Fixed Point Theory and Applications Volume 2011, Article ID 945051,24pages doi:10.1155/2011/945051
Research Article
A Viscosity of Ces `aro Mean Approximation
Methods for a Mixed Equilibrium, Variational
Inequalities, and Fixed Point Problems
Thanyarat Jitpeera, Phayap Katchang, and Poom Kumam
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand
Correspondence should be addressed to Poom Kumam,[email protected]
Received 6 September 2010; Accepted 15 October 2010
Academic Editor: Qamrul Hasan Ansari
Copyrightq2011 Thanyarat Jitpeera 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 introduce a new iterative method for finding a common element of the set of solutions for mixed equilibrium problem, the set of solutions of the variational inequality for a β -inverse-strongly monotone mapping, and the set of fixed points of a family of finitely nonexpansive mappings in a real Hilbert space by using the viscosity and Ces`aro mean approximation method. We prove that the sequence converges strongly to a common element of the above three sets under some mind conditions. Our results improve and extend the corresponding results of Kumam and Katchang2009, Peng and Yao2009, Shimizu and Takahashi1997, and some authors.
1. Introduction
Throughout this paper, we assume thatHis a real Hilbert space with inner product and norm are denoted by·,·and · , respectively and letCbe a nonempty closed convex subset of
H. A mappingT :C → Cis callednonexpansiveifTx−Ty ≤ x−y, for allx, y ∈C. We useFTto denote the set offixed points ofT, that is,FT {x ∈ C : Tx x}. It is assumed throughout the paper thatT is a nonexpansive mapping such thatFT/∅. Recall that a self-mappingf :C → Cis acontractiononCif there exists a constantα∈0,1and
x, y∈Csuch thatfx−fy ≤αx−y.
Letϕ:C → R∪{∞}be a proper extended real-valued function andφbe a bifunction ofC×CintoR, whereRis the set of real numbers. Ceng and Yao1considered the following
mixed equilibrium problemfor findingx∈Csuch that
The set of solutions of1.1is denoted by MEPφ, ϕ. We see thatxis a solution of problem
1.1implies thatx ∈ domϕ {x ∈ C| ϕx < ∞}. If ϕ 0, then the mixed equilibrium problem1.1becomes the followingequilibrium problemis to findx∈Csuch that
φx, y≥0, ∀y∈C. 1.2
The set of solutions of 1.2 is denoted by EPφ. 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 of1.2. Some methods have been proposed to solve the equilibrium problemsee2–14.
Let B : C → H be a mapping. The variational inequality problem, denoted by VIC, B, is to findx∈Csuch that
Bx, y−x≥0, 1.3
for ally∈C.The variational inequality problem has been extensively studied in the literature. See, for example, 15, 16 and the references therein. A mapping B of C into H is called
monotoneif
Bx−By, x−y≥0, 1.4
for allx, y∈C.Bis calledβ-inverse-strongly monotoneif there exists a positive real number
β >0 such that for allx, y∈C
Bx−By, x−y≥βBx−By2. 1.5
LetAbe a strongly positive linear bounded operator onH: that is, there is a constantγ >0 with property
Ax, x ≥γx2, ∀x∈H. 1.6
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH:
min
x∈FT
1
2Ax, x −hx, 1.7
whereAis strongly positive linear bounded operator andhis a potential function forγfi.e.,
hx γfxforx∈H. Moreover, it is shown in17that the sequence{xn}defined by the scheme
xn1nγfxn l−nATxn, 1.8
In 1997, Shimizu and Takahashi18originally studied the convergence of an iteration process{xn}for a family of nonexpansive mappings in the framework of a real Hilbert space. They restate the sequence{xn}as follows:
xn1αnx 1−αn
1
n1
n
j0
Tjx
n, forn0,1,2, . . . , 1.9
wherex0andxare all elements ofCandαnis an appropriate in0,1.They proved that{xn}
converges strongly to an element of fixed point ofTwhich is the nearest tox.
In 2007, Plubtieng and Punpaeng19proposed the following iterative algorithm:
φun, y
1
rn
y−un, un−xn
≥0, ∀y∈H,
xn1nγfxn I−nATun.
1.10
They proved that if the sequence{n}and{rn}of parameters satisfy appropriate condition, then the sequences{xn}and{un}both converge to the unique solutionzof the variational inequality
A−γfz, x−z≥0, ∀x∈FT∩EPφ, 1.11
which is the optimality condition for the minimization problem
min
x∈FT∩EPφ
1
2Ax, x −hx, 1.12
wherehis a potential function forγfi.e.,hx γfxforx∈H.
