Volume 2009, Article ID 531308,20pages doi:10.1155/2009/531308
Research Article
An Iterative Method for Generalized
Equilibrium Problems, Fixed Point Problems
and Variational Inequality Problems
Qing-you Liu,
1Wei-you Zeng,
2and Nan-jing Huang
21State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University,
Chengdu, Sichuan 610500, China
2Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Correspondence should be addressed to Nan-jing Huang,[email protected]
Received 11 January 2009; Accepted 28 May 2009
Recommended by Fabio Zanolin
We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of generalized equilibrium problems, the set of common fixed points of infinitely many nonexpansive mappings, and the set of solutions of the variational inequality forα-inverse-strongly monotone mappings in Hilbert spaces. We give some strong-convergence theorems under mild assumptions on parameters. The results presented in this paper improve and generalize the main result of Yao et al.2007.
Copyrightq2009 Qing-you Liu 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.
1. Introduction
LetCbe a nonempty closed convex subset of a real Hilbert spaceHand letΦ:C×C → R
be a bifunction, whereRis the set of real numbers. LetΨ:C → Hbe a nonlinear mapping. The generalized equilibrium problemGEPforΦ:C×C → RandΨ:C → His to find
u∈Csuch that
Φu, v Ψu, v−u ≥0 ∀v∈C. 1.1
The set of solutions for the problem1.1is denoted byΩ, that is,
IfΨ 0 in1.1, then GEP1.1reduces to the classical equilibrium problemEPand
Ωis denoted by EPΦ, that is,
EPΦ {u∈C:Φu, v≥0,∀v∈C}. 1.3
IfΦ 0 in1.1, then GEP1.1reduces to the classical variational inequality andΩis denoted by VIΨ, C, that is,
VIΨ, C {u∗∈C:Ψu∗, v−u∗ ≥0,∀v∈C}. 1.4
It is well known that GEP1.1 contains as special cases, for instance, optimization problems, Nash equilibrium problems, complementarity problems, fixed point problems, and variational inequalitiessee, e.g., 1–6and the reference therein.
A mappingA : C → H is calledα-inverse-strongly monotone 7, if there exists a positive real numberαsuch that
Ax−Ay, x−y ≥αAx−Ay2 1.5
for allx, y∈C. It is obvious that anyα-inverse-strongly monotone mappingAis monotone and Lipschitz continuous. A mappingS:C → Cis called nonexpansive if
Sx−Sy ≤ x−y 1.6
for allx, y ∈C. We denote byFSthe set of fixed points ofS, that is,FS {x∈C :x Sx}. IfC⊂His bounded, closed and convex andSis a nonexpansive mappings ofCinto itself, thenFSis nonemptysee 8.
In 1997, Fl˚am and Antipin 9 introduced an iterative scheme of finding the best approximation to initial data when EPΦ is nonempty and proved a strong convergence theorem. In 2003, Iusem and Sosa 10presented some iterative algorithms for solving equi-librium problems in finite-dimensional spaces. They have also established the convergence of the algorithms. Recently, Huang et al. 11studied the approximate method for solving the equilibrium problem and proved the strong convergence theorem for approximating the solutions of the equilibrium problem.
On the other hand, for finding an element ofFS∩VIA, C, Takahashi and Toyoda
12introduced the following iterative scheme:
xn1αnxn 1−αnSPCxn−λnAxn, n0,1,2, . . . , 1.7
wherex0 ∈ C,PC is metric projection ofHontoC,{αn}is a sequence in0,1and{λn}is
a sequence in0,2α. Further, Iiduka and Takahashi 13introduced the following iterative scheme:
xn1αnuβnxnγnSPCxn−λnAxn, 1.8
whereu, x0∈C, and proved the strong convergence theorems for iterative scheme1.8under
iterative scheme by the viscosity approximation method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert spaces. They also proved a strong convergence theorem which is connected with Combettes and Hirstoaga’s result 3 and Wittmann’s result 15. Tada and W. Takahashi
16introduced the Mann type iterative algorithm for finding a common element of the set of solutions of the EPΦ and the set of common fixed points of nonexpansive mapping and obtained the weak convergence of the Mann type iterative algorithm. Yao et al. 17 introduced an iteration process for finding a common element of the set of solutions of the EPΦand the set of common fixed points of infinitely many nonexpansive mappings in Hilbert spaces. They proved a strong-convergence theorem under mild assumptions on parameters. Very recently, Moudafi 18 proposed an iterative algorithm for finding a common element ofΩ∩FS, whereΨ:C → His anα-inverse-strongly monotone mapping, and obtained a weak convergence theorem. There are some related works, we refer to 19–22 and the references therein.
