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Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2009, Article ID 374815, 32 pages
doi:10.1155/2009/374815
Research Article
A Hybrid Extragradient Viscosity
Approximation Method for Solving Equilibrium
Problems and Fixed Point Problems of Infinitely
Many Nonexpansive Mappings
Chaichana Jaiboon and Poom Kumam
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi,
Bangkok 10140, Thailand
Correspondence should be addressed to Poom Kumam,
Received 25 December 2008; Accepted 4 May 2009
Recommended by Wataru Takahashi
We introduce a new hybrid extragradient viscosity approximation method for finding the common
element of the set of equilibrium problems, the set of solutions of fixed points of an infinitely
many nonexpansive mappings, and the set of solutions of the variational inequality problems for
β-inverse-strongly monotone mapping in Hilbert spaces. Then, we prove the strong convergence
of the proposed iterative scheme to the unique solution of variational inequality, which is the
optimality condition for a minimization problem. Results obtained in this paper improve the
previously known results in this area.
Copyright q 2009 C. Jaiboon 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.
1. Introduction
Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Recall
that a mapping T of H into itself is called nonexpansive see 1 if Tx − Ty≤x − y
for all x, y ∈ H. We denote by FT{x ∈ C : Tx x} the set of fixed points of T. Recall
also that a self-mapping f : H → H is a contraction if there exists a constant α ∈ 0, 1 such
that fx − fy≤αx − y, for all x, y ∈ H. In addition, let B : C → H be a nonlinear
mapping. Let P
C
be the projection of H onto C. The classical variational inequality which is
denoted by VIC, B is to find u ∈ C such that
Bu, v − u
≥ 0, ∀v ∈ C. 1.1
2 Fixed Point Theory and Applications
For a given z ∈ H, u ∈ C satisfies the inequality
u − z, v − u
≥ 0, ∀v ∈ C, 1.2
if and only if u P
C
z. It is well known that P
C
is a nonexpansive mapping of H onto C and
satisfies
x − y, P
C
x − P
C
y
≥
P
C
x − P
C
y
2
, ∀x, y ∈ H. 1.3
Moreover, P
C
x is characterized by the following properties: P
C
x ∈ C and for all x ∈ H, y ∈ C,
x − P
C
x, y − P
C
x
≤ 0, 1.4
x − y
2
≥
x − P
C
x
2
y − P
C
x
2
. 1.5
It is easy to see that the following is true:
u ∈ VI
C, B
⇐⇒ u P
C
u − λBu
,λ>0. 1.6
One can see that the variational inequality 1.1 is equivalent to a fixed point problem.
The variational inequality has been extensively studied in literature; see, for instance, 2–
6. This alternative equivalent formulation has played a significant role in the studies of the
variational inequalities and related optimization problems. Recall the following.
1 A mapping B of C into H is called monotone if
Bx − By, x − y
≥ 0, ∀x, y ∈ C. 1.7
2 A mapping B is called β-strongly monotone see 7, 8 if there exists a constant
β>0 such that
Bx − By, x − y
≥ β
x − y
2
, ∀x, y ∈ C. 1.8
3 A mapping B is called k-Lipschitz continuous if there exists a positive real number
k such that
Bx − By
≤ k
x − y
, ∀x, y ∈ C. 1.9
4 A mapping B is called β-inverse-strongly monotone see 7, 8 if there exists a
constant β>0 such that
Bx − By, x − y
≥ β
Bx − By
2
, ∀x, y ∈ C. 1.10
Fixed Point Theory and Applications 3
Remark 1.1. It is obvious that any β-inverse-strongly monotone mapping B is monotone and
1/β-Lipschitz continuous.
