Fixed Point Theory and Applications Volume 2010, Article ID 264628,13pages doi:10.1155/2010/264628
Research Article
Hybrid Viscosity Iterative Method for Fixed Point,
Variational Inequality and Equilibrium Problems
Yi-An Chen and Yi-Ping Zhang
College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
Correspondence should be addressed to Yi-An Chen,[email protected]
Received 27 December 2009; Revised 3 May 2010; Accepted 1 June 2010
Academic Editor: Simeon Reich
Copyrightq2010 Y.-A. Chen and Y.-P. Zhang. 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 an iterative scheme by the viscosity iterative method for finding a common element of the solution set of an equilibrium problem, the solution set of the variational inequality, and the fixed points set of infinitely many nonexpansive mappings in a Hilbert space. Then we prove our main result under some suitable conditions.
1. Introduction
LetH be a real Hilbert space with the inner product and the norm being denoted by·,· and · , respectively. LetCbe a nonempty, closed, and convex subset ofHand letF be a bifunction ofC×CintoR, whereRdenotes the real numbers. The equilibrium problem for
F:C×C → Ris to findx∈Csuch that
Fx, y≥0, ∀y∈C. 1.1
The solution set of1.1is denoted by EPF.
Let A : C → H be a mapping. The classical variational inequality, denoted by VIA, C, is to findx∗∈Csuch that
The variational inequality has been extensively studied in the literaturesee, e.g.,1–3. The
mappingAis calledα-inverse-strongly monotone if
Au−Av, u−v ≥αAu−Av2, ∀u, v∈C, 1.3
whereαis a positive real number.
A mappingT :C → Cis called strictly pseudocontractive if there existskwith 0 ≤
k <1 such that
Tx−Ty2≤x−y2kI−Tx−I−Ty2, ∀x, y∈C. 1.4
It is easy to know thatI −T is1−k/2-inverse-strongly-monotone. Ifk 0, then T is nonexpansive. We denote byFTthe fixed points set ofT.
In 2003, for x0 ∈ C, Takahashi and Toyoda 4 introduced the following iterative scheme:
xn1αnxn 1−αnSPCxn−λnAxn, n≥0, 1.5
where{αn}is a sequence in0,1,Ais anα-inverse-strongly monotone mapping,{λn}is a
sequence in0,2α, andPC is the metric projection. They proved that ifFSVIA, C/∅, then{xn}converges weakly to somez∈FSVIA, C.
Recently, S. Takahashi and W. Takahashi5introduced an iterative scheme for finding a common element of the solution set of 1.1 and the fixed points set of a nonexpansive mapping in a Hilbert space. IfFis bifunction which satisfies the following conditions:
A1Fx, x 0 for allx∈C;
A2Fis monotone, that is,Fx, y Fy, x≤0 for allx, y∈C; A3for eachx, y, z∈C,limt→0Ftz 1−tx, y≤Fx, y;
A4for eachx∈C, y→Fx, yis convex and lower semicontinuous,
then they proved the following strong convergence theorem.
Theorem Asee5. LetCbe a closed and convex subset of a real Hilbert spaceH. LetF:C×C →
Rbe a bifunction which satisfies conditionsA1–A4.
LetT :C → Hbe a nonexpansive mapping such thatFTEPF/∅and letf:H → H be a contraction; that is, there is a constantk∈0,1such that
fx−f
y≤kx−y, ∀x, y∈H, 1.6
and let{xn}and{un}be sequences generated byx1∈Cand
Fun, yr1 n
y−un, un−xn≥0, ∀y∈C,
xn1αnfxn 1−αnTun, n≥1,
where{αn} ⊂ 0,1and{rn} ⊂0,∞satisfylimn→ ∞αn 0,∞n1αn ∞, ∞
n1|αn1−αn| <
∞,lim infn→ ∞rn>0, and∞n1|rn1−rn|<∞.
Then,{xn}and{un}converge strongly toz∈FTEPF,wherezPFTEPFfz. Let{Tn}∞n1 be a sequence of nonexpansive mappings ofCinto itself and{λn}∞n1 a sequence of nonnegative numbers in 0,1. For each n ≥ 1, define a mapping Wn of C into itself as follows:
Un,n1I,
Un,nλnTnUn,n1 1−λnI,
Un,n−1λn−1Tn−1Un,n 1−λn−1I,
.. .
Un,kλkTkUn,k1 1−λkI,
Un,k−1λk−1Tk−1Un,k 1−λk−1I,
.. .
