Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2009, Article ID 362191, 21 pages
doi:10.1155/2009/362191
Research Article
Strong Convergence of an Iterative
Method for Equilibrium Problems and
Variational Inequality Problems
HongYu Li1 and HongZhi Li2
1
2
Department of Mathematics, TianJin Polytechnic University, TianJin 300160, China
Department of Mathematics, Agricultural University of Hebei, BaoDing 071001, China
Correspondence should be addressed to HongYu Li, lhy x1976@eyou.com
Received 26 August 2008; Revised 11 November 2008; Accepted 9 January 2009
Recommended by Massimo Furi
We introduce an iterative method for finding a common element of the set of solutions of
equilibrium problems, the set of solutions of variational inequality problems, and the set of fixed
points of finite many nonexpansive mappings. We prove strong convergence of the iterative
sequence generated by the proposed iterative algorithm to the unique solution of a variational
inequality, which is the optimality condition for the minimization problem.
Copyright q 2009 H. Li and H. Li. 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 H be a real Hilbert space with inner product · , · and norm · , respectively. Suppose
that C is nonempty, closed convex subset of H and F is a bifunction from C × C to R, where
R is the set of real number. The equilibrium problem is to find a x ∈ C such that
Fx, y ≥ 0,
∀y ∈ C.
1.1
The set of such solutions is denoted by EPf. Numerous problems in physics, optimization,
and economics reduce to find a solution of equilibrium problem. Some methods have been
proposed to solve the equilibrium problems in Hilbert space, see, for instance, Blum and
Oettli 1, Combettes and Hirstoaga 2, and Moudafi 3.
A mapping A : C → H is called monotone if Au − Av, u − v ≥ 0. A is called relaxed
u, v-cocoercive, if there exist constants u > 0 and v > 0 such that
Ax − Ay, x − y ≥ −uAx − Ay2 vx − y2 ,
∀x, y ∈ C,
1.2
2
Fixed Point Theory and Applications
when u 0, A is called v-strong monotone; when v 0, A is called relaxed u-cocoercive.
Let A : C → H be a monotone operator, the variational inequality problem is to find a point
u ∈ C, such that
Au, v − u ≥ 0,
∀v ∈ C.
1.3
The set of solutions of variational inequality problem is denoted by VIC, A. The variational
inequality problem has been extensively studied in literatures, see, for example, 4, 5 and
references therein.
Let B be a strong positive bounded linear operator on H with coefficient γ, that is,
there exists a constant γ > 0 such that Bx, x ≥ γx2 , for all x ∈ H. A typical problem is to
minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on
a real Hilbert space H:
min Ax, x − x, b,
1.4
x∈FT where T is a nonexpansive mapping on H and b is a point on H.
A mapping T from C into itself is called nonexpansive, if T x−T y ≤ x−y, ∀x, y ∈ C.
The set of fixed points of T is denoted by FT . Let {Ti }N
i1 be a finite family of nonexpansive
FT
∅,
define
the
mappings
mappings and F N
i /
i1
Un,1 λn,1 T1 1 − λn,1 I,
Un,2 λn,2 T2 Un,1 1 − λn,2 I,
..
.
1.5
Un,N−1 λn,N−1 TN−1 Un,N−2 1 − λn,N−1 I,
Wn Un,N λn,N TN Un,N−1 1 − λn,N I,
where {λn,i }N
generated by
i1 ⊂ 0, 1 for all n ≥ 1. Such a mapping Wn is called W-mapping
N
N
T1 , T2 , . . . , TN and {λn,i }i1 . We know that Wn is nonexpansive and FWn i1 FTi , see 6.
Let S : C → C be a nonexpansive mapping and f : C → C is a contractive with
coefficient α ∈ 0, 1. Marino and Xu 7 considered the following general iterative scheme:
xn1 αn γf xn 1 − αn B Sxn .
1.6
They proved that {xn } converges strongly to z PFS I − B γfz, where PFS is the metric
projection from H onto FS.
Fixed Point Theory and Applications
3
By combining equilibrium problems and 1.6, Plutbieng and Pumpaeng 8 proposed
the following algorithm:
1
y − un , un − xn ≥ 0, ∀y ∈ H,
F un , y rn
xn1 αn γf xn I − αn B Sun .
1.7
They proved that if the sequences {αn } and {rn } satisfy some appropriate conditions, then
sequence {xn } convergence to the unique solution z of the variational inequality
B − γfz, x − z ≥ 0,
∀x ∈ FS ∩ EPF.