5 An operator A is strongly positive on H if there exists a constant
γ>0withthe
property
Ax, x
≥
γ
x
2
, ∀x ∈ H. 1.11
6 A set-valued mapping T : H → 2
H
is called monotone if for all x, y ∈ H, f ∈ Tx,
and g ∈ Tyimply x−y,f−g≥0. A monotone mapping T : H → 2
H
is maximal if the graph
of GT of T is not properly contained in the graph of any other monotone mapping. It is
known that a monotone mapping T is maximal if and only if for x, f ∈ H ×H, x−y, f −g≥
0 for every y, g ∈ GT implies f ∈ Tx.LetB be a monotone map of C into H, and let N
C
v
be the normal cone to C at v ∈ C,thatis,N
C
v {w ∈ H : u − v, w≥0, for all u ∈ C}, .
Tv
⎧
⎨
⎩
Bv N
C
v, v ∈ C,
∅,v
/
∈ C.
1.12
Then T is the maximal monotone and 0 ∈ Tv if and only if v ∈ VIC, B;see9.
7 Let F be a bifunction of C × C into R, where R is the set of real numbers. The
equilibrium problem for F : C × C → R is to find x ∈ C such that
F
x, y
≥ 0, ∀y ∈ C. 1.13
The set of solutions of 1.13 is denoted by EPF. Given a mapping T : C → H, let
Fx, yTx,y − x for all x, y ∈ C. Then, z ∈
EPF if and only if Tz,y − z≥0
for all y ∈ C. Numerous problems in physics, saddle point problem, fixed point problem,
variational inequality problems, optimization, and economics are reduced to find a solution
of 1.13. Some methods have been proposed to solve the equilibrium problem; see, for
instance, 10–16. Recently, Combettes and Hirstoaga 17 introduced an iterative scheme
of finding the best approximation to the initial data when EPF is nonempty and proved a
strong convergence theorem.
In 1976, Korpelevich 18 introduced the following so-called extragradient method:
x
0
x ∈ C,
y
n
P
C
x
n
− λBx
n
,
x
n1
P
C
x
n
− λBy
n
1.14
for all n ≥ 0, where λ ∈ 0, 1/k,Cis a closed convex subset of R
n
, and B is a monotone and
k-Lipschitz continuous mapping of C into R
n
. He proved that if VIC, B is nonempty, then
the sequences {x
n
} and {y
n
}, generated by 1.14, converge to the same point z ∈ VIC, B.
For finding a common element of the set of fixed points of a nonexpansive mapping and
4 Fixed Point Theory and Applications
the set of solution of variational inequalities for β-inverse-strongly monotone, Takahashi and
Toyoda 19 introduced the following iterative scheme:
x
0
∈ C chosen arbitrary,
x
n1
α
n
x
n
1 − α
n
SP
C
x
n
− λ
n
Bx
n
, ∀n ≥ 0,
1.15
where B is β-inverse-strongly monotone, {α
n
} is a sequence in 0, 1,and{λ
n
} is a sequence
in 0, 2β. They showed that if FS ∩VIC, B is nonempty, then the sequence {x
n
} generated
by 1.15 converges weakly to some z ∈ FS ∩ VIC, B . Recently, Iiduka and Takahashi 20
proposed a new iterative scheme as follows:
x
0
x ∈ C chosen arbitrary,
x
n1
α
n
x
1 − α
n
SP
C
x
n
− λ
n
Bx
n
, ∀n ≥ 0,
1.16
where B is β-inverse-strongly monotone, {α
n
} is a sequence in 0, 1,and{λ
n
} is a sequence
in 0, 2β. They showed that if FS ∩VIC, B is nonempty, then the sequence {x
n
} generated
by 1.16 converges strongly to some z ∈ FS ∩ VIC, B.
