Volume 2010, Article ID 370197,25pages doi:10.1155/2010/370197
Research Article
A General Iterative Method of
Fixed Points for Mixed Equilibrium Problems and
Variational Inclusion Problems
Phayap Katchang
1and Poom Kumam
1, 21Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi,
(KMUTT), Bangmod, Bangkok 10140, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand Correspondence should be addressed to Poom Kumam,[email protected]
Received 27 October 2009; Accepted 16 March 2010
Academic Editor: Jong Kyukkyungnam Kim
Copyrightq2010 P. Katchang and P. Kumam. 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.
The purpose of this paper is to investigate the problem of finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mappings and inverse-strongly monotone mappings, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. We propose a new iterative scheme for finding the common element of the above three sets. Our results improve and extend the corresponding results of the works by Zhang et al.2008, Peng et al.2008, Peng and Yao2009, as well as Plubtieng and Sriprad2009and some well-known results in the literature.
1. Introduction
Throughout this paper, we assume thatHis a real Hilbert space with inner product and norm being denoted by·,·and · , respectively, 2Hdenoting the family of all subsets ofHand letingCbe a closed convex subset of H.AmappingS : C → Cis called nonexpansiveif
Sx−Sy ≤ x−y,for allx, y∈C. We useFSto denote the set of fixed points ofS, that is,FS {x ∈ C : Sx x}. It is assumed throughout the paper thatSis a nonexpansive mapping such thatFS/∅. Recall that a self-mappingf :C → CiscontractiononCif there exists a constantα∈0,1andx, y∈Csuch thatfx−fy ≤αx−y. LetAbe a strongly positive bounded linear operator onH: that is, there is a constantγ >0 with property
LetA : H → H be a single-valued nonlinear mapping and letM : H → 2H be a set-valued mapping. We consider the followingvariational inclusion problem, which is to find a pointu∈Hsuch that
θ∈Au Mu, 1.2
whereθis the zero vector inH. The set of solutions of problem1.2is denoted byIA, M. IfM ∂φ : H → 2H, whereφ : H → R∪ {∞}is a proper convex lower semi-continuous function and∂φis the subdifferential ofφ, then the variational inclusion problem
1.2is equivalent to findu∈Hsuch that
Au, v−uφy−φu≥0, ∀v, y∈H, 1.3
which is called themixed quasivariational inequalitysee1.
IfM∂δC, whereCis a nonempty closed convex subset ofHandδC :H → 0,∞is
the indicator function ofC, that is,
δCx
0, x∈C,
∞, x /∈C, 1.4
then the variational inclusion problem1.2is equivalent to findu∈Csuch that
Au, v−u ≥0, ∀v∈H. 1.5
This problem is calledHartman-Stampacchia variational problemsee2–4.
A set-valued mappingM :H → 2His called monotone if, for allx, y ∈H,f ∈Mx andg∈Myimply thatx−y, f−g ≥0. A monotone mappingM:H → 2His maximal if the graph ofGMofMis not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if, for x, f ∈ H×H,
x−y, f−g ≥0 for everyy, g∈GMimplies thatf ∈Mx.
Let the set-valued mappingM : H → 2H be a maximal monotone. We define the
resolvent operatorJM,λthat is associate withMandλas follows:
JM,λu IλM−1u, u∈H, 1.6
whereλis a positive number. It is worth mentioning that the resolvent operatorJM,λis
single-valued, nonexpansive, and 1-inverse-strongly monotonesee5,6.
Let ϕ : C → R∪ {∞} be a proper extended real-valued function and letF be a bifunction ofC×CintoR, whereRis the set of real numbers. Ceng and Yao7considered the followingmixed equilibrium problemfor findingx∈Csuch that
The set of solutions of1.7is denoted by MEPF, ϕ. We see thatxis a solution of problem
1.7implying thatx∈domϕ {x∈C|ϕx <∞}. Ifϕ 0, then the mixed equilibrium problem1.7becomes the followingequilibrium problemthat is to findx∈Csuch that
Fx, y≥0 ∀y∈C. 1.8
The set of solutions of 1.8 is denoted by EP(F). Given a mapping T : C → H, let
Fx, y Tx, y−x for allx, y ∈ C. Then z ∈ EPF if and only ifTz, y−z ≥ 0 for ally∈C,that is,zis a solution of the variational inequality. The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilibrium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of1.8. Some methods have been proposed to solve the equilibrium problemsee8–21.
In 2008, Zhang et al.6introduced an iterative scheme for finding a common element of the set of solutions to the variational inclusion problem with a multivalued maximal monotone mapping and an inverse-strongly monotone mapping and the set of fixed points of nonexpansive mapping in Hilbert spaces. The iterative scheme isx0x∈H,and:
ynJM,λxn−λAxn,
xn1αnx 1−αnSyn
1.9
for alln≥0. They proved the strong convergence theorem under some mind conditions. In the same year, Peng et al.22introduced an iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of a variational inclusion with set-valued maximal monotone mapping and inverse-strongly monotone mapping, the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping in Hilbert spaces. The sequence{xn}is generated as follows:
x1∈H,
Fun, yr1
ny−un, un−xn ≥0, ∀y∈C,
ynJM,λun−λAun,
xn1αnfxn 1−αnSyn, ∀n∈N.