1.8
Motivated by 8, Colao et al. 9 introduced an iterative method for equilibrium problem
and finite family of nonexpansive mappings
1
F un , y y − un , un − xn ≥ 0, ∀y ∈ H,
rn
xn1 αn γf xn βxn 1 − βI − αn B Wn un ,
1.9
and proved that {xn } converges strongly to a point x∗ ∈ F ∩ EPF and x∗ also solves the
variational inequality 1.8. For equilibrium problems, also see 10, 11.
On the other hand, let A : C → C be a α-cocoercive mapping, for finding common
element of the solution of variational inequality problems and the set of fixed point of
nonexpansive mappings, Takahashi and Toyoda 12 introduced iterative scheme
xn1 αn xn 1 − αn SPC I − λn A xn .
1.10
They proved that {xn } converges weakly to z ∈ FS ∩ VIC, A. Inspired by 1.10 and 13,
Y. Yao and J.-C. Yao 14 given the following iterative process:
yn PC I − λn A xn ,
xn1 αn u βn xn γn SPC I − λn A yn ,
1.11
and proved that {xn } converges strongly to z ∈ FS ∩ VIC, A. By combining viscosity
approximation method and 1.10, Chen et al. 15 introduced the process
xn1 αn f xn βn SPC I − λn A xn ,
1.12
4
Fixed Point Theory and Applications
and studied the strong convergence of sequence {xn } generated by 1.12. Motivated by 1.6,
1.11, and 1.12, Qin et al. 16 introduced the following general iterative process
yn PC I − sn A xn ,
xn1 αn γf Wn xn I − αn B Wn PC I − tn A yn ,
1.13
and established a strong convergence theorem of {xn } to an element of N
i1 FTi ∩ VIC, A.
The purpose of this paper is to introduce the iterative process: x1 ∈ H and
1
y − un , un − xn ≥ 0, ∀y ∈ C,
rn
yn bn un 1 − bn Wn PC I − sn A un ,
Fun , y 1.14
xn1 αn γf Wn xn βxn 1 − βI − αn B Wn PC I − tn A yn ,
where Wn is defined by 1.5, A is u, v-cocoercive, and B is a bounded linear operator. We
should show that the sequences {xn } converge strongly to an element of N
i1 FTi ∩VIC, A∩
EPF. Our result extends the corresponding results of Qin et al. 16 and Colao et al. 9, and
many others.
2. Preliminaries
Let H be a real Hilbert space and C a nonempty, closed convex subset of H. We denote strong
convergence of {xn } to x by xn → x and weak convergence by xn x. Let PC : C → H
is a mapping such that for every point x ∈ H, there exists a unique PC x ∈ C satisfying
x − PC x ≤ x − y, for all y ∈ C. PC is called the metric projection of H onto C. It is known
that PC is a nonexpansive mapping from H onto C. It is also known that PC x ∈ C and
x − PC x, PC x − y ≥ 0,
∀x ∈ H, y ∈ C,
2
x − y, Pc x − PC y ≥ PC x − PC y ,
∀x, y ∈ H.
2.1
2.2
Let A : C → H be a monotone mapping of C into H, then u ∈ VIC, A if and only if
u PC u − λAu, for all λ > 0. The following result is useful in the rest of this paper.
Lemma 2.1 see 17. Assume {an } is a sequence of nonegative real number such that
an1 ≤ 1 − αn an δn ,
∀n ≥ 0,
where {αn } is a sequence in 0, 1, and {δn } is a sequence in R such that
1 ∞
n0 αn ∞,
2 lim supn → ∞ δn /αn ≤ 0 or ∞
n0 |δn | < ∞.
Then, limn → ∞ an 0.
2.3
Fixed Point Theory and Applications
5
Lemma 2.2 see 18. Let {xn }, {un } be bounded sequences in Banach space E satisfying xn1 τn xn 1 − τn un for all n ≥ 0 and lim infn → ∞ un1 − un − xn1 − xn ≤ 0. Let τn be a sequence
in 0, 1 with 0 < lim infn → ∞ τn ≤ lim supn → ∞ τn < 1. Then, limn → ∞ xn − un 0.
Lemma 2.3. For all x, y ∈ H, there holds the inequality
x y ≤ x2 2y, x y.
2.4
Lemma 2.4 see 7. Assume that A is a strong positive linear bounded operator on a Hilbert space
H with coefficient γ > 0 and 0 < ρ ≤ A−1 . Then I − ρA ≤ 1 − ργ.
For solving the equilibrium problem for a bifunction F : C × C → R, we assume that
F satisfies the following conditions:
A1 Fx, x 0 for all x ∈ C;
A2 F is monotone: Fx, y Fy, x ≤ 0 for all x, y ∈ C;
A3 for all x, y, z ∈ C, lim supt↓0 Ftz 1 − tx, y ≤ Fx, y;
A4 for all x ∈ C, Fx, · is convex and lower semicontinuous.