Iterative methods for nonexpansive mappings have recently been applied to solve
convex minimization problems; see, for example, 21–24 and t he references therein. Convex
minimization problems have a great impact and influence in the development of almost all
branches of pure and applied sciences. 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
x∈C
1
2
Ax, x
−
x, b
, 1.17
where A is a linear b ounded operator, C is the fixed point set of a nonexpansive mapping
S on H, and b is a given point in H. Moreover, it is shown in 25 that the sequence {x
n
}
defined by the scheme
x
n1
n
γf
x
n
1 −
n
A
Sx
n
1.18
converges strongly to z P
FS
I − A γfz. Recently, Plubtieng and Punpaeng 26
proposed the following iterative algorithm:
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ H,
x
n1
n
γf
x
n
I −
n
A
Su
n
.
1.19
They prove that if the sequences {
n
} and {r
n
} of parameters satisfy appropriate condition,
then the sequences {x
n
} and {u
n
} both converge to the unique solution z of the variational
inequality
A − γf
q, q − p≥0,p∈ F
S
∩ EP
F
, 1.20
Fixed Point Theory and Applications 5
which is the optimality condition for the minimization problem
min
x∈FS∩EPF
1
2
Ax, x−h
x
, 1.21
where h is a potential function for γf i.e., hxγfx for x ∈ H.
Furthermore, for finding approximate common fixed points of an infinite countable
family of nonexpansive mappings {T
n
} under very mild conditions on the parameters.
Wangkeeree 27 introduced an iterative scheme for finding a common element of the set of
solutions of the equilibrium problem 1.13 and the set of common fixed points of a countable
family of nonexpansive mappings on C. Starting with an arbitrary initial x
1
∈ C, define a
sequence {x
n
} recursively by
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C,
y
n
P
C
u
n
− λ
n
Bu
n
,
x
n1
α
n
f
x
n
β
n
x
n
γ
n
S
n
P
C
u
n
− λ
n
By
n
, ∀n ≥ 1,
1.22
where {α
n
}, {β
n
}, and {γ
n
} are sequences in 0, 1. It is proved that under certain appropriate
conditions imposed on {α
n
}, {β
n
}, {γ
n
}, and {r
n
}, the sequence {x
n
} generated by 1.22
strongly converges to the unique solution q ∈∩
∞
n1
FS
n
∩ VIC, B ∩ EPF, where p
P
∩
∞
n1
FS
n
∩VIC,B∩EP F
fq which extend and improve the result of Kumam 14.
Definition 1.2 see 21.Let{T
n
} be a sequence of nonexpansive mappings of C into itself,
and let {μ
n
} be a sequence of nonnegative numbers in 0,1. For each n ≥ 1, define a mapping
W
n
of C into itself as follows:
U
n,n1
I,
U
n,n
μ
n
T
n
U
n,n1
1 − μ
n
I,
U
n,n−1
μ
n−1
T
n−1
U
n,n
1 − μ
n−1
I,
.
.
.
U
n,k
μ
k
T
k
U
n,k1
1 − μ
k
I,
U
n,k−1
μ
k−1
T
k−1
U
n,k
1 − μ
k−1
I,
.
.
.
U
n,2
μ
2
T
2
U
n,3
1 − μ
2
I,
W
n
U
n,1
μ
1
T
1
U
n,2
1 − μ
1
I.
1.23
Such a mapping W
n
is nonexpansive from C to C, and it is called the W-mapping generated
by T
1
,T
2
, ,T
n
and μ
1
,μ
2
, ,μ
n
.
6 Fixed Point Theory and Applications
On the other hand, Colao et al. 28 introduced and considered an iterative scheme for
finding a common element of the set of solutions of the equilibrium problem 1.13 and t he
set of common fixed points of infinitely many nonexpansive mappings on C. Starting with an
arbitrary initial x
0
∈ C, define a sequence {x
n
} recursively by
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥0, ∀y ∈ H,
x
n1
n
γf
x
n
βx
n
1 − β
I −
n
A
W
n
u
n
,
1.24
where {
n
} is a sequence in 0, 1. It is proved 28 that under certain appropriate conditions
imposed on {
n
} and {r
n
}, the sequence {x
n
} generated by 1.24 strongly converges to z ∈
∩
∞
n1
FT
n
∩ EPF, where z is an equilibrium point for F and is the unique solution of the
variational inequality 1.20,thatis,z P
∩
∞
n1
FT
n
∩EPF
I − A − γfz.
In this paper, motivated by Wangkeeree 27, Plubtieng and Punpaeng 26,Marino
and Xu 25, and Colao, et al. 28, we introduce a new iterative scheme in a Hilbert space H
which is mixed by the iterative schemes of 1.18, 1.19, 1.22,and1.24 as follows.
Let f be a contraction of H into itself, A a strongly positive bounded linear operator on
H with coefficient
γ>0, and B a β-inverse-strongly monotone mapping of C into H; define
sequences {x
n
}, {y
n
}, {k
n
}, and {u
n
} recursively by
x
1
x ∈ C chosen arbitrary,
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥0, ∀y ∈ C,
y
n
P
C
u
n
− λ
n
Bu
n
,
k
n
α
n
u
n
1 − α
n
P
C
u
n
− λ
n
By
n
,
x
n1
n
γf
x
n
β
n
x
n
1 − β
n
I −
n
A
W
n
k
n
, ∀n ≥ 1,
1.25
where {W
n
} is the sequence generated by 1.23, {
n
}, {α
n
}, and {β
n
}⊂0, 1 and {r
n
}⊂
0, ∞ satisfying appropriate conditions. We prove that the sequences {x
n
}, {y
n
}, {k
n
} and
{u
n
} generated by the above iterative scheme 1.25 converge strongly to a common element
of the set of solutions of the equilibrium problem 1.13, the set of common fixed points of
infinitely family nonexpansive mappings, and the set of solutions of variational inequality
1.1 for a β-inverse-strongly monotone mapping in Hilbert spaces. The results obtained in
this paper improve and extend the recent ones announced by Wangkeeree 27, Plubtieng
and Punpaeng 26,MarinoandXu25, Colao, et al. 28, and many others.
2. Preliminaries
We now recall some well-known concepts and results.
Let H be a real Hilbert space, whose inner product and norm are denoted by ·, · and
·, respectively. We denote weak convergence and strong convergence by notations and
→ , respectively.
Fixed Point Theory and Applications 7
A space H is said to satisfy Opial’s condition 29 if for each sequence {x
n
} in H which
converges weakly to point x ∈ H, we have
lim inf
n →∞
x
n
− x
< lim inf
n →∞
x
n
− y
, ∀y ∈ H, y
/
x. 2.1
Lemma 2.1 see 25. Let C be a nonempty closed convex subset of H, let f be a contraction of
H into itself with α ∈ 0, 1, and let A be a strongly positive linear bounded operator on H with
coefficient
γ>0.Then,for0 <γ<γ/α,
x − y,
A − γf
x −
A − γf
y
≥
γ − αγ
x − y
2
,x,y∈ H. 2.2
That is, A − γf is strongly monotone with coefficient
γ − γα.
Lemma 2.2 see 25. Assume that A is a strongly positive linear bounded operator on H with
coefficient
γ>0 and 0 <ρ≤A
−1
.ThenI − ρA≤1 − ργ.
For solving the equilibrium problem for a bifunction F : C × C → R, let us assume
that F satisfies the following conditions:
A1 Fx, x0 for all x ∈ C;
A2 F is monotone, that is, Fx, yFy,x ≤ 0 for all x,y ∈ C;
A3 for each x,y,z ∈ C, lim
t↓0
Ftz 1 − tx, y ≤ Fx, y;
A4 for each x ∈ C, y → Fx, y is convex and lower semicontinuous.
The following lemma appears implicitly in 30.
Lemma 2.3 see 30. Let C be a nonempty closed convex subset of H and let F be a bifunction of
C × C into R satisfying (A1)–(A4). Let r>0 and x ∈ H. Then, there exists z ∈ C such that
F
z, y
1
r
y − z, z − x
≥ 0 ∀y ