞n1FTn∩VIA, C, where
x∗P ∩∞
n1FTn∩VIA,Cfx
∗.
Acknowledgment
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