1.10
They proved that if {αn} and {rn} satisfy appropriate conditions, then {xn} converges
strongly toz∈FS∩IA, M, wherezPFS∩IA,Mfz.
problem in Hilbert spaces. Starting with an arbitraryx1 ∈H, define the sequences{xn},{yn},
and{un}by
Fun, yr1
ny−un, un−xn ≥0, ∀y∈C,
ynJM,λun−λAun,
xn1αnγfxn 1−αnBSyn, ∀n∈N,
1.11
whereAis an inverse-strongly monotone mapping andBis a bounded linear operator onH. They proved that if the sequences{αn}and{rn}of parameters satisfy appropriate conditions,
then{xn}is generated by1.11converging strongly to the unique solution of the variational
inequality
B−γfz, z−x ≤0, ∀x∈FS∩EPF∩IA, M, 1.12
which is the optimality condition for the minimization problem
min
x∈C
1
2Ax, x −hx, 1.13
wherehis a potential function forγf, that is,h1x γfx, for allx∈H.
LetTi :C → C, wherei1,2, . . . , N, be a family of finitely nonexpansive mappings.
Let the mappingWn:C → Cbe defined by
Un,0I,
Un,1λn,1T1Un,0 1−λn,1I,
Un,2λn,2T2Un,1 1−λn,2I,
.. .
Un,N−1λn,N−1TN−1Un,N−2 1−λn,N−1I,
WnUn,Nλn,NTNUn,N−1 1−λn,NI,
1.14
where{λn,1},{λn,2}, . . . ,{λn,N} ∈0,1. Such a mappingWnis called theW-mappinggenerated
byT1, T2, . . . , TNandλn,1, λn,2, . . . , λn,N. Nonexpansivity of eachTiensures the nonexpansivity
ofWn. Moreover, in24, Lemma 3.1, it is shown thatFWn Ni1FTi.
The concept ofW-mappings was introduced in 25, 26. It is now one of the main tools in studying convergence of iterative methods for approaching common fixed points of nonlinear mappings; more recent progresses can be found in24,27,28and the references cited therein.
nonexpansive mappings, and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. The sequence{xn}is generated by
x1x∈C,
Fun, yϕy−ϕun Bxn, y−unr1
ny−un, un−xn ≥0, ∀y∈C,
ynPCun−λAun,
xn1αnxn 1−αnWnPC
un−λAyn, ∀n∈N,
1.15
where A is monotone and Lipschitz continuous mapping and B is an inverse-strongly monotone mapping. They proved the weak convergence theorem if the sequences{αn}and
{rn}of parameters satisfy appropriate conditions.
In this paper, motivated by the above results and the iterative schemes considered by Zhang et al. in6, Peng et al. in 22, Peng and Yao in 29, and Plubtieng and Sriprad in
23, we present a new general iterative scheme for finding a common element of the set of solutions for mixed equilibrium problems, the set of solutions of the variational inclusions with set-valued maximal monotone mapping and inverse-strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in the setting of Hilbert spaces. Then, we prove strong convergence theorem under some mind conditions. Furthermore, by using above result, an iterative algorithm for solution of an optimization problem was obtained. The results presented in this paper extend and improve the results of Zhang et al.6, Peng et al.22, Peng and Yao29, Plubtieng and Sriprad23, and some authors.
2. Preliminaries
LetHbe a real Hilbert space with norm · and inner product·,·and letCbe a closed convex subset ofH. When{xn}is a sequence inH,xn xmeans that{xn}converges weakly
toxandxn → xmeans the strong convergence. In a real Hilbert spaceH, we have
x−y2x2−y2−2x−y, y, λx 1−λy2λx2 1−λy2−λ
1−λx−y2
2.1
for allx, y∈Handλ∈0,1. For every pointx∈H, there exists a unique nearest point inC, denoted byPCx, such that
x−PCx ≤x−y ∀y∈C. 2.2
PC is called the metric projection ofH ontoC. It is well known that PC is a nonexpansive
mapping ofHontoCand satisfies
for everyx, y∈H.Moreover,PCxis characterized by the following properties:
PCx∈C,
x−PCx, y−PCx ≤0,
x−y2 ≥ x−P
Cx2y−PCx2
2.4
for allx∈H, y∈C.
Recall that a mappingAofH into itself is calledβ-inverse-strongly monotone if there exists a positive real numberβsuch that
Au−Av, u−v ≥αAu−Av2, 2.5
for allu, v ∈ H.It is obvious that any β-inverse-strongly monotone mapping A is1/β -Lipschitz monotone and continuous mapping. We also have that, for allu, v∈Handλ >0,
I−λAu−I−λAv2u−v−λAu−Av2
u−v2−2λu−v, Au−Avλ2Au−Av2
≤ u−v2λλ−2αAu−Av2.
2.6
So, ifλ≤2α,thenI−λAis a nonexpansive mapping fromHinto itself.
It is also known that Hsatisfies the Opial condition 30, that is, for any sequence
{xn} ⊂Hwithxn x, the inequality
lim inf
n→ ∞ xn−x<lim infn→ ∞ xn−y 2.7
holds for everyy∈Hwithx /y.