Inspired and motivated by the works mentioned above, in this paper, we introduce an iterative process for finding a common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of GEP1.1, and the solution set of the variational inequality problem for anα-inverse-strongly monotone mapping in real Hilbert spaces. We give some strong-convergence theorems under mild assumptions on parameters. The results presented in this paper improve and generalize the main result of Yao et al. 17.
2. Preliminaries
LetHbe a real Hilbert space with inner product·,·and norm · , and letCbe a closed convex subset ofH. Then, for anyx∈H, there exists a unique nearest point inC, denoted by
PCx, such that
x−PCx ≤ x−y ∀y∈C. 2.1
PC is called the metric projection ofH ontoC. It is well known thatPC is a nonexpansive mapping and satisfies
PCx−PCy, x−y ≥PCx−PCy2 2.2
for allx, y∈H. Furthermore,PCx∈Cis characterized by the following properties:
x−PCx, y−PCx ≤0,
x−PCx 2y−PCx2 ≤x−y2
2.3
for allx∈Handy∈C. It is easy to see that
u∈VIA, C⇐⇒uPCu−λAu, 2.4
A set-valued mappingT :H → 2His called monotone if for allx, y∈H,p∈Txand q∈Tyimplyx−y, p−q ≥0. A monotone mappingT :H → 2His maximal if the graph GTofTis not properly contained in the graph of any other monotone mappings. It is known that a monotone mappingT is maximal if and only if forx, p∈ H×H,x−y, p−q ≥0 for ally, q∈GTimpliesp∈Tx. LetA:C → Hbe a monotone,L-Lipschitz continuous mappings and letNCube the normal cone toCatu∈C, that is,NCu{w∈H:u−v, w ≥
0,∀v∈C}. Define
Tu
⎧ ⎨ ⎩
AuNCu, u∈C,
∅, u /∈C. 2.5
ThenT is the maximal monotone and 0∈Tuif and only ifu∈VIA, C; see 23.
Let{Tn}∞n1be a sequence of nonexpansive mappings ofCinto itself and let{πn}∞n1be
a sequence of nonnegative numbers in 0,1. For anyn ≥ 1, define a mappingSnofCinto itself as follows:
Un,n1I,
Un,nπnTnUn,n1 1−πnI,
Un,n−1πn−1Tn−1Un,n 1−πn−1I,
.. .
Un,kπkTkUn,k1 1−πkI,
Un,k−1πk−1Tk−1Un,k 1−πk−1I,
.. .
Un,2π2T2Un,3 1−π2I,
SnUn,1π1T1Un,2 1−π1I.
2.6
Such a mappingSnis called theS-mapping generated byTn, Tn−1, . . . , T1andπn, πn−1, . . . , π1
see 24. It is obvious thatSnis nonexpansive and ifxTnxthenxUn,kSnx.
Lemma 2.1see 24. LetC be a nonempty closed convex subset of a real Hilbert spaceH. Let {Tn}∞n1 be a sequence of nonexpansive mappings ofC into itself such that∞n1FTn/∅ and let
{πn}∞n1be a sequence in0, σfor someσ∈0,1. Then, for everyx∈Candk∈N{1,2, . . .}, the limitlimn→ ∞Un,kxexists.
Remark 2.2see 17. It can be known fromLemma 2.1that ifD is a nonempty bounded subset ofC, then for >0, there existsn0≥1 such that for alln > n0
sup
x∈D
UsingLemma 2.1, one can define a mappingSofCinto itself as follows:
Sx lim
n→ ∞Snxnlim→ ∞Un,1x 2.8
for every x ∈ C. Such a mapping S is called the S-mapping generated byT1, T2, . . . and
π1, π2, . . . .SinceSnis nonexpansive,S :C → Cis also nonexpansive. If{xn}is a bounded
sequence inC, then we putD {xn :n ≥0}. Hence, it is clear fromRemark 2.2that for an arbitrary >0 there existsN0∈Nsuch that for alln > N0
Snxn−Sxn Un,1xn−U1xn ≤sup
x∈D
Un,1x−U1x ≤. 2.9
This implies that
lim
n→ ∞ Snxn−Sxn 0. 2.10
SinceTiandUn,iare nonexpansive, we deduce that, for eachn≥1,
Sn1xn−Snxn π1T1Un1,2xn−π1T1Un,2xn
≤π1 Un1,2xn−Un,2xn
π1 π2T2Un1,3xn−π2T2Un,3xn
≤π1π2 Un1,3xn−Un,3xn
.. .
