Volume 2010, Article ID 642584,16pages doi:10.1155/2010/642584
Research Article
Hierarchical Convergence of a
Double-Net Algorithm for Equilibrium Problems
and Variational Inequality Problems
Yonghong Yao,
1Yeong-Cheng Liou,
2and Chia-Ping Chen
31Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
2Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan 3Department of Computer Science and Engineering, National Sun Yat-sen University,
Kaohsiung 80424, Taiwan
Correspondence should be addressed to Chia-Ping Chen,[email protected]
Received 21 May 2010; Accepted 22 December 2010
Academic Editor: Satit Saejung
Copyrightq2010 Yonghong Yao et al. 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 consider the following hierarchical equilibrium problem and variational inequality problem abbreviated as HEVP: find a point x∗ ∈ EPF, B such that Ax∗, x−x∗ ≥ 0, for all x ∈ EPF, B, whereA,Bare two monotone operators and EPF, Bis the solution of the equilibrium problem of findingz ∈ C such thatFz, y Bz, y−z ≥ 0, for ally ∈ C. We note that the problemHEVPincludes some problems, for example, mathematical program and hierarchical minimization problems as special cases. For solvingHEVP, we propose a double-net algorithm which generates a net{xs,t}. We prove that the net{xs,t}hierarchically converges to the solution of HEVP; that is, for each fixedt∈0,1, the net{xs,t}converges in norm, ass → 0, to a solution
xt ∈EPF, Bof the equilibrium problem, and ast → 0, the net{xt}converges in norm to the
unique solutionx∗ofHEVP.
1. Introduction
LetH be a real Hilbert space with inner product·,·and norm · , respectively, and let Cbe a nonempty closed convex subset ofH. Recall that a mappingAofCintoHis called monotone if
Au−Av, u−v ≥0, 1.1
for allu, v∈CandA:C → His calledα-inverse strongly monotone mapping if there exists a positive real numberαsuch that
for allu, v∈C. It is obvious that anyα-inverse strongly monotone mappingAis monotone and 1/α-Lipschitz continuous.
Recently, the following problem has attracted much attention: find hierarchically a fixed point of a nonexpansive mappingTwith respect to a nonexpansive mappingP, namely,
Find x∈FixTsuch thatx−Px, x−x ≤0, ∀x∈FixT. 1.3
Some algorithms for solving the hierarchical fixed point problem1.3have been introduced by many authors. For related works, please see, for instance,1–9and the references therein.
Remark 1.1. It is not hard to check that solving1.3is equivalent to the fixed point problem
Findx∈Csuch thatxprojFixT·Px, 1.4
where projFixTstands for the metric projection on the closed convex set FixT. By using the definition of the normal cone to FixT, that is,
NFixT:x−→
⎧ ⎨ ⎩
u∈H|u, y−x≤0, ∀y∈FixT if x∈FixT,
∅, otherwise, 1.5
we easily prove that1.3is equivalent to the variational inequality
0∈I−PxNFixTx. 1.6
At this point, we wish to point out the link with some monotone variational inequalities and convex programming problems as follows.
Example 1.2. SettingP I−γA, whereA isη-Lipschitzian andk-strongly monotone with γ∈0,2k/η2, then1.3reduces to
Findx∈FixTsuch thatAx, x −x ≥ 0, ∀x∈FixT, 1.7
a variational inequality studied by Yamada and Ogura10.
Example 1.3. LetAbe a maximal monotone operator. TakingT JλA : IλA−1andP I−γ∇ψ, whereψis a convex function such that∇ψisη-Lipschitzianwhich is equivalent to the fact that∇ψisη−1cocoercivewithγ ∈0,2/η, and FixJA
to the following mathematical program with generalized equation constraint:
min
0∈Axψx, 1.8
a problem considered by Luo et al.11.