Un,2λ2T2Un,3 1−λ2I, WnUn,1λ1T1Un,2 1−λ1I.
1.8
Such a mappingWnis called theW-mapping generated byTn, Tn−1, . . . , T1andλn, λn−1, . . . , λ1(see [6]).
In this paper, we introduced a new iterative scheme generated byx1 ∈Cand findun such that
Fun, yr1 n
y−un, un−xn≥0, ∀y∈C,
ynβnfxn 1−βnxn, n≥1,
xn1αnyn 1−αnWnPCun−δnAun,
1.9
where {αn} and {βn} are sequences in 0,1,{rn} and{δn}are sequences in 0,∞,f is a
fixed contractive mapping with contractive coefficientk ∈0,1,Ais anα-inverse-strongly monotone mapping ofCtoH,F is a bifunction which satisfies conditionsA1–A4, and
{Wn} is generated by 1.8. Then we proved that the sequences {xn} and {un} converge
2. Preliminaries
LetHbe a real Hilbert space and letCbe a closed and convex subset ofH. PC is the metric projection fromHontoC, that is, for anyx∈H,x−PCx ≤ x−yfor ally∈C.It is easy to see thatPCis nonexpansive and
u∈VIA, C⇐⇒uPCu−λAu, λ >0. 2.1
IfAis anα-inverse-strongly monotone mapping ofCtoH, then it is obvious thatAis1/α -Lipschitz continuous. We also have that for allx, y∈Candλ >0,
I−λAx−I−λAy2x−y2−
2λx−y, Ax−Ayλ2Ax−Ay2
≤x−y2λλ−
2αAx−Ay2.
2.2
So, ifλ≤2α, thenI−λAis nonexpansive.
Lemma 2.1see7. Let{xn}and{zn}be bounded sequences in a Banach spaceE, and let{βn}be
a sequence in0,1with0<lim infn→ ∞βn≤lim supn→ ∞βn<1. Supposexn1 1−βnznβnxn
for alln≥1andlim supn→ ∞zn1−zn − xn1−xn≤0. Then,limn→ ∞zn−xn0.
Lemma 2.2see8. Assume that{an}is a sequence of nonnegative real numbers such that
an1≤1−αnanδn, n≥1, 2.3
where{αn}is a sequence in0,1and{δn}is a sequence inRsuch that
∞
n1
αn∞; lim sup n→ ∞
δn
αn ≤0 or
∞
n1
|δn|<∞. 2.4
Thenlimn→ ∞αn0.
Lemma 2.3see9. LetCbe a nonempty, closed, and convex subset ofH andF a bifunction of
C×CintoRthat satisfies conditionsA1–A4. Letr >0andx∈H. Then, there existsz∈Csuch that
Fz, y1ry−z, z−x≥0, ∀y∈C. 2.5
Lemma 2.4see9. Assume thatF :C×C → Rsatisfies conditionsA1–A4. Forr >0and
x∈H, define a mappingTr:H → Cas follows:
Trx
z∈C:Fz, y1
r
y−z, z−x≥0,∀y∈C
Then, the following holds: iTr is single-valued;
iiTr is firmly nonexpansive, that is,
Trx−Try2≤T
rx−Try, x−y, ∀x, y∈H; 2.7
iiiFTr EPF;
ivEPFis closed and convex.
Lemma 2.5Opial’s theorem10. Each Hilbert spaceH satisfies Opial’s condition; that is, for any sequence{xn} ⊂Hwithxn x, the inequality
lim inf
n→ ∞ xn−x<lim infn→ ∞ xn−y 2.8
holds for eachy∈Hwithx /y.
Let{Tn}∞n1be a sequence of nonexpansive self-mappings onC, whereCis a nonempty, closed and convex subset of a real Hilbert spaceH. Given a sequence{λn}∞n1in0,1, one defines a sequence
{Wn}∞n1of self-mappings onCgenerated by1.8. Then one has the following results.
Lemma 2.6see6. LetCbe a nonempty, closed, and convex subset of a real Hilbert spaceH. Let
{Tn}∞n1 be a sequence of nonexpansive self-mappings onCsuch that ∞
n1FTn/∅and{λn}is a sequence in0, bfor someb∈0,1. Then, for everyx∈Candk≥1the limitlimn→ ∞Un,kxexists.