The following result is in Blum and Oettli 1.
Lemma 2.5 see 1. Let C be a nonempty closed convex subset of a Hilbert space E, let F be a
bifunction from C × C into R satisfying (A1)–(A4), let r > 0, and let x ∈ H. Then there exists z ∈ C
such that
1
Fz, y y − z, z − x ≥ 0,
r
∀y ∈ C.
2.5
We also know the following lemmas.
Lemma 2.6 see 19. Let C be a nonempty closed convex subset of Hilbert space H, let F be a
bifunction from C × C to R satisfying (A1)–(A4), let r > 0, and let x ∈ H, define a mapping Tr :
H → C as follows:
1
Tr x z ∈ C : Fz, y y − z, z − x ≥ 0, ∀y ∈ C ,
r
2.6
for all x ∈ H. Then, the following holds:
1 Tr is single-valued;
2 Tr is firmly nonexpansive-type mapping, that is, for all x, y ∈ H,
Tr x − Tr y2 ≤ Tr x − Tr y, x − y ;
3 FTr EPF;
4 EPF is closed and convex.
2.7
6
Fixed Point Theory and Applications
A monotone operator T : H → 2H is said to be maximal monotone if its graph GT {u, v : v ∈ T u} is not properly contained in the graph of any other monotone operators. Let
A be a monotone mapping of C into H and let NC v be the normal cone for C at a point
v ∈ C, that is
NC v x ∈ H : v − y, x ≥ 0, ∀y ∈ C .
2.8
Define
Tv ⎧
⎨Av NC v, v ∈ C,
⎩∅,
v/
∈ C.
2.9
It is known that in this case T is maximal monotone, and 0 ∈ T v if and only if v ∈ VIC, A.
3. Strong Convergence Theorem
Theorem 3.1. Let H be a real Hilbert space and C be a nonempty closed convex subset of H. {Ti }N
i1
a finite family of nonexpansive mappings from C into itself and F : C × C → R a bifunction satisfying
(A1)–(A4). Let A : C → H be relaxed u, v-cocoercive and μ-Lipschitzian. Let f : C → C be an
α-contraction with 0 ≤ α < 1 and B a strong positive linear bounded operator with coefficient γ > 0,
γ is a constant with 0 < γ < γ/α. Let sequences {αn }, {bn } be in 0, 1 and {rn } be in 0, ∞, β is a
constant in 0, 1. Assume C0 N
/ ∅ and
i1 FTi ∩ VIC, A ∩ EPF ∞
i limn → ∞ αn 0, n1 αn ∞;
ii limn → ∞ |rn1 − rn | 0, lim infn → ∞ rn > 0;
iii {sn }, {tn } ∈ a, b for some a, b with 0 ≤ a ≤ b ≤ 2v − uμ2 /μ2 and limn → ∞ |sn1 − sn | limn → ∞ |tn1 − tn | 0;
iv limn → ∞ |λn1 − λn | limn → ∞ |bn1 − bn | 0.
Then the sequence {xn } generated by 1.14 converges strongly to x∗ ∈ C0 and x∗ solves the
variational inequality x∗ PC0 I − B − γfx∗ , that is,
γfx∗ − Bx∗ , x − x∗ ≤ 0,
∀x ∈ C0 .
3.1
Proof. Without loss of generality, we can assume αn ≤ 1 − βB−1 . Then from Lemma 2.4 we
know
1 − βI − αn B 1 − βI − αn B ≤ 1 − β 1 − αn γ 1 − β − αn γ.
3.2
1−β 1−β
Since A is relaxed u, v-cocoercive and μ-Lipschitzian and iii holds, we know from 14
that for all x, y ∈ C and n ≥ 1, the following holds:
I − sn A x − I − sn A y ≤ x − y,
I − tn A x − I − tn A y ≤ x − y.
3.3
Fixed Point Theory and Applications
7
We divide the proof into several steps.
Step 1. {xn } is bounded.
Take p ∈ C0 , notice that un Trn xn and form Lemma 2.62 that Trn is nonexpansive,
we have
un − p Tr xn − Tr p ≤ xn − p.
n
n
3.4
Since p Wn PC p − sn Ap, we have
yn − p bn un 1 − bn Wn PC un − sn Aun − p
≤ bn un − p 1 − bn Wn PC un − sn Aun − p
≤ bn un − p 1 − bn un − sn Aun − p − sn Ap ≤ bn un − p 1 − bn un − p
≤ xn − p.