Example 1.4. TakingA∂ϕ, where∂ϕis the subdifferential of a lower semicontinuous convex function, then1.8reduces to the following hierarchical minimization problem considered in Cabot12and Solodov13:
min
x∈arg minϕψx. 1.9
LetB : C → H be a nonlinear mapping, and letF be a bifunction ofC×CintoR. Consider the following equilibrium problem of findingz∈Csuch that
Fz, yBz, y−z≥0, ∀y∈C. 1.10
IfB0, then1.10reduces to
Fz, y≥0, ∀y∈C. 1.11
The solution set of equilibrium problems1.10and1.11are denoted by EPF, Band EPF, respectively. The equilibrium problem1.10is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, fixed point problems, minimax problems, Nash equilibrium problem in noncooperative games, and others. We remind the readers to refer to14–30and the references therein.
Motivated and inspired by the above works, in this paper, we consider the following hierarchical equilibrium problem and variational inequality problem: find a point x∗ ∈ EPF, Bsuch that
Ax∗, x−x∗ ≥0, ∀x∈EPF, B, 1.12
whereA, Bare two monotone operators. The solution set of1.12is denoted byΩ.
Remark 1.5. It is clear that the hierarchical variational inequality problem and equilibrium problem1.12includes the variational inequality problem studied by Yamada and Ogura
10, mathematical program studied by Luo et al.11, hierarchical minimization problem considered by Cabot12and Solodov13, as special cases.
For solving1.12, we propose a double-net algorithm which generates a net{xs,t}.
We prove that the net{xs,t}hierarchically converges to the solution of1.12; that is, for each fixedt∈0,1, the net{xs,t}converges in norm, ass → 0, to a solutionxt ∈EPF, Bof the
equilibrium problem, and ast → 0, the net{xt}converges in norm to the unique solution
2. Preliminaries
Let H be a real Hilbert space. Throughout this paper, let us assume that a bifunctionF : H×H → R satisfies the following conditions:
F1Fx, x 0 for allx∈H;
F2Fis monotone, that is,Fx, y Fy, x≤0 for allx, y∈H;
F3for eachx, y, z∈H, lim supt0Ftz 1−tx, y≤Fx, y;
F4for eachx∈H,y→Fx, yis convex and lower semicontinuous.
On the equilibrium problems, we have the following important lemma. You can find it in
31.
Lemma 2.1. LetH be a real Hilbert space, and letF be a bifunction ofH×H into Rsatisfying conditions (F1)–(F4). Letr >0, andx∈H. Then, there existsz∈Hsuch that
Fz, y1ry−z, z−x≥0, ∀y∈H. 2.1
Further, ifTrx {z∈H|Fz, y 1/ry−z, z−x ≥0, for ally ∈H}, then the following
hold:
1Tr is single-valued;
2Tr is firmly nonexpansive; that is, for anyx, y∈H,
Trx−Try2≤T
rx−Try, x−y, 2.2
3FixTr EPF;
4EPFis closed and convex.
Below we gather some basic facts that are needed in the argument of the subsequent sections.
Lemma 2.2see32. LetHbe a real Hilbert space. Let the mappingA :H → H beα-inverse strongly monotone, and letλ >0be a constant. Then, one has
I−λAx−I−λAy2≤x−y2λλ−2αAx−Ay2, ∀x, y∈H. 2.3
In particular, if0≤λ≤2α, thenI−λAis nonexpansive.
Lemma 2.3 demiclosedness principle for nonexpansive mappings, see 33. Let C be a nonempty closed convex subset of a real Hilbert space H and letT : C → Cbe a nonexpansive mapping with FixT/∅. If {xn} is a sequence inCweakly converging tox, and if {I −Txn}
Lemma 2.4. Let H be a real Hilbert space. Let f : H → H be a ρ-contraction with coefficient ρ ∈ 0,1. Let the mapping A : H → H beα-inverse strongly monotone. Letλ ∈ 0,2α, and t∈0,1. Then the variational inequality
x∗∈EPF, B, tfz 1−tI−λAz−z, x∗−z≥0, ∀z∈EPF, B 2.4
is equivalent to the dual variational inequality
x∗∈EPF, B, tfx∗ 1−tI−λAx∗−x∗, x∗−z≥0, ∀z∈EPF, B. 2.5
Proof. Assume thatx∗∈EPF, Bsolves2.4. For allz∈EPF, B, set
xx∗sz−x∗∈EPF, B, 0< s <1. 2.6
We note that
tfx 1−tI−λAx−x, x∗−x≥0. 2.7
Hence, we have
tfx∗sz−x∗ 1−tI−λAx∗sz−x∗−x∗−sz−x∗, sx∗−z≥0, 2.8
which implies that
tfx∗sz−x∗ 1−tI−λAx∗sz−x∗−x∗−sz−x∗, x∗−z≥0. 2.9
Lettings → 0, we have
tfx∗ 1−tI−λAx∗−x∗, x∗−z≥0, 2.10
which is exactly2.5.
