Volume 2008, Article ID 417089,15pages doi:10.1155/2008/417089
Research Article
A New Hybrid Iterative Algorithm for
Fixed-Point Problems, Variational Inequality
Problems, and Mixed Equilibrium Problems
Yonghong Yao,1 Yeong-Cheng Liou,2 and Jen-Chih Yao3
1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China 2Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan 3Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
Correspondence should be addressed to Jen-Chih Yao,[email protected]
Received 29 August 2007; Accepted 6 February 2008
Recommended by Tomonari Suzuki
We introduce a new hybrid iterative algorithm for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the variational inequality of a monotone mapping, and the set of solutions of a mixed equilibrium problem. This study, proves a strong convergence theorem by the proposed hybrid iterative algorithm which solves fixed-point problems, variational inequality problems, and mixed equilibrium problems.
Copyrightq2008 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.
1. Introduction
Let Cbe a nonempty closed convex subset of a real Hilbert space H. Recall that a mapping
f : C → Cis called contractive if there exists a constantα ∈ 0,1such thatfx−fy ≤
αx−yfor allx, y∈C. A mappingT :C→Cis said to be nonexpansive ifTx−Ty ≤ x−y for allx, y ∈C. Denote the set of fixed points ofT byFT.
Letϕ:C→Rbe a real-valued function andΘ:C×C→Rbe an equilibrium bifunction, that is,Θu, u 0 for eachu∈C. The mixed equilibrium problemfor short, MEPis to find
x∗∈Csuch that
MEP: Θx∗, yϕy−ϕx∗≥0 ∀y∈C. 1.1 In particular, ifϕ≡ 0, this problem reduces to the equilibrium problemfor short, EP, which is to findx∗∈Csuch that
Denote the set of solutions of MEP by Ω. The mixed equilibrium problems include fixed-point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and the equilibrium problems as special cases see, e.g., 1–5 . Some methods
have been proposed to solve the MEP and EP see, e.g., 5–14 . In 1997, Combettes and
Hirstoaga13 introduced an iterative method of finding the best approximation to the initial data and proved a strong convergence theorem. Subsequently, S. Takahashi and W. Takahashi 8 introduced another iterative scheme for finding a common element of the set of solutions of EP and the set of fixed-point points of a nonexpansive mapping. Yao et al.12 considered an iterative scheme for finding a common element of the set of solutions of EP and the set of common fixed points of an infinite nonexpansive mappings. Very recently, Zeng and Yao 14 considered a new iterative scheme for finding a common element of the set of solutions of MEP and the set of common fixed points of finitely many nonexpansive mappings. Their results extend and improve many results in the literature.
LetAofCintoHbe a nonlinear mapping. It is well known that the variational inequality problem is to findu∈Csuch that
Au, v−u ≥0 ∀v ∈C. 1.3
The set of solutions of the variational inequality problem is denoted byV IC, A. A mapping
A: C→H is calledβ-inverse-strongly monotone if there exists a positive real numberβsuch that
Au−Av, u−v ≥βAu−Av2 ∀u, v∈C. 1.4
Recently, some authors have proposed new iterative algorithms to approximate a common element of the set of fixed points of a nonxpansive mapping and the set of solutions of the variational inequality. For the details, see15,16 and the references therein.
Motivated by the recent works, in this paper we introduce a new hybrid iterative algorithm for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the variational inequality of a monotone mapping, and the set of solutions of a mixed equilibrium problem. We prove a strong convergence theorem by the proposed hybrid iterative algorithm which solves fixed-point problems, variational inequality problems, and mixed equilibrium problems.
2. Preliminaries
LetHbe a real Hilbert space with inner product·,·and norm·. LetCbe a nonempty closed convex subset ofH. Then for anyx ∈H, there exists a unique nearest point inC, denoted by
PCxsuch that
x−PCx≤ x−y ∀y∈C. 2.1
Such aPCis called the metric projection ofHontoC. It is well known thatPCis a nonexpansive
mapping and satisfies
Moreover,PCis characterized by the following properties:
x−PCx, y−PCx≤0,
x−y2≥x−P
Cx2y−PCx2 ∀x∈H, y∈C.