≤n
i1
πi Un1,n1xn−Un,n1xn
≤M n
i1
πi
2.11
for some constantM≥0.
Lemma 2.3see 24. LetC be a nonempty closed convex subset of a real Hilbert spaceH. Let {Tn}∞n1 be a sequence of nonexpansive mappings of Cinto itself such that
∞
n1FTn/∅,and let
{πn}∞n1be a sequence in0, σfor someσ∈0,1. Then,FS ∞n1FTn.
For solving the generalized equilibrium problem, we assume that the bifunctionΦ :
C×C → Rsatisfies the following conditions:
a1 Φu, u 0 for allu∈C;
a2 Φis monotone, that is,Φu, v Φv, u≤0 for allu, v∈C;
a3for eachu, v, w∈C, limt↓0Φtw 1−tu, v≤Φu, v;
The following lemma appears implicitly in 1.
Lemma 2.4see 1. LetCbe a nonempty closed convex subset ofH,and letΦbe a bifunction from
C×CintoRsatisfying (a1)–(a4). Letr >0andx∈H. Then, there existsu∈Csuch that
Φu, v 1
rv−u, u−x ≥0 ∀v∈C. 2.12
The following lemma was also given in 3.
Lemma 2.5see 3. Assume thatΦ:C×C → Rsatisfies (a1)–(a4). Forr >0, define a mapping
Tr :H → Cas follows:
Trx
u∈C:Φu, v 1
rv−u, u−x ≥0,∀v∈C 2.13
for allx∈H. Then, the following hold:
b1Tr is single-valued;
b2Tr is firmly nonexpansive, that is, for anyx, y∈H, Trx−Try 2 ≤ Trx−Try, x−y;
b3FTr EPΦ;
b4EPΦis closed and convex.
Remark 2.6. Replacingxwithx−rΨx∈Hin2.12, then there existsu∈Csuch that
Φu, v Ψx, v−u1
rv−u, u−x ≥0 ∀v∈C. 2.14
The following lemmas will be useful for proving the convergence result of this paper.
Lemma 2.7see 25. Let{xn}and{zn}be bounded sequences in Banach spaceE,and let{βn}be a sequence in 0,1. Suppose
xn11−βnznβnxn 2.15
for all integersn≥1. If
0<lim inf
n→ ∞ βn≤lim supn→ ∞ βn<1, lim sup
n→ ∞ zn1−zn − xn1−xn ≤0,
2.16
thenlimn→ ∞ zn−xn 0.
Lemma 2.8see 26. Assume{an}is a sequence of nonnegative real numbers such that
where{αn}is a sequence in (0,1) and{δn}is a sequence inRsuch that
1∞n1αn∞;
2lim supn→ ∞δn/αn≤0or∞n1|δn|<∞. Thenlimn→ ∞an0.
3. Main Results
In this section, we deal with an iterative scheme by the approximation method for finding a common element of the set of common fixed points of infinitely many nonexpansive mappings, the set of solutions of GEP1.1, and the solution set of the variational inequality problem for anα-inverse-strongly monotone mapping in real Hilbert spaces.
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let Φ be a bifunction from C×C into R satisfying (a1)–(a4), Ψ : C → H an inverse-strongly monotone mapping with constantφ > 0,A :C → Han inverse-strongly monotone mapping with constant
>0,f :C → Ca contraction mapping with constantα∈ 0,1. LetSn :C → Cbe aS-mapping
generated byT1, T2, . . .and π1, π2, . . .and ∞n1FTn∩Ω∩VIA, C/∅, where sequence{Tn}is
nonexpansive and{πn}is a sequence in0, σfor someσ ∈ 0,1. Forx1 ∈C, suppose that{xn},
{yn},and{un}are generated by
Φun, v Ψxn, v−un 1
rnv−un, un−xn ≥0, ∀v∈C,
ynPCun−λnAun,
xn1αnfxn βnxnγnSnyn
3.1
for alln∈N, where{αn},{βn}, and{γn}are three sequences in 0,1,{λn}is a sequence in0, bfor
some0< b <2and{rn} ⊂0, dfor some0< d <2φsatisfying
iαnβnγn1;
iilimn→ ∞αn0and∞n1αn∞;
iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1;
ivlim infn→ ∞λn>0andlimn→ ∞|λn1−λn|0;
vlim infn→ ∞rn>0andlimn→ ∞|rn1−rn|0. Then{xn},{yn}, and{un}converge strongly to the pointz0 ∈
∞
n1FTn∩Ω∩VIA, C, where
z0P∞n1FTn∩Ω∩VIA,Cfz0.
Proof. For anyx, y∈Candr ∈0,2φ, we have
I−rΨx−I−rΨy2
x−y−rΨx−Ψy2
x−y2−2rx−y,Ψx−Ψyr2Ψx−Ψy2
≤x−y2rr−2φΨx−Ψy2
≤x−y2,
which implies thatI−rΨis nonexpansive.Remark 2.6implies that the sequences{un}and
{xn}are well defined. In view of the iterative sequence3.1, we have
0≤Φun, v Ψxn, v−un 1
rnv−un, un−xn
Φun, v 1
rnv−un, un−xn−rnΨxn.
3.3
It follows fromLemma 2.5thatun Trnxn−rnΨxnfor alln≥1. Letz∗ ∈ ∞
n1FTn∩Ω∩
VIA, C. For eachn≥1, we havez∗Snz∗ Trnz∗−rnΨz∗. ByLemma 2.5,
un−z∗ 2 Trnxn−rnΨxn−Trnz
∗−rnΨz∗ 2
≤ un−z∗,xn−rnΨxn−z∗−rnΨz∗
1
2
un−z∗ 2 xn−rnΨxn−z∗−rnΨz∗ 2
− un−z∗−xn−rnΨxn−z∗−rnΨz∗ 2
3.4
and so3.2implies that
un−z∗ 2≤ xn−rnΨxn−z∗−rnΨz∗ 2− un−xn−rnΨz∗−Ψxn 2
≤ xn−z∗ 2− un−xn−rnΨz∗−Ψxn 2
≤ xn−z∗ 2.
3.5
For z∗ ∈ VIA, C, we have z∗ PCz∗ −λnAz∗ from 2.4. Since PC is a nonexpansive mapping andAis an inverse-strongly monotone mapping with constant >0, by3.1, we have
yn−z∗2
PCun−λnAun−PCz∗−λnAz∗ 2
≤ un−λnAun−z∗−λnAz∗ 2
≤ un−z∗ 2λnλn−2 Aun−Az∗ 2
≤ un−z∗ 2.
3.6
Thus,3.5and3.6imply that
and so
xn1−z∗ αnfxn βnxnγnSnyn−z∗
≤αn fxn−z∗ βn xn−z∗ γn Snyn−z∗
≤αn fxn−fz∗ fz∗−z∗ βn xn−z∗ γn Snyn−Snz∗
≤αnα xn−z∗ fz∗−z∗ βn xn−z∗ γn yn−z∗
≤αnα xn−z∗ fz∗−z∗ βn xn−z∗ γn xn−z∗
1−αn1−α xn−z∗ αn1−α fz
∗−z∗ 1−α
≤max
x1−z∗ ,
fz∗−z∗
1−α .
3.8
This implies that{xn}is bounded. Therefore,{un},{yn},{Ψxn},{Aun},and{Snyn}are also bounded.
FromunTrnxn−rnΨxnandun1Trn1xn1−rn1Ψxn1, we have
Φun, v Ψxn, v−un 1
rnv−un, un−xn ≥0 ∀v∈C, 3.9
Φun1, v Ψxn1, v−un1 1
rn1v−un1, un1−xn1 ≥0 ∀v∈C. 3.10
Puttingvun1in3.9andvunin3.10, we get
Φun, un1 Ψxn, un1−un 1
rnun1−un, un−xn ≥0,
Φun1, un Ψxn1, un−un1 1
rn1un−un1, un1−xn1 ≥0.