3.5
Then we have
xn1 − p αn γf Wn xn βxn 1 − βI − αn B Wn PC I − tn A yn − p
αn γf Wn xn − Bp β xn − p
1 − βI − αn B Wn PC I − tn A yn − p αn γf Wn xn − Bp βxn − p 1 − β − αn γ yn − p.
3.6
Thus From 3.5 we have
xn1 − p ≤ αn γ f Wn xn − f p αn γfp − Bp βxn − p
1 − β − αn γ xn − p
≤ αn γαxn − p αn γfp − Bp 1 − αn γxn − p
1 − αn γ − αγ xn − p αn γfp − Bp
γf p − Bp
1 − αn γ − αγ xn − p αn γ − αγ ·
γ − αγ
γf p − Bp
≤ max xn − p,
γ − αγ
γf p − Bp
≤ max x1 − p,
,
γ − αγ
hence {xn } is bounded, so is {un }, {yn }.
3.7
8
Fixed Point Theory and Applications
Step 2. limn → ∞ xn1 − xn 0.
Let xn1 βxn 1 − βzn , for all n ≥ 0, where
zn 1 αn γf Wn xn 1 − βI − αn B Wn PC yn − tn Ayn .
1−β
3.8
Then we have
zn1 − zn 1 γ αn1 f Wn1 xn1 − αn f Wn xn
1−β
1 − βI − αn1 B Wn1 PC yn1 − tn1 Ayn1
− 1 − βI − αn B Wn PC yn − tn Ayn γ 1 − β αn1 f Wn1 xn1 − αn f Wn xn
Wn1 PC yn1 − tn1 Ayn1 − Wn PC yn − tn Ayn
1 −
αn1 BWn1 PC yn1 − tn1 Ayn1
1−β
− αn BWn PC yn − tn Ayn 3.9
≤ Wn1 PC yn1 − tn1 Ayn1 − Wn1 PC yn − tn1 Ayn Wn1 PC yn − tn1 Ayn − Wn1 PC yn − tn Ayn Wn1 PC yn − tn Ayn − Wn PC yn − tn Ayn K1 ,
where
K1 αn1 γ f Wn1 xn1 BWn1 PC yn1 − tn1 Ayn1 1−β
αn γ f Wn xn BWn PC yn − tn Ayn .
1−β
3.10
Next we estimate Wn1 PC yn1 −tn1 Ayn1 −Wn1 PC yn −tn1 Ayn , Wn1 PC yn −tn1 Ayn −
Wn1 PC yn − tn Ayn and Wn1 PC yn − tn Ayn − Wn PC yn − tn Ayn . At first
Wn1 PC yn − tn1 Ayn − Wn1 PC yn − tn Ayn ≤ tn Ayn − tn1 Ayn tn1 − tn · Ayn .
3.11
Fixed Point Theory and Applications
9
Put vn PC yn − tn Ayn , we have
Wn1 PC yn − tn Ayn − Wn PC yn − tn Ayn Wn1 vn − Wn vn Un1,N vn − Un,N vn λn1,N TN Un1,N−1 vn 1 − λn1,N vn − λn,N TN Un,N−1 vn − 1 − λn,N vn ≤ λn1,N TN Un1,N−1 vn − λn,N TN Un,N−1 vn λn1,N − λn,N · vn ≤ λn1,N Un1,N−1 vn − Un,N−1 vn λn1,N − λn,N · TN Un,N−1 vn λn1,N − λn,N · vn ≤ Un1,N−1 vn − Un,N−1 vn λn1,N − λn,N TN Un,N−1 vn vn .
3.12
By recursion we get
N Wn1 PC yn − tn Ayn − Wn PC yn − tn Ayn ≤ M ·
λn1,i − λn,i ,
3.13
i1
for some M > 0. Similarly, we also get
N Wn1 PC un − sn Aun − Wn PC un − sn Aun ≤ M ·
λn1,i − λn,i .
3.14
i1
Since
1
F un , y y − un , un − xn ≥ 0,
rn
∀y ∈ C,
1 F un1 , y y − un1 , un1 − xn1 ≥ 0,
rn1
3.15
∀y ∈ C.
Put y un1 in the first inequality and y un in the second one, we have
1
F un , un1 un1 − un , un − xn ≥ 0,
rn
F un1 , un
1 un − un1 , un1 − xn1 ≥ 0.
rn1
3.16
Adding both inequality, by A2 we have
1
1 un1 − un ,
un − xn −
un1 − xn1
≥ 0,
rn
rn1
3.17
10
Fixed Point Theory and Applications
therefore, we have
rn1 − rn un1 − xn1
un1 − un , un − un1 xn1 − xn ≥ 0,
rn1
3.18
which implies that
rn1 − rn un1 − un , xn1 − xn un1 − xn1
rn1
|rn1 − rn | xn1 − xn un1 − xn1 .