For solving the mixed equilibrium problem, let us give the following assumptions for the bifunctionF,ϕ, and the setC.
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.
A5For eachy∈C, x→Fx, yis weakly upper semicontinuous.
B1For eachx ∈Handr >0, there exist a bounded subsetDx ⊆Candyx ∈ C,such
that for anyz∈C\Dx,
Fz, yxϕyx1ryx−z, z−x< ϕz 2.8
We need the following lemmas for proving our main result.
Lemma 2.1Peng and Yao31. Let C be a nonempty closed convex subset of H. LetF:C×C → R
be a bifunction satisfying (A1)–(A5) and letϕ :C → R∪ {∞}be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. Forr >0andx∈H, define a mapping
Tr :H → Cas follows:
Trx z∈C:Fz, yϕy1ry−z, z−x ≥ϕz, ∀y∈C
2.9
for allz∈H. Then, the following hold. 1For eachx∈H, Trx/∅.
2Tr is single valued.
3Tr is firmly nonexpansive, that is, for anyx, y∈H, Trx−Try2≤ Trx−Try, x−y.
4FTr MEPF, ϕ.
5MEPF, ϕis closed and convex.
Lemma 2.2Xu32. Assume that{an}is a sequence of nonnegative real numbers such that
an1≤1−αnanδn, n≥0, 2.10
where{αn}is a sequence in0,1and{δn}is a sequence inRsuch that
1∞n1αn∞,
2lim supn→ ∞δn/αn≤0or∞n1|δn|<∞.
Thenlimn→ ∞an0.
Lemma 2.3 Osilike and Igbokwe 33. Let E,·,·be an inner product space. Then for all
x, y, z∈Eandα, β, γ∈0,1withαβγ 1,one has
αxβyγz2 αx2βy2γz2−αβx−y2−αγx−z2−βγy−z2. 2.11
Lemma 2.4Colao et al.28. LetCbe a nonempty convex subset of a Banach space. Let{Ti}Ni1 be a family of finitely nonexpansive mappings ofC into itself and let{λn,1},{λn,2}, . . . ,{λn,N} be
sequences in 0,1such thatλn,i → λi i 1, . . . , N. Moreover for every integern ≥ 1, letW
andWn be theW-mappings generated byT1, T2, . . . , TN and λ1, λ2, . . . , λN andT1, T2, . . . , TN and
{λn,1},{λn,2}, . . . ,{λn,N}, respectively. Then for everyx∈C, it follows that
lim
n→ ∞Wnx−Wx0. 2.12
Lemma 2.5Suzuki34. Let{xn}and{yn}be bounded sequences in a Banach spaceX and let
{βn}be a sequence in 0,1with0 < lim infn→ ∞βn ≤ lim supn→ ∞βn < 1.Suppose thatxn1 1−βnynβnxnfor all integersn ≥ 0andlim supn→ ∞yn1−yn − xn1−xn≤ 0.Then,
Lemma 2.6Marino and Xu35. Assume thatAis a strongly positive linear bounded operator on a Hilbert spaceHwith coefficientγ >0and0< ρ≤ A−1. ThenI−ρA ≤1−ργ.
Lemma 2.7Br´ezis5. LetM:H → 2Hbe a maximal monotone mapping andA:H → Hbe a Lipschitz continuous mapping. Then the mappingSMA:H → 2H is a maximal monotone mapping.
Remark 2.8. Lemma 2.7implies that IA, M is closed and convex if M : H → 2H is a maximal monotone mapping andA:H → His a Lipschitz continuous mapping.
Lemma 2.9Zhang et al.6. u ∈ H is a solution of variational inclusion 1.2 if and only if
uJM,λu−λAu,for allλ >0,that is,
IA, M FJM,λI−λA, ∀λ >0. 2.13
3. Main Result
In this section, we prove a strong convergence theorem for finding a common element of the set of fixed points of a family of finitely nonexpansive mappings, the set of solutions of a mixed equilibrium problem, and the set of solutions of a variational inclusion problem for an inverse-strongly monotone mapping in a real Hilbert space.
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert Space H. Let F be a bifunction ofC×Cinto real numbersRsatisfying (A1)–(A5) and letϕ:C → R∪ {∞}be a proper lower semicontinuous and convex function. Letf be a contraction ofH into itself with coefficient
α∈ 0,1. LetAbe anβ-inverse-strongly monotone mapping ofHinto itself, M: H → 2H be a maximal monotone mapping, and letBbe a strongly bounded linear operator onH with coefficient
γ >0and0< γ < γ/α. LetT1, T2, . . . , TNbe a family of finitely nonexpansive mappings ofCintoH
such thatΩ : ∩Nn1FTi∩IA, M∩MEPF, ϕ/∅and letWn be theW-mapping generated by
T1, T2, . . . , TNandλn,1, λn,2, . . . , λn,N. Assume that either (B1) or (B2) holds. Let{xn}be a sequence
generated byx1∈Hand
Fun, yϕy−ϕun r1
ny−un, un−xn ≥0, ∀y∈C,
ynJM,λun−λAun,
vnJM,λyn−λAyn,
xn1 αnγfxn βnxn1−βnI−αnBWnvn
3.1
for every n ≥ 1, where{αn},{βn} ⊂ 0,1,{rn} ⊂ 0,∞, andλ ∈ 0,2βsatisfy the following
conditions:
i∞n0αn∞andlimn→ ∞αn0,
iilim infn→ ∞rn>0 and limn→ ∞|rn1−rn|0, iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1,
Then{xn}converges strongly toz∈Ω, wherezPΩI−Bγfz, which is the unique solution
of the variational inequality
B−γfz, z−x ≤0, x∈Ω. 3.2
Proof. Since limn→ ∞αn 0, we may assume, without loss of generality, thatαn<1−βnB−1
for alln. We assume thatI−B ≤1−γ. SinceBis linear bounded self-adjoint operator onH, we have
Bsup{|Bx, x|:x∈H,x1}. 3.3
Observe that
1−βnI−αnBx, x1−βn−αnBx, x ≥1−βn−αnB ≥0, 3.4
this shows that1−βnI−αnBis positive. It follows that
1−βnI−αnBsup1−βnI−αnBx, x:x∈H,x1
sup1−βn−αnBx, x:x∈H,x1
≤1−βn−αnγ.
