Volume 2008, Article ID 134148,17pages doi:10.1155/2008/134148
Research Article
An Extragradient Approximation Method for
Equilibrium Problems and Fixed Point Problems of
a Countable Family of Nonexpansive Mappings
Rabian Wangkeeree
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Correspondence should be addressed to Rabian Wangkeeree,[email protected]
Received 28 February 2008; Accepted 13 July 2008
Recommended by Huang Nanjing
We introduce a new iterative scheme for finding the common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings, and the problem of finding a zero of a monotone operator. This main theorem extends a recent result of Yao et al.2007and many others.
Copyrightq2008 Rabian 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
LetH be a real Hilbert space with inner product·,·and norm·, and letCbe a closed convex subset ofH. LetFbe a bifunction ofC×CintoR, whereRis the set of real numbers. The equilibrium problem forφ:C×C→Ris to findx∈Csuch that
φx, y≥0 ∀y∈C. 1.1
The set of solutions of1.1is denoted by EPφ.Given a mappingT :C→ H, letφx, y
Tx, y−xfor allx, y ∈C. Thenz ∈EPφif and only ifTz, y−z ≥0 for ally ∈C,that is,zis a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of1.1. In 1997, Fl˚am and Antipin1introduced an iterative scheme of finding the best approximation to initial data when EPφis nonempty and proved a strong convergence theorem.
LetA:C→Hbe a mapping. The classical variational inequality, denoted by VIA, C, is to findx∗∈Csuch that
for allv ∈ C.The variational inequality has been extensively studied in the literature. See, for example,2,3and the references therein. A mappingAofCintoHis calledα -inverse-strongly monotone4,5if there exists a positive real numberαsuch that
Au−Av, u−v ≥αAu−Av2 1.3
for allu, v∈C. It is obvious that anyα-inverse-strongly monotone mappingAis monotone and Lipschitz continuous. A mappingSofCinto itself is called nonexpansive if
Su−Sv ≤ u−v 1.4
for all u, v ∈ C. We denote byFSthe set of fixed points ofS. For finding an element of
FS∩VIA, C, under the assumption that a setC ⊆ His nonempty, closed, and convex, a mappingS : C → Cis nonexpansive and a mappingA : C → H isα-inverse-strongly monotone, Takahashi and Toyoda6introduced the following iterative scheme:
xn1αnxn1−αnSPCxn−λnAxn 1.5
for everyn0,1,2, . . . ,wherex0x∈C, {αn}is a sequence in0, 1, and{λn}is a sequence
in0,2α. They proved that ifFS∩VIA, C/∅, then the sequence{xn}generated by1.5
converges weakly to somez∈FS∩VIA, C. Recently, motivated by the idea of Korpeleviˇc’s extragradient method 7, Nadezhkina and Takahashi 8 introduced an iterative scheme for finding an element ofFS∩VIA, Cand the weak convergence theorem is presented. Moreover, Zeng and Yao 9 proposed some new iterative schemes for finding elements in FS∩ VIA, C and obtained the weak convergence theorem for such schemes. Very recently, Yao et al.10introduced the following iterative scheme for finding an element of
FS∩VIA, Cunder some mild conditions. LetCbe a closed convex subset of a real Hilbert spaceH, A: C→ Ha monotone,L-Lipschitz continuous mapping, andSa nonexpansive mapping ofCinto itself such thatFS∩VIA, C/∅.Suppose thatx1u∈Cand{xn}, {yn}
are given by
ynPCxn−λnAxn,
xn1αnuβnxnγnSPCxn−λnAyn ∀n∈N,
1.6
where {αn},{βn},{γn} ⊆ 0,1 and {λn} ⊆ 0,1 satisfy some parameters controlling
conditions. They proved that the sequence {xn} defined by 1.6 converges strongly to a
common element ofFS∩VIA, C.
On the other hand, S. Takahashi and W. Takahashi11introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution
1.1and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Let S :
C→ Cbe a nonexpansive mapping. Starting with arbitrary initialx1 ∈C,define sequences
{xn}and{un}recursively by
φun, yr1 n
y−un, un−xn≥0 ∀y∈C,
xn1αnfxn1−αnSun ∀n∈N.
1.7
They proved that under certain appropriate conditions imposed on {αn} and {rn}, the
Moreover, Aoyama et al.12introduced an iterative scheme for finding a common fixed point of a countable family of nonexpansive mappings in Banach spaces and obtained the strong convergence theorem for such scheme.
In this paper, motivated by Yao et al.10, S. Takahashi and W. Takahashi11and Aoyama et al. 12, we introduce a new extragradient method 4.2 which is mixed the iterative schemes considered in10–12for finding a common element of the set of common fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem, and the solution set of the classical variational inequality problem for a monotone L-Lipschitz continuous mapping in a real Hilbert space. Then, the strong convergence theorem is proved under some parameters controlling conditions. Further, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings, and the problem of finding a zero of a monotone operator. The results obtained in this paper improve and extend the recent ones announced by Yao et al. results10and many others.
2. Preliminaries
LetH be a real Hilbert space with norm ·and inner product ·,·and letCbe a closed convex subset ofH. For every pointx∈H, there exists a unique nearest point inC, denoted byPCx, such that
x−PCx≤ x−y ∀y∈C. 2.1
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 2.2
for everyx, y∈H.Moreover,PCxis characterized by the following properties:PCx∈Cand
x−PCx, y−PCx≤0, 2.3
x−y2≥x−P
Cx2y−PCx2 2.4
for allx∈H, y∈C. For more details, see13. It is easy to see that the following is true:
u∈VIA, C⇐⇒uPCu−λAu, λ >0. 2.5
A set-valued mappingT :H →2His called monotone if for allx, y∈H, f ∈Tx,and
g∈Tyimplyx−y, f−g ≥0. A monotone mappingT :H→2His maximal if the graph of
GTofTis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingT is maximal if and only if forx, f∈H×H, x−y, f−g ≥0 for everyy, g ∈GTimpliesf ∈Tx. LetBbe a monotone map ofCintoH,L-Lipschitz continuous mapping and letNCvbe the normal cone toCatv∈C, that is,NCv{w∈H:
u−v, w ≥0 for allu∈C}. Define
Tv
⎧ ⎨ ⎩
BvNCv, v∈C;
∅, v /∈C. 2.6
The following lemmas will be useful for proving the convergence result of this paper.
Lemma 2.1see15. LetE,·,·be an inner product space. Then for allx, y, z∈Eandα, β, γ ∈
0,1withαβγ1,one has
αxβyγz2αx2βy2γz2−αβx−y2−αγx−z2−βγy−z2. 2.7
Lemma 2.2see16. Let{xn}and{zn}be bounded sequences in a Banach spaceEand let{βn}be
a sequence in0,1with0<lim infn→∞βn≤lim supn→∞βn<1.Supposexn1 1−βnznβnxn
for all integersn≥1andlim supn→∞zn1−zn − xn1−xn≤0.Then,limn→∞zn−xn0.
Lemma 2.3see17. Assume{an}is a sequence of nonnegative real numbers such that
an1≤
1−αnanδn, n≥1, 2.8
where{αn}is a sequence in0,1and{δn}is a sequence inRsuchthat
i∞n1αn∞and
iilim supn→∞δn/αn≤0or∞n1|δn|<∞.
Thenlimn→∞an 0.
Lemma 2.4see12, Lemma 3.2. LetCbe a nonempty closed subset of a Banach space and let
{Sn}be a sequence of mappings ofCinto itself. Suppose that∞n1sup{Sn1z−Snz:z∈C}<∞.
Then, for eachy ∈C,{Sny}converges strongly to some point ofC. Moreover, letSbe a mapping of
Cinto itself defined by
Sy lim
n→∞Sny ∀y∈C. 2.9
Thenlimn→∞sup{Sz−Snz:z∈C}0.
For solving the equilibrium problem for a bifunctionφ:C×C→R, let us assume that
φsatisfies the following conditions:
A1φx, x 0 for allx∈C;
A2φis monotone, that is,φx, y φy, x≤0 for allx, y∈C;
A3for eachx, y, z∈C, limt→0φtz 1−tx, y≤φx, y;
A4for eachx∈C, y→φx, yis convex and lower semicontinuous.
The following lemma appears implicitly in18.
Lemma 2.5see18. LetCbe a nonempty closed convex subset ofHand letφbe a bifunction of C×CintoRsatisfying (A1)–(A4). Letr >0andx∈H. Then, there existsz∈Csuch that
φz, y 1
ry−z, z−x ≥0 ∀y∈C. 2.10
Lemma 2.6see1. Assume thatφ:C×C→Rsatisfies (A1)–(A4). Forr >0andx∈H, define a mappingTr :H→Cas follows:
Trx
z∈C:φz, y 1
ry−z, z−x ≥0∀y∈C
2.11
for allz∈H. Then, the following hold:
iTr is single-valued;
iiTr is firmly nonexpansive, that is, for anyx, y∈H, Trx−Try2≤ Trx−Try, x−y;
iiiFTr EPφ;
ivEPφis closed and convex.
3. Main results
In this section, we prove a strong convergence theorem.
Theorem 3.1. LetCbe a closed convex subset of a real Hilbert spaceH. Letφbe a bifunction fromC× CtoRsatisfying (A1)–(A4),A:C→Ha monotoneL-Lipschitz continuous mapping and let{Sn}
be a sequence of nonexpansive mappings ofCinto itself such that∩∞n1FSn∩VIA, C∩EPφ/∅. Let the sequences{xn}, {un}, and{yn}be generated by
x1x∈Cchosen arbitrarily,
φun, yr1 n
y−un, un−xn≥0 ∀y∈C,
ynPCun−λnAun,
xn1αnfxnβnxnγnSnPCun−λnAyn ∀n≥1,
3.1
where{αn},{βn},{γn} ⊆0,1,{λn} ⊆0,1, and{rn} ⊆0,∞satisfy the following conditions:
C1αnβnγn1,
C2limn→∞αn0, ∞n1αn∞,
C30<lim infn→∞βn≤lim supn→∞βn<1,
C4limn→∞λn0,
C5lim infn→∞rn>0, ∞n1|rn1−rn|<∞.
Suppose that ∞n1sup{Sn1z− Snz : z ∈ B} < ∞ for any bounded subset B of C. Let
S be a mapping of C into itself defined by Sy limn→∞Snyfor ally ∈ C and suppose that
FS ∩∞
n1FSn. Then the sequences{xn}, {un}, and{yn}converge strongly to the same point
q∈ ∩∞
n1FSn∩VIA, C∩EPφ, whereqP∩∞n1FSn∩VIA,C∩EPφfq. Proof. LetQP∩∞
n1FSn∩VIA,C∩EPφ. Sincefis a contraction withα∈0,1, we obtain Qfx−Qfy≤fx−fy≤αx−y ∀x, y∈C. 3.2
Therefore,Qfis a contraction ofCinto itself, which implies that there exists a unique element
Step 1{xn}is bounded. Indeed, puttnPCun−λnAynfor alln≥1. Letx∗∈ ∩∞n1FSn∩
VIA, C∩EPφ. From2.5we havex∗PCx∗−λnAx∗. Also it follows from2.4that
tn−x∗2≤u
n−λnAyn−x∗2−un−λnAyn−tn2
un−x∗2−2λnAyn, un−x∗λ2nAyn2−un−tn2
2λnAyn, un−tn−λ2nAyn2
un−x∗22λnAyn, x∗−tn−un−tn2
un−x∗2−un−tn22λnAyn−Ax∗, x∗−yn
2λnAx∗, x∗−yn2λnAyn, yn−tn.
3.3
SinceAis monotone andx∗is a solution of the variational inequality problem VIA, C, we have
Ayn−Ax∗, x∗−yn≤0, Ax∗, x∗−yn≤0. 3.4
This together with3.3implies that
tn−x∗2≤un−x∗2−un−tn22λnAyn, yn−tn
un−x∗2−un−yn yn−tn22λnAyn, yn−tn
un−x∗2−un−yn2−2un−yn, yn−tn−yn−tn22λnAyn, yn−tn
un−x∗2−un−yn2−yn−tn22un−λnAyn−yn, tn−yn.
3.5
From2.3, we have
un−λnAun−yn, tn−yn≤0, 3.6
so that
un−λnAyn−yn, tn−ynun−λnAun−yn, tn−ynλnAun−λnAyn, tn−yn
≤λnAun−λnAyn, tn−yn
≤λnAun−Ayntn−yn
≤λnLun−yntn−yn.
3.7
Hence it follows from3.5and3.7that
tn−x∗2≤un−x∗2−un−yn2−yn−tn22λnLun−yntn−yn
≤un−x∗2−un−yn2−yn−tn2λnLun−yn2yn−tn2
un−x∗2λnL−1un−yn2λnL−1yn−tn2.
Sinceλn → 0 asn → ∞, there exists a positive integerN0 such thatλnL−1 ≤ −1/3, when
n≥N0. Hence it follows from3.8that
tn−x∗≤un−x∗. 3.9
Observe that
un−x∗Trnxn−Trnx∗≤xn−x∗, 3.10
and hence
tn−x∗≤xn−x∗. 3.11
Thus, we can calculate
xn1−x∗αnfxnβnxnγnSntn−x∗
≤αnfxn−x∗βnxn−x∗γntn−x∗
≤αnfxn−fx∗αnfx∗−x∗βnxn−x∗γnxn−x∗
≤1−αn1−αxn−x∗αnfx∗−x∗
1−αn1−αxn−x∗αn1−α
f
x∗−x∗
1−α .
3.12
It follows from induction that
xn−x∗≤max
x1−x∗,
fx∗−x∗
1−α
, n≥N0. 3.13
Therefore,{xn}is bounded. Hence, so are{tn}, {Sntn}, {Aun}, {Ayn}, and{fxn}.
Step 2limn→∞xn1−xn0. Indeed, we observe that for anyx, y∈C,
I−λnA
x−I−λnAy2 x−y−λnAx−Ay2
x−y2−2λ
nx−y, Ax−Ayλ2nAx−Ay2
≤ x−y2λ2
nL2x−y2
1λ2nL2x−y2,
3.14
which implies that
I−λnA
x−I−λnAy≤1λnLx−y. 3.15
Thus
tn1−tn≤PC
un1−λn1Ayn1
−PCun−λnAyn
≤un1−λn1Ayn1−
un−λnAyn
un1−λn1Aun1
−un−λn1Aunλn1
Aun1−Ayn1−AunλnAyn
≤un1−λn1Aun1
−un−λn1Aun
λn1Aun1Ayn1AunλnAyn
≤1λn1Lun1−unλn1Aun1Ayn1AunλnAyn.
On the other hand, fromunTrnxnandun1Trn1xn1,we note that
φun, yr1 n
y−un, un−xn≥0 ∀y∈C , 3.17
φun1, y
r1
n1
y−un1, un1−xn1
≥0 ∀y∈C. 3.18
Puttingyun1in3.17andyunin3.18, we have
φun, un1
1
rn
un1−un, un−xn≥0,
φun1, unr1 n1
un−un1, un1−xn1
≥0.
3.19
So, fromA2, we have
un1−un,
un−xn
rn −
un1−xn1
rn1
≥0 3.20
and hence
un1−un, un−un1un1−xn−rrn n1
un1−xn1
≥0. 3.21
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−unxn1−xn1−rrn n1
un1−xn1
3.22
and hence
un1−un≤xn1−xnr1 n1
rn1−rnun1−xn1
≤xn1−xn1crn1−rnM,
3.23
whereMsup{un−xn:n∈N}. It follows from3.16and the last inequality that
tn1−tn≤1λn1Lxn1−xn1λn1L
1
crn1−rnM
λn1Aun1Ayn1AunλnAyn.
Settingzn αnfxn γnSntn/1−βn, we obtainxn1 1−βnznβnxnfor alln∈N. Thus,
we have
zn1−znαn1f
xn1
γn1Sn1tn1 1−βn1 −
αnfxnγnSntn
1−βn
αn1
1−βn1
fxn1
γn1
1−βn1
Sn1tn1−Sntn− αn
1−βnf
xn
−
1− αn 1−βn
Sntn
1− αn1 1−βn1
Sntn
≤ αn1 1−βn1
fxn1
−Sntn αn
1−βn
Sntn−fxn
γn1
1−βn1
Sn1tn1−Sntn.
3.25
It follows from3.24that
Sn1tn1−Sntn≤Sn1tn1−Sn1tnSn1tn−Sntn
≤tn1−tnSn1tn−Sntn
≤1λn1Lxn1−xn1λn1L
1
crn1−rnM
λn1Aun1Ayn1AunλnAynSn1tn−Sntn.
3.26
Combining3.25and3.26, we have
zn1−zn−xn1−xn≤
αn1 1−βn1
fxn1
−Sntn αn
1−βn
Sntn−fxn
γn1
1−βn1
1λn1Lxn1−xn
γn1 1−βn1
1λn1L
1
c|rn1−rn|M
γn1
1−βn1
λn1Aun1Ayn1Aun γn1
1−βn1
λnAyn
γn1
1−βn1
Sn1tn−Sntn−xn1−xn
≤ αn1 1−βn1
fxn1
−Sntn 1−αnβ n
Sntn−fxn
γn1
1−βn1
λn1Lxn1−xn1−γnβ1 n1
1λn1L
1
c|rn1−rn|M
γn1
1−βn1
λn1Aun1Ayn1Aun
γn1
1−βn1
λnAyn γn1
1−βn1
supSn1t−Snt:t∈tn.
This together withC1–C5and limn→∞sup{Sn1t−Snt:t∈ {tn}}0 implies that
lim sup
n→∞
zn1−zn−xn1−xn≤0. 3.28
Hence, byLemma 2.2, we obtainzn−xn →0 asn→ ∞. It then follows that
lim
n→∞xn1−xnnlim→∞
1−βnzn−xn0. 3.29
By3.23and3.24, we also have
lim
n→∞tn1−tnnlim→∞un1−un0. 3.30
Step 3limn→∞Stn−tn0. Indeed, pick anyx∗∈ ∩∞n1FSn∩VIA, C∩EPφ, to obtain
un−x∗2Trnxn−Trnx∗
2≤T
rnxn−Trnx∗, xn−x∗
un−x∗, xn−x∗
1
2
un−x∗2xn−x∗2−xn−un2
. 3.31
Therefore,un−x∗2 ≤ xn−x∗2− xn−un2. FromLemma 2.1and3.9, we obtain, when
n≥N0, that
xn1−x∗2α
nfxnβnxnγnSntn−x∗2
≤αnfxn−x∗2βnxn−x∗2γnSntn−x∗2
≤αnfxn−x∗2βnxn−x∗2γntn−x∗2
≤αnfxn−x∗2βnxn−x∗2γnun−x∗2
≤αnfxn−x∗2βnxn−x∗2γnxn−x∗2−xn−un2
≤αnfxn−x∗21−αnxn−x∗2−γnxn−un2
3.32
and hence
γnxn−un2≤αnfxn−x∗2xn−x∗2−xn1−x∗2
≤αnfxn−x∗2xn−xn1xn−x∗xn1−x∗.
3.33
It now follows from the last inequality,C1,C2,C3and3.29, that
lim
n→∞xn−un0. 3.34
Noting that
yn−xnPCun−λnAun−xn≤un−xnλnAun−→0 asn−→ ∞,
yn−tnPC
un−λnAun−PCun−λnAyn≤λnAun−Ayn−→0 asn−→ ∞.
Thus
tn−xn≤tn−ynyn−xn−→0 asn−→ ∞. 3.36
We note that
Snyn−xn1≤Snyn−SntnSntn−xn1
≤yn−tnαnSntn−fxnβnSntn−xn
≤yn−tnαnSntn−fxnβnSntn−SnxnβnSnxn−xn
≤yn−tnαnSntn−fxnβntn−xnβnSnxn−xn.
3.37
Using3.37, we have
Snxn−xn≤Snxn−SnynSnyn−xn1xn1−xn
≤xn−ynyn−tnαnSntn−fxnβntn−xn
βnSnxn−xnxn1−xn,
3.38
so that
1−βnSnxn−xn≤xn−ynyn−tnαnSntn−fxnβntn−xnxn1−xn.
3.39
This implies that
lim
n→∞Snxn−xn0. 3.40
It now follows from3.36and3.40that
Sntn−tn≤Sntn−SnxnSnxn−xnxn−tn ≤2tn−xnSnxn−xn−→0 asn−→ ∞.
3.41
ApplyingLemma 2.4and3.41, we have
Stn−tn≤Stn−SntnSntn−tn
≤supSt−Snt:t∈tnSntn−tn−→0.
3.42
It follows from the last inequality and3.36that
xn−Stn≤xn−tntn−Stn−→0 asn−→ ∞. 3.43
Step 4lim supn→∞fq−q, xn−q ≤0. Indeed, we choose a subsequence{tni}of{tn}such
that
lim sup
n→∞
fq−q, Stn−qilim
→∞
fq−q, Stni−q
Without loss of generality, we may assume that{tni}converges weakly toz∈C. FromStn−
tn →0,we obtainStni z.Now, we will show thatz∈FS∩VIA, C∩EPφ. Firstly, we
will showz∈EPφ. Indeed, we observe thatunTrnxn, and
φun, yr1 n
y−un, un−xn≥0 ∀y∈C. 3.45
FromA2, we also have
1
rn
y−un, un−xn≥φy, un, 3.46
and hence
y−uni,
uni−xni
rni
≥φy, uni
. 3.47
Fromun−xn →0, xn−Stn →0,andStn−tn →0,we getuni z. Sinceuni−xni/rni →
0, it follows by A4that 0 ≥ φy, z for ally ∈ C.For twith 0 < t ≤ 1 and y ∈ C,let
ytty 1−tz.Sincey∈Candz∈C,we haveyt∈Cand henceφyt, z≤0.So, fromA1
andA4, we have
0φyt, yt≤tφyt, y 1−tφyt, z≤tφyt, y 3.48
and hence 0≤φyt, y. FromA3, we have 0≤φz, yfor ally∈C, and hencez∈EPφ.By
the Opial’s condition, we can obtain thatz∈FS.Next we will show thatz∈VIA, C. Let
Tv
⎧ ⎨ ⎩
AvNCv, v∈C;
∅, v /∈C. 3.49
ThenTis maximal monotonesee14. Letv, w∈GT. Sincew−Av∈NCvandtn∈C,
we havev−tn, w−Av ≥0. On the other hand, fromtnPCun−λnAyn, we have
v−tn, tn−un−λnAyn≥0, 3.50
that is,
v−tn,tnλ−un
n Ayn
≥0. 3.51
Therefore, we obtain
v−tni, w
≥v−tni, Av
≥v−tni, Av
−
v−tni,
tni−uni
λni
Ayni
v−tni, Av−Ayni−
tni−uni
λni
v−tni, Av−Atni
v−tni, Atni−Ayni −
v−tni,
tni−uni
λni
≥v−tni, Atni
−
v−tni,
tni −uni
λni
Ayni
v−tni, Atni−Ayni
−
v−tni,
tni−uni
λni
.
Noting thattni−yni →0, tni−uni →0 asn→ ∞,Ais Lipschitz continuous and3.52,
we obtain
v−z, w ≥0. 3.53
SinceT is maximal monotone, we havez ∈T−10, and hencez∈VIA, C. Hencez∈FS∩ VIA, C∩EPφ.The property of the metric projection implies that
lim sup
n→∞
fq−q, xn−qlim sup n→∞
fq−q, Stn−q
lim
i→∞
fq−q, Stni−q
fq−q, z−q≤0.
3.54
Step 5limn→∞xn−q0. Indeed, we observe that
xn1−q2
αnfxnβnxnγnSntn, xn1−q
αnfxn−q, xn1−q
βnxn−q, xn1−q
γnSntn−q, xn1−q
≤ 1
2βn
xn−q2xn1−q2
1
2γn
tn−q2xn1−q2
αnfxn−fq, xn1−qαnfq−q, xn1−q
≤ 1
2
1−αnxn−q2xn1−q2
1
2αn
fxn−fq2xn1−q2
αnfq−q, xn1−q
≤ 1
2
1−αn1−α2xn−q2121−αnxn1−q2
1
2αnxn1−q 2α
nfq−q, xn1−q
,
3.55
which implies that
xn1−q2≤
1−αn1−α2xn−q22αnfq−q, xn1−q
. 3.56
Setting δn 2αnfq−q, xn1 −q, we have lim supn→∞δn/αn1− α2 ≤ 0. Applying
Lemma 2.3to3.56, we conclude that{xn}converges strongly toq. Consequently,{un}and
{yn}converge strongly toq. This completes the proof.
As in12, Theorem 4.1, we can generate a sequence{Sn}of nonexpansive mappings
satisfying condition∞n1sup{Sn1z−Snz : z ∈ B} < ∞for any bounded subsetBofC
by using convex combination of a general sequence{Tk}of nonexpansive mappings with a
common fixed point.
{βk
n}be a family of nonnegative numbers with indicesn, k∈Nwithk≤nsuch that
ink1βkn1for alln∈N;
iilimn→∞βkn>0for everyk∈N; iii∞n1nk1|βkn1−βkn|<∞.
Let{Tk}be a sequence of nonexpansive mappings ofCinto itself with∩∞k1FTk∩VIA, C∩
EPφ/∅.Letx1x∈Cand{xn}, {yn}and{un}be the sequences generated by
x1x∈Cchosen arbitrary,
φun, yr1 n
y−un, un−xn≥0 ∀y∈C,
ynPCun−λnAun,
xn1αnfxnβnxnγn n
k1
βk
nTkPCun−λnAyn ∀n≥1,
3.57
where{αn},{βn},{γn} ⊆0,1, {λn} ⊆0,1, and{rn} ⊆0,∞satisfy the following conditions:
C1αnβnγn1,
C2limn→∞αn0, ∞n1αn∞,
C30<lim infn→∞βn≤lim supn→∞βn<1,
C4limn→∞λn0,
C5lim infn→∞rn>0, ∞n1|rn1−rn|<∞.
Then the sequences {xn},{un}, and {yn} converge strongly to the same point q ∈ ∩∞k1FTk∩
VIA, C∩EPφ, whereqP∩∞
k1FTk∩VIA,C∩EPφfq.
SettingSn≡Sandf :uinTheorem 3.1, we have the following result.
Corollary 3.3see10, Theorem 3.1. LetCbe a closed convex subset of a real Hilbert spaceH. Let A:C→Hbe a monotone,L-Lipschitz continuous mapping, and letSbe a nonexpansive mapping of Cinto itself such thatFS∩VIA, C/∅.Supposex1u∈Cand{xn},{yn}are given by
ynPCxn−λnAxn,
xn1αnuβnxnγnSPCxn−λnAyn,
3.58
where{λn},{αn},{βn},{γn}are sequences in0,1satisfying the following conditions:
iαnβnγn1,
iilimn→∞αn0, ∞n1αn∞,
iii0<lim infn→∞βn≤lim supn→∞βn<1,
ivlimn→∞λn0.
Then{xn}converges strongly toPFS∩VIA,Cu.
Proof. Putφx, y 0 for allx, y ∈ Cand{rn} 1 inTheorem 3.1. Thus, we haveun xn.
4. Applications
In this section, we consider the problem of finding a zero of a monotone operator. A multivalued operator S : H → 2H with domain DS {z ∈ H : Sz /∅} and range
RS {Sz :z ∈ DT}is said to be monotone if for eachxi ∈DSandyi ∈Sxi, i 1,2,
we havex1−x2, y1−y2 ≥ 0. A monotone operatorS is said to be maximal if its graph
GS {x, y : y ∈ Sx} is not properly contained in the graph of any other monotone operator. Let I denote the identity operator on H and let S : H → 2H be a maximal monotone operator. Then we can define, for each r > 0, a nonexpansive single-valued mappingJr :H→HbyJr IrS−1. It is called the resolventor the proximal mappingof
S. We also define the Yosida approximationArbyAr I−Jr/r. We know thatArx∈SJrx
andArx ≤inf{y:y∈Sx}for allx∈H. We also know thatS−10FJrfor allr >0; see,
for instance, Rockafellar19or Takahashi20.
Lemma 4.1the resolvent identity. Forλ, μ >0, there holds the identity
JλxJμ
μ
λ
1− μ
λ
Jλx
, x∈H. 4.1
By usingTheorem 3.1andLemma 4.1, we may obtain the following improvement.
Theorem 4.2. LetS:H→2Hbe a maximal monotone operator. Letφbe a bifunction fromC×Cto
Rsatisfying (A1)–(A4),A:C→Ha monotoneL-Lipschitz continuous mapping ofCintoHsuch thatS−10∩VIA, C∩EPφ/∅andfa contraction ofCinto itself with coefficientα∈0,1. Let
the sequences{xn}, {un}and{yn}be generated by
x1x∈Cchosen arbitrary,
φun, yr1
ny−un, un−xn ≥0 ∀y∈C,
ynPCun−λnAun,
xn1αnfxnβnxnγnJrnPC
un−λnAyn ∀n≥1,
4.2
where{αn},{βn},{γn} ⊆0,1, {λn} ⊆0,1, and{rn} ⊆0,∞satisfy the following conditions:
C1αnβnγn1,
C2limn→∞αn0, ∞n1αn∞,
C30<lim infn→∞βn≤lim supn→∞βn<1,
C4limn→∞λn0,
C5lim infn→∞rn>0, ∞n1|rn1−rn|<∞.
Then{xn}converges strongly toqPS−10∩VIA,C∩EPφfq.
Proof. We first verify that∞n1sup{Jrn1z−Jrnz:z∈B}<∞for any bounded subsetBof
C. LetBbe a bounded subset ofC. SinceS−10 FJr
nfor eachn∈N, {Jrnz:z∈B, n∈N}
is bounded. It follows fromLemma 4.1that
Jrn1zJrn r
n
rn1
z
1− rn
rn1
Jrn1z
Thus
Jrn1z−Jrnz≤
rn1−rn
rn1
Jrn1z−z
≤Mrn1−rn
4.4
for eachz∈Bandn∈N,whereMsup{Jrn1z−z:z∈B, n∈N}/inf{rn:n∈N}. Hence
we get
∞
n1
supJrn1z−Jrnz:z∈B
≤M∞
n1
rn1−rn<∞. 4.5
By the assumption that∞n1|rn1−rn|<∞, we obtainrn→rfor somer >0. SinceJrz−Jrnz ≤
|r−rn|/rz−Jrz, we obtain that limn→∞Jrnz Jrzfor allz∈C. SinceFJμ S−10for
allμ >0, we haveFJr ∩∞n1FJrn S−10/∅. Therefore, byTheorem 3.1,{xn}converges
strongly toqPS−10∩VIA,C∩EPφfq.
Acknowledgments
The author would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper. This research was partially supported by the Commission on Higher Education.
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