≤ un1 − un ·
rn1
un1 − un 2 ≤
3.19
Hence we have
un1 − un ≤ xn1 − xn |rn1 − rn | un1 − xn1 ,
rn1
3.20
so, by 3.20 and the property I − tn Ax − I − tn Ay ≤ x − y, we arrive at
Wn1 PC yn1 − tn1 Ayn1 − Wn1 PC yn − tn1 Ayn ≤ yn1 − yn bn1 un1 1 − bn1 Wn1 PC I − sn1 A un1
− bn un − 1 − bn Wn PC I − sn A un bn1 un1 − un bn1 − bn un
1 − bn1 Wn1 PC un1 − sn1 Aun1 − Wn1 PC un − sn1 Aun
1 − bn1 Wn1 PC un − sn1 Aun − Wn1 PC un − sn Aun
× 1 − bn1 Wn1 PC un − sn Aun − Wn PC un − sn Aun
bn − bn1 Wn PC un − sn Aun ≤ bn1 un1 − un |bn1 − bn | · un 1 − bn1 un1 − un 1 − bn1 |sn − sn1 | · Aun 1 − bn1 Wn1 PC un − sn Aun − Wn PC un − sn Aun bn − bn1 · Wn PC un − sn Aun un1 − un 1 − bn1 Wn1 PC un − sn Aun − Wn PC un − sn Aun K2 ,
3.21
Fixed Point Theory and Applications
11
where
K2 : 1 − bn1 sn − sn1 · Aun bn1 − bn Wn PC un − sn Aun vn .
3.22
Therefore, by 3.14 and 3.20 we get
Wn1 PC yn1 − tn1 Ayn1 − Wn1 PC yn − tn1 Ayn |rn1 − rn | un1 − xn1 ≤ xn1 − xn rn1
3.23
N λn1,i − λn,i K2 .
1 − bn1 M ·
i1
Now submitting 3.11, 3.13, and 3.23 into 3.9, we have
zn1 − zn xn1 − xn |rn1 − rn | un1 − xn1 rn1
N λn1,i − λn,i K2
1 − bn1 M ·
i1
3.24
N λn1,i − λn,i K1 .
tn1 − tn · Ayn M ·
i1
Thus conditions ii, iii, and iv imply that
lim sup zn1 − zn − xn1 − xn ≤ 0.
3.25
lim xn1 − xn lim 1 − βxn − zn 0.
3.26
n→∞
Then, Lemma 2.2 yields
n→∞
n→∞
Step 3. limn → ∞ Ayn − Ap limn → ∞ Aun − Ap 0 for p ∈ C0 .
12
Fixed Point Theory and Applications
Note that
yn − p2 bn un 1 − bn Wn PC I − sn A un − p2
2 2
≤ bn un − p 1 − bn un − sn Aun − p − sn Ap 2 bn un − p 1 − bn
2
2 × un − p − 2sn un − p, Aun − Ap s2n Aun − Ap
2 ≤ un − p 1 − bn
3.27
2
2
2 × 2sn uAun − Ap − 2sn vun − p s2n Aun − Ap
2 2
2sn v un − p 1 − bn 2sn u s2n − 2 Aun − Ap ,
μ
vn − p2 PC yn − tn Ayn − PC p − tn Ap 2
2
≤ yn − p − tn Ayn − Ap 2
2
yn − p − 2tn yn − p, Ayn − Ap t2n Ayn − Ap
2
2
2
≤ yn − p 2tn uAyn − Ap − 2tn vyn − p t2n Ayn − Ap
2
2tn v 2
≤ yn − p 2tn u tn − 2 Ayn − Ap,
μ
3.28
hence by 2tn u t2n − 2tn v/μ2 < 0, we know
vn − p2 ≤ yn − p2 ≤ un − p2 1 − bn 2sn u s2n − 2sn v Aun − Ap2 .