3.5
Letp ∈Ω, let{Trn}be a sequence of mappings defined as inLemma 2.1, and letun Trnxn.
For anyn∈N, we have
un−pTrnxn−Trnp≤xn−p. 3.6
Sincep∈Ω, we havepJM,λp−λAp. FromJM,λandI−λAbeing nonexpansive, then we
have
vn−pJM,λyn−λAyn−JM,λp−λAp
≤yn−λAyn−p−λAp
≤yn−p
JM,λun−λAun−JM,λp−λAp
≤un−λAun−p−λAp
≤un−p
≤xn−p
for alln∈N. It follows that
xn1−pαnγfxn βnxn1−βnI−αnBWnvn−p
αnγfxn−Bpβnxn−p1−βnI−αnBWnvn−p
≤αnγfxn−Bpβnxn−p1−βn−αnγvn−p
≤αnγfxn−Bpβnxn−p1−βn−αnγxn−p
≤αnγfxn−γfpαnγfp−Bp1−αnγxn−p
≤αnγαxn−pαnγfp−Bp1−αnγxn−p
1−γ−γααnxn−pγ−γααn
γf
p−Bp γ−γα
3.8
for everyn∈N. It follows by mathematical induction that
xn1−p≤max
x1−p,
γfp−Bp γ−γα
, n≥1. 3.9
Therefore,{xn}is bounded. We also obtain that{un}, {Wnvn}, {fxn}, {yn},and{vn}, are
all bounded.
Next, we show that limn→ ∞xn1−xn 0.Observing thatun Trnxn ∈ domϕand
un1Trn1xn1∈domϕ, we get
Fun, yϕy−ϕun r1 n
y−un, un−xn≥0 ∀y∈C, 3.10
Fun1, y
ϕy−ϕun1
1
rn1
y−un1, un1−xn1 ≥0 ∀y∈C. 3.11
Takeyun1in3.10andyunin3.11; by using conditionA2, we obtain
un1−un,unr−xn
n −
un1−xn1
rn1
≥0. 3.12
Thusun1−un, un−un1xn1−xn1−rn/rn1un1−xn1 ≥0. Without loss of generality,
let us assume that there exists a real numbercsuch thatrn> c,for alln≥1. Then, we have
un1−un2≤ un1−un xn1−xn1−rrn
n1
un1−xn1
and hence
un1−un ≤ xn1−xnr1
n1|
rn1−rn|un1−xn1
≤ xn1−xn1c|rn1−rn|M1,
3.14
whereM1sup{un−xn:n∈N}.
On the other hand, again sinceJM,λandI−λAare nonexpansive, we obtain
vn1−vnJM,λ
yn1−λAyn1
−JM,λyn−λAyn
≤yn1−λAyn1
−yn−λAyn
≤yn1−yn
JM,λun1−λAun1−JM,λun−λAun ≤un1−λAun1−
un−λAyn
≤ un1−un.
3.15
It follows from3.14and3.15that
vn1−vn ≤ xn1−xn 1c|rn1−rn|M1. 3.16
Define the sequence{zn}byxn1 1−βnznβnxn,for alln≥1. Then, observe that
zn xn1−βnxn
1−βn
αnγfxn βnxn
1−βnI−αnBWnvn−βnxn
1−βn
αnγfxn
1−βnI−αnBWnvn
1−βn .
It follows that
zn1−zn αn1γfxn1
1−βn1
I−αn1B
Wn1vn1
1−βn1
−αnγfxn
1−βnI−αnBWnvn
1−βn
αn1γfxn1
1−βn1
1−βn1
Wn1vn1
1−βn1 −
αn1BWn1vn1
1−βn1
−αnγfxn
1−βn −
1−βnWnvn
1−βn
αnBWnvn
1−βn
αn1
1−βn1
γfxn1−BWn1vn1
αn
1−βn
BWnvn−γfxnWn1vn1−Wnvn.