In 2008, Peng and Yao20introduced an iterative algorithm based on extragradient method which solves the problem of finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a family of finitely nonexpansive mappings and the set of the variational inequality for a monotone, Lipschitz continuous mapping in a real Hilbert space. The sequences generated byv∈C,
x1x∈C,
φun, y
ϕy−ϕun 1 rn
y−un, un−xn
≥0, ∀y∈C,
ynPC
un−γnBun
,
xn1αnvβnxn
1−αn−βn
WnPC
un−λnByn
,
1.13
for alln≥1, whereWnisW-mapping. They proved the strong convergence theorems under
some mind conditions.
In this paper, motivated by the above results and the iterative schemes considered in
approximation method for finding a common element of the set of fixed points of a family of finitely nonexpansive mappings, the set of solutions of the variational inequality problem for aβ-inverse-strongly monotone mapping and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Then, we prove strong convergence theorems which are connected with5,21–24. We extend and improve the corresponding results of Kumam and Katchang9, Peng and Yao20, Shimizu and Takahashi18and some authors.
2. Preliminaries
LetHbe a real Hilbert space with inner product·,·and norm · and letCbe a nonempty closed convex subset ofH. Then
x−y2x2−y2−2x−y, y, 2.1
λx 1−λy2
λx2 1−λy2−λ1−λx−y2, 2.2
for allx, y∈Handλ∈0,1. For every pointx∈H, there exists a uniquenearest pointin
C, denoted byPCx, such that
x−PCx ≤ x−y, ∀y∈C. 2.3
PC is called the metric projection ofH ontoC. It is well known that PC is a nonexpansive
mapping ofHontoCand satisfies
x−y, PCx−PCy
≥PCx−PCy2, 2.4
for everyx, y∈H.Moreover,PCxis characterized by the following properties:PCx∈Cand
x−PCx, y−PCx
≤0, 2.5
x−y2
≥ x−PCx2y−PCx2, 2.6
for allx∈H, y∈C. LetBbe a monotone mapping ofCintoH. In the context of the variational inequality problem the characterization of projection2.5implies the following:
u∈VIC, B⇐⇒uPCu−λBu, λ >0. 2.7
It is also known thatHsatisfies the Opial condition25, that is, for any sequence{xn} ⊂H
withxn x, the inequality
lim inf
n→ ∞ xn−x<lim infn→ ∞ xn−y, 2.8
A set-valued mappingU:H → 2His calledmonotoneif for allx, y∈H,f∈Uxand
g∈Uyimplyx−y, f−g ≥0. A monotone mappingU:H → 2Hismaximalif the graph of
GUofUis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingUis maximal if and only if forx, f∈H×H,x−y, f−g ≥0 for everyy, g∈GUimpliesf∈Ux. LetBbe a monotone mapping ofCintoHand letNCy
be thenormal conetoCaty∈C, that is,NCy{w∈H:u−y, w ≤0,∀u∈C}and define
Uy
⎧ ⎨ ⎩
ByNCy, y∈C,
∅, y /∈C. 2.9
ThenUis themaximal monotoneand 0∈Uyif and only ify∈VIC, B; see26.
For solving the mixed equilibrium problem, let us give the following assumptions for a bifunctionφ:C×C → Rand a proper extended real-valued functionϕ:C → R∪ {∞}
satisfies the following conditions:
A1φx, x 0 for allx∈C;
A2φis monotone, that is,φx, y φy, x≤0 for allx, y∈C;
A3for eachx, y, z∈C, limt→0φtz 1−tx, y≤φx, y;
A4for eachx∈C,y→φx, yis convex and lower semicontinuous;
A5for eachy∈C,x→φx, yis weakly upper semicontinuous;
B1for eachx∈Handr >0, there exist a bounded subsetDx⊆Candyx∈Csuch that
for anyz∈C\Dx,
φz, yx
ϕyx
1
r
yx−z, z−x
< ϕz; 2.10
B2Cis a bounded set.
We need the following lemmas for proving our main results.
Lemma 2.1Peng and Yao20. LetCbe a nonempty closed convex subset of H. Letφ:C×C → R
be a bifunction satisfies (A1)–(A5) and letϕ:C → R∪ {∞}be a proper lower semicontinuous and
convex function. Assume that either (B1) or (B2) holds. For r > 0 andx ∈ H, define a mapping
Tr :H → Cas follows:
Trx
z∈C:φz, yϕy1 r
y−z, z−x≥ϕz, ∀y∈C
, 2.11
for allx∈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 MEPφ, ϕ;
Lemma 2.2Xu27. Assume{an}is a sequence of nonnegative real numbers such that
an1≤1−αnanδn, n≥0, 2.12
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 28. Let C,·,· be an inner product space. Then for all
x, y, z∈Candα, β, γ∈0,1withαβγ 1,we have
αxβyγz2
αx2βy2γz2−αβx−y2−αγx−z2−βγy−z2. 2.13
Lemma 2.4 Suzuki 29. Let {xn} and {yn} be bounded sequences in a Banach space X and
let{βn} be a sequence in0,1 with 0 < lim infn→ ∞βn ≤ lim supn→ ∞βn < 1.Supposexn1
1−βnynβnxnfor all integersn ≥ 0andlim supn→ ∞yn1−yn − xn1−xn≤ 0.Then,
limn→ ∞yn−xn0.