3.11
Adding the above two inequalities, the monotonicity ofΦimplies that
Ψxn1−Ψxn, un−un1
un−un1,un1−xn1
rn1 −
un−xn
rn
≥0, 3.12
and so
0≤
un−un1, rnΨxn1−Ψxn rn
rn1un1−xn1−un−xn
un1−un,un−un1
1− rn
rn1
un1 xn1−rnΨxn1−xn−rnΨxn−xn1 rn
rn1xn1
un1−un, un−un1
1− rn
rn1
un1−xn1 xn1−rnΨxn1−xn−rnΨxn
.
It follows from3.2that
un1−un 2≤ un1−un 1− rn
rn1
un1−xn1 xn1−xn , 3.14
and hence
un1−un ≤1− rn
rn1
un1−xn1 xn1−xn . 3.15
From3.1,
yn1−yn PCun1−λn1Aun1−PCun−λnAun
≤ un1−λn1Aun1−un−λnAun
≤ un1−λn1Aun1−un−λnAun1 |λn1−λn| Aun
≤ un1−un |λn1−λn| Aun .
3.16
Putting
zn αn
1−βnfxn γn
1−βnSnyn, 3.17
we have
xn1βnxn1−βnzn. 3.18
Obviously, we get
zn1−zn 1αn−βn1
1fxn1
γn1
1−βn1Sn1yn1−
αn
1−βnfxn γn
1−βnSnyn
≤ αn1
1−βn1 fxn1−fxn
αn1
1−βn1 −
αn
1−βn
fxn
γn1
1−βn1 Sn1yn1−Snyn
γn1
1−βn1 −
γn
1−βn
Snyn
≤ αn1
1−βn1α xn1−xn
αn1
1−βn1 −
αn
1−βn
fxn
αn
1−βn −
αn1
1−βn1
Snyn
1− αn1 1−βn1
Sn1yn1−Snyn .
From2.11and3.16, we have
Sn1yn1−Snyn ≤ Sn1yn1−Sn1yn Sn1yn−Snyn
≤ yn1−yn M
n
i1
πi
≤ un1−un |λn1−λn| Aun M
n
i1
πi
3.20
for some constantM≥0. Combining3.15,3.19, and3.20, we deduce
zn1−zn − xn1−xn
≤ αn1α−1
1−βn1 xn1−xn
αn1
1−βn1 −
αn
1−βn
fxn Snyn
γn1
1−βn1
M n
i1
πi|λn1−λn| Aun |rn1−rn|
rn1 un1−xn1
.
3.21
It is easy to check that
lim
n→ ∞
αn1
1−βn1 −
αn
1−βn
0, lim
n→ ∞ n
i1
πi0, lim
n→ ∞|rn1−rn|0, 3.22
and so
lim sup
n→ ∞ zn1−zn − xn1−xn ≤0. 3.23
Thus, byLemma 2.7, we obtain limn→ ∞ zn−xn 0. It then follows that
lim
n→ ∞ xn1−xn nlim→ ∞
1−βn zn−xn 0. 3.24
By3.15and3.16, we have
lim
Sincexn1αnfxn βnxnγnSnyn, we get
xn1− βnxn1−γnSnyn
αn
αnfxn
1− 1 1−αn
βnxnγnSnyn
αnfxn−βnxnγnSnyn
1−αn
−→0.
3.26
On the other hand, forαnβnγn1,
xn−βnxnγnSnyn
1−αn γn
1−αn
xn−Snyn. 3.27
It follows that
γn xn−Snyn 1−αnxn−βnxnγnSnyn
1−αn
≤1−αn
xn−xn1 xn1−βnxn1−γnSnyn
αn
−→0.
3.28
It is easy to see that lim infn→ ∞γn>0 and hence limn→ ∞ xn−Snyn 0. From3.5and3.6, we obtain
xn1−z∗ 2≤αnfxn−z∗2βn xn−z∗ 2γnSnyn−z∗2
≤αnfxn−z∗2βn xn−z∗ 2γnyn−z∗2
≤αnfxn−z∗2βn xn−z∗ 2γn un−z∗ 2λnλn−2 Aun−Az∗ 2
≤αnfxn−z∗2 xn−z∗ 2γnλnλn−2 Aun−Az∗ 2.
