Volume 2011, Article ID 173430,15pages doi:10.1155/2011/173430
Research Article
An Application of Hybrid Steepest Descent
Methods for Equilibrium Problems and Strict
Pseudocontractions in Hilbert Spaces
Ming Tian
College of Science, Civil Aviation University of China, Tianjin 300300, China
Correspondence should be addressed to Ming Tian,[email protected]
Received 9 December 2010; Accepted 13 February 2011
Academic Editor: Shusen Ding
Copyrightq2011 Ming Tian. 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 use the hybrid steepest descent methods for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudocontraction mapping in the setting of real Hilbert spaces. We proved strong convergence theorems of the sequence generated by our proposed schemes.
1. Introduction
LetHbe a real Hilbert space andCa closed convex subset ofH, and letφbe a bifunction of C×CintoR, whereRis the set of real numbers. The equilibrium problem forφ:C×C → R is to findx∈Csuch that
EP :φx, y≥0 ∀y∈C 1.1
denoted the set of solution by EPφ. Given a mappingT :C → H, letφx, y Tx, y−x for allx, y ∈C, thenz ∈ EPφif and only ifTz, y−z ≥ 0 for all y ∈ C, that is, z is a solution of the variational inequality. Numerous problems in physics, optimizations, and economics reduce to find a solution of1.1. Some methods have been proposed to solve the equilibrium problem, see, for instance,1,2.
Recall that an operatorAis strongly positive if there exists a constantγ >0 with the property
Ax, x ≥γx2, ∀x∈H. 1.2
In 2006, Marino and Xu6introduced the general iterative method and proved that for a givenx0 ∈H, the sequence{xn}is generated by the algorithm
xn1αnγfxn I−αnATxn, n≥0, 1.3
whereTis a self-nonexpansive mapping onH,fis a contraction ofHinto itself withβ∈0,1 and {αn} ⊂ 0,1 satisfies certain conditions, andA is a strongly positive bounded linear operator onHand converges strongly to a fixed-pointx∗ofT which is the unique solution to the following variational inequality:
γf−Ax∗, x−x∗ ≤ 0, forx ∈FT, and is also the optimality condition for some
minimization problem. A mappingS : C → H is said to bek-strictly pseudocontractive if there exists a constantk∈0,1such that
Sx−Sy2
≤x−y2kI−Sx−I−Sy2, ∀x, y∈C. 1.4
Note that the class ofk-strict pseudo-contraction strictly includes the class of nonex-pansive mapping, that is,Sis nonexpansive if and only ifSis 0-srictly pseudocontractive; it is also said to be pseudocontractive ifk1. Clearly, the class ofk-strict pseudo-contractions falls into the one between classes of nonexpansive mappings and pseudo-contractions.
The set of fixed points ofS is denoted by FS. Very recently, by using the general approximation method, Qin et al.7obtained a strong convergence theorem for finding an element ofFS. On the other hand, Ceng et al.8proposed an iterative scheme for finding an element of EPφ∩FSand then obtained some weak and strong convergence theorems. Based on the above work, Y. Liu9introduced two iteration schemes by the general iterative method for finding an element of EPφ∩FS.
In 2001, Yamada10introduced the following hybrid iterative method for solving the variational inequality:
xn1Txn−μλnFTxn, n≥0, 1.5
whereFisk-Lipschitzian andη-strongly monotone operator withk >0,η >0, 0< μ <2η/k2, then he proved that if {λn} satisfyies appropriate conditions, the {xn} generated by 1.5
converges strongly to the unique solution of variational inequality
Fx, x −x ≥0, ∀x∈FixT, x∈FixT. 1.6
2. Preliminaries
Throughout this paper, we always assume thatCis a nonempty closed convex subset of a Hilbert spaceH. We writexn xto indicate that the sequence{xn}converges weakly to x.xn → ximplies that{xn}converges strongly tox. For anyx∈ H, there exists a unique
nearest point inC, denoted byPCx, such that
x−PCx ≤x−y, ∀y∈C. 2.1
Such aPCxis called the metric projection ofHontoC. It is known thatPCis nonexpansive.
Furthermore, forx∈Handu∈C,upcx,⇔ x−u, u−y ≥0, for ally∈C.
It is widely known thatH satisfies Opial’s condition11, that is, for any sequence
{xn}withxn x, the inequality
lim inf
n→ ∞ xn−x<lim infn→ ∞ xn−y, 2.2
holds for everyy∈Hwithy /x. In order to solve the equilibrium problem for a bifunction φ:C×C → R, let us assume thatφsatisfies the following conditions:
A1φx, x 0, for allx∈C,
A2φis monotone, that is,φx, y φy, x≤0, for allx, y∈C,
A3For allx, y, z∈C.
lim
t↓0φ
tz 1−tx, y≤φx, y; 2.3
A4For each fixedx∈C, the functiony→φx, yis convex and lower semicontinuous. Let us recall the following lemmas which will be useful for our paper.
Lemma 2.1see12. Letφbe a bifunction fromC×CintoRsatisfying (A1), (A2),(A3) and (A4) then, for anyr >0andx∈H, there existsz∈Csuch that
φz, y1ry−z, z−x ≥0, ∀y∈C. 2.4
Further, ifTrx{z∈C:φz, y 1/ry−z, z−x ≥0,∀y∈C}, then the following hold: 1Tr is single-valued,
2Tr is firmly nonexpansive, that is,
Trx−Try2 ≤T
rx−Try, x−y, ∀x, y∈H; 2.5
3FTr EPφ,
Lemma 2.2see13. IfS:C → His ak-strict pseudo-contraction, then the fixed-point setFS is closed convex, so that the projectionPFSis well difened.
Lemma 2.3see14. LetS : C → H be ak-strict pseudo-contraction. DefineT :C → Hby Tx λx 1−λSx for eachx ∈ C, then, asλ ∈ k,1, T is nonexpansive mapping such that FT FS.
Lemma 2.4see15. In a Hilbert spaceH, there holds the inequality
xy2
≤ x22y,xy, ∀x, y∈H. 2.6
Lemma 2.5see16. Assume that{an}is a sequence of nonnegative real numbers such that
an1≤
1−γnanγnδn, n≥0, 2.7
where{γn}is a sequence in (0,1) and{δn}is a sequence inÊ, such that
i∞n1γn∞,
iilim supn→ ∞δn≤0or∞n1|δnγn|<∞. Thenlimn→ ∞an0.
3. Main Results
Throughout the rest of this paper, we always assume thatFis aL-lipschitzian continuous and η-strongly monotone operator withL, η >0 and assume that 0< μ <2η/L2.τ μη−μL2/2.
Let {Tλn} be mappings defined asLemma 2.1. Define a mappingSn : C → H bySnx
βnx 1−βnSx, for allx∈C, whereβn∈k,1, then, byLemma 2.3,Snis nonexpansive. We
consider the mappingGnonHdefined by
GnxI−αnμFSnTλnx, x∈H, n∈N, 3.1
whereαn∈0,1. By Lemmas2.1and 2.3, we have
Gnx−Gny≤1−αnτTλ
nx−Tλny
≤1−αnτx−y.
3.2
It is easy to see thatGnis a contraction. Therefore, by the Banach contraction principle, Gnhas a unique fixed-pointxFn ∈Hsuch that
xF n
For simplicity, we will writexnforxFn provided no confusion occurs. Next, we prove
that the sequence{xn}converges strongly to aq∈FS∩EPφwhich solves the variational
inequality
Fq, p−q≥0, ∀p∈FS∩EPφ. 3.4
Equivalently,qPFS∩EPφI−μFq.
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H and φ a bifunction from C×C into R satisfying (A1), (A2), (A3), and (A4). Let S : C → H be a k -strictly pseudocontractive nonself mapping such thatFS∩EPφ/φ. LetF : H → H be an L-Lipschitzian continuous andη-strongly monotone operator onHwithL, η >0and0< μ <2η/L2,
τ μη−μL2/2. Let{x
n}be asequence generated by
φun, yλ1
ny−un, un−xn ≥0, ∀y∈C, ynβnun1−βnSun,
xnI−αnμFyn, ∀n∈N,
3.5
whereun Tλnxn,yn Snun, and{λn} ⊂ 0,∞satisfylim infn→ ∞λn > 0if {αn} and{βn}
satisfy the following conditions:
i{αn} ⊂0,1,limn→ ∞αn0,
ii0≤k≤βn≤λ <1andlimn→ ∞βnλ,
then{xn} converges strongly to a pointq ∈ FS∩EPφwhich solves the variational inequality 3.4.
Proof. First, takep ∈FS∩EPφ. Sinceun Tλnxnandp Tλnp, fromLemma 2.1, for any
n∈N, we have
un−pTλnxn−Tλnp≤xn−p. 3.6
Then, sinceSnpp, we obtain that
yn−pSnun−Snp≤un−p≤xn−p. 3.7
Further, we have
xn−p−αnμFp
I−μαnFyn−I−μαnFp
≤αn−μFp 1−αnτyn−p.
3.8
Hence,{xn}is bounded, and we also obtain that{un}and{yn}are bounded. Notice
that
un−yn≤ un−xnxn−yn
un−xnαn−μFyn.
3.9
ByLemma 2.1, we have
un−p2T
λnxn−Tλnp
2≤x
n−p, un−p
1
2 xn−p
2u
n−p2− un−xn2
. 3.10
It follows that
un−p2
≤xn−p2− xn−un2. 3.11
Thus, fromLemma 2.4,3.7, and3.11, we obtain that
xn−p2α
n−μFpI−μαnFyn−I−μαnFp2
≤1−αnτ2yn−p22αn−μFp, xn−p
≤1−αnτ2un−p22αn−μFp, xn−p
≤1−αnτ2 xn−p2− xn−un2
2αn−μFpxn−p
1−2αnτ αnτ2xn−p2
−1−αnτ2xn−un22αn −μFpxn−p
≤xn−p2 αnτ2xn−p2−1−αnτ2xn−un22αn−μFpxn−p. 3.12
It follows that
1−αnτ2xn−un2 ≤αnτ2xn−p22αnμFpxn−p. 3.13
Sinceαn → 0, therefore
lim
n→ ∞xn−un0. 3.14
From3.9, we derive that
lim
DefineT :C → HbyTx λx 1−λSx, thenT is nonexpansive withFT FS
byLemma 2.3. We note that
Tun−un ≤Tun−ynyn−un≤λ−βnun−Sunyn−un. 3.16
So by3.15andβn → λ, we obtain that
lim
n→ ∞Tun−un0. 3.17
Since{un}is bounded, so there exists a subsequence{uni}which converges weakly to
q. Next, we show thatq∈FS∩EPφ. SinceCis closed and convex,Cis weakly closed. So we haveq∈C. Let us show thatq∈FS. Assume thatq∈FT, Sinceuni qandq /Tq, it
follows from the Opial’s condition that
lim inf
n→ ∞ uni−q<lim infn→ ∞ uni−Tq
≤lim inf
n→ ∞
uni−TuniTuni−Tq
≤lim inf
n→ ∞ uni−q.
3.18
This is a contradiction. So, we getq∈FTandq∈FS.
Next, we show thatq∈EPφ. SinceunTλnxn, for anyy∈C, we obtain
φun, y λ1 n
y−un, un−xn≥0. 3.19
FromA2, we have
1 λn
y−un, un−xn≥φy, un. 3.20
Replacingnbyni, we have
y−uni,
uni−xni
λni
≥φy, uni
. 3.21
Sinceuni −xni/λni → 0 anduni q, it follows fromA4that 0 ≥ φy, q, for all
y ∈ C. Letzt ty 1−tqfor all t ∈ 0,1andy ∈ C, then we havezt ∈ Cand hence φzt, q≤0. Thus, fromA1andA4, we have
and hence 0 ≤ φzt, y. FromA3, we have 0 ≤ φq, yfor ally ∈ Cand henceq ∈EPφ.
Therefore,q∈FS∩EPφ. On the other hand, we note that
xn−q−αnμFqI−μαnFyn−I−μαnFq. 3.23
Hence, we obtain
xn−q2−α
nμFq, xn−qI−μαnFyn−I−μαnFq, xn−q
≤αn−μFq, xn−q 1−αnτxn−q2.
3.24
It follows that
xn−q2
≤ τ1−μFq, xn−q. 3.25
This implies that
xn−q2≤
−μFq, xn−q
τ . 3.26
In particular,
xni−q2≤
−μFq, xni −q
τ . 3.27
Sincexni q, it follows from3.27thatxni → qasi → ∞. Next, we show thatq
solves the variational inequality3.4. As a matter of fact, we have
xn I−αnμFyn I−αnμFSnTλnxn,
3.28
and we have
μFxn−α1 n
I−SnTλnxn−μαnFxn−FSnTλnxn
. 3.29
Hence, forp∈FS∩EPφ,
μFxn, xn−p−α1 n
I−SnTλnxn−μαnFxn−FSnTλnxn
, xn−p
− 1
αn
I−SnTλnxn−I−SnTλnp, xn−p
μFxn−FSnTλnxn, xn−p
.
SinceI−SnTλn is monotonei.e.,x−y,I−SnTλnx−I−SnTλny ≥0, for allx, y∈H. This
is due to the nonexpansivity ofSnTλn.
Now replacingnin3.30withniand lettingi → ∞, we obtain
μFq, q−p lim
i→ ∞
μFxni, xni−p
≤ lim
i→ ∞μ
Fxni −FSnTλnxni, xni−p
0.
3.31
That is,q∈FS∩EPφis a solution of3.4. To show that the sequence{xn}converges
strongly toq, we assume thatxnk → x. Similiary to the proof above, we derivex ∈FS∩
EPφ. Moreover, it follows from the inequality3.31that
μFq, q−x≤0. 3.32
Interchangeqandxto obtain
μFx, x−q≤0. 3.33
Adding up3.32and3.33yields
μηq−x2≤q−x, μFq−μFx≤0. 3.34
Hence,qx, and thereforexn → qasn → ∞,
I−μFq−q, q−p≥0,∀p∈FS∩EPφ. 3.35
This is equivalent to the fixed-point equation
PFS∩EPφI−μFqq. 3.36
Theorem 3.2. LetCbe a nonempty closed convex subset of a real Hilbert spaceHandφa bifunction from C × C into R satisfying (A1), (A2), (A3) and (A4). Let S : C → H be a k-strictly pseudocontractive nonself mapping such thatFS∩EPφ/φ. LetF:H → Hbe anL-Lipschitzian continuous andη-strongly monotone operator onH withL, η > 0. Suppose that0 < μ < 2η/L2, τ μη−μL2/2. Let{x
n}and{un}be sequences generated byx1∈Hand
φun, y λ1 n
y−un, un−xn≥0, ∀y∈C, ynβnun1−βnSun,
xn1
I−αnμFyn, ∀n∈N,
whereunTλnxn,ynSnunif{αn},{βn}, and{λn}satisfy the following conditions:
i{αn} ⊂0,1,limn→ ∞αn0,∞n1αn∞,
∞
n1|αn1−αn|<∞, ii0≤k≤βn≤λ <1andlimn→ ∞βnλ,∞n1|βn1−βn|<∞, iii{λn} ∈0,∞,limn→ ∞λn>0and∞n1|λn1−λn|<∞,
then{xn}and{un}converge strongly to a pointq∈FS∩EPφwhich solves the variational inequality3.4.
Proof. We first show that{xn}is bounded. Indeed, pick anyp∈FS∩EPφto derive that
xn1−p−αnμFp
I−μαnFyn−I−μαnFp
≤αn−μFp 1−αnτxn−p
≤1−αnτxn−pαn−μFp.
3.38
By induction, we have
xn−p ≤max
x1−p,
1 τ −μF
p
, ∀n∈N, 3.39
and hence {xn} is bounded. From3.6 and 3.7, we also derive that {un} and {yn} are
bounded. Next, we show thatxn1−xn → 0. We have
xn1−xnI−αnμF
yn−I−αn−1μFyn−1
I−αnμFyn−I−αnμFyn−1I−αnμFyn−1−I−αn−1μFyn−1
≤1−αnτyn−yn−1|αn−αn−1|μFyn−1
≤1−αnτyn−yn−1K|αn−αn−1|,
3.40
where
KsupμFyn:n∈N<∞. 3.41
On the other hand, we have
yn−yn−1Snun−Sn−1un−1
≤ Snun−Snun−1Snun−1−Sn−1un−1
≤ un−un−1Snun−1−Sn−1un−1.
Fromun1Tλn1xn1andun Tλnxn, we note that
φun1, y
1
λn1
y−un1, un1−xn1
≥0, ∀y∈C, 3.43
φun, yλ1 n
y−un, un−xn≥0, ∀y∈C. 3.44
Puttingyunin3.43andyun1in3.44, we have
φun1, un λ1
n1un−un1, un1−xn1 ≥
0,
φun, un1 λ1
nun1−un, un−xn ≥0.
3.45
So, fromA2, we have
un1−un,unλ−xn
n −
un1−xn1
λn1
≥0 , 3.46
and hence
un1−un, un−un1un1−xn−λλn
n1un1−xn1
≥0. 3.47
Since limn→ ∞λn > 0, without loss of generality, let us assume that there exists a real
number a such thatλn> a >0 for alln∈N. Thus, we have
un1−un2≤
un1−un, xn1−xn
1− λn λn1
un1−xn1
≤ un1−un
xn1−xn
1− λn
λn1
un1−xn1
un1−un ≤ xn1−xna1|λn1−λn|M0,
3.48
whereM0sup{un−xn:n∈N}. Next, we estimateSnun−1−Sn−1un−1. Notice that
Snun−1−Sn−1un−1βnun−11−βnSun−1−βn−1un−11−βn−1Sun−1
≤βn−βn−1un−1−Sun−1.
3.49
From3.48,3.49, and3.42, we obtain that
yn−yn−1≤ xn−xn−1M0
a |λn−λn−1|βn−βn−1un−1−Sun−1
≤ xn−xn−1|λn−λn−1|M1βn−βn−1M1,
whereM1is an appropriate constant such that
M1≥ Ma0 un−1−Sun−1, ∀n∈N. 3.51
From3.41and3.50, we obtain
xn1−xn ≤K|αn−αn−1| 1−αnτ
xn−xn−1|λn−λn−1|M1βn−βn−1M1
≤1−αnτxn−xn−1M|αn−αn−1||λn−λn−1|βn−βn−1,
3.52
whereMmaxK, M1. Hence, few byLemma 2.5, we have
lim
n→ ∞xn1−xn0. 3.53
From3.48and3.50,|λn−λn−1| → 0 and|βn−βn−1| → 0, we have
lim
n→ ∞un1−un0, nlim→ ∞yn1−yn0. 3.54
Since
xn1
I−αnμFyn, 3.55
it follows that
xn−yn≤ xn−xn1xn1−yn
xn−xn1αn−μFyn.
3.56
Fromαn → 0 and3.53, we have
lim
n→ ∞xn−yn0. 3.57
Forp∈FS∩EPφ, we have
un−p2Tλnxn−Tλnp
2≤x
n−p, un−p
1
2 xn−p
2u
n−p2− un−xn2
. 3.58
This implies that
un−p2≤x
Then, from3.7and3.59, we derive that
xn1−p2−μα
nFpI−μαnFyn−I−μαnFp2
≤1−αnτ2yn−p2αn2−μFp22αn−μFpyn−p
≤un−p2α2n−μFp22αn−μFpyn−p
≤xn−p2− xn−un2αn2−μFp22αn−μFpyn−p.
3.60
Sinceαn → 0,xn−xn1 → 0, we have
lim
n→ ∞xn−un0. 3.61
From3.57and3.61, we obtain that
un−yn ≤ un−xnxn−yn → 0, asn → ∞. 3.62
DefineT :C → HbyTx λx 1−λSx, thenT is nonexpansive withFT FS
byLemma 2.3. Notice that
Tun−un ≤Tun−ynyn−un
≤λ−βnun−Sunyn−un.
3.63
By3.62andβn → λ, we obtain that
lim
n→ ∞Tun−un0. 3.64
Next, we show that lim supn→ ∞μFq, q−xn ≤0, whereq PFS∩EPφI−μFqis a
unique solution of the variational inequality3.4. Indeed, take a subsequence{xni}of{xn}
such that
lim
i→ ∞
μFq, q−xni
lim sup
n→ ∞
μFq, q−xn. 3.65
Since{xni} is bounded, there exists a subsequence {xnij} of {uni} which converges
weakly tow.
Without loss of generality, we can assume thatuni w. From3.61and3.64, we
obtainxni wandTuni w. By the same argument as in the proof ofTheorem 3.1, we have
w∈FS∩EPφ. SinceqPFS∩EPφI−μFq, it follows that
lim sup
n→ ∞
Fromxn1−q−αnμFq I−μαnFyn−I−μαnFq, we have
xn1−q2 ≤I−μα
nFyn−I−μαnFq22αn−μFq, xn1−q
≤1−αnτ2xn−q22αn−μFq, xn1−q
.
3.67
This implies that
xn1−q2≤
1−2αnτ αnτ2xn−q22αn−μFq, xn1−q
1−2αnτxn−q2 αnτ2xn−q22αn−μFq, xn1−q
1−2αnτxn−q22αnτ
αnτ2
2τ M
∗1
τ
−μFq, xn1−q
1−γnxn−q2γnδn,
3.68
whereM∗sup{xn−q2:n∈N},γn2αnτ, andδn αnτ2/2τM∗1/τ−μFq, xn1−q.
It is easy to see thatγn → 0,∞n1γn∞, and lim supn→ ∞δn ≤0 by3.66. Hence by
Lemma 2.5, the sequence{xn}converges strongly toq.
Acknowledgments
M. Tian was supported in part by the Science Research Foundation of Civil Aviation University of Chinano. 2010kys02. He was also Supported in part by The Fundamental Research Funds for the Central UniversitiesGrant no. ZXH2009D021.
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