Lemma 2.5Marino and Xu17. AssumeAis a strongly positive linear bounded operator on a
Hilbert spaceHwith coefficientγ >0and0< ρ≤ A−1. ThenI−ρA ≤1−ργ.
Lemma 2.6Bruck30. LetCbe a nonempty bounded closed convex subset of a uniformly convex
Banach spaceEandT : C → Ca nonexpansive mapping. For eachx ∈ Cand the Ces`aro means
Tnx 1/n1ni0Tix,thenlim supn→ ∞Tnx−TTnx0.
3. Main Results
In this section, we show 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 mixed equilibrium problem and the set of solutions of a variational inequality problem for a
β-inverse-strongly monotone mapping in a real Hilbert space by using the viscosity of Ces`aro mean approximation method.
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let φ be a
bifunction of C×Cinto real numbersRsatisfying (A1)–(A5) and let ϕ : C → R∪ {∞} be a
proper lower semicontinuos and convex function. LetTi:C → Cbe a nonexpansive mappings for all
into itself with coefficientα∈0,1and letBbe aβ-inverse-strongly monotone mapping ofCintoH.
LetAbe a strongly positive bounded linear self-adjoint onHwith coefficientγ >0and0< γ < γ/α.
Assume that eitherB1orB2holds. Let{xn},{yn}and{un}be sequences generated byx0∈C,un∈C
and
φun, y
ϕy−ϕun 1 rn
y−un, un−xn
≥0, ∀y∈C,
ynδnun 1−δnPCun−λnBun,
xn1 αnγfxn βnxn
1−βn
I−αnA 1
n1
n
i0
Tiyn, ∀n≥0,
3.1
where{αn},{βn}andδn⊂0,1,{λn} ⊂0,2βand{rn} ⊂0,∞satisfy the following conditions:
i∞n0αn∞andlimn→ ∞αn0,
iilimn→ ∞δn 0,
iii0<lim infn→ ∞βn≤lim supn→ ∞βn<1,
iv{λn} ⊂a, b,∃a, b∈0,2βandlimn→ ∞|λn1−λn|0,
vlim infn→ ∞rn>0andlimn→ ∞|rn1−rn|0.
Then,{xn}converges strongly toz∈Θ,wherezPΘI−Aγfz, which is the unique solution
of the variational inequality
A−γfz, x−z≥0, ∀x∈Θ. 3.2
Proof. Now, we have1−βnI−αnA ≤1−βn−αnγ see9, page 479. Sinceλn ∈0,2β
andBis aβ-inverse-strongly monotone mapping. For anyx, y∈C, we have
I−λnBx−I−λnBy2
x−y−λnBx−By2
x−y2−2λn
x−y, Bx−Byλ2nBx−By2
≤x−y2−2λnβBx−By2λ2nBx−By 2
x−y2λn
λn−2βBx−By2
≤x−y2.
3.3
Letx∗∈Θ, Trn be a sequence of mapping defined as inLemma 2.1andunTrnxn, for alln≥0,we have
un−x∗Trnxn−Trnx∗ ≤ xn−x∗. 3.4
By the fact thatPCandI−λnBare nonexpansive andx∗PCx∗−λnBx∗, we get
yn−x∗δnun 1−δnPCun−λnBun−x∗
≤δnun−x∗ 1−δnPCun−λnBun−PCx∗−λnBx∗
≤δnun−x∗ 1−δnI−λnBun−I−λnBx∗
≤δnun−x∗ 1−δnun−x∗
un−x∗
≤ xn−x∗.
3.5
LetTn 1/n1ni0Ti; it follows that
Tnx−Tny
1
n1
n
i0
Tix− 1 n1
n
i0
Tiy
≤ 1
n1
n
i0
Tix−Tiy
≤ 1
n1
n
i0 x−y
n1
n1x−y
x−y,
3.6
which implies thatTnis nonexpansive. Sincex∗∈Θ,we haveTnx∗ 1/n1 n
i0Tix∗ 1/n1ni0x∗x∗, for allx, y∈C.By3.5and3.6, we have
xn1−x∗αn
γfxn−Ax∗βnxn−x∗
1−βn
I−αnA
Tnyn−x∗
≤αnγfxn−Ax∗βnxn−x∗
1−βn−αnγ
≤αnγfxn−Ax∗βnxn−x∗
1−βn−αnγ
xn−x∗
≤αnγfxn−fx∗αnγfx∗−Ax∗
1−αnγ
xn−x∗
≤αnγαxn−x∗αnγfx∗−Ax∗
1−αnγ
xn−x∗
1−αn
γ−γαxn−x∗αn
γ−γαγfx
∗−Ax∗
γ−γα
≤max
x0−x∗,
γfx∗−Ax∗
γ−γα
.
