Volume 2009, Article ID 656534,17pages doi:10.1155/2009/656534
Research Article
Fixed Points Approximation and Solutions of Some
Equilibrium and Variational Inequalities Problems
Bashir Ali
1, 21Mathematics Stream, African University of Science and Technology, Abuja, Nigeria
2Department of Mathematical Sciences, Bayero University, Kano, P.M.B. 3011, Nigeria
Correspondence should be addressed to Bashir Ali,[email protected]
Received 28 September 2009; Accepted 16 December 2009
Recommended by Asao Arai
We prove a new strong convergence theorem for an element in the intersection of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of some variational inequality problems, and the set of solutions of some equilibrium problems using a new iterative scheme. Our theorem generalizes and improves some recent results.
Copyrightq2009 Bashir Ali. 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; a mapping B : DB → H is said to be monotone if for all
x, y∈DB
Bx−By, x−y ≥0. 1.1
For someλ >0,the mappingBis said to beλ-inverse strongly monotone if
Bx−By, x−y ≥λBx−By2. 1.2
Aλ-inverse strongly monotone map is some time calledλ-cocoercive. A mapBis said to be
relaxedλ-cocoercive if there exists a constantλ >0 such that
Bis said to be relaxedλ, γ-cocoercive, if there existλ, γ >0 such that
Bx−By, x−y ≥ −λBx−By2γx−y2. 1.4
A mapB:H → His said to beλ-Lipschitzian if there exists a real numberλ≥0 such that
Bx−By ≤λx−y ∀x, y∈H. 1.5
Bis a contraction map, if in the above inequalityλ∈0,1and nonexpansive ifλ1.
LetKbe a nonempty, closed, and convex subset of a real Hilbert spaceH.A variational inequality problem is searched forx∗∈Ksuch that
Bx∗, y−x∗≥0 ∀y∈K, 1.6
whereB is some nonlinear mapping of K intoH. Inequality1.6 is called the variational inequality.
Recall that for eachx ∈ H there exists a unique nearest point inK toxdenoted by
PKx.That is,x−PKx ≤ x−yfor ally∈K.PKis called a metric projection ofHontoK.
The mappingPKis nonexpansive in this setting, that is,PKx−PKy ≤ x−yfor allx, y∈H. It is also known thatPKsatisfies the following inequalityPKx−PKy2≤ x−y, PKx−PKy.
The solution set of the problem1.6is denoted by VIK, B.It is well knownsee1 thatx∗∈VIK, Bif and only if
x∗PKx∗−λBx∗, ∀λ >0. 1.7
A monotone mapBis said to be maximal if the graphΓBofBis not properly contained in the graph of any other monotone map, whereΓB {x, y ∈ H×H : y ∈ Bx}for a multivalued mapB.It is also known thatBis maximal monotone if and only if forx, f ∈
H×H,x−y, f−g ≥0 for everyy, g∈ΓBimpliesf∈Bx.LetBbe a monotone mapping defined fromKintoHand letNKqbe a normal cone toKatq∈K,that is,NKq{p∈H:
q−u, p ≥, for allu∈K}.Define a mapMby
Mq
⎧ ⎨ ⎩
BqNKq, q∈K,
∅, q /∈K. 1.8
Then,Mis maximal monotone andx∗∈M−10⇔x∗∈VIK, B,see, for example,2.
LetG : K×K → Rbe a bifunction on a closed convex nonempty subsetK of a real Hilbert spaceH.An equilibrium problem is searched forx∗∈Ksuch that
Gx∗, y≥0 ∀y∈K. 1.9
Several physical problemssuch as the theories of lubrications, filterations and flows, moving boundary problems, see, e.g., 1, 3 can be reduced to variational inequality or equilibrium problems. Consequently, these problems have solutions as the solutions of these resultant variational inequality or equilibrium problems.
Maing´e 4 introduced a Halpern-type scheme and proved a strong convergence theorem for family of nonexpansive mappings in Hilbert space.
Recently, S. Takahashi and W. Takahashi5introduced an iterative scheme which they used to study the problem of approximating a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping.
More recently, Kumam and Katchang6, W. Kumam and P. Kumam7, Li and Su8 and many otherssee, e.g.9–12and the references contained in themstudied the problem of fixed point approximations and solutions of some equilibrium and, or solutions of some variational inequalities problems.
In this paper we introduce a new iterative scheme for approximation of a common element in the intersection of the set of fixed points of some countable family of nonexpansive mappings, the set of solutions of some equilibrium problem, and the set of solutions of some variational inequality problem and prove a new theorem. Our theorem generalizes and improves some recent results.
2. Preliminaries
For a sequence {xn} the notation xn → x∗ and xn x∗ means that the sequence {xn} converges strongly and weakly to x∗, repectively. A Banach space E is said to satisfy an Opial’s conditionor in other words is an Opial’s space if for a sequence{xn}in Ewith xn x∗,then
lim inf
n→ ∞ xn−x
∗<lim inf
n→ ∞ xn−y for any y∈E y /x
∗.
2.1
It is well known that Hilbert spaces are Opial’s spacessee13. In the sequel we shall make use of the following results.
Lemma 2.1see14. LetKbe a nonempty closed convex subset ofHand letGbe a bifunction of
K×KintoR satisfying
A1Gx, x 0 forallx∈K;
A2Gis monotone, i.e.Gx, y Gy, x≤0 forallx, y∈K;
A3forallx, y, z∈K,limsupt→0Gtz 1tx,y≤Gx,y;
A4forall x∈K,Gx, .is convex and lower semicontinuous.
Letr >0 andx∈H.Then there existsz∈Ksuch that
Lemma 2.2see15. LetKbe a nonempty closed convex subset ofHand letGbe a bifunction of
K×KintoRsatisfyingA1–A4.Forr >0 andx∈Hdefine a mapTr :H → Kby
Trx
z∈K:Gz, y1
ry−z, z−x ≥0, ∀y∈K
. 2.3
Then, the following holds:
1Tr is single-valued;
2Tr is firmly nonexpansive, that is, for anyx, y∈H
Trx−Try2 ≤Trx−Try, x−y; 2.4
3FixTr EPG;
4EPGis closed and convex.
Lemma 2.3see16. LetHbe a real inner product space. Then, the following inequality holds:
xy2≤ x22y, xy ∀x, y∈H. 2.5
Lemma 2.4see17. Let{xn}and{yn}be bounded sequences in a Banach spaceEand let{βn}
be a sequence in0,1with 0<lim infβn ≤lim supβn <1. Supposexn1 βnyn 1−βnxnfor
all integersn≥0 and lim supyn1−yn − xn1−xn≤0. Then, limyn−xn0.
Lemma 2.5see18. Let{an}be a sequence of nonnegative real numbers satisfying the following
relation:
an1≤1−αnanαnσnγn, n≥0, 2.6
Wherei{αn} ⊂ 0,1,αn ∞;iilim supσn ≤ 0;iiiγn ≥ 0;n ≥ 0,γn < ∞.Then,
an → 0 asn → ∞.
3. Main Results
In the sequel we assume that the sequences{αn},{σi,n}i⊂0,1satisfyi≥1σi,n1−αn.
Theorem 3.1. LetKbe a nonempty, closed, and convex subset of a real Hilbert spaceH.LetGbe a
bifunction fromK×KtoRwhich satisfies conditionsA1–A4.Let{Ti, i1,2,3, . . .}be family
KintoHsuch thatF:∞i1FixTi∩EPG∩VIK, B/∅.For an arbitrary but fixedδ ∈0,1,
let{xn}and{yn}be sequences generated by
x1, u∈H,
Gyn, ηrn1η−yn, yn−xn ≥0, ∀η∈K,
xn1αnu 1−δ1−αnxnδ
i≥1
σinTiPKI−snByn, n≥1,
3.1
where{αn}and{σin}are sequences in0,1and{rn}and{sn}are sequences in0,∞satisfying
C1limn→ ∞αn0, C2∞n1αn∞,
C3{sn} ⊂a, bfor somea, bsatisfying 0≤a≤b≤2γ−λμ2/μ2,
C4limn→ ∞|sn1−sn|0, limn→ ∞|rn1−rn|0, limn→ ∞i≥1|σi,n1−σi,n|0, C5lim infn→ ∞rn>0.
Then, both{xn}and{yn}converge strongly toPFu.
Proof. First, we show thatI−snBis nonexpansive, actually, using the property ofBwe have
forx, y∈H,
I−snBx−I−snBy2x−y−snBx−By2
x−y2−2snx−y, Bx−Bys2
nBx−By2
≤ x−y2−2s n
−λBx−By2γx−y2s2
nBx−By2
≤ x−y22s
nμ2λx−y2−2snγx−y2μ2s2nx−y2 12snμ2λ−2snγμ2s2nx−y2
≤ x−y2,
3.2
and thusI−snBis nonexpansive.
Letx∗ ∈F; sinceyn Trnxn,n∈ Nand the fact thatTrn is firmly nonexpansiveand
hence nonexpansivewe have the following:
yn−x∗Trnxn−Trnx∗ ≤ xn−x∗. 3.3
We claim that{xn}satisfies
We prove this by induction. Cearly the result is true forn1.Assume that the result holds fornkfor somek∈N.Then, fornk1 we have
xk1−x∗αku−x∗ 1−δ1−αkxk−x∗ δ
i≥1
σikTiPKI−skByk−x∗
≤αku−x∗ 1−δ1−αkxk−x∗
δ i≥1
σikTiPKI−skByk−TiPKI−skBx∗
≤αku−x∗ 1−αkxk−x∗
≤max{u−x∗,xk−x∗}.
3.5
Hence the result, and so{xn}is bounded. Furthermore,{yn},{TiPKyn−snByn}and{Byn}
are each bounded.
We now show that limn→ ∞xn1−xn 0.Note thatyn Trnxn,yn1 Trn1xn1,so
that
Gyn, ηrn1η−yn, yn−xn ≥0 ∀η∈K,
Gyn1, η
r1 n1
η−yn1, yn1−xn1 ≥0 ∀η∈K.
3.6
Using3.6andA2,we have
yn1−yn,ynr−xn
n −
yn1−xn1 rn1
≥0. 3.7
which implies that
yn1−yn, yn−yn1yn1−xn−rrn n1
yn1−xn1
≥0. 3.8
from which we get
yn1−yn2≤ yn1−yn
xn1−xn1−rrn n1
yn1−xn1
If, without loss of generality M,m are real numbers such that rn > m > 0 for all nand
M:supn,i{yn−xn,TiPKI−snByn,Byn}we then have
yn1−yn ≤ xn1−xn1−rrn n1
yn1−xn1
≤ xn1−xnMm|rn1−rn|.
3.10
Now, define two sequences{βn}and{zn}byβn: 1−δαnδandzn: xn1−xnβnxn/βn.
Then,
zn
αnuδi≥1σi,nTiPKI−snByn
βn . 3.11
Observe that{zn}is bounded and that
zn1−zn − xn1−xn ≤αnβ 1 n1 −
αn βn
u
δ1β−αn1 n1
yn1−yn − xn1−xn
βδ n1
i≥1
|σi,n1−σi,n|TiPKI−sn1Byn
β δ n1βn
βn−βn1 i≥1
σi,n1TiPKI−sn1Byn
δ1−αβnByn
n |sn−sn1|.
3.12
Using3.10we have that
zn1−zn − xn1−xn ≤αnβ 1 n1 −
αn βn
u
δ1β−αn1 n1 −
1xn1−xnδ
1−αn1M
βn1m |
rn1−rn|
βδM n1
i≥1
|σi,n1−σi,n| δM βn1βn
βn−βn1δ
1−αnM
βn |sn−sn1|. 3.13
This implies
lim sup
and by Lemma2.4, limn→ ∞zn−xn0.Hence,
xn1−xnβnzn−xn −→0 asn−→ ∞. 3.15
Using3.10we have
yn1−yn −→0 as n−→ ∞. 3.16
We now have
yn−x∗2 Trnxn−Trnx∗2
≤ Trnxn−Trnx∗, xn−x∗
yn−x∗, xn−x∗
1
2
yn−x∗2xn−x∗2− xn−yn2
,
3.17
so that
yn−x∗2≤ xn−x∗2− xn−yn2. 3.18
But,
xn1−x∗2
αnu−x∗ 1−δ1−αnxn−x∗
δ
i≥1
σinTiPKI−snByn−TiPKI−snBx∗ 2
≤αnu−x∗2 1−δ1−αnxn−x∗2 δ
i≥1
σinTiPKI−snByn−TiPKI−snBx∗2
≤αnu−x∗2 1−δ1−αnxn−x∗2δ1−αnyn−x∗2.
3.19
Putting3.18in3.19, we have
δ1−αnxn−yn2≤αnu−x∗2xn−x∗2− xn1−x∗2
αnu−x∗2 xn−x∗ − xn1−x∗xn−x∗xn1−x∗
≤αnu−x∗2 xn−xn
1xn−x∗xn1−x∗,
and hence,
lim
n→ ∞xn−yn0. 3.21
Observe also that ifwn:PKI−snByn,then
wn−x∗2PKI−snByn−PKI−snBx∗2
≤ yn−x∗−snByn−Bx∗2
yn−x∗2−2snyn−x∗, Byn−Bx∗s2
nByn−Bx∗2
≤ yn−x∗2−2sn−λByn−Bx∗2γyn−x∗2s2
nByn−Bx∗2 yn−x∗22snλByn−Bx∗2−2snγyn−x∗2s2nByn−Bx∗2
≤ yn−x∗2
2snλs2n−2snγ
μ2
Byn−Bx∗2.
3.22
Also,
xn1−x∗2
αnu−x∗ 1−δ1−αnxn−x∗
δ
i≥1
σinTiPKI−snByn−TiPKI−snBx∗ 2
≤αnu−x∗2 1−δ1−αnxn−x∗2
δ i≥1
σinTiPKI−snByn−TiPKI−snBx∗2
αnu−x∗2 1−δ1−αnxn−x∗2δ
i≥1
σinTiwn−x∗2
≤αnu−x∗2 1−δ1−αnxn−x∗2δ1−αnwn−x∗2
≤αnu−x∗2 1−δ1−αnxn−x∗2
δ1−αn
yn−x∗2
2snλs2n−2snγμ2
Byn−Bx∗2
≤αnu−x∗2 1−αnxn−x∗2δ1−αn
2snλs2n−2 snγ μ2
which implies
−δ1−αn
2snλs2n−2snγ
μ2
Byn−Bx∗2≤αnu−x∗2xn−x∗2− xn
1−x∗2 3.24
so that
lim
n→ ∞Byn−Bx
∗0.
3.25
We go further to prove that for eachi∈N,
lim
n→ ∞TiPKI−snByn−yn0. 3.26
Consider the following estimates:
TiPKI−snByn−yn2TiPKI−snByn−x∗x∗−yn2
x∗−yn22TiPKI−snByn−x∗, x∗−yn TiPKI−snByn−x∗2
≤2x∗−yn22TiPKI−snByn−ynyn−x∗, x∗−yn
2yn−x∗22TiPKI−snByn−yn, x∗−yn −2yn−x∗2 2TiPKI−snByn−yn, x∗−yn.
3.27
Using3.1, we have
xn1−x∗, yn−x∗αnu−x∗, yn−x∗ 1−αn1−δxn−x∗, yn−x∗ δ
i≥1
σi,nTiPKI−snByn−ynyn−x∗, yn−x∗
αnu−x∗, yn−x∗ 1−αn1−δxn−ynyn−x∗, yn−x∗
δ i≥1
σi,nTiPKI−snByn−yn, yn−x∗δ1−αnyn−x∗2
αnu−x∗, yn−x∗ 1−αn1−δxn−yn, yn−x∗
δ i≥1
σi,nTiPKI−snByn−yn, yn−x∗ 1−αn1−δyn−x∗2
δ1−αnyn−x∗2,
which implies
δ
i≥1
σi,nTiPKI−snByn−yn, x∗−ynαnu−x∗, yn−x∗
1−αn1−δxn−yn, yn−x∗
1−αnyn−x∗, yn−x∗
− xn1−x∗, yn−x∗ αnu−xn1, yn−x∗
1−αn1−δxn−yn, yn−x∗
1−αnyn−xn1, yn−x∗.
3.29
Using this and3.27, we get
δ
2
i≥1
σi,nTiPKI−snByn−yn2≤αnu−xn1, yn−x∗
1−αn1−δxn−yn, yn−x∗
1−αnyn−xn1, yn−x∗.
3.30
Since{xn},{yn}are each bounded and using3.21, we have that
lim
n→ ∞TiPKI−snByn−yn0 ∀i∈N. 3.31
Using this and3.21, we also have
lim
n→ ∞TiPKI−snByn−xn0 ∀i∈N. 3.32
Next we show that lim supn→ ∞u−w, xn−w ≤0,wherew P∞
i1FixTi∩EPG∩VIK,Bu.Let
{xnj}be a subsequence of{xn}such that lim supn→ ∞u−w, xn−wlimj→ ∞u−w, xnj−w.
Since{yn}is bounded, there exists a subsequence {ynj}of {yn}such that ynj z.
Then,z∈∞i1FixTiPKI−snB,otherwise, fori∈N,we have
lim inf
j→ ∞ ynj−z<lim infj→ ∞
yn
j−TiPK
I−snjB
z
≤lim inf
j→ ∞
y
nj−TiPK
I−snjB
ynj TiPK
I−snjB
ynj−TiPK
I−snjB
z
≤lim inf
j→ ∞ ynj−z,
3.33
Next we show thatz∈EPG.Sinceyn Trnxn,we have
Gyn, η r1
nη−yn, yn−xn ≥0, ∀η∈K. 3.34
It follows fromA2that
η−yn,ynr−xn n
≥Gη, yn, 3.35
and so
η−yni,
yni−xni
rni
≥Gη, yni
. 3.36
Sinceyni −xni/rni → 0,yni zand usingA4,we haveGη, z ≤0 for allη ∈K.For a
real numbert, 0< t≤1 andη∈K,letηttη 1−tz.Clearlyηt∈K,so that usingA1and
A4,we have
0Gηt, ηt≤tGηt, η 1−tGηt, z≤tGηt, η. 3.37
This impliesGηt, η≥0,and using this andA3we have thatGz, η≥0 for allη∈Kand
hencez∈EPG.
Next we show thatz∈VIK, B. Letx∗∈Fthen, we have the following:
wn−x∗2PKI−snByn−PKI−snBx∗2
≤ yn−snByn−x∗−snBx∗, wn−x∗
yn−x∗, wn−x∗ −snByn−Bx∗, wn−x∗
≤ 1
2
wn−x∗2yn−x∗2− wn−yn2−snByn−Bx∗, wn−x∗,
3.38
so that
We then have
xn1−x∗2αnu−x∗ 1−αn1−δxn−x∗ δ
i≥1
σi,nTiwn−x∗2
≤αnu−x∗2 1−αn1−δxn−x∗2 1−αnδwn−x∗2
≤αnu−x∗2 1−αn1−δxn−x∗2
1−αnδyn−x∗2− wn−yn2−2snByn−Bx∗, wn−x∗
≤αnu−x∗2 1−αn1−δxn−x∗2
1−αnδxn−x∗2− wn−yn2−2snByn−Bx∗, wn−x∗
αnu−x∗2 1−αnxn−x∗2−δ1−αnwn−yn2
−2snδ1−αnByn−Bx∗, wn−x∗,
3.40
so that
δ1−αnwn−yn2≤αnu−x∗2xn−x∗2− xn1−x∗2
−2δsn1−αnByn−Bx∗, wn−x∗,
3.41
and asαn → 0,xn−xn1 → 0 andByn−Bx∗ → 0 asn → ∞,we get
lim
n→ ∞yn−wn0. 3.42
Using this and3.21we also have
lim
n→ ∞xn−wn0. 3.43
AsBis a relaxedλ, γ-cocoercive and using conditionC3we have forx, y∈H
and soBis monotone. If
Mq
⎧ ⎨ ⎩
BqNKq, q∈K,
∅, q /∈K, 3.45
thenMis maximal monotone. LetΓMdenote the graph ofM.
Letq, p∈ΓM.Sincep−Bq∈NKqandwn ∈K,we havep−Bq, q−wn ≥0 by definition ofNKq.Also, aswn PKyn−snByn using property of the projectionPK, we
have
wn−yn−snByn, q−wn ≥0, 3.46
and hence
wn−yn
sn Byn, q−wn
≥0. 3.47
Using this, we obtain the following estimates:
p, q−wni ≥ Bq, q−wni
≥ Bq, q−wni −
wn
i−yni
sni
Byni, q−wni
Bq−wni−yni
sn −Byni, q−wni
Bq−Bwni, q−wniBwni−Byni, q−wni −
wn
i −yni
sn , q−wni
≥ Bwni−Byni, q−wni −
w
ni −yni
sn , q−wni
,
3.48
which impliesp, q−z ≥0lettingi → ∞.
SinceMis maximal monotone, we obtained thatz∈M−10and hencez∈VIK, B. We mention here that since we proved that z ∈ ∞i1FixTiPKI −snB and z ∈
VIK, B,thenz∈∞i1FixTi.So, clearlyz∈∞i1FixTi∩EPG∩VIK, B. SincewP∞
i1FixTi∩EPG∩VIK,Bu, we have
lim sup
n→ ∞ u−w, xn−wilim→ ∞u−w, xni−wu−w, z−w ≤0. 3.49
From the recursion formula3.1and Lemma2.3, we have
xn1−w2
αnu−w 1−αn1−δxn−w
δ i≥1
σi,nTiPKI−snByn−w
2
≤ 1−αn1−δxn−w δ
i≥1
σi,nTiPKI−snByn−w 2
2αnu−w, xn1−w
≤
1−αn1−δxn−wδ
i≥1
σi,nTiPKI−snByn−w
2
2αnu−w, xn1−w
≤1−αnxn−w22αnu−w, xn1−w.
3.50
Using Lemma2.5, we have that{xn}and consequently{yn}converge towand the proof is complete.
The following corollaries follow from Theorem3.1.
Corollary 3.2. LetK, GandBbe as in Theorem3.1and let{Ti, i1,2,3, . . . , N}be finite family of
nonexpansive mappings ofKintoH.LetF:Ni1FixTi∩EPG∩VIK, B/∅. For an arbitrary
but fixedδ∈0,1,let{xn}and{yn}be sequences generated by
x1, u∈H,
Gyn, ηrn1η−yn, yn−xn ≥0, ∀η∈K,
xn1αnu 1−δ1−αnxnδ N
i1
σinTiPKI−snByn, n≥1,
3.51
where{αn}and{σin}are sequences in0,1and{rn}and{sn}are sequences in0,∞satisfying
C1limn→ ∞αn0,Ni1σin 1−αn, C2∞n1αn∞,
C3{sn} ⊂a, bfor somea, bsatisfying 0≤a≤b≤2γ−λμ2/μ2,
C4limn→ ∞|sn1−sn|0, limn→ ∞|rn1−rn|0, limn→ ∞Ni1|σi,n1−σi,n|0, C5lim infn→ ∞rn>0.
Corollary 3.3. LetK andGbe as in Theorem3.1. LetT be a nonexpansive map ofK intoH. Let
F : FixT∩EPG/∅. For an arbitrary but fixed δ ∈ 0,1, let{xn} and {yn} be sequences
generated by
x1, u∈H,
Gyn, ηrn1η−yn, yn−xn ≥0, ∀η∈K,
xn1αnu 1−δ1−αnxnδ1−αnTyn, n≥1,
3.52
where{αn}is a sequence in0,1and{rn}is a sequences in0,∞satisfying
C1limn→ ∞αn0 C2∞n1αn∞,
C3limn→ ∞|rn1−rn|0, C4lim infn→ ∞rn>0.
Then, both{xn}and{yn}converge strongly toPFu.
Remark 3.4. Prototypes of the sequences{αn}and{σi,n}in our theorem are the following:
αn: n11, σi,n: n
2in1 ∀i∈N. 3.53
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