Assume thatx∗solves2.5. Hence,
tfx∗ 1−tI−λAx∗−x∗, x∗−z≥0. 2.11
Noting thatI−fandAare monotone, we have
I−fz−I−fx∗, z−x∗≥0,
Az−Ax∗, z−x∗ ≥0. 2.12
It follows that
which implies that
tfz 1−tI−λAz−z, x∗−z≥tfx∗ 1−tI−λAx∗−x∗, x∗−z≥0. 2.14
This implies thatx∗solves2.4. The proof is completed.
3. Main Results
In this section, we first introduce our double-net algorithm.
Let H be a real Hilbert space. Let f : H → H be a ρ-contraction with coefficient ρ ∈ 0,1. Let the mappingsA, B : H → Hbe α-inverse strongly monotone andβ-inverse strongly monotone, respectively. LetFbe a bifunction fromH×H → R ,and letλ∈0,2α andr∈0,2βbe two constants. Fors, t∈0,1, we define the following mapping:
x−→Ws,tx:stfx 1−tx−λAx 1−sTrx−rBx, 3.1
whereTrxis defined byLemma 2.1. We note that the mappingWs,tis a contraction. As a matter of fact, we have
Ws,tx−Ws,tys
tfx 1−tx−λAx 1−sTrx−rBx
−stfy 1−ty−λAy−1−sTry−rBy ≤stfx−fys1−tx−λAx−y−λAy
1−sTrx−rBx−Try−rBy
≤stρx−ys1−tx−y 1−sx−y
1−1−ρstx−y,
3.2
which implies that the mappingWs,tis contractive. Hence, by Banach’s contraction principle, Ws,thas a unique fixed point which is denotedxs,t∈H; that is,xs,tis the unique solution in
Hof the fixed point equation
xs,tstfxs,t 1−txs,t−λAxs,t 1−sTrxs,t−rBxs,t, s, t∈0,1. 3.3
Below is our main result of this paper which displays the behavior of the net{xs,t}ass → 0
andt → 0 successively.
Theorem 3.1. LetHbe a real Hilbert space. Letf : H → Hbe a ρ-contraction with coefficient ρ∈0,1. Let the mappingsA, B:H → Hbeα-inverse strongly monotone andβ-inverse strongly monotone, respectively. Letλ ∈0,2αandr ∈0,2βbe two constants. LetF be a bifunction from H×H → Rsatisfying (F1)–(F4). Suppose the solution setΩof 1.12is nonempty. Let, for each
s, t∈0,12,xs,tbe defined implicitly by3.3. Then, the net{xs,t}hierarchically converges to the
xt∈EPF, Bof the equilibrium problem1.10. Moreover, ast → 0, the net{xt}converges in norm
to the unique solutionx∗∈Ω. Furthermore,x∗also solves the following variational inequality:
x∗∈Ω, I−fx∗, x−x∗≥0, ∀x∈Ω. 3.4
We divide our detailed proofs into several conclusions as follows. Throughout, we assume all assumptions ofTheorem 3.1are satisfied.
Conclusion 1. For each fixedt∈0,1, the net{xs,t}is bounded.
Proof. Take anyz ∈EPF, B. It is clear thatz Trz−rBz. Setus,t Trxs,t−rBxs,tfor all
s, t∈0,1. SinceTr,I−λAandI−rBare nonexpansiveby Lemmas2.1and2.2, we have from3.3that
xs,t−zstfxs,t 1−tI−λAxs,t 1−sTrxs,t−rBxs,t−z
≤stfxs,t 1−tI−λAxs,t−z 1−sTrxs,t−rBxs,t−Trz−rBz
≤stfxs,t−fztfz−z
1−tI−λAxs,t−I−λAz 1−tI−λAz−z 1−sxs,t−z ≤stρxs,t−ztfz−z 1−txs,t−z 1−tλAz 1−sxs,t−z
1−1−ρstxs,t−zstfz−zs1−tλAz.
3.5
This implies that
xs,t−z ≤ 1
1−ρt
tfz−z 1−tλAz
≤ 1
1−ρtmaxfz−z, λAz .
3.6
It follows that for each fixedt∈0,1,{xs,t}is bounded, so are the nets{fxs,t},{I−λAxs,t}
and {us,t}. Note that we use Mt as a positive constant which bounds all bounded terms
appearing in the following.
Conclusion 2. xs,t → xt∈EPF, Bass → 0.
Proof. FromLemma 2.2, we have
xs,t−λAxs,t−z−λAz2≤ xs,t−z2λλ−2αAxs,t−Az2,
us,t−z2Trxs,t−rBxs,t−Trz−rBz2
≤ xs,t−rBxs,t−z−rBz2
≤ xs,t−z2rr−2βBxs,t−Bz2.
By3.3, we have
xs,t−z2stfxs,t−fz, xs,t−zstfz−z, xs,t−z
s1−tI−λAxs,t−I−λAz, xs,t−z
s1−tI−λAz−z, xs,t−z
1−sTrxs,t−rBxs,t−Trz−rBz, xs,t−z
≤stfxs,t−fzxs,t−zstfz−z, xs,t−z
s1−tI−λAxs,t−I−λAzxs,t−z −s1−tλAz, xs,t−z
1−sTrxs,t−rBxs,t−Trz−rBzxs,t−z
≤stρxs,t−z2stfz−z, xs,t−z−s1−tλAz, xs,t−z
s1−tI−λAxs,t−I−λAzxs,t−z
1−sI−rBxs,t−I−rBzxs,t−z
≤stρxs,t−z2stfz−z, xs,t−z−s1−tλAz, xs,t−z
s1−t
2
I−λAxs,t−I−λAz2xs,t−z2
1−s
2
I−rBxs,t−I−rBz2xs,t−z2
.
3.8
This together with3.7implies that
xs,t−z2≤stρxs,t−z2stfz−z, xs,t−z−s1−tλAz, xs,t−z
s1−t
2
xs,t−z2λλ−2αAxs,t−Az2xs,t−z2
1−s
2
xs,t−z2rr−2βBxs,t−Bz2xs,t−z2
1−1−ρstxs,t−z2stfz−z, xs,t−z−s1−tλAz, xs,t−z
s1−t
2 λλ−2αAxs,t−Az
21−s
2 r
r−2βBxs,t−Bz2.
3.9
It follows that
1−sr2β−rBxs,t−Bz2
≤ −21−ρstxs,t−z22stfz−zxs,t−z
−2s1−tλAzxs,t−zs1−tλλ−2αAxs,t−Az2 −→0 ass−→0 for each fixedt∈0,1.
Therefore
lim
s→0Bxs,t−Bz0. 3.11
UsingLemma 2.1, we obtain
us,t−z2 Trxs,t−rBxs,t−Trz−rBz2
≤ xs,t−rBxs,t−z−rBz, us,t−z
1
2
xs,t−rBxs,t−z−rBz2us,t−z2
−xs,t−z−rBxs,t−Bz−us,t−z2
≤ 1 2
xs,t−z2us,t−z2− xs,t−us,t−rBxs,t−Bz2
1
2
xs,t−z2us,t−z2− xs,t−us,t2
2rxs,t−us,t, Bxs,t−Bz −r2Bxs,t−Bz2
,
3.12
which implies that
us,t−z2≤ xs,t−z2− xs,t−us,t22rxs,t−us,t, Bxs,t−Bz −r2Bxs,t−Bz2
≤ xs,t−z2− xs,t−us,t22rxs,t−us,tBxs,t−Bz.
3.13
From3.3, we have
xs,t−z1−sus,t−z stfxs,t 1−txs,t−λAxs,t−z
≤ us,t−zsMt.
3.14
Hence,
xs,t−z2≤ us,t−z2sMt
≤ xs,t−z2− xs,t−us,t2MtBxs,t−BzsMt.
3.15
It follows that
Next, we show that, for each fixed t ∈ 0,1, the net{xs,t} is relatively norm-compact as
s → 0. It follows from3.8that
xs,t−z2stfxs,t−fz, xs,t−zstfz−z, xs,t−z
s1−tI−λAxs,t−I−λAz, xs,t−z
s1−tI−λAz−z, xs,t−z
1−sTrxs,t−rBxs,t−Trz−rBz, xs,t−z
≤stρxs,t−z2stfz−z, xs,t−zs1−txs,t−z2
s1−tI−λAz−z, xs,t−z 1−sxs,t−z2
1−1−ρstxs,t−z2stfz−z, xs,t−z−s1−tλAz, xs,t−z.
3.17
It turns out that
xs,t−z2≤ 1
1−ρt
tfz 1−tI−λAz−z, xs,t−z, z∈EPF, B. 3.18
Assume that{sn} ⊂0,1is such thatsn → 0 asn → ∞. By3.18, we conclude immediately that
xsn,t−z2≤
1
1−ρt
tfz 1−tI−λAz−z, xsn,t−z
, z∈EPF, B. 3.19
Since{xsn,t}is bounded, without loss of generality, we may assume that assn → 0,{xsn,t} converges weakly to a pointxt. Note that{usn,t}also converges weakly to a pointxt.
Now we show thatxt∈EP. Sinceusn,tTrxsn,t−rBxsn,t, for anyy∈H, we have
Fusn,t, y
1
r
y−usn,t, usn,t−xsn,t−rBxsn,t
≥0. 3.20
From the monotonicity ofF, we have
1 r
y−usn,t, usn,t−xsn,t−rBxsn,t
≥Fy, usn,t
, ∀y∈H. 3.21
Hence,
y−usni,t,usni,t−rxsni,t Bxsni,t
≥Fy, usni,t
Putzkky 1−kxtfor allk∈0,1andy∈H. From3.22, we have
zk−usni,t, Bzk
≥zk−usni,t, Bzk
−
zk−usni,t,usni,t−rxsni,t Bxsni,t
Fzk, usni,t
zk−usni,t, Bzk−Busni,t
zk−usni,t, Busni,t−Bxsni,t
−
zk−usni,t,
usni,t−xsni,t
r
Fzk, usni,t
.
3.23
Note thatBusni,t−Bxsni,t ≤ 1/βusni,t−xsni,t → 0. Further, from monotonicity ofB, we havezk−usni,t, Bzk−Busni,t ≥0. Lettingi → ∞in3.23, we have
zk−xt, Bzk ≥Fzk, xt. 3.24
FromF1,F4, and3.24, we also have
0Fzk, zk≤kFzk, y 1−kFzk, xt
≤kFzk, y 1−kzk−xt, Bzk
kFzk, y 1−kky−xt, Bzk,
3.25
and hence
0≤Fzk, y 1−kBzk, y−xt. 3.26
Lettingk → 0 in3.26, we have, for eachy∈H,
0≤Fxt, yy−xt, Bxt. 3.27
This implies thatxt∈EPF, B.
We can then substitutextforzin3.19to get
xsn,t−xt
2≤ 1
1−ρt
tfxt 1−tI−λAxt−xt, xsn,t−xt
. 3.28
Consequently, the weak convergence of{xsn,t}toxtactually implies thatxsn,t → xtstrongly. This has proved the relative norm-compactness of the net{xs,t}ass → 0.
Now we return to3.19and take the limit, asn → ∞, to get
xt−z2≤ 1
1−ρt
In particular,xtsolves the following variational inequality:
xt∈EPF, B, tfz 1−tI−λAz−z, xt−z≥0, ∀z∈EPF, B, 3.30
or the equivalent dual variational inequalityseeLemma 2.4
xt∈EPF, B, tfxt 1−tI−λAxt−xt, xt−z≥0, ∀z∈EPF, B. 3.31
Notice that3.31is equivalent to the fact thatxt PEPF,Btf 1−tI−λAxt. That is,
xtis the unique element in EPF, Bof the contractionPEPF,Btf 1−tI−λA. Clearly,
this is sufficient to conclude that the entire net{xs,t}converges in norm toxt ∈EPF, Bas
s → 0.
Conclusion 3. The net{xt}is bounded.
Proof. In3.31, we take anyy∈Ωto deduce
tfxt 1−tI−λAxt−xt, xt−y≥0. 3.32
By virtue of the monotonicity ofAand the fact thaty∈Ω, we have
I−λAxt−xt, xt−y≤I−λAy−y, xt−y≤0. 3.33
It follows from3.32and3.33that
fxt−xt, xt−y≥0, ∀y∈Ω. 3.34
Hence,
xt−y2≤x
t−y, xt−yfxt−xt, xt−y
fxt−fy, xt−yfy−y, xt−y
≤ρxt−y2fy−y, xt−y.
3.35
Therefore,
xt−y2≤ 1
1−ρ
fy−y, xt−y, ∀y∈Ω. 3.36
In particular,
xt−y≤ 1 1−ρf
y−y, ∀t∈0,1, 3.37
Conclusion 4. The netxt → x∗∈Ωwhich solves the variational inequality VI3.4.
Proof. First, we note that the solution of the variational inequality VI3.4is unique.
We next prove thatωwxt ⊂ Ω; namely, if tnis a null sequence in0,1such that
xtn → xweakly asn → ∞, thenx∈Ω. To see this, we use3.31to get
λAxt, z−xt ≥ 1−t tI−fxt, xt−z, z∈EPF, B. 3.38
However, sinceAis monotone,
Az, z−xt ≥ Axt, z−xt. 3.39
Combining the last two relations yields
λAz, z−xt ≥ 1−t tI−fxt, xt−z, z∈EPF, B. 3.40
Lettingttn → 0 asn → ∞in3.40, we get
Az, z−x≥0, z∈EPF, B, 3.41
which is equivalent to its dual variational inequality
Ax, z−x≥0, z∈EPF, B. 3.42
Namely,xis a solution of VI1.12; hence,x∈Ω.
We further prove thatxx∗, the unique solution of VI3.4. As a matter of fact, we have by3.36
xtn −x2≤ 1
1−ρ
fx−x, x tn−x
, x∈Ω. 3.43
Therefore, the weak convergence toxof{xtn}implies thatxtn → xin norm. Now we can let ttn → 0 in3.36to get
fx−x, y−x≤0, ∀y∈Ω. 3.44
It turns out thatx ∈Ωsolves VI3.4. By uniqueness, we havex x∗. This is sufficient to guarantee thatxt → x∗in norm, ast → 0. The proof is complete.
TakeB0. Then1.12reduces to the following: find a pointx∗∈EPFsuch that
Ax∗, x−x∗ ≥0, ∀x∈EPF. 3.45
The solution of3.45is denoted byΩ1.
Corollary 3.2. LetHbe a real Hilbert space. Letf : H → Hbe aρ-contraction with coefficient ρ ∈ 0,1. Let the mapping A : H → H beα-inverse strongly monotone. Letλ ∈ 0,2αbe a constant. LetFbe a bifunction fromH×H → Rsatisfying (F1)–(F4). Suppose the solution setΩ1
is nonempty. Let, for eachs, t∈0,12,xs,tbe defined implicitly by
xs,tstfxs,t 1−txs,t−λAxs,t 1−sTrxs,t, s, t∈0,1. 3.46
Then, the net{xs,t}hierarchically converges to the unique solutionx∗of the hierarchical equilibrium problem and variational inequality problem3.45. That is to say, for each fixedt ∈ 0,1, the net {xs,t}converges in norm, as s → 0, to a solutionxt ∈ EPFof the equilibrium problem1.11.
Moreover, ast → 0, the net{xt}converges in norm to the unique solutionx∗∈Ω1. Furthermore,x∗
solves the following variational inequality:
x∗∈Ω 1,
I−fx∗, x−x∗≥0, ∀x∈Ω
1. 3.47
TakingA0 inTheorem 3.1, we have the following corollary.
Corollary 3.3. LetHbe a real Hilbert space. Letf : H → Hbe aρ-contraction with coefficient ρ ∈ 0,1. Let the mappingB : H → H be β-inverse strongly monotone. Letr ∈ 0,2β be a constant. LetFbe a bifunction fromH×H → Rsatisfying (F1)–(F4). Suppose that the solution set EPF, Bof1.10is nonempty. Let, for eachs, t∈0,12,xs,tbe defined implicitly by
xs,tstfxs,t 1−txs,t 1−sTrxs,t−rBxs,t, s, t∈0,1. 3.48
Then, the net {xs,t}hierarchically converges to the unique solution x∗ of the equilibrium problem
1.10. That is to say, for each fixedt∈0,1, the net{xs,t}converges in norm, ass → 0, to a solution xt∈EPF, Bof the equilibrium problem1.10. Moreover, ast → 0, the net{xt}converges in norm
to the unique solutionx∗∈EPF, B. Furthermore,x∗solves the following variational inequality:
x∗∈EPF, B, I−fx∗, x−x∗≥0, ∀x∈EPF, B. 3.49
TakingAB0 inTheorem 3.1, we have the following corollary.
Corollary 3.4. LetHbe a real Hilbert space. Letf : H → Hbe aρ-contraction with coefficient ρ ∈ 0,1. LetFbe a bifunction fromH×H → Rsatisfying (F1)–(F4). Suppose the solution set EPFof1.11is nonempty. Let, for eachs, t∈0,12,xs,tbe defined implicitly by
Then, the net {xs,t}hierarchically converges to the unique solution x∗ of the equilibrium problem
1.11. That is to say, for each fixed t ∈ 0,1, the net{xs,t} converges in norm, ass → 0, to a solutionxt∈EPFof the equilibrium problem1.11. Moreover, ast → 0, the net{xt}converges in
norm to the unique solutionx∗∈EPF. Furthermore,x∗solves the following variational inequality:
x∗∈EPF, I−fx∗, x−x∗≥0, ∀x∈EPF. 3.51
Acknowledgment
The work of the second author was partially supported by the Grant NSC 98-2923-E-110-003-MY3 and the work of the third author was partially supported by the Grant NSC 98-2221-E-110-064.
References
1 P.-E. Maing´e and A. Moudafi, “Strong convergence of an iterative method for hierarchical fixed-point problems,”Pacific Journal of Optimization, vol. 3, no. 3, pp. 529–538, 2007.
2 L. C. Ceng and A. Petrus¸el, “Krasnoselski-Mann iterations for hierarchical fixed point problems for a finite family of nonself mappings in Banach spaces,”Journal of Optimization Theory and Applications, vol. 146, no. 3, pp. 617–639, 2010.
3 A. Moudafi, “Krasnoselski-Mann iteration for hierarchical fixed-point problems,”Inverse Problems, vol. 23, no. 4, pp. 1635–1640, 2007.
4 A. Moudafi and P.-E. Maing´e, “Towards viscosity approximations of hierarchical fixed-point problems,”Fixed Point Theory and Applications, vol. 2006, Article ID 95453, 10 pages, 2006.
5 Y. Yao and Y.-C. Liou, “Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problems,”Inverse Problems, vol. 24, no. 1, Article ID 015015, 8 pages, 2008.
6 F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, “On a two-step algorithm for hierarchical fixed point problems and variational inequalities,”Journal of Inequalities and Applications, vol. 2009, Article ID 208692, 13 pages, 2009.
7 F. Cianciaruso, V. Colao, L. Muglia, and H.-K. Xu, “On an implicit hierarchical fixed point approach to variational inequalities,”Bulletin of the Australian Mathematical Society, vol. 80, no. 1, pp. 117–124, 2009.
8 X. Lu, H.-K. Xu, and X. Yin, “Hybrid methods for a class of monotone variational inequalities,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 1032–1041, 2009.
9 Y. Yao, R. Chen, and H.-K. Xu, “Schemes for finding minimum-norm solutions of variational inequalities,”Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 7-8, pp. 3447–3456, 2010. 10 I. Yamada and N. Ogura, “Hybrid steepest descent method for variational inequality problem over
the fixed point set of certain quasi-nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 25, no. 7-8, pp. 619–655, 2004.
11 Z.-Q. Luo, J.-S. Pang, and D. Ralph,Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, UK, 1996.
12 A. Cabot, “Proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimization,”SIAM Journal on Optimization, vol. 15, no. 2, pp. 555–572, 2005.
13 M. Solodov, “An explicit descent method for bilevel convex optimization,”Journal of Convex Analysis, vol. 14, no. 2, pp. 227–237, 2007.
14 L.-C. Ceng, S. Al-Homidan, Q. H. Ansari, and J.-C. Yao, “An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings,”Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 967–974, 2009.
16 Y. Yao, M. A. Noor, and Y.-C. Liou, “On iterative methods for equilibrium problems,”Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 497–509, 2009.
17 J.-W. Peng and J.-C. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems,”Taiwanese Journal of Mathemat-ics, vol. 12, no. 6, pp. 1401–1432, 2008.
18 J.-W. Peng and J.-C. Yao, “Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings,”Taiwanese Journal of Mathematics, vol. 13, no. 5, pp. 1537–1582, 2009.
19 L. C. Ceng, A. Petrus¸el, and J. C. Yao, “Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,”Journal of Optimization Theory and Applications, vol. 143, no. 1, pp. 37–58, 2009.
20 Y. Yao, Y.-C. Liou, and J.-C. Yao, “A new hybrid iterative algorithm for fixed-point problems, vari-ational inequality problems, and mixed equilibrium problems,”Fixed Point Theory and Applications, vol. 2008, Article ID 417089, 15 pages, 2008.
21 J.-W. Peng, Y.-C. Liou, and J.-C. Yao, “An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions,”Fixed Point Theory and Applications, vol. 2009, Article ID 794178, 21 pages, 2009.
22 L. C. Ceng, G. Mastroeni, and J. C. Yao, “Hybrid proximal-point methods for common solutions of equilibrium problems and zeros of maximal monotone operators,”Journal of Optimization Theory and Applications, vol. 142, no. 3, pp. 431–449, 2009.
23 Y. Yao, Y.-C. Liou, and J.-C. Yao, “Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings,”Fixed Point Theory and Applications, vol. 2007, Article ID 64363, 12 pages, 2007.
24 Y. Yao, M. A. Noor, S. Zainab, and Y.-C. Liou, “Mixed equilibrium problems and optimization problems,”Journal of Mathematical Analysis and Applications, vol. 354, no. 1, pp. 319–329, 2009. 25 Y. Yao, Y. J. Cho, and R. Chen, “An iterative algorithm for solving fixed point problems, variational
inequality problems and mixed equilibrium problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3363–3373, 2009.
26 G. Marino, V. Colao, L. Muglia, and Y. Yao, “Krasnoselski-Mann iteration for hierarchical fixed points and equilibrium problem,”Bulletin of the Australian Mathematical Society, vol. 79, no. 2, pp. 187–200, 2009.
27 L.-C. Ceng and J.-C. Yao, “A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem,”Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1922–1937, 2010.
28 L. C. Zeng, Y. C. Lin, and J. C. Yao, “Iterative schemes for generalized equilibrium problem and two maximal monotone operators,”Journal of Inequalities and Applications, vol. 2009, Article ID 896252, 34 pages, 2009.
29 T. Honda, W. Takahashi, and J.-C. Yao, “Nonexpansive retractions onto closed convex cones in Banach spaces,”Taiwanese Journal of Mathematics, vol. 14, no. 3B, pp. 1023–1046, 2010.
30 L.-C. Zeng, Q. H. Ansari, D. S. Shyu, and J.-C. Yao, “Strong and weak convergence theorems for common solutions of generalized equilibrium problems and zeros of maximal monotone operators,”
Fixed Point Theory and Applications, vol. 2010, Article ID 590278, 33 pages, 2010.
31 P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.
32 W. Takahashi and M. Toyoda, “Weak convergence theorems for nonexpansive mappings and monotone mappings,”Journal of Optimization Theory and Applications, vol. 118, no. 2, pp. 417–428, 2003.