3.7
Hence{xn}is bounded and also{un},{yn}and{Tnyn}are bounded.
Next, we show that limn→ ∞xn1−xn 0.Observing thatun Trnxn ∈domϕand
un1Trn1xn1∈domϕ, we get
φun, y
ϕy−ϕun 1 rn
y−un, un−xn
≥0, ∀y∈C, 3.8
φun1, yϕy−ϕun1 r1 n1
y−un1, un1−xn1≥0, ∀y∈C. 3.9
Takeyun1in3.8andyunin3.9, by using conditionA2; it follows that
un1−un,
un−xn
rn −
un1−xn1 rn1
≥0. 3.10
Thusun1−un, un−un1xn1−xn1−rn/rn1un1−xn1 ≥0. Without loss of generality,
let us assume that there exists a nonnegative real numbercsuch thatrn > c, for alln ≥ 1.
Then, we have
un1−un2≤ un1−un
xn1−xn1−rrn n1
un1−xn1
3.11
and hence
un1−un ≤ xn1−xn 1
rn1|rn1−rn|un1−xn1
≤ xn1−xn1c|rn1−rn|M1,
whereM1 sup{un−xn :n∈N}. On the other hand, letvn PCun−λnBun; it follows
from the definition of{yn}that
yn1−yn{δn1un1 1−δn1PCun1−λn1Bun1} − {δnun 1−δnPCun−λnBun}
δn1un1−un δn1−δnun 1−δn1PCun1−λn1Bun1 −1−δn1PCun−λnBun 1−δn1PCun−λnBun −1−δnPCun−λnBun
δn1un1−un δn1−δnun
1−δn1{PCun1−λn1Bun1−PCun−λnBun}
δn−δn1PCun−λnBun
≤δn1un1−un|δn1−δn|unvn
1−δn1un1−λn1Bun1−un−λnBun δn1un1−un|δn1−δn|unvn
1−δn1un1−λn1Bun1−un−λn1Bun
un−λn1Bun−un−λnBun ≤δn1un1−un|δn1−δn|unvn
1−δn1{un1−un|λn1−λn|Bun}
|δn1−δn|unvn un1−un 1−δn1|λn1−λn|Bun ≤ |δn1−δn|unvn un1−un|λn1−λn|Bun
≤ |δn1−δn|unvn xn1−xn 1
c|rn1−rn|M1
|λn1−λn|Bun.
3.13
We compute that
Tn1yn1−Tnyn ≤ Tn1yn1−Tn1ynTn1yn−Tnyn
≤ yn1−yn
1
n2
n1
i0
Tiyn− 1
n1
n
i0
Tiyn
yn1−yn
1
n2
n
i0
Tiyn 1
n2T
n1y n− 1
n1
n
i0
yn1−yn −
1
n1n2
n
i0
Tiyn 1
n2T
n1y n
≤ yn1−ynn11n2 n
i0
Tiyn 1 n2T
n1yn
≤ yn1−yn 1
n1n2
n
i0
Tiyn−Tix∗x∗
1
n2
Tn1yn−Tn1x∗x∗
≤ yn1−ynn11n2 n
i0
yn−x∗x∗
1
n2
yn−x∗x∗
≤ yn1−ynnn1n12yn−x∗x∗
1
n2yn−x ∗ 1
n2x ∗
yn1−ynn2 2yn−x∗ 2
n2x ∗
≤ |δn1−δn|unvn xn1−xn1
c|rn1−rn|M1
|λn1−λn|Bunn22yn−x∗ 2
n2x ∗.
3.14
Letxn1 1−βnznβnxn; it follows that
zn
xn1−βnxn
1−βn
αnγfxn
1−βn
I−αnA
Tnyn
1−βn ,
3.15
and hence
zn1−zn
αn1γfxn1 1−βn1I−αn1ATn1yn1
1−βn1
− αnγfxn
1−βn
I−αnA
Tnyn
1−βn
αn1γfxn1
1−βn1
1−βn1Tn1yn1
1−βn1
−αn1ATn1yn1
1−βn1 −
αnγfxn
1−βn
−
1−βn
Tnyn
1−βn
αnATnyn
1−βn
αn1
1−βn1
γfxn1−ATn1yn11α−nβ n
ATnyn−γfxn
Tn1yn1−Tnyn
≤ αn1
1−βn1γfxn1−ATn1yn1
αn
1−βn
ATnyn−γfxnTn1yn1−Tnyn
≤ αn1
1−βn1γfxn1−ATn1yn1
αn
1−βn
ATnyn−γfxn
|δn1−δn|unvn xn1−xn 1c|rn1−rn|M1
|λn1−λn|Bunn22yn−x∗
2
n2x ∗.