Remark 2.7. It can be shown fromLemma 2.6that ifDis a nonempty and bounded subset of
C, then forε >0 there existsn0≥ksuch that supx∈DUn,kx−Un−1,kx ≤εfor alln > n0.
Remark 2.8. UsingLemma 2.6, we can define a mappingW :C → Cas follows:
Wx lim
n→ ∞Wnxnlim→ ∞Un,1x 2.9
for allx∈C.Such aW is called theW-mapping generated byT1, T2, . . .andλ1, λ2, . . . .Since Wnis nonexpansive,W:C → Cis also nonexpansive. Indeed, observe that for eachx, y∈C,
Wx−Wy lim
n→ ∞Wnx−Wny≤x−y. 2.10
Let{xn}be a bounded sequence inCandD{xn:n≥0}. Then, it is clear fromRemark 2.7
that forε >0 there existsN0≥1 such that for alln > N0,
Wnxn−WxnUn,1xn−U1xn ≤sup
x∈DUn,1x−U1x ≤ε. 2.11
This implies that limn→ ∞Wnxn−Wxn0.
Lemma 2.9see6. LetCbe a nonempty, closed, and convex subset of a real Hilbert spaceH. Let
{Tn}∞n1 be a sequence of nonexpansive self-mappings on C such that ∞
3. Strong Convergence Theorem
Theorem 3.1. LetHbe a Hilbert space. LetCbe a nonempty, closed, and convex subset ofH. Let
F : C×C → R be a bifunction which satisfies conditionsA1–A4,A an α-inverse-strongly monotone mapping ofCtoH,fa contraction ofCinto itself, and{Tn}∞n1a sequence of nonexpansive self-mappings onCsuch thatF /∅. Suppose that{αn},{βn}, and{λn}are sequences in0,1, and
{rn}and{δn}are sequences in0,∞which satisfies the following conditions:
i0<lim infn→ ∞αn≤lim supn→ ∞αn<1;
iilimn→ ∞βn0;∞n1βn ∞;
iiilim infn→ ∞rn>0,∞n1|rn1−rn|<∞;
ivδn ∈0, b, b <2α, limn→ ∞δn0;
vλn∈0, c, c∈0,1.
Then{xn}and{un}generated by1.9converge strongly tox∗∈F, wherex∗PFfx∗.
Proof. Letp∈F. It follows fromLemma 2.4and1.9thatunTrnxn, and hence,
un−pTrnxn−Trnp≤xn−p, 3.1
for alln∈N. Letzn PCun−δnAun. SinceI−δnAis nonexpansive andpPCp−δnAp,
we have
zn−p≤un−δnAun−
p−δnAp≤un−p≤xn−p, 3.2
yn−p≤βnfxn−p
1−βnxn−p
≤βnfxn−fpβnfp−p1−βnxn−p
≤1−βn1−kxn−pβnfp−p.
3.3
Thus,
xn1−pαnyn 1−αnWnzn−p
≤αnyn−p 1−αnzn−p
≤αn1−βn1−kxn−pαnβnfp−p 1−αnxn−p
1−αnβn1−kxn−pαnβn1−k
f
p−p
1−k
≤max
xn−p,fp−p
1−k
.
3.4
Next, we claim that limn→ ∞xn1−xn0.Indeed, assume thatxn1ρnxn1−ρntn,
whereρnαn1−βn,n≥0. Then,
tn1−tn αn1βn1fxn11 1−ρ −αn1Wn1zn1
n1 −
αnβnfxn 1−αnWnzn
1−ρn
αn1βn1fxn1 1−ρn1 −
αnβnfxn
1−ρn
1−αn1
1−ρn1Wn1zn1−Wn1zn
1−αn1
1−ρn1Wn1zn− 1−αn 1−ρnWnzn
≤ αn1βn1fxn1 1−ρn1 −
αnβnfxn
1−ρn
1−αn1
1−ρn1zn1−zn
Wn1zn− α1n−1ρβn1
n1Wn1zn−Wnzn αnβn
1−ρnWnzn,
3.5
zn1−zn ≤ un1−δn1Aun1−un−δnAun
≤ I−δn1Aun1−I−δn1AunI−δn1Aun−I−δnAun
≤ un1−unδn1−δnAun.
3.6
Using1.8and the nonexpansivity ofTi, we deduce that
Wn1zn−Wnznλ1T1Un1,2zn−λ1T1Un,2zn
≤λ1Un1,2zn−Un,2zn
≤λ1λ2T2Un1,3zn−λ2T2Un,3zn
≤λ1λ2Un1,3zn−Un,3zn
.. .
≤
n
i1 λi
Un1,n1zn−Un,n1zn
≤Mn
i1 λi,
3.7
for some constantM≥0.On the other hand, fromunTrnxnandun1Trn1xn1, we obtain
Fun, yr1 n
y−un, un−xn≥0, ∀y∈C, 3.8
Fun1, y
r1
n1
y−un1, un1−xn1
Settingyun1in3.8andyunin3.9, we get
Fun, un1 1
rnun1−un, un−xn ≥0,
Fun1, un r1
n1un−un1, un1−xn1 ≥ 0.
3.10
FromA2, we have
un1−un,unr−xn n −
un1−xn1 rn1
≥0, 3.11
and hence
un1−un, un−un1un1−xn−rrn
n1un1−xn1
≥0. 3.12
Without loss of generality, we may assume that there exists a real numberrsuch thatrn> r >
0 for alln≥0.Then
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−xn1−rrn n1
un1−xn1
≤ xn1−xn 1
r|rn1−rn|L,
3.14
whereLsup{un−xn:n≥0}. It follows from3.5,3.6,3.7, and3.14that
tn1−tn − xn1−xn ≤
αn1βn1 1−ρn1
fxn1Wn1zn
αnβn
1−ρnfxnWnzn
1−αn1 1−ρn1
xn1−xn Lr|rn1−rn||δn1−δn|Aun
Mn
i1
≤ αn1βn1 1−ρn1
fxn1Wn1zn
αnβn 1−ρn
fxnWnzn
1−αn1 1−ρn1
L
r|rn1−rn||δn1−δn|Aun
Mn
i1 λi.
3.15
Therefore, lim supn→ ∞tn1−tn − xn1−xn≤0.
Since 0<lim infn→ ∞αn≤lim supn→ ∞αn<1 and limn→ ∞βn0, hence,
0<lim inf
n→ ∞ ρn≤lim supn→ ∞ ρn<1. 3.16
Lemma 2.1yields that limn→ ∞tn−xn0. Consequently, limn→ ∞xn1−xnlimn→ ∞1−
ρntn−xn0.
Forp∈F, we obtain
un−p2T
rnxn−Trnp
2
≤ Trnxn−Trnp, xn−p
un−p, xn−p
1 2
un−p2xn−p2− xn−un2
,
3.17
and hence
un−p2≤x
n−p2− xn−un2. 3.18
This together with3.2yields that
xn1−p2 ≤α
nyn−p2 1−αnzn−p2
≤αnβnfxn−p 1−βnxn−p2 1−αnun−p2
≤αnβnfxn−p2αn1−βnxn−p2
1−αnxn−p2− un−xn2,
3.19
and hence,
1−αnun−xn2 ≤αnβnfxn−p21−αnβnxn−p2−xn1−p2
≤αnβnfxn−p2−xn−p2
xn1−xnxn−pxn1−p.
Soun−xn → 0note that limn→ ∞βn0 and limn→ ∞xn1−xn0. Since
Wnun−un ≤ Wnun−WnxnWnxn−xnxn−un
≤2xn−unWnxn−xn,
xn−Wnxn ≤ xn−xn1xn1−Wnxn
≤ xn−xn1αnβnfxn−Wnxn
αn1−βnxn−Wnxn 1−αnWnzn−Wnxn
≤ xn−xn1αnβnfxn−Wnxn
αn1−βnxn−Wnxn 1−αnPCun−δnAun−PCxn
≤ xn−xn1αnβnfxn−Wnxnαn
1−βnxn−Wnxn
1−αnun−xn 1−αnδnAun,
3.21
we obtain limn→ ∞xn−Wnxn0,and hence limn→ ∞un−Wnun0. Thus,un−Wun ≤
un−WnunWnun−Wun → 0.
LetQPF.ThenQfis a contraction ofHinto itself. In fact, there existsk∈0,1such
thatfx−fy ≤kx−yfor allx, y∈H. So
Qfx−Qf
y≤fx−fy≤kx−y 3.22
for all x, y ∈ H.SoQf is a contraction by Banach contraction principle 11. SinceH is a complete space, there exists a unique elementx∗∈C⊂Hsuch thatx∗Qfx∗.
Next we show that
lim sup
n→ ∞ fx
∗−x∗, x
n−x∗ ≤0, 3.23
wherex∗Qfx∗.To show this inequality, we choose a subsequence{uni}of{un}such that
lim sup
n→ ∞
fx∗−x∗, u
n−x∗nlim→ ∞fx∗−x∗, uni−x∗
. 3.24
Since{uni}is bounded, there exists a subsequence of{uni}which converges weakly to some
ω∈C, that is,uni ω. FromWun−un → 0, we obtain thatWuni ω. Now we will show
thatω∈FWVIA, CEPF. First, we will showω∈EPF. FromunTrnxn,we have
Fun, yr1 n
y−un, un−xn≥0, ∀y∈C. 3.25
ByA2, we also have
1
rn
and hence
y−uni,
uni−xni rni
≥Fy, uni
. 3.27
Sinceuni−xni/rni → 0 anduni ω, it follows fromA4that 0≥Fy, ωfor ally∈C.
For any 0< t≤1 andy∈C, letytty 1−tω.Sincey∈Candω∈C, then we haveyt∈C and henceFyt, ω≤0. This together withA1andA4yields that
0Fyt, yt≤tFyt, y 1−tFyt, ω≤tFyt, y, 3.28
and thus 0 ≤Fyt, y. FromA3, we have 0 ≤Fω, yfor ally ∈ Cand henceω ∈ EPF.
Now, we show thatω∈FW. Indeed, we assume thatω /∈FW; from Opial’s condition, we have
lim inf
i→ ∞ uni−ω<lim infi→ ∞ uni−Wω
≤lim inf
i→ ∞ uni−WuniWuni−Wω
≤lim inf
i→ ∞ uni−ω.
3.29
This is a contradiction. Thus, we obtain that ω ∈ FW. Finally, by the same argument as in the proof of 3, Theorem 3.1, we can show that ω ∈ VIA, C. Hence ω ∈
FWVIA, CEPF.Hence,
lim sup
n→ ∞
fx∗−x∗, x
n−x∗lim sup n→ ∞
fx∗−x∗, u
n−x∗
lim
i→ ∞
fx∗−x∗, u
ni−x∗
fx∗−x∗, ω−x∗≤0.
3.30
Now we show that limn→ ∞xn−x∗0.
From1.9, we have
xn1−x∗2
αnβnfxn αn1−βnxn 1−αnWnzn−x∗, xn1−x∗
αnβnfxn−x∗, xn1−x∗αn
1−βnxn−x∗, xn1−x∗
1−αnWnzn−x∗, xn1−x∗
≤αnβnkxn−x∗xn1−x∗αnβnfx∗−x∗, xn1−x∗
αn1−βnxn−x∗xn1−x∗ 1−αnxn−x∗xn1−x∗
≤1−αnβn1−kxn−x
∗2x
n1−x∗2 2
αnβnfx∗−x∗, xn1−x∗
,
and hence,
xn1−x∗2≤
1−αnβn1−kxn−x∗2
αnβn1−k1−2kfx∗−x∗, xn1−x∗
. 3.32
Using3.23andLemma 2.2, we conclude that{xn}converges strongly tox∗.Consequently,
{un}converges strongly tox∗.This completes the proof.
UsingTheorem 3.1, we prove the following theorem.
Theorem 3.2. Let H, C, F, f, and {Tn} be given as in Theorem 3.1 and let S be an α-strictly
pseudocontractive mapping such thatF /∅. Suppose thatδn ∈0, b, b <1−αandlimn→ ∞δn0.
Let{xn}and{un}be the sequences and findunsuch that
Fun, yr1 n
y−un, un−xn≥0, ∀y∈C,
ynβnfxn 1−βnxn, n≥1,
xn1 αnyn 1−αnWn1−δnunδnSun,
3.33
where{αn},{βn},{rn}, and{λn}are given as inTheorem 3.1. Then{xn}and{un}converge strongly
tox∗∈F, wherex∗PFfx∗.
Proof. Put A I −S. Then Ais 1−α/2-inverse-strongly-monotone. We haveFS VIC, Aand putPCun−δnun 1−δnunδnSun. So byTheorem 3.1we obtain the desired
result.
Acknowledgments
The author would like to express his thanks to Professor Simeon Reich, Technion-Israel Institute of Technology, Israel, and the anonymous referees for their valuable comments and suggestions on a previous draft, which resulted in the present version of the paper. This work was supported by the Natural Science Foundation of China10871217and Grant KJ080725 of the Chongqing Municipal Education Commission.
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