2
μ
3.29
By Lemma 2.3, we have
xn1 − p2 αn γf Wn xn βxn 1 − βI − αn B Wn vn − p2
2
1 − β Wn vn − p β xn − p αn γf Wn xn − BWn vn 2
≤ 1 − β Wn vn − p β xn − p 2αn γf Wn xn − BWn vn , xn1 − p
2
2
≤ 1 − βWn vn − p βxn − p 2αn γf Wn xn − BWn vn , xn1 − p
2
2
≤ 1 − βvn − p βxn − p 2αn M1 ,
3.30
Fixed Point Theory and Applications
13
for some M1 > 0. Submitting 3.28 into 3.30, we have
xn1 − p2 ≤ 1 − β yn − p2 2tn u t2n − 2tn v Ayn − Ap
μ2
2
βxn − p 2αn M1
2
2tn v ≤ xn − p 1 − β 2tn u t2n − 2 Ayn − Ap 2αn M1 ,
μ
3.31
which implies
2tn v
2 1 − β
− 2tn u − tn Ayn − Ap
μ2
2 2
≤ xn − p − xn1 − p 2αn M1
2 2
xn − xn1 xn1 − p − xn1 − p 2αn M1
3.32
2
xn − xn1 2 xn − xn1 , xn1 − p 2αn M1
2
≤ xn − xn1 2xn − xn1 · xn1 − p 2αn M1 ,
hence from conditions i, iii, and 3.26, we have
lim Ayn − Ap 0.
n→∞
3.33
Similarly, submitting 3.29 into 3.30, we also have
lim Aun − Ap 0.
n→∞
3.34
Step 4. limn → ∞ Wn vn − vn 0.
By 2.2 we have
vn − p2 PC yn − tn Ayn − PC p − tn Ap 2
≤ yn − tn Ayn − p − tn Ap , vn − p
1 yn − tn Ayn − p − tn Ap 2 vn − p2
2
2 − yn − vn − tn Ayn − Ap 1 yn − p2 vn − p2 − yn − vn − tn Ayn − Ap 2
2
1 yn − p2 vn − p2 − yn − vn 2 − t2n Ayn − Ap2
2
2tn yn − vn , Ayn − Ap ,
≤
3.35
14
Fixed Point Theory and Applications
hence
vn − p2 ≤ xn − p2 − yn − vn 2 − t2n Ayn − Ap2
2tn yn − vn · Ayn − Ap.
3.36
Submitting 3.36 into 3.30, we have
xn1 − p2 ≤ 1 − β xn − p2 − yn − vn 2 − t2n Ayn − Ap2 2tn yn − vn · Ayn − Ap
2
βxn − p 2αn M1 .
3.37
This implies that
2 2
2
2 1 − βyn − vn ≤ xn − p − xn1 − p − t2n Ayn − Ap
2tn yn − vn · Ayn − Ap 2αn M1 .
3.38
Hence by Step 3 we get
lim yn − vn 0.
n→∞
3.39
Put yn bn un 1 − bn Wn qn , where qn PC I − sn Aun . We have
Wn qn − p2 ≤ PC I − sn A un − PC I − sn A p2
≤ un − sn Aun − p − sn Ap , Wn qn − p
≤
1 un − sn Aun − p − sn Ap 2 Wn qn − p2
2
2 − un − Wn qn − sn Aun − Ap 3.40
1 un − p2 Wn qn − p2 − un − Wn qn 2
2
2
− s2n Ayn − Ap 2sn un − Wn qn , Aun − Ap .
Hence
Wn qn − p2 ≤ un − p2 − un − Wn qn 2 − s2n Ayn − Ap2
2sn un − Wn qn · Aun − Ap,
3.41
Fixed Point Theory and Applications
15
so, we get
yn − p2 ≤ bn un − p2 1 − bn Wn qn − p2
2 2
≤ un − p − 1 − bn un − Wn qn 3.42
2
1 − bn − s2n Ayn − Ap 2sn un − Wn qn · Aun − Ap .
Submitting into 3.30, we have
xn1 − p2 ≤ 1 − β un − p2 − 1 − bn un − Wn qn 2
2
1 − bn − s2n Ayn − Ap 2sn un − Wn qn · Aun − Ap
2
βxn − p 2αn M1
2
2
≤ xn − p − 1 − β 1 − bn un − Wn qn K3 ,
3.43
where
2
K3 : 1 − bn − s2n Ayn − Ap 2sn un − Wn qn · Aun − Ap 2αn M1 ,
3.44
which implies
lim un − Wn qn 0.
3.45
lim yn − un lim 1 − bn un − Wn qn 0,
3.46
n→∞
So, we have
n→∞
n→∞
which together with 3.39 gives
lim un − vn 0.
n→∞
3.47
Since un Trn xn and Trn is firmly nonexpansive, we have
un − p2 Tr xn − Tr p
n
n
≤ Trn xn − Trn p, xn − p
1 un − p2 xn − p2 − Tr xn − xn 2 ,
n
2
3.48
16
Fixed Point Theory and Applications
which implies
un − p2 ≤ xn − p2 − un − xn 2 ,
3.49
which together with 3.30 gives
xn1 − p2 ≤ 1 − βvn − p2 βxn − p2 2αn M1
2
2 ≤ 1 − β vn − un un − p 2 vn − un , un − p
2
βxn − p 2αn M1
3.50
2 2
2 ≤ 1 − βvn − un 1 − β xn − p − un − xn 2
21 − βvn − un · un − p βxn − p 2αn M1 .