3.16
Therefore,
zn1−zn − xn1−xn ≤ 1α−n1β
n1γfxn1−ATn1yn1
αn
1−βnATnyn−γfxn
|δn1−δn|unvn 1crn1−rnM1
|λn1−λn|Bun 2
n2yn−x ∗ 2
n2x ∗.
3.17
It follows fromn → ∞and the conditionsi–v, that
lim sup
n→ ∞ zn1−zn − xn1−xn≤0. 3.18
FromLemma 2.4and3.18, we obtain limn→ ∞zn−xn0 and also
lim
n→ ∞xn1−xnnlim→ ∞
1−βn
Next, we show thatxn−un → 0 asn → ∞.Forx∗∈Θ,we obtain
un−x∗2Trnxn−Trnx
∗2
≤ Trnxn−Trnx∗, xn−x∗ un−x∗, xn−x∗
1
2
un−x∗2xn−x∗2− un−x∗−xnx∗2
1
2
un−x∗2xn−x∗2− un−xn2
,
3.20
and hence
un−x∗2≤ xn−x∗2− un−xn2. 3.21
Sinceyn−x∗ ≤ un−x∗and fromLemma 2.3and3.21, we obtain
xn1−x∗2αnγfxn βnxn
1−βn
I−αnA
Tnyn−x∗2
αn
γfxn−Ax∗βnxn−x∗
1−βn
I−αnA
Tnyn−x∗2
≤αnγfxn−Ax∗2βnxn−x∗2
1−βn−αnγyn−x∗2
≤αnγfxn−Ax∗2βnxn−x∗2
1−βn−αnγ
un−x∗2
≤αnγfxn−Ax∗2βnxn−x∗2
1−βn−αnγ
xn−x∗2− xn−un2
≤αnγfxn−Ax∗2xn−x∗2−
1−βn−αnγ
xn−un2.
3.22
Then, we have
1−βn−αnγ
xn−un2≤αnγfxn−Ax∗2xn−x∗2− xn1−x∗2
αnγfxn−Ax∗2xn−xn1xn−x∗xn1−x∗. 3.23
By limn→ ∞xn1−xn0,iandiv, imply that
lim
n→ ∞un−xn0. 3.24
Since lim infn→ ∞rn>0,we obtain
lim
n→ ∞ xn−un
rn lim
n→ ∞
1
rn
Next, we show that limn→ ∞Tnyn−xn0.Indeed, observe that
xn−Tnyn ≤ xn−xn1xn1−Tnyn
xn−xn1αnγfxn βnxn
1−βn
I−αnA
Tnyn−Tnyn
xn−xn1αnγfxn−αnATnynαnATnynβnxn−βnTnynβnTnyn
1−βn
I−αnA
Tnyn−Tnyn
≤ xn−xn1αnγfxn−ATnynβnxn−Tnyn
3.26
and then
xn−Tnyn ≤
1 1−βn
xn−xn1
αn
1−βn
γfxn−ATnyn. 3.27
Since limn→ ∞xn1−xn0,iandiv, we get limn→ ∞xn−Tnyn0.
Next, we show that limn→ ∞yn−vn0,wherevnPCun−λnBun.FromLemma 2.3
and3.3, we obtain
xn1−x∗2≤αnγfxn−Ax∗2βnxn−x∗2
1−βn
I−αnATnyn−x∗2
≤αnγfxn−Ax∗2βnxn−x∗2
1−βn−αnγyn−x∗2
αnγfxn−Ax∗2βnxn−x∗2
1−βn−αnγ
δnun−x∗ 1−δn{PCun−λnBun−PCx∗−λnBx∗}2
≤αnγfxn−Ax∗2βnxn−x∗2
1−βn−αnγ
δnun−x∗2
1−βn−αnγ
1−δnun−λnBun−x∗−λnBx∗2
αnγfxn−Ax∗2βnxn−x∗2
1−βn−αnγ
δnun−x∗2
1−βn−αnγ
1−δnun−x∗−λnBun−Bx∗2
≤αnγfxn−Ax∗2βnxn−x∗2
1−βn−αnγ
1−βn−αnγ
1−δnxn−x∗2λn
λn−2β
Bun−Bx∗2
≤αnγfxn−Ax∗2
1−αnγ
xn−x∗2
1−βn−αnγ
1−δnλn
λn−2β
Bun−Bx∗2
≤αnγfxn−Ax∗2xn−x∗2 1−βn−αnγ
1−δnab−2βBun−Bx∗2.
3.28
It follows that
0≤1−βn−αnγ
1−δna2β−bBun−Bx∗2
≤αnγfxn−Ax∗2xn−x∗2− xn1−x∗2
≤αnγfxn−Ax∗2xn1−xnxn−x∗xn1−x∗.
3.29
Sinceαn → 0 andxn1−xn → 0,asn → ∞,we obtainBun−Bx∗ → 0 asn → ∞.Using 2.1, we have
vn−x∗2PCun−λnBun−PCx∗−λnBx∗2 ≤ un−λnBun−x∗−λnBx∗, vn−x∗
1
2
un−λnBun−x∗−λnBx∗2vn−x∗2
−1
2
un−λnBun−x∗−λnBx∗−vn−x∗2
1
2
un−x∗2vn−x∗2− un−vn−λnBun−Bx∗2
1
2
un−x∗2vn−x∗2
−un−vn2λ2nBun−Bx∗2−2λnun−vn, Bun−Bx∗
≤ 1
2
un−x∗2vn−x∗2− un−vn2−λn2Bun−Bx∗2
2λnun−vn, Bun−Bx∗
,
3.30
so, we obtain
and hence
xn1−x∗2≤αnγfxn−Ax∗2βnxn−x∗2
1−βn−αnγyn−x∗2
αnγfxn−Ax∗2βnxn−x∗2
1−βn−αnγ
δnun 1−δnvn−x∗2
≤αnγfxn−Ax∗2βnxn−x∗2
1−βn−αnγ
δnun−x∗2
1−βn−αnγ
1−δnvn−x∗2
≤αnγfxn−Ax∗2βnxn−x∗2
1−βn−αnγ
δnun−x∗2
1−βn−αnγ
vn−x∗2
≤αnγfxn−Ax∗2βnxn−x∗2
1−βn−αnγ
δnun−x∗2
1−βn−αnγ
un−x∗2− un−vn2−λ2nBun−Bx∗2
2λnun−vn, Bun−Bx∗
≤αnγfxn−Ax∗2βnxn−x∗2
1−βn−αnγ
δnun−x∗2
1−βn−αnγ
xn−x∗2−
1−βn−αnγ
un−vn2
−1−βn−αnγ
λ2nBun−Bx∗2
1−βn−αnγ
2λnun−vn, Bun−Bx∗
≤αnγfxn−Ax∗2xn−x∗2
1−βn−αnγ
δnun−x∗2
−1−βn−αnγ
un−vn2
−1−βn−αnγ
λ2nBun−Bx∗2
1−βn−αnγ
2λnun−vnBun−Bx∗, 3.32
which implies that
1−βn−αnγ
un−vn2≤αnγfxn−Ax∗2xn−x∗2− xn1−x∗2 1−βn−αnγ
δnun−x∗2
−1−βn−αnγ
λ2nBun−Bx∗2 1−βn−αnγ
≤αnγfxn−Ax∗2xn−xn1xn−x∗xn1−x∗ 1−βn−αnγ
δnun−x∗2−
1−βn−αnγ
λ2
nBun−Bx∗2 1−βn−αnγ
2λnun−vnBun−Bx∗.
3.33
SinceBun −Bx∗ → 0,xn1−xn → 0 asn → 0 and the conditioni–iii, we have un−vn → 0 asn → 0.At the same time, we note that
yn−vnδnun−vnδnun−vn, 3.34
sinceδn → 0, we haveyn−vn → 0 asn → ∞.Consequently, we observe that
Tnyn−yn ≤ Tnyn−xnxn−unun−vnvn−yn −→0, asn−→ ∞. 3.35
It is easy to see thatPΘI−Aγfzis a contradiction ofHinto itself. HenceHis complete, there exists a unique fixed pointz∈H, such thatzPΘI−Aγfz.
Next, we show that
lim sup
n→ ∞
A−γfz,z−xn
≤0. 3.36
Indeed, we can choose a subsequence{yni}of{yn},such that
lim
i→ ∞
A−γfz, z−yni
lim sup
n→ ∞
A−γfz, z−yn
. 3.37
Since{yni}is bounded, there exists a subsequence{ynij}of{yni}which converge weakly to
y ∈ C.Without loss of generality, we can assume thatyni y.FromTnyn−yn → 0, we obtainTnyni y.
Let us showy∈MEPφ, ϕ.Sinceun Trnxn∈domϕ, we obtain
φun, y
ϕy−ϕun 1 rn
y−un, un−xn
≥0, ∀y∈C. 3.38
FromA2, we also have
ϕy−ϕun 1 rn
y−un, un−xn
≥φy, un
, ∀y∈C 3.39
and hence
ϕy−ϕun
y−uni,
uni−xni
rni
≥φy, uni
Fromun−xn → 0,xn−Tnyn → 0 andTnyn−yn → 0,we getuni → y. Sinceuni −
xni/rni → 0,thus that fromA4and the weakly lower semicontinuity ofϕthatφy, y
ϕy−ϕy≤0, for ally∈C.Fortwith 0< t≤1 andx∈C,letxttx 1−ty. Sincex∈C
andy∈C, we havext∈Cand henceφxt, y ϕy−ϕxt≤0.So, fromA1,A4and the
convexity ofϕ, we have
0φxt, xt ϕxt−ϕxt ≤tφxt, x 1−tφ
xt, y
tϕx 1−tϕy−ϕxt
tφxt, x ϕx−ϕxt
1−tφxt, y
ϕy−ϕxt
≤tφxt, x ϕx−ϕxt
.
3.41
Dividing by t, we get φxt, x ϕx − ϕxt ≥ 0. From A3 and the weakly lower
semicontinuity ofϕ,we haveφy, yϕy−ϕy≥0, for ally∈Cand hencey∈MEPφ, ϕ.
Next, we show that y ∈ FTn 1/n1i0n FTi. Assume that y /∈1/n
1ni0FTi, sincey
ni yandTny /y. From Opial’s condition, we have
lim inf
i→ ∞ yni−y<lim inf
i→ ∞ yni−Tny ≤lim inf
i→ ∞
yni −TnyniTnyni−Tny
≤lim inf
i→ ∞ yni−y,
3.42
which is a contradiction. Thus, we obtainy∈FTn 1/n1ni0FTi.
Now, let us show that v ∈ VIC, B.Let U : H → 2H be a set-valued mapping is
defined by
Uv
⎧ ⎨ ⎩
BvNCv, v∈C,
∅, v /∈C,
3.43
whereNCvis the normal cone toCatv∈C.We haveUis maximal monotone and 0∈Uv
if and only ifv ∈ VIC, B.Letv, w ∈ GU,hencew−Bv ∈ NCvandvn ∈ C,we have v−vn, w−Bv ≥0.On the other hand, fromvnPCun−λnBun,we have
v−vn, vn−un−λnBun ≥0, 3.44
that is
v−vn,
vn−un
λn
Bun
Therefor, we have
v−vni, w ≥ v−vni, Bv
≥ v−vni, Bv −
v−vni,
vni −uni
λni
Buni
v−vni, Bv−
vni−uni
λni
−Buni
v−vni, Bv−Bvniv−vni, Bvni−Buni −
v−vni,
vni−uni
λni
≥ v−vni, Bvni −Buni −
v−vni,
vni−uni
λni
≥ v−vniBvni−Buni − v−vni
vni−uni
λni .
3.46
Noting thatvni −uni → 0 as i → ∞and Bisβ-inverse-strongly monotone, hence from 3.46, we obtainv−y, w ≥0 asi → ∞. SinceUis maximal monotone, we havey∈U−10,
and hencey∈VIC, B. Therefore,y∈Θ:ni1FTi∩VIC, B∩MEPφ, ϕ.
SincezPΘI−Aγfz, we have
lim sup
n→ ∞
γf−Az, xn−z
lim sup
n→ ∞
γf−Az, Tnyn−z
lim sup
i→ ∞
γf−Az, Tnyni−z
γf−Az, y−z≤0.
3.47
Finally, we show that{xn}converge strongly toz,we obtain that
xn1−z2αnγfxn βnxn
1−βn
I−αnA
Tnyn−z2
αnγfxn−Az βnxn−z
1−βn
I−αnA
Tnyn−z2
α2
nγfxn−Az
2βnx n−z
1−βn
I−αnA
2βnxn−z
1−βn
I−αnA
Tnyn−z
, αn
γfxn−Az
≤α2nγfxn−Az2βnxn−z
1−βn−αnγ
yn−z 2
2αnβn
xn−z, γfxn−Az
2αn
1−βn−αnγ
Tnyn−z, γfxn−Az
≤α2nγfxn−Az2βnxn−z
1−βn−αnγ
xn−z 2
2αnβn
xn−z, γfxn−γfz
2αnβn
xn−z, γfz−Az
2αn
1−βn−αnγ
Tnyn−z, γfxn−γfz
2αn
1−βn−αnγ
Tnyn−z, γfz−Az
≤α2nγfxn−Az21−αnγ 2
xn−z22αnβnγxn−zfxn−fz
2αnβn
xn−z, γfz−Az
2αn
1−βn−αnγ
γTnyn−zfxn−fz
2αn
1−βn−αnγ
Tnyn−z, γfz−Az
≤α2nγfxn−Az21−αnγ 2
xn−z22αnβnγαxn−z2
2αnβnxn−z, γfz−Az2αn
1−βn−αnγ
γαxn−z2
2αn
1−βn−αnγ
Tnyn−z, γfz−Az
α2nγfxn−Az21−2αnγα2nγ22αnγα−2α2nγγα
xn−z2
2αnβnxn−z, γfz−Az2αn
1−βn−αnγ
Tnyn−z, γfz−Az
≤1−αn
2γ−αnγ2−2γα2αnγγα
xn−z2α2nγfxn−Az 2
2αnβnxn−z, γfz−Az2αn
1−βn−αnγ
Tnyn−z, γfz−Az
≤1−αn
2γ−αnγ2−2γα2αnγγα
xn−z2αnσn,
3.48
whereσnαnγfxn−Az22βnxn−z, γfz−Az21−βn−αnγTnyn−z, γfz−Az.
By3.47,iandiii, we get lim supn→ ∞σn≤0.ApplyingLemma 2.2to3.48we conclude
thatxn → z.This completes the proof.
UsingTheorem 3.1, we obtain the following corollaries.
Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert Space H. Let φ be a
with coefficientα ∈ 0,1and letT : C → Cbe a nonexpansive mapping such thatΘ : FT∩
VIC, B∩EPφ/∅. LetBbe aβ-inverse-strongly monotone mapping ofCintoH.Let{xn},{yn}
and{un}be sequences generated byx0∈C,un∈Cand
φun, y
1
rn
y−un, un−xn
≥0, ∀y∈C,
ynPCun−λnBun,
xn1αnγfxn βnxn
1−βn−αn
Tyn, ∀n≥0,
3.49
where{αn},{βn} ⊂0,1,{λn} ⊂0,2β, for alln≥0satisfy the condition (i), (iii)–(v). Then,{xn}
converges strongly toz∈ΘwherezPΘz.
Proof. TakingTi Tfori0,1, . . . , n,δ
n0,AIandϕ≡0 inTheorem 3.1, we can conclude
the desired conclusion easily.
Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert Space H. Let φ be a
bifunction ofC×Cinto real numbersRsatisfying (A1)–(A5) and letT :C → Cbe a nonexpansive
mapping such thatΘ : FT ∩ VIC, B ∩ EPφ/∅. Let Bbe a β-inverse-strongly monotone
mapping ofCintoH.Let{xn},{yn}and{un}be sequences generated byx0 ∈C,un∈Cand
φun, y
1
rn
y−un, un−xn
≥0, ∀y∈C,
ynPCun−λnBun,
xn1αnvβnxn
1−βn−αn
Tyn, ∀n≥0,
3.50
where{αn},{βn} ⊂0,1,{λn} ⊂0,2β, for alln≥0satisfy the condition (i), (iii)–(v). Then,{xn}
converges strongly toz∈Θ,wherezPΘz.
Proof. Takingγ 1 andfx v, for allx∈CinCorollary 3.2, we can conclude the desired
conclusion easily.
4. Applications to Optimization Problem
LetHbe a real Hilbert space,Ca nonempty closed convex subset ofH,Aa strongly positive linear bounded operator on Hwith a constantγ > 0, andT : C → Cbe a nonexpansive mapping. In this section we will utilize the results present in section main results to study the followingoptimization problem:
min
x∈FT
1
2Ax, x −hx, 4.1
Theorem 4.1. LetCbe a nonempty closed convex subset of a real Hilbert SpaceH. LetT :C → C
be a nonexpansive mappings and let f be a contraction ofCinto itself with coefficientα ∈ 0,1.
LetAbe a strongly positive linear bounded operator onHwith coefficientγ >0and0< γ < γ/α. Let
{αn},{βn} ⊂ 0,1satisfying condition (i) and (iii) inTheorem 3.1. IfFTis a nonempty compact
subset ofC,then for eachn≥0there is a uniquexn∈Csuch that
xn1αnγfxn βnxn
1−βn
I−αnA
Txn, ∀n≥0, 4.2
and the sequence {xn} converges strongly to some point z ∈ FT, which solves the following
optimization problem4.1.
Proof. Takingφ ≡ 0,B ≡0 inCorollary 3.2, we getPC Iand we also haveyn xn.Hence
the sequence{xn}converges strongly to some pointz∈FTwhich is the unique solution of the following variational inequality:
A−γfz, x−z≥0, x∈FT. 4.3
SinceT is nonexpansive, thenFTis convex. Again by the assumption thatFTis compact, therefore, it is a compact and convex subset ofC, and
1
2Ax, x −hx:C−→ R 4.4
is a continuous mapping. By virtue of the well-know Weierstrass theorem, there exists a point
z∗ ∈FTwhich is a minimal point of optimization problem4.1. As is know to all,4.3is the optimality necessary conditionsee Xu31for the optimization problem4.1. Then, we also have
A−γfz∗, x−z∗≥0, x∈FT. 4.5
Since z is the unique solution of 4.3, therefor, z z∗. This complete the proof of
Theorem 4.1.
Acknowledgments
The authors would like to thank the Faculty of science KMUTT. Poom Kumam was supported by the Thailand Research Fund and the Commission on Higher Education under Grant no. MRG5380044.
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