3.29
Since lim inf rn>0, without loss of generality, we may assume that there exists a real number
a >0 such thatrn> afor alln∈N. Therefore, we have
γna2−b Aun−Az∗ 2
≤αnfxn−z∗2 xn−z∗ 2− xn1−z∗ 2
≤αnfxn−z∗2 xn−xn1 xn−z∗ xn1−z∗ .
Sinceαn → 0, xn−xn1 → 0 and{xn}is bounded,3.30implies that Aun−Az∗ → 0 as
n → ∞. From2.2, we have
yn−z∗ 2 PCun−λnAun−PCz∗−λnAz∗ 2
≤ un−λnAun−z∗−λnAz∗, yn−z∗
1
2
un−λnAun−z∗−λnAz∗ 2yn−z∗2
−un−yn−λnAun−Az∗2
≤ 1
2
un−z∗ 2yn−z∗2
−un−yn2−2λnun−yn, Aun−Az∗λ2n Aun−Az∗ 2
,
3.31
and so
yn−z∗2≤
un−z∗ 2−un−yn22λnun−yn, Aun−Az∗ −λ2n Aun−Az∗ 2. 3.32
It follows that
xn1−z∗ ≤αnfxn−z∗2βn xn−z∗ 2γnyn−z∗2
≤αnfxn−z∗2βn xn−z∗ 2
γn un−z∗ 2−un−yn22λnun−yn, Aun−Az∗ −λ2n Aun−Az∗ 2
≤αnfxn−z∗2 xn−z∗ 2−γnun−yn2
2γnλnun−yn, Aun−Az∗ −γnλ2n Aun−Az∗ 2
3.33
which implies that
γnun−yn2≤αnfxn−z∗2 xn−z∗ 2− xn1−z∗ 2
2γnλnun−yn, Aun−Az∗
≤αnfxn−z∗2 xn−xn1 xn−z∗ xn1−z∗
2γnλn Aun−Az∗ un−yn .
Sinceαn → 0, xn−xn1 → 0, Aun−Az∗ → 0, and the sequences{xn},{yn}and{un}are
bounded, it follows from3.34that limn→ ∞ un−yn 0. On the other hand, from3.5, we have
xn1−z∗ 2≤αnfxn−z∗2βn xn−z∗ 2γnyn−z∗2
≤αnfxn−z∗2βn xn−z∗ 2γn un−z∗ 2
≤αnfxn−z∗2βn xn−z∗ 2
γn xn−z∗ 2rnrn−2φ Ψxn−Ψz∗ 2
≤αnfxn−z∗2 xn−z∗ 2−γnrn2φ−rn Ψxn−Ψz∗ 2.
3.35
The same as in3.30, we have Ψxn−Ψz∗ → 0 asn → ∞. Likewise, using3.5, we find
xn1−z∗ 2≤αnfxn−z∗2βn xn−z∗ 2γn un−z∗ 2
≤αnfxn−z∗2βn xn−z∗ 2
γn xn−z∗ 2− un−xn 22rnxn−un,Ψxn−Ψz∗ −rn2 Ψxn−Ψz∗ 2
≤αnfxn−z∗2 xn−z∗ 22γnrnxn−un,Ψxn−Ψz∗ −γn un−xn 2.
3.36
The same as in3.34, we have limn→ ∞ un−xn 0. Since
xn−Snxn ≤ xn−Snyn Snyn−Snun Snun−Snxn
≤ xn−Snyn yn−un un−xn ,
3.37
we get xn−Snxn → 0 asn → ∞. From2.10and
Snyn−yn ≤ Snyn−Snxn Snxn−xn xn−yn
≤ Snxn−xn 2 xn−yn
≤ Snxn−xn 2 xn−un un−yn −→0,
3.38
we get limn→ ∞ Syn−yn 0.
Next, we show lim supn→ ∞fz0−z0, xn−z0 ≤0, wherez0 P∞n1FTn∩Ω∩VIA,Cfz0.
To show this inequality, we choose a subsequence{yni}of{yn}such that
lim sup
n→ ∞
fz0−z0, Syn−z0 lim
i→ ∞fz0−z0, Syni−z0. 3.39
Since{yni}is bounded, there exists a subsequence{ynij}of {yni}which converges weakly
obtainSyni z. We now show thatz ∈ ∞
n1FTn∩Ω∩VIA, C. Indeed, we observe that
unTrnxn−rnΨxnand
Φun, v Ψxn, v−un 1
rnv−un, un−xn ≥0 ∀v∈C. 3.40
Froma2, we deduce that
Ψxn, v−un 1
rnv−un, un−xn ≥ −Φun, v Φv, un, 3.41
and hence
Ψxni, v−uni
v−uni,
uni−xni
rni
≥Φv, uni. 3.42
Form limn→ ∞ un−yn 0, we getuni z. Putzttv 1−tzfor allt∈0,1andv∈C.
Consequently, we getzt∈C. From3.42, it follows that
Ψzt, zt−uni ≥ Ψzt, zt−uni − Ψxni, zt−uni −
zt−uni,
uni−xni
rni
Φzt, uni
Ψzt−Ψuni, zt−uniΨuni−Ψxni, zt−uni
−
zt−uni,
uni−xni
rni
Φzt, uni.
3.43
From the Lipschitz continuous ofΨand limn→ ∞ un−xn 0, we obtain Ψuni−Ψxni → 0.
SinceΨis monotone, we know thatΨzt−Ψuni, zt−uni ≥0. Further,uni−xni/rni → 0. It
follows froma4that
Φzt, z≤ lim
i→ ∞Φzt, uni≤ilim→ ∞Ψzt, zt−uniΨzt, zt−z. 3.44
Owing toa1anda4, we get that
0 Φzt, zt≤tΦzt, v 1−tΦzt, z
≤tΦzt, v 1−tΨzt, zt−z
≤tΦzt, v 1−ttΨzt, v−z,
3.45
and hence
Lettingt → 0, we have
Φz, v Ψz, v−z ≥0. 3.47
This implies thatz∈Ω.
Furthermore, we prove thatz∈FS ∞n1FTn. Assumez /Sz, since limn→ ∞ un−
xn 0, we havexniz. From Opial’s theorem 27, we get
lim inf
i→ ∞ xni−z <lim inf
i→ ∞ xni−Sz ≤lim inf
i→ ∞ xni−Sxni Sxni−Sz
≤lim inf
i→ ∞ xni−z .
3.48
This is a contradiction. Hence,z∈FS ∞n1FTn.
Now, we will show thatz∈VIA, C. Let
Tu
⎧ ⎨ ⎩
AuNCu, u∈C,
∅, uis not in C.
3.49
ThenTis a maximal monotone 23. Letu, w∈GT, sincew−Au∈NCuandyn∈C, we
haveu−yn, w−Au ≥0. FromynPCun−λnAun, we have
u−yn, yn−un−λnAun ≥0. 3.50
This is,
u−yn,yn−un
λn Aun
≥0. 3.51
Therefore, we obtain
u−yni, w ≥ u−yni, Au
≥ u−yni, Au −
u−yni,
yni −uni
λni
Auni
u−yni, Au−Auni−
yni−uni
λni
u−yni, Au−Ayniu−yni, Ayni−Auni −
u−yni,
yni−uni
λni
≥u−yni, Ayni−Auni
−
u−yni,
yni −uni
λni
.
Noting that limn→ ∞ un−yn 0 andAis Lipschitz continuous, we obtain
u−z, w ≥0. 3.53
SinceTis maximal monotone, we havez∈T−10 and soz∈VIA, C. Thus,z∈∞
n1FTn∩
Ω∩VIA, C. The property of the metric projection implies that
lim sup
n→ ∞ fz0−z0, xn−z0lim supn→ ∞ fz0−z0, Snyn−z0
lim sup
n→ ∞ fz0−z0, Syn−z0
lim
i→ ∞fz0−z0, Syni−z0
fz0−z0, z−z0 ≤0.
3.54
From3.1we obtain
xn1−z0 2αnfxn βnxnγnSnyn−z0, xn1−z0
αnfxn−z0, xn1−z0βnxn−z0, xn1−z0γnSnyn−z0, xn1−z0
≤αnfxn−fz0, xn1−z0αnfz0−z0, xn1−z0
1
2βn
xn−z0 2 xn1−z0 2
1
2γn
Snyn−z02 xn1−z0 2
≤ 1
21−αn
xn−z0 2 xn1−z0 2
αnfz0−z0, xn1−z0
1
2αn
fxn−fz02 xn1−z0 2
≤ 1
2
1−αn1−α2 xn−z0 21
2 xn1−z0
2
αnfz0−z0, xn1−z0
3.55
which implies that
xn1−z0 2 ≤
1−αn1−α2 xn−z0 22αnfz0−z0, xn1−z0. 3.56
Setting
δn2αnfz0−z0, xn1−z0, 3.57
we have
lim sup
n→ ∞
δn
ApplyingLemma 2.8to3.56, we conclude that{xn}converges strongly toz0. Consequently,
{un}and{yn}converge strongly toz0. This completes the proof.
As direct consequences ofTheorem 3.1, we have the following two corollaries.
Corollary 3.2. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Letf :C → Cbe a contraction mapping with Lipschitz constantα∈ 0,1,and letA:C → Hbe an inverse-strongly monotone mapping with constant >0. Supposex1∈Cand{xn}generated by
xn1αnfxn βnxnγnPCxn−λnAxn 3.59
for alln∈N, where{αn},{βn},{γn}are three sequences in 0,1,{λn}is a sequence in0, bfor some
0 < b <2satisfying conditions (i)–(iv). Then{xn}converges strongly to the pointx∗ ∈VIA, C,
wherex∗PVIA,Cfx∗.
Proof. LetΦu, v 0 andΨx0 for allu, v, x∈Candrn1 inTheorem 3.1. Thenunxnfor
n1,2, . . . .LettingTnIthe identity mappingfor alln∈N, thenSnIforn1,2, . . . .It is
easy to see that all conditions ofTheorem 3.1hold. Therefore, we know that the sequence{xn}
generated by3.59converges strongly tox∗PVIA,Cfx∗. This completes the proof.
Remark 3.3. FromCorollary 3.2, we can get an iterative scheme for finding the solution of the variational inequality involving theα-inverse-strongly monotone mappingA.
Corollary 3.4see 17, Theorem 3.5. LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetΦbe a bifunction fromC×CintoRsatisfying (a1)–(a4),f : C → Ca contraction mapping with constantα ∈ 0,1. LetSn : C → Cbe anS-mapping generated byT1, T2, . . .and
π1, π2, . . .and∞n1FTn∩EPΦ/∅, where sequence{Tn}is nonexpansive and{πn}is a sequence
in0, σfor someσ∈0,1. Supposex1∈Cand{xn},{un}are generated by
Φun, v 1
rnv−un, un−xn ≥0 ∀v∈C,
xn1αnfxn βnxnγnSnun
3.60
for alln∈N, where{αn},{βn},{γn}are three sequences in 0,1, and{rn}is a sequence in0,∞
satisfying conditions (i)–(iii) and (v). Then, the sequences{xn} and{un} converge strongly to the pointx∗∈∞n1FTn∩EPΦ, wherex∗P∞n1FTn∩EPΦfx∗.
Proof. Letλn1 forn1,2, . . .andΨx 0 andAx 0 for allx∈CinTheorem 3.1. Since
un∈C, we get thatun PCun. It follows fromTheorem 3.1that the sequences{xn}and{un}
converge strongly to the pointx∗P∞
n1FTn∩EPΦfx∗. This completes the proof.
Our Theorem 3.1 improves and extends Theorem 3.5 of Yao et al. 17 in the following aspects:
1the equilibrium problem is extended to the generalized equilibrium problem;
2our iterative process3.1is different from Yao et al. iterative process3.60because there are a project operator and anα-inverse-strongly monotone mapping;
3our iterative process3.1is more general than Yao et al. iterative process 3.60 because it can be applied to solving the problem of finding a common element of the set of solutions of generalized equilibrium problems, the set of common fixed points of infinitely many nonexpansive mappings, and the set of solutions of the variational inequality forα-inverse-strongly monotone mapping.
Acknowledgments
This work was supported by the National Natural Science Foundation of China50674078, 50874096, 10671135, 70831005and the Open Fund of State Key Laboratory of Oil and Gas Reservoir Geology and ExploitationSouthwest Petroleum University.
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