So
2 2
2
2 1 − βun − xn ≤ 1 − βvn − un xn − p − xn1 − p
21 − βvn − un · un − p 2αn M1 .
3.51
Now 3.47 and condition i imply that
lim un − xn 0.
n→∞
3.52
Since
xn − Wn vn ≤ xn − xn1 xn1 − Wn vn xn − xn1 αn γf Wn xn βxn 1 − βI − αn B Wn vn − Wn vn xn − xn1 αn γf Wn xn − BWn vn β xn − Wn vn ≤ xn − xn1 αn γ f Wn xn BWn vn βxn − Wn vn .
3.53
Then we get
xn − Wn vn ≤
1 xn − xn1 αn γ fWn xn BWn vn ,
1−β
1−β
3.54
hence
lim xn − Wn vn 0.
n→∞
3.55
Fixed Point Theory and Applications
17
Note that
Wn vn − vn ≤ Wn vn − xn xn − un un − vn ,
3.56
thus from 3.47–3.55, we have
lim vn − Wn vn 0.
n→∞
3.57
Step 5. lim supn → ∞ γfx∗ − Bx∗ , xn1 − x∗ ≤ 0, where x∗ is the unique solution of variational
inequality γfx∗ − Bx∗ , x − x∗ ≤ 0, for all x ∈ C0 .
Take a subsequence {xnj } of {xn1 }, such that
lim sup γf x∗ − Bx∗ , xn1 − x∗ lim γf x∗ − Bx∗ , xnj − x∗ .
j →∞
n→∞
3.58
Since {xn } is bounded, without loss of generality, we assume {xnj } itself converges weakly to
a point p. We should prove p ∈ C0 N
i1 FTi ∩ VIC, A ∩ EPF.
First, let
Tv ⎧
⎨Av NC v, v ∈ C,
⎩∅,
v/
∈ C,
3.59
with the same argument as used in 14, we can derive p ∈ T −1 0, since T is maximal monotone,
we know p ∈ VIC, A.
Next, from A2, for all y ∈ C we have
1
y − un , un − xn ≥ F y, un ,
rn
3.60
unj − xnj ≥ F y, unj .
y − unj ,
rnj
3.61
in particular
Condition A4 implies that F is weakly semicontinuous, then from 3.52 and let j → ∞ we
have
Fy, p ≤ 0,
∀y ∈ C.
3.62
Replacing y by yt : ty 1 − tp with t ∈ 0, 1, using A1 and A4, we get
0 F yt , yt tF yt , y 1 − tF yt , p ≤ tF yt , y .
3.63
18
Fixed Point Theory and Applications
Divide by t in both side yields Fty 1 − tp, y ≥ 0, let t → 0 , by A3 we conclude
Fp, y ≥ 0, for all y ∈ C. Therefore, p ∈ EPF.
∈ N
Finally, from xn − vn → 0 we know that vnj p j → ∞. Assume p /
i1 FTi ,
p,
for
all
n
∈
N.
Since
Hilbert
space
satisfies
Opial’s
condition,
we
have
that is, p W
/ n
lim infvnj − p ≤ lim infvnj − Wnj p
j →∞
j →∞
≤ lim inf vnj − Wnj vnj Wnj vnj − Wnj p
j →∞
3.64
≤ lim infvnj − p,
j →∞
this is a contradiction, thus p ∈
we know
N
i1 FTi ,
therefore, p ∈ C0 N
i1 FTi ∩ VIC, A ∩ EPF.
So
lim sup γf x∗ − Bx∗ , xn1 − x∗ lim γf x∗ − Bx∗ , xnj − x∗
n→∞
j →∞
γf x∗ − Bx∗ , p − x∗ ≤ 0.
3.65
Step 6. The sequence {xn } converges strongly to x∗ .
From the definition of {xn } and Lemmas 2.3, and 2.4, we have
xn1 − x∗ 2 1 − βI − αn B Wn vn − x∗ β xn − x∗ αn γf Wn xn − Bx∗ 2
2
≤ 1 − βI − αn B Wn vn − x∗ β xn − x∗ 2αn γf Wn xn − Bx∗ , xn1 − x∗
1 − βI − αn B 2
2
∗ ≤ 1 − β
Wn vn − x βxn − x∗ 1−β
2αn γ f Wn xn − f x∗ , xn1 − x∗ 2αn γf x∗ − Bx∗ , xn1 − x∗
≤
1 − β − αn γ2 Wn vn − x∗ 2 βxn − x∗ 2
1−β
2 2 ααn γ xn − x∗ xn1 − x∗ 2αn γf x∗ − Bx∗ , xn1 − x∗
2
1 − β − αn γ2
β ααn γ xn − x∗ ≤
1−β
2
ααn γ xn1 − x∗ 2αn γf x∗ − Bx∗ , xn1 − x∗
α2n γ 2 xn − x∗ 2
1−β
2
ααn γ xn1 − x∗ 2αn γf x∗ − Bx∗ , xn1 − x∗ ,
1 − 2γ − αγαn 3.66
Fixed Point Theory and Applications
19
which implies that
xn1 − x∗ 2 ≤
α2n γ 2
1 − 2γ − αγαn
xn − x∗ 2
1 − ααn γ
1 − β1 − ααn γ
2αn ∗ γf x − Bx∗ , xn1 − x∗
1 − ααn γ
2αn γ − αγ xn − x∗ 2
1−
1 − ααn γ
αn
1 − ααn γ
Since from condition i we have
lim sup
n→∞
3.67
αn γ 2 xn − x∗ 2 2 γf x∗ − Bx∗ , xn1 − x∗ .
1−β
∞
n1 2αn γ
− αγ/1 − ααn γ ∞ and
αn γ 2
xn − x∗ 2 2 γfx∗ − Bx∗ , xn1 − x∗ ≤ 0,
1−β
3.68
so, by Lemma 2.1, we conclude xn → x∗ . This completes the proof.
Putting F ≡ 0 and bn β 0 for all n ≥ 1 in Theorem 3.1, we obtain the following
corollary.
Corollary 3.2. Let H be a real Hilbert space and C be a nonempty closed convex subset of H. {Ti }N
i1 a
finite family of nonexpansive mappings from C into itself. Let A : C → H be relaxed u, v-cocoercive
and μ-Lipschitzian. Let f : C → C be an α-contraction with 0 ≤ α < 1 and B a strong positive linear
bounded operator with coefficient γ > 0, γ be a constant with 0 < γ < γ/α. Let sequences {αn }, {bn }
∅ and
in 0, 1 and β be a constant in 0, 1. Assume C0 N
i1 FTi ∩ VIC, A /
∞
i limn → ∞ αn 0, n1 αn ∞;
ii {sn }, {tn } ∈ a, b for some a, b with 0 ≤ a ≤ b ≤ 2v − uμ2 /μ2 and limn → ∞ |sn1 −
sn | limn → ∞ |tn1 − tn | 0;
iii limn → ∞ |λn1 − λn | 0.
Then the sequence {xn } generated by x1 ∈ C and
xn1
yn bn xn 1 − bn Wn PC I − sn A xn ,
αn γf Wn xn βxn 1 − βI − αn B Wn PC I − tn A yn ,
3.69
converges strongly to x∗ ∈ C0 and x∗ solves the variational inequality x∗ PC0 I − B − γfx∗ , that
is,
γfx∗ − Bx∗ , x − x∗ ≤ 0,
∀x ∈ C0 .
3.70
Putting PC I − sn A PC I − tn A I and N 1, T1 S, β bn 0 in Theorem 3.1,
we obtain the following corollary.
20
Fixed Point Theory and Applications
Corollary 3.3. Let H be a real Hilbert space and C be a nonempty closed convex subset of H. S a
finite family of nonexpansive mappings from C into itself and F : C × C → R a bifunction satisfying
(A1)–(A4). Let f : C → C be an α-contraction with 0 ≤ α < 1 and B a strong positive linear bounded
operator with coefficient γ > 0, γ is a constant with 0 < γ < γ/α. Let sequences {αn }, {bn } in 0, 1
and {rn } in 0, ∞. Assume C0 FS ∩ EPF / ∅ and
i limn → ∞ αn 0,
∞
n1 αn
∞;
ii limn → ∞ |rn1 − rn | 0, lim infn → ∞ rn > 0.
Then the sequence {xn } generated by x1 ∈ C and
1
F un , y y − un , un − xn ≥ 0, ∀y ∈ H,
rn
xn1 αn γf Sxn 1 − αn B Sun ,
3.71
converges strongly to x∗ ∈ C0 and x∗ solves the variational inequality x∗ PC0 I − B − γfx∗ , that
is,
γfx∗ − Bx∗ , x − x∗ ≤ 0,
∀x ∈ C0 .
3.72
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