2.3
It is clear that
u∈V IC, A⇐⇒uPCu−λAu ∀λ >0. 2.4
In this paper, for solving the mixed equilibrium problems for an equilibrium bifunction Θ:C×C→R, we assume thatΘsatisfies the following conditions:
H1 Θis monotone, that is,Θx, y Θy, x≤0 for allx, y ∈C;
H2for each fixedy∈C,x→Θx, yis concave and upper semicontinuous; H3for eachx∈C,y→Θx, yis convex.
A mappingη:C×C→ His called Lipschitz continuous if there exists a constantλ >0 such that
ηx, y≤λx−y ∀x, y ∈C. 2.5
A differentiable functionK:C→Ron a convex setCis called: iη-convex if
Ky−Kx≥Kx, ηy, x ∀x, y∈C, 2.6
whereKis the Fr´echet derivative ofKatx;
iiη-strongly convex if there exists a constantσ >0 such that
Ky−Kx−Kx, ηy, x≥σ 2
x−y2 ∀x, y∈C. 2.7
LetCbe a nonempty closed convex subset of a real Hilbert spaceH,ϕ:C→Rbe a real-valued function, andΘ: C×C →Rbe an equilibrium bifunction. Letrbe a positive number. For a given pointx∈C, the auxiliary problem for MEP consists of findingy∈Csuch that
Θy, z ϕz−ϕy 1rKy−Kx, ηz, y≥0 ∀z∈C. 2.8
LetSr :C→Cbe the mapping such that for eachx∈C,Srxis the solution set of the auxiliary
problem MEP, that is,
Srx y∈C:Θy, z ϕz−ϕy 1rKy−Kx, ηz, y≥0∀z∈C
∀x∈C.
Lemma 2.1see14 . LetC be a nonempty closed convex subset of a real Hilbert space H and let
ϕ : C → Rbe a lower semicontinuous and convex functional. LetΘ : C×C → Rbe an equilibrium bifunction satisfying conditions (H1)–(H3). Assume that
iη:C×C→His Lipschitz continuous with constantλ >0such that aηx, y ηy, x 0for allx, y∈C,
bη·,·is affine in the first variable,
cfor each fixedy ∈C,x→ ηy, xis sequentially continuous from the weak topology to the weak topology;
iiK : C → Risη-strongly convex with constantσ > 0and its derivative K is sequentially continuous from the weak topology to the strong topology;
iiifor eachx∈C, there exist a bounded subsetDx⊂Candzx∈Csuch that for anyy∈C\Dx,
Θy, zxϕzx−ϕy 1rKy−Kx, ηzx, y<0. 2.10
Then there hold the following:
iSris single-valued;
iiSris nonexpansive ifKis Lipschitz continuous with constantν >0such thatσ ≥λνand
Kx
1
−Kx
2
, ηu1, u2
≥Ku
1
−Ku
2
, ηu1, u2
∀x1, x2
∈C×C, 2.11
whereuiSrxifori1,2;
iiiFSr Ω;
vi Ωis closed and convex.
We also need the following lemmas for proving our main results.
Lemma 2.2see17 . Let{xn}and{zn}be bounded sequences in a Banach spaceXand let{βn}be
a sequence in0,1 with0< lim infn→∞βn ≤lim supn→∞βn <1. Supposexn1 1−βnznβnxn
for all integersn≥0andlim supn→∞zn1−zn − xn1−xn≤0. Thenlimn→∞zn−xn0.
Lemma 2.3 see 18 . Assume {an} is a sequence of nonnegative real numbers such that an1 ≤
1−γnanδn, where{γn}is a sequence in0,1and{δn}is a sequence such that
1∞n1γn∞;
2lim supn→∞δn/γn≤0or∞n1|δn|<∞.
Thenlimn→∞an0.
3. Iterative algorithm and strong convergence theorems
infinite mappings ofCinto itself and letξ1, ξ2, . . .be real numbers such that 0≤ξi≤1 for every i∈N. For anyn∈N, define a mappingWnofCinto itself as follows:
Un,n1I,
Un,n ξnTnUn,n1
1−ξnI, Un,n−1ξn−1Tn−1Un,n
1−ξn−1
I,
.. .
Un,kξkTkUn,k1
1−ξkI, Un,k−1ξk−1Tk−1Un,k 1−ξk−1I,
.. .
Un,2ξ2T2Un,3 1−ξ2I,
WnUn,1 ξ1T1Un,2
1−ξ1
I.
3.1
SuchWnis called theW-mapping generated byTn, Tn−1, . . . , T2, T1andξn, ξn−1, . . . , ξ2, ξ1. For the
iterative algorithm for a finite family of nonexpansive mappings, we refer the reader to19 .
We have the following crucial Lemmas3.1and3.2concerningWnwhich can be found in
20 . Now we only need the following similar version in Hilbert spaces.
Lemma 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T1, T2, . . . be
nonexpansive mappings of C into itself such that ∞n1FTn is nonempty, and let ξ1, ξ2, . . . be real
numbers such that 0 < ξi ≤ b < 1 for any i ∈ N. Then for every x ∈ C and k ∈ N, the limit
limn→∞Un,kxexists.
Lemma 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T1, T2, . . . be
nonexpansive mappings of C into itself such that ∞n1FTn is nonempty, and let ξ1, ξ2, . . . be real
numbers such that0< ξi≤b <1for anyi∈N. ThenFW ∞n1FTn.
The following remark12 is important to prove our main results.
Remark 3.3. Using Lemma 3.1, one can define a mapping W of C into itself as Wx limn→∞Wnx limn→∞Un,1x for every x ∈ C. If {xn} is a bounded sequence in C, then we
have
lim
n→∞Wxn−Wnxn0. 3.2 Throughout this paper, we will assume that 0< ξi≤b <1 for everyi∈N.
Now we introduce the following iteration algorithm.
contraction ofCinto itself with coefficientα∈0,1and givenx0∈Carbitrarily. Suppose that
the sequences{xn}and{yn}are generated iteratively by
Θzn, xϕx−ϕzn1rKzn−Kxn, ηx, zn≥0 ∀x∈C, ynPCzn−λnAzn,
xn1αnfWnxnβnxnγnWnPCyn−λnAyn ∀n≥0,
3.3
where{αn},{βn}, and{γn}are three sequences in0,1, and{λn}is a sequence in0,2β .
Now we study the strong convergence of the hybrid iterative algorithm3.3.
Theorem 3.5. LetCbe a nonempty closed convex subset of a real Hilbert spaceH and letϕ: C→ R be a lower semicontinuous and convex functional. LetΘ : C×C → Rbe an equilibrium bifunction satisfying conditions (H1)–(H3) and letT1, T2, . . .be an infinite family of nonexpansive mappings ofC
into itself. LetA:C→ Hbe aβ-inverse-strongly monotone mapping such that∩∞n1FTn∩V IA, C∩
Ω /∅. Suppose{αn},{βn}, and{γn}are three sequences in0,1withαnβnγn1, n≥0. Assume
that
iη:C×C→His Lipschitz continuous with constantλ >0such that aηx, y ηy, x 0for allx, y∈C,
bη·,·is affine in the first variable,
cfor each fixedy ∈C,x→ ηy, xis sequentially continuous from the weak topology to the weak topology;
iiK : C → R is η-strongly convex with constant σ > 0 and its derivative K is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constantν >0such thatσ ≥λν;
iiifor eachx∈C; there exist a bounded subsetDx⊂Candzx∈Csuch that for anyy∈C\Dx,
Θy, zxϕzx−ϕy 1rKy−Kx, ηzx, y<0; 3.4
ivlimn→∞αn 0, ∞n0αn ∞, 0 < lim infn→∞βn ≤ lim supn→∞βn < 1, λn ∈ a, b ⊂ 0,2β, andlimn→∞λn1−λn 0.
Letf be a contraction ofCinto itself and givenx0 ∈Carbitrarily. Then the sequence{xn}generated
by3.3converges strongly tox∗PΓfx∗, whereΓ ∩n∞1FTn∩V IA, C∩Ωprovided thatSr is
firmly nonexpansive.
Proof. We first note thatfis a contraction with coefficientα∈0,1. ThenPΓfx−PΓfy ≤
fx−fy ≤ αx−yfor allx, y ∈C. ThereforePΓf is a contraction ofCinto itself which
implies that there exists a unique elementx∗∈Csuch thatx∗PΓfx∗.
Next we divide the following proofs into several steps.
Step 1 {xn}, {yn}, and {zn}are bounded. Let x∗ ∈ Γ. From the definition ofSr, we know that znSrxn. It follows that
zn−x∗S
For allx, y ∈Candλn∈0,2β , we note that
I−λnA
x−I−λnAy2x−y−λnAx−Ay2
x−y2−2λ
nAx−Ay, x−yλ2nAx−Ay2
≤ x−y2λ
nλn−2βAx−Ay2,
3.6
which implies thatI−λnAis nonexpansive.
Setun PCyn−λnAynfor all n ≥ 0. From 2.4, we have that x∗ PCx∗−λnAx∗. It
follows from3.6that yn−x∗P
Czn−λnAzn−PCx∗−λnAx∗
≤zn−λnAzn−x∗−λnAx∗≤zn−x∗≤xn−x∗,
un−x∗P
Cyn−λnAyn−PCx∗−λnAx∗
≤yn−λnAyn−x∗−λnAx∗≤yn−x∗≤xn−x∗.
3.7
Hence we obtain that xn1−x∗α
nfWnxn−x∗βnxn−x∗γnWnun−x∗
≤αnfWnxn−x∗βnxn−x∗γnWnun−x∗
≤αnfWnxn−fx∗αnfx∗−x∗βnxn−x∗γnun−x∗
≤αnβxn−x∗αnfx∗−x∗βnxn−x∗γnxn−x∗
1−βαn
f
x∗−x∗
1−β
1−1−βαnxn−x∗
≤max xn−x∗,
f
x∗−x∗ 1−β
≤max x0−x∗,
f
x∗−x∗ 1−β
.
3.8
Therefore{xn}is bounded, so are{yn}and{zn}.
Step 2xn1−xn →0. Settingxn1βnxn 1−βnVnfor alln≥0. It follows that
Vn1−Vn
xn2−βn1xn1
1−βn1 −
xn1−βnxn
1−βn
αn1f
Wn1xn1
γn1Wn1un1
1−βn1 −
αnfWnxnγnWnun
1−βn
αn1f
Wn1xn1
1−βn1 −
αnfWnxn
1−βn γn1
1−βn1
Wn1un1−Wn1un
γ
n1
1−βn1 −
γn
1−βn
Wn1un1−γnβ n
Wn1un−Wnun,
which implies that
Vn1−Vn≤ αn1
1−βn1
f
Wn1xn1Wn1un
αn 1−βn
f
WnxnWn1un
γn1 1−βn1
un1−un γn
1−βn
Wn1un−Wnun.
3.10
Now we estimateun1−unandWn1un−Wnun.
From3.1, sinceTiandUn,iare nonexpansive, we have
Wn1un−Wnunξ1T1Un1,2un−ξ1T1Un,2un
≤ξ1Un1,2un−Un,2un
ξ1ξ2T2Un1,3un−ξ2T2Un,3un ≤ξ1ξ2Un1,3un−Un,3un
≤ · · ·
≤ξ1ξ2· · ·ξnUn1,n1un−Un,n1un
≤Mn i1
ξi,
3.11
whereMis a constant such thatsup{Un1,n1un−Un,n1un, n≥0} ≤M.
At the same time, we observe that yn1−ynPC
zn1−λn1Azn1
−PCzn−λnAzn
≤zn1−λn1Azn1
−zn−λnAzn
zn1−λn1Azn1
−zn−λn1Aznλn−λn1
Azn
≤zn1−λn1Azn1
−zn−λn1Aznλn−λn1Azn
≤zn1−znλn−λn1Azn,
un1−unPC
yn1−λn1Ayn1
−PCyn−λnAyn
≤yn1−λn1Ayn1
−yn−λnAyn
yn1−λn1Ayn1
−yn−λn1Ayn
λn−λn1
Ayn
≤yn1−ynλn−λn1Ayn
≤zn1−znλn−λn1AynAzn.
SinceznSrxnandzn1Srxn1, from the nonexpansivity ofSr, we get
zn1−zn≤xn1−xn. 3.13 Substituting3.11–3.13into3.10, we have
Vn1−Vn−xn1−xn≤ αn1
1−βn1
f
Wn1xn1Wn1un
αn 1−βn
f
WnxnWn1unM n
i1
ξi
λn−λn1AynAzn.
3.14
Sinceαn→0, λn1−λn→0, andξi∈a, b , we have
lim sup
n→∞
Vn1−Vn−xn1−xn≤0. 3.15
Hence byLemma 2.2, we have
lim
n→∞Vn−xn0. 3.16 Consequently,
lim
n→∞xn1−xn0. 3.17 Step 3un−Wun →0. Note thatxn1−xnαnfWnxn−xn γnWnun−xn. Then we have
xn−Wnun≤ 1
γn
xn−xn1αnf
Wnxn−xn−→0. 3.18
Forx∗∈Γ, noting thatSris firmly nonexpansive, we have
zn−x∗2S
rxn−Srx∗2
≤Srxn−Srx∗, xn−x∗
zn−x∗, xn−x∗
1
2zn−x
∗2x
n−x∗2−xn−zn2,
3.19
and hence
zn−x∗2≤x
n−x∗2−xn−zn2. 3.20
So, we have
xn1−x∗2≤α
nfWnxn−x∗2βnxn−x∗2γnWnun−x∗2
≤αnfWnxn−x∗2βnxn−x∗2γnun−x∗2
≤αnfWnxn−x∗2βnxn−x∗2γnzn−x∗2
≤αnfWnxn−x∗2βnxn−x∗2γnxn−x∗2−xn−zn2
≤αnfWnxn−x∗2xn−x∗2−γnxn−zn2,
that is,
xn−zn2≤ 1
γn
αnfWnxn−x∗2xn−x∗2−xn1−x∗2
≤ γ1
n
αnfWnxn−x∗2xn1−xnxn−x∗xn1−x∗
−→0.
3.22
From3.6, we obtain that
xn1−x∗2 ≤α
nfWnxn−x∗2βnxn−x∗2γnyn−x∗2
≤αnfWnxn−x∗2βnxn−x∗2γnzn−λnAzn−x∗−λnAx∗2
≤αnfWnxn−x∗2βnxn−x∗2γnzn−x∗2λnλn−2βAzn−Ax∗2
≤αnfWnxn−x∗2xn−x∗2γnab−2βAzn−Ax∗2.
3.23 Then we have
−γnab−2βAzn−Ax∗2≤αnfWnxn−x∗2xn−x∗2−xn1−x∗2−→0, 3.24
which implies that
lim
n→∞Azn−Ax
∗0. 3.25
We note that yn−x∗2P
Czn−λnAzn−PCx∗−λnAx∗2
≤zn−λnAzn−x∗−λnAx∗, yn−x∗
1 2
zn−λnAzn−x∗−λnAx∗2yn−x∗2
−zn−λnAzn−x∗−λnAx∗−yn−x∗2
≤ 1
2
zn−x∗2yn−x∗2−zn−yn−λnAzn−Ax∗2
1 2
zn−x∗2yn−x∗2−zn−yn22λnAzn−Ax∗, zn−yn−λ2nAzn−Ax∗2
.
3.26 Then we derive
yn−x∗2≤z
n−x∗2−zn−yn22λnAzn−Ax∗, zn−yn−λ2nAzn−Ax∗2
≤xn−x∗2−zn−yn22λnAzn−Ax∗, zn−yn.
Hence
xn1−x∗2
≤αnfWnxn−x∗2βnxn−x∗2γnxn−x∗2−zn−yn22λnAzn−Ax∗, zn−yn,
3.28 which implies that
zn−yn≤γ1 n
αnfWnxn−x∗2xn−x∗2−xn1−x∗22γnλnAzn−Ax∗zn−yn−→0.
3.29 Sinceyn−unPCzn−λnAzn−PCyn−λnAyn ≤ zn−yn, we have
Wun−un≤Wun−WnunWnun−un
≤Wun−WnunWnun−xnxn−znzn−ynyn−un
≤Wun−WnunWnun−xnxn−zn2zn−yn.
3.30
Combining the above inequality,3.18–3.29, andRemark 3.3, we have lim
n→∞Wun−un0. 3.31 Step 4limn→∞fx∗−x∗, xn−x∗, wherex∗ PΓfx∗. To show the above inequality, we can
choose a subsequence{uni}of{un}such that
lim
j→∞
fx∗−x∗, u
nj−x∗
lim sup
n→∞
fx∗−x∗, u
n−x∗. 3.32
Since{unj}is bounded, there exists a subsequence{unji}of{unj}which converges weakly tow. Without
loss of generality, we can assume thatunj w. FromWun−un →0, we obtainWunj w.
First, we show w ∈ FW ∩∞n1FTn. Assume that ω /∈ FW. Since unj w and w /Ww, by Opial’s condition, we have
lim inf
j→∞ unj −w<lim infj→∞ unj −Ww
≤lim inf
j→∞ unj −WunjWunj−Ww
≤lim inf
j→∞ unj −w,
3.33
which is a contradiction. Hence we getw ∈FW. By the same argument as that in the proof of [21, Theorem 3.1], we can prove thatw∈V IA, C; and by the same argument as that in the proof of [14, Theorem 4.1], we also can prove thatw∈Ω. Hencew∈Γ.
Sincex∗PΓfx∗∈Γandun−xn →0, we have
lim sup
n→∞
fx∗−x∗, x
n−x∗lim j→∞
fx∗−x∗, x
nj−x∗
lim
j→∞
fx∗−x∗, u
nj −x∗
Step 5xn→x∗, wherex∗PΓfx∗. From3.3, we have
xn1−x∗2≤β
nxn−x∗γnWnun−x∗22αnfWnxn−x∗, xn1−x∗
≤βnxn−x∗γnun−x∗22αnfWnxn−fx∗, xn1−x∗
2αnfx∗−x∗, xn1−x∗
≤1−αn2xn−x∗22αnfWnxn−fx∗xn1−x∗
2αnfx∗−x∗, xn1−x∗
≤1−αn2xn−x∗22ααnxn−x∗xn1−x∗2αnfx∗−x∗, xn1−x∗
≤1−αn2xn−x∗2ααnxn−x∗2xn1−x∗2
2αnfx∗−x∗, xn1−x∗,
3.35 that is,
xn1−x∗2≤
1−21−α 1−ααnαn
xn−x∗2211−αα−α nαn
αn
21−αxn−x∗
2 1
1−α
fx∗−x∗, x
n1−x∗.
3.36 It is easy to see that∞n021−α/1−ααnαn∞and
lim sup
n→∞
αn
21−αxn−x∗
2 1
1−α
fx∗−x∗, x
n1−x∗
≤0. 3.37
ApplyingLemma 2.3and3.34to3.36, we conclude thatxn → x∗ asn → ∞. This completes the
proof.
ConcerningSr, we give the following remark.
Remark 3.6. For eachx1, x2∈C, we denoteu1Srx1andu2 Srx2. Then for ally∈C, we
have
rΘu1, y
ϕy−ϕu1
Ku
1
−Kx
1
, ηy, u1
≥0, 3.38
rΘu2, y
ϕy−ϕu2
Ku
2
−Kx
2
, ηy, u2
≥0. 3.39 Takingyu2in3.38andyu1in3.39, and adding up these two inequalities, we obtain
rΘu1, u2
ϕu2
−ϕu1
Ku
1
−Kx
1
, ηu2, u1
rΘu2, u1
ϕu1
−ϕu2
Ku2
−Kx2
, ηu1, u2
≥0.
3.40
Note thatηu1, u2 ηu2, u1 0 andΘu1, u2 Θu2, u1≤0. Hence from3.40, we deduce
Kx
1
−Ku
1
, ηu1, u2
Ku
2
−Kx
2
, ηu1, u2
which implies that
Kx1
−Kx2
, ηu1, u2
≥Ku1
−Ku2
, ηu1, u2
. 3.42
SinceK : C → Hisη-strongly monotone with constantμ > 0, then from3.42, we conclude
that
Kx
1
−Kx
2
, ηu1, u2
≥μu1−u22. 3.43
TakeKx x2/2,ηy, x y−x, andμ1. Then from3.43, we have
x1−x2, u1−u2
≥u1−u22. 3.44
This indicates thatSris firmly nonexpansive.
Corollary 3.7. LetCbe a nonempty closed convex subset of a real Hilbert spaceHand letϕ:C→ R be a lower semicontinuous and convex functional. LetΘ : C×C → Rbe an equilibrium bifunction satisfying conditions (H1)–(H3). LetA:C→ Hbe aβ-inverse-strongly monotone mapping such that
V IA, C∩Ω /∅. Suppose{αn}, {βn}, and{γn}are three sequences in0,1withαnβnγn
1, n≥0. Assume that
iη:C×C→His Lipschitz continuous with constantλ >0such that
aηx, y ηy, x 0for allx, y∈C, bη·,·is affine in the first variable,
cfor each fixedy ∈C,x→ ηy, xis sequentially continuous from the weak topology to the weak topology;
iiK : C → R is η-strongly convex with constant σ > 0 and its derivative K is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constantν >0such thatσ ≥λν;
iiifor eachx∈C, there exist a bounded subsetDx⊂Candzx∈Csuch that, for anyy∈C\Dx,
ivlimn→∞αn 0,∞n0αn ∞, 0 < lim infn→∞βn ≤ lim supn→∞βn < 1, λn ∈ a, b ⊂ 0,2β, andlimn→∞λn1−λn 0.
Letf be a contraction ofCinto itself and givenx0 ∈Carbitrarily. Let the sequences{xn},{yn}, and
{zn}be generated iteratively by
Θzn, xϕx−ϕzn1rKzn−Kxn, ηx, zn≥0 ∀x∈C, ynPCzn−λnAzn,
xn1αnfxnβnxnγnPCyn−λnAyn ∀n≥0.
3.46
Then the sequence{xn}generated by3.46converges strongly tox∗PΓfx∗, whereΓ V IA, C∩
Ωprovided thatSris firmly nonexpansive.
Proof. TakeTnx xfor alln 1,2, . . ., and for allx∈ Cin3.1. ThenWnx xfor all x∈C.
The conclusion follows immediately fromTheorem 3.5. This completes the proof.
Corollary 3.8. LetCbe a nonempty closed convex subset of a real Hilbert space H. LetT1, T2, . . .be
an infinite family of nonexpansive mappings ofCinto itself. LetA : C → H be a β-inverse-strongly monotone mapping such that ∩∞n1FTn∩V IA, C/∅. Suppose {αn}, {βn}, and {γn} are three
sequences in0,1withαnβnγn1,n≥0. Assume that
ilimn→∞αn0and∞n0αn∞;
ii0<lim infn→∞βn≤lim supn→∞βn<1;
iiiλn∈a, b ⊂0,2βandlimn→∞λn1−λn 0.
Let f be a contraction of Cinto itself and given x0 ∈ Carbitrarily. Then the sequence {xn}, generated iteratively by
ynPCxn−λnAxn,
xn1αnfWnxnβnxnγnWnPCyn−λnAyn ∀n≥0,
3.47
converges strongly tox∗PΓfx∗, whereΓ ∩∞n1FTn∩V IA, C.
Proof. Setϕx 0 and Θx, y 0 for allx, y ∈ C and putr 1. TakeKx x2/2 and
ηy, x y−xfor allx, y ∈C. Then we havezn PCxn xn. Hence the conclusion follows.
This completes the proof.
Acknowledgments
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