3.18
From the definition ofWn, sinceTiandUn,i, i1,2, . . . , N,are nonexpansive, we have
Wn1vn−Wnvnλn1,NTNUn1,N−1vn 1−λn1,Nvn−λn,NTNUn,N−1vn−1−λn,Nvn
≤ |λn1,N−λn,N|vnλn1,NTNUn1,N−1vn−λn,NTNUn,N−1vn
≤ |λn1,N−λn,N|vnλn1,NTNUn1,N−1vn−TNUn,N−1vn |λn1,N−λn,N|TNUn,N−1vn
≤2M2|λn1,N−λn,N|λn1,NUn1,N−1vn−Un,N−1vn,
3.19
where M2 is an approximate constant such that M2 ≥ max{supn≥1{vn}, {supn≥1{TmUn,m−1vn} |m1,2, . . . , N}}.
Since 0< λni ≤1 for alln≥1 andi1,2, . . . , N, we compute
Un1,N−1vn−Un,N−1vn
λn1,N−1TN−1Un1,N−2vn 1−λn1,N−1vn−λn,N−1TN−1Un,N−2vn−1−λn,N−1vn
≤ |λn1,N−1−λn,N−1|vnλn1,N−1TN−1Un1,N−2vn−λn,N−1TN−1Un,N−2vn
≤ |λn1,N−1−λn,N−1|vnλn1,N−1TN−1Un1,N−2vn−TN−1Un,N−2vn
|λn1,N−1−λn,N−1|TN−1Un,N−2vn
≤2M2|λn1,N−1−λn,N−1|Un1,N−2vn−Un,N−2vn.
It follows that
Un1,N−1vn−Un,N−1vn ≤2M2|λn1,N−1−λn,N−1|2M2|λn1,N−2−λn,N−2| Un1,N−3vn−Un,N−3vn
≤2M2
N−1
i2
|λn1,i−λn,i|Un1,1vn−Un,1vn
2M2
N−1
i2
|λn1,i−λn,i|
λn1,1T1vn 1−λn1,1vn−λn,1T1vn−1−λn,1vn
≤2M2
N−1
i1
|λn1,i−λn,i|.
3.21
Substituting3.21into3.19yields that
Wn1vn−Wnvn ≤2M2|λn1,N−λn,N|2λn1,NM2
N−1
i1
|λn1,i−λn,i|
≤2M2
N
i1
|λn1,i−λn,i|,
3.22
and hence
Wn1vn1−Wnvn ≤ Wn1vn1−Wn1vnWn1vn−Wnvn
≤ vn1−vn2M2
N
i1
|λn1,i−λn,i|.
3.23
Combining3.16and3.23, we obtain
zn1−zn ≤ αn1
1−βn1
γfxn1BWn1vn1
αn
1−βn
BWnvnγfxn
Wn1vn1−Wnvn
≤ αn1
1−βn1
γfxn1BWn1vn1
αn
1−βn
BWnvnγfxn
vn1−vn2M2
N
i1
|λn1,i−λn,i|
≤ αn1
1−βn1
γfxn1BWn1vn1
αn
1−βn
BWnvnγfxn
xn1−xn1c|rn1−rn|M12M2
N
i1
|λn1,i−λn,i|.
So
zn1−zn − xn1−xn ≤ αn1
1−βn1
γfxn1BWn1vn1
αn
1−βn
BWnvnγfxn
1c|rn1−rn|M12M2
N
i1
|λn1,i−λn,i|.
3.25
Conditionsi–ivimply that
lim sup
n→ ∞ zn1−zn − xn1−xn≤0. 3.26
Hence, byLemma 2.5, we have
lim
n→ ∞zn−xn0. 3.27
Consequently,
lim
n→ ∞xn1−xnnlim→ ∞
1−βnzn−xn0. 3.28
Fromii,3.14,3.16, and3.28, we also haveun1−un → 0 and vn1−vn → 0 as
n → ∞.We note that
xn1−xnαnγfxn βnxn1−βnI−αnBWnvn−xn
αnγfxn−αnBxnαnBxnβnxn1−βnI−αnBWnvn
−1−βnI−αnBxn1−βnI−αnBxn−xn
αnγfxn−Bxn1−βnI−αnBWnvn−xn
≤αnγfxn−Bxn1−βn−αnγWnvn−xn
3.29
It follows that
Fromi,iii, and3.28, we obtain
lim
n→ ∞xn−Wnvn0. 3.31
Next, we shall show that limn→ ∞xn −un 0. For any p ∈ Ω, since Trn is firmly
nonexpansive, we have
un−p2Trnxn−Trnp 2≤T
rnxn−Trnp, xn−p
un−p, xn−p
1
2
un−p2xn−p2− un−xn2
.
3.32
It follows that
un−p2≤xn−p2− xn−un2. 3.33
Therefore, we have
xn1−p2αnγfxn βnxn1−βnI−αnBWnvn−p2
αnγfxn−Bp βnxn−p 1−βnI−αnBWnvn−p2
≤αnγfxn−Bp2βnxn−p21−βn−αnγvn−p2
≤αnγfxn−Bp2βnxn−p21−βn−αnγun−p2
≤αnγfxn−Bp2βnxn−p21−βn−αnγxn−p2− xn−un2
αnγfxn−Bp21−αnγxn−p2−1−βn−αnγxn−un2.
3.34
Then, we obtain
1−βn−αnγxn−un2≤αnγfxn−Bp21−αnγxn−p2−xn1−p2
Byi,iii, and3.28imply that
lim
n→ ∞xn−un0. 3.36
Since lim infn→ ∞rn>0,we have
lim
n→ ∞
xn−un
rn
lim
n→ ∞
1
rnxn−un0. 3.37
We note that, by3.34, nonexpansiveness ofJM,λand the inverse-strong monotonicity
ofAimply that
xn1−p2≤α
nγfxn−Bp2βnxn−p21−βn−αnγvn−p2
αnγfxn−Bp2βnxn−p2
1−βn−αnγJM,λyn−λAyn−JM,λp−λAp2
≤αnγfxn−Bp2βnxn−p2
1−βn−αnγyn−λAyn−p−λAp
≤αnγfxn−Bp2βnxn−p2
1−βn−αnγyn−p2λλ−2βAyn−Ap2
≤αnγfxn−Bp2βnxn−p2
1−βn−αnγxn−p21−βn−αnγλλ−2βAyn−Ap2
≤αnγfxn−Bp2xn−p21−βn−αnγλλ−2βAyn−Ap2.
3.38
It follows fromi,iii, and3.28that
0≤1−βn−αnγλ2β−λAyn−Ap2
≤αnγfxn−Bp2xn−p2−xn1−p2
≤αnγfxn−Bp2xn−xn1xn−pxn1−p−→0,
3.39
which implies that
On the other hand, sinceJM,λis firmly nonexpansive, we have
vn−p2J
M,λyn−λAyn−JM,λp−λAp2
≤yn−λAyn−p−λAp, vn−p
1
2
yn−λAyn−p−λAp2vn−p2
−yn−λAyn−p−λAp−vn−p2
≤ 1
2
yn−p2vn−p2−yn−vn−λAyn−Ap2
1
2
yn−p2vn−p2−yn−vn2
2λyn−vn, Ayn−Ap−λ2Ayn−Ap2
≤ 1
2
yn−p2vn−p2−yn−vn2
2λyn−vnAyn−Ap−λ2Ayn−Ap2
,
3.41
which yields that
vn−p2≤yn−p2−yn−vn22λyn−vnAyn−Ap. 3.42
From3.34and3.42, we obtain
xn1−p2 ≤α
nγfxn−Bp2βnxn−p21−βn−αnγvn−p2
≤αnγfxn−Bp2βnxn−p2
1−βn−αnγyn−p2−yn−vn22λyn−vnAyn−Ap
≤αnγfxn−Bp2βnxn−p21−βn−αnγxn−p2
−1−βn−αnγyn−vn2
2λ1−βn−αnγyn−vnAyn−Ap
≤αnγfxn−Bp2xn−p2−1−βn−αnγyn−vn2
2λ1−βn−αnγyn−vnAyn−Ap.
Hence, we get
1−βn−αnγyn−vn2≤αnγfxn−Bp2xn−p2−xn1−p2 2λ1−βn−αnγyn−vnAyn−Ap
≤αnγfxn−Bp2xn−xn1xn−pxn1−p 2λ1−βn−αnγyn−vnAyn−Ap.
3.44
Byi,iii,3.28, and3.40, we have
lim
n→ ∞yn−vn0. 3.45
Similarly, we can prove that
lim
n→ ∞Aun−Ap0. 3.46
By the same idea in3.42,3.45and using3.46, then we obtain that
lim
n→ ∞un−yn0. 3.47
From
Wnvn−vn ≤ Wnvn−xnxn−unun−ynyn−vn, 3.48
hence
lim
n→ ∞Wnvn−vn0, 3.49
and also
vn−xn ≤vn−ynyn−unun−xn −→0 asn−→ ∞. 3.50
Observe thatPΩI−Bγfis a contraction ofHinto itself. Indeed, for allx, y∈H, we have
PΩI−Bγfx−PΩI−Bγfy≤I−Bγfx−I−Bγfy ≤ I−Bx−yγfx−fy
≤1−γx−yγαx−y
1−γ−γαx−y.
SinceHis complete, there exists a unique fixed pointz ∈Hsuch thatz PΩI−Bγfz. Next, we show that
lim sup
n→ ∞
B−γfz, z−xn ≤0. 3.52
Indeed, we can choose a subsequence{vni}of{vn}such that
lim
i→ ∞
B−γfz, z−vni
lim sup
n→ ∞
B−γfz, z−vn. 3.53
Since{vni}is bounded, there exists a subsequence{vnij}of{vni}which converges weakly to
v∈C. Without loss of generality, we can assume thatvni v.FromWnvn−vn → 0,we
obtainWnvni v. Let us show thatv∈MEPF, ϕ.SinceunTrnxn∈domϕ, we have
Fun, yϕy−ϕun r1
ny−un, un−xn ≥0, ∀y∈C. 3.54
FromA2, we also have
ϕy−ϕun r1
ny−un, un−xn ≥F
y, un, ∀y∈C, 3.55
and hence
ϕy−ϕun
y−uni,
uni−xni
rni
≥Fy, uni
, ∀y∈C. 3.56
From xn−un → 0, xn−Wnvn → 0, andWnvn−vn → 0,we get uni v. Since uni−xni/rni → 0,it follows byA4and the weakly lower semicontinuity ofϕthat
Fy, vϕv−ϕy≤0, ∀y∈C. 3.57
Fortwith 0< t≤1 andy ∈C,letytty 1−tv.Sincey∈Candv∈C,we haveyt∈C,
and henceFyt, v ϕv−ϕyt≤0.So, fromA1,A4, and the convexity ofϕ, we have
0Fyt, ytϕyt−ϕyt
≤tFyt, y 1−tFyt, vtϕy 1−tϕv−ϕyt
≤tFyt, yϕy−ϕyt.
3.58
Dividing by t, we get Fyt, y ϕy − ϕyt ≥ 0. From A3 and the weakly lower
Next, we show thatv∈FWn Nn1FTi. Assume thatv /∈ N
n1FTi. Sincevni v
andWnv /v, from Opial’s condition, we have
lim inf
i→ ∞ vni−v<lim infi→ ∞ vni−Wnv
≤lim inf
i→ ∞ vni−WnvniWnvni−Wnv
≤lim inf
i→ ∞ vni−v,
3.59
which is a contradiction. Thus, we obtainv∈FWn nN1FTi.
Next, we show thatv ∈ IA, M. In fact, havingAasβ-inverse-strongly monotone, implies that Ais1/β-Lipschitz continuous monotone mapping and that domain of Ais equal toH. It follows fromLemma 2.7thatMAis a maximal monotone. Lety, g∈GM
A, that is,g−Ay∈My. SincevniJM,λyni−λAyni, we haveyni−λAyni ∈IλMvni,
that is,
1
λ
yni−vni−λAyni
∈Mvni. 3.60
WithMAbeing a maximal monotone, we have
y−vni, g−Ay−
1
λ
yni−vni−λAyni
≥0, 3.61
and so
y−vni, g
≥
y−vni, Ay
1
λ
yni−vni−λAyni
y−vni, Ay−AvniAvni−Ayni
1
λ
yni−vni
≥0y−vni, Avni−Ayni
y−vni,
1
λ
yni−vni
.
3.62
It follows fromyn−vn → 0,Ayn−Avn → 0, andvni vthat
lim
i→ ∞y−vni, gy−v, g ≥0. 3.63
It follows from the maximal monotonicity ofMAthat 0∈MAv, that is,v∈IA, M. This implies thatv∈Ω.
SincezPΩI−Bγfz, it follows that
lim sup
n→ ∞
B−γfz, z−xnlim sup
n→ ∞
B−γfz, z−vn
lim
i→ ∞
B−γfz, z−vni
B−γfz, z−v≤0.
By3.49,3.50, and the last inequality, we have
lim sup
n→ ∞
γfz−Bz, Wnvn−z≤0. 3.65
Finally, we show that{xn}converges strongly toz. Indeed, from3.1, we have
xn1−z2αnγfxn βnxn1−βnI−αnBWnvn−z2
αnγfxn−Bz βnxn−z 1−βnI−αnBWnvn−z2
α2
nγfxn−Bz2βnxn−z 1−βnI−αnBWnvn−z2
2βnxn−z 1−βnI−αnBWnvn−z, αnγfxn−Bz
≤α2
nγfxn−Bz2βnxn−z1−βn−αnγvn−z2
2αnβnxn−z, γfxn−Bz2αn1−βn−αnγWnvn−z, γfxn−Bz
≤α2
nγfxn−Bz2βnxn−z1−βn−αnγxn−z2
2αnβnxn−z, γfxn−γfz2αnβnxn−z, γfz−Bz
2αn1−βn−αnγWnvn−z, γfxn−γfz
2αn1−βn−αnγWnvn−z, γfz−Bz
≤α2
nγfxn−Bz21−αnγ2xn−z2
2αnβnγxn−zfxn−fz2αnβnxn−z, γfz−Bz
2αn1−βn−αnγγWnvn−zfxn−fz
2αn1−βn−αnγWnvn−z, γfz−Bz
≤α2
nγfxn−Bz21−αnγ2xn−z2
2αnβnγαxn−z22αnβnxn−z, γfz−Bz
2αn1−βn−αnγγαxn−z22αn1−βn−αnγWnvn−z, γfz−Bz
α2
nγfxn−Bz2
1−2αnγα2nγ22αnγα−2α2nγγα
xn−z2
2αnβnxn−z, γfz−Bz2αn1−βn−αnγWnvn−z, γfz−Bz
1−αn
2γ−αnγ2−2γα2αnγγα
xn−z2αnσn,
3.66
whereσnαnγfxn−Bz22βnxn−z, γfz−Bz21−βn−αnγWnvn−z, γfz−Bz.
By3.65, we get lim supn→ ∞σn≤0. Hence byLemma 2.2to3.66, we conclude thatxn → z.
UsingTheorem 3.1, we obtain the following corollaries.
Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert SpaceH. Let F be a bifunction ofC×Cinto real numbersRsatisfying (A1)–(A5) and letf be a contraction ofH into itself with coefficientα ∈ 0,1. LetAbe anβ-inverse-strongly monotone mapping ofHinto itself and letM:H → 2Hbe a maximal monotone mapping such thatΘ:FT∩IA, M∩EPF/∅. Let{xn}be a sequence generated byx1∈Hand
Fun, yr1
ny−un, un−xn ≥0, ∀y∈C,
ynJM,λun−λAun,
xn1αnγfxn βnxn
1−βn−αnTJM,λyn−λAyn
3.67
for everyn ≥ 1, where {αn},{βn} ⊂ 0,1,{rn} ⊂ 0,∞,and λ ∈ 0,2βsatisfy the conditions
(i)–(iii) inTheorem 3.1. Then{xn}converges strongly tozPΘγfz.
Proof. TakingTiTfori1,2, . . . , N,BI,andϕ0, inTheorem 3.1, we can conclude the
desired conclusion easily. This completes the proof.
Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert SpaceH. Let F be a bifunction ofC×Cinto real numbersRsatisfying (A1)–(A5) and letϕ:C → R∪ {∞}be a proper lower semicontinuous and convex function. Letf be a contraction ofH into itself with coefficient
α ∈ 0,1. LetAbe anβ-inverse-strongly monotone mapping ofCintoHand letBbe a strongly bounded linear operator onHwith coefficientγ >0and0< γ < γ/α. LetT1, T2, . . . , TNbe a family of finitely nonexpansive mappings ofCintoHsuch thatΥ:Nn1FTi∩VIC, A∩MEPF, ϕ/∅
and letWnbe theW-mapping generated byT1, T2, . . . , TNandλn,1, λn,2, . . . , λn,N. Assume that either
(B1) or (B2) holds. Let{xn}be a sequence generated byx1 ∈Hand
Fun, yϕy−ϕun r1
ny−un, un−xn ≥0, ∀y∈C,
ynPCun−λAun,
vnPCyn−λAyn,
xn1 αnγfxn βnxn1−βnI−αnBWnvn
3.68
For every n ≥ 1. where {αn},{βn} ⊂ 0,1,{rn} ⊂ 0,∞,andλ ∈ 0,2βsatisfy the condition
(i)–(iv) inTheorem 3.1. Then{xn}converges strongly toz ∈Υwhich is the unique solution of the
variational inequality
B−γfz, z−x≤0, x∈Υ. 3.69
Equivalently, one haszPΥI−Bγfz.
Proof. From Theorem 3.1putM ∂δC; thenJM,λ PC. So we haveyn PCun −λAun
andvn PCyn−λAyn. The conclusion ofCorollary 3.3can be obtained fromTheorem 3.1
4. Application
In this section, we study a kind of optimization problem by using the result of this paper. We will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions:
minhx, x∈C, 4.1
wherehxis a convex and lower semicontinuous functional defined convex subsetCof a Hilbert spaceH. We denote byMhthe set of solutions of4.1. LetF : C×C → Rbe a bifunction defined byFx, y hy−hx. We consider the equilibrium problem1.8; it is obvious that EPF Minh. Therefore, fromTheorem 3.1, we give the following corollary.
Corollary 4.1. Let C be a nonempty closed convex subset of a real Hilbert SpaceH. Let F be a bifunction ofC×Cinto real numbersRsatisfying (A1)–(A5) and leth:C → R∪ {∞}be a lower semicontinuous and convex function. Letfbe a contraction ofHinto itself with coefficientα∈0,1.
LetAbe anβ-inverse-strongly monotone mapping ofHinto itself, letM :H → 2Hbe a maximal monotone mapping, and letBbe a strongly bounded linear operator onHwith coefficientγ > 0and
0< γ < γ/α. LetT1, T2, . . . , TNbe a family of finitely nonexpansive mappings ofCintoHsuch that
Φ:Nn1FTi∩IA, M∩Minh/∅and letWnbe theW-mapping generated byT1, T2, . . . , TN
andλn,1, λn,2, . . . , λn,N. Let{xn}be a sequence generated byx1∈Hand
hy−hun r1
ny−un, un−xn ≥0, ∀y∈C,
ynJM,λun−λAun,
vnJM,λyn−λAyn,
xn1 αnγfxn βnxn1−βnI−αnBWnvn
4.2
for every n ≥ 1, where{αn},{βn} ⊂ 0,1,{rn} ⊂ 0,∞, andλ ∈ 0,2βsatisfy the following
conditions:
i∞n0αn∞andlimn→ ∞αn0.
iilim infn→ ∞rn>0 and limn→ ∞|rn1−rn|0.
iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1.
ivlimn→ ∞|λn,i−λn−1,i|0for alli1,2, . . . , N.
Then{xn}converges strongly toz∈Φ, wherezPΦI−Bγfz, which is the unique solution of the variational inequality
B−γfz, z−x≤0, x∈Φ. 4.3
Proof. From Theorem 3.1 put Fun, y hy − hun and ϕ ≡ 0. The conclusion of
Acknowledgments
The authors would like to thank the Centre of Excellence in Mathematics, under the Commission on Higher Education, Ministry of Education, Thailand. Mr. Phayap Katchang was supported by King Mongkut’s Diamond scholarship for fostering special academic skills by KMUTT for Ph.D. Program at KMUTT. Moreover, the authors are also very grateful to Professor Yeol Je Cho and Professor Jong Kyu Kim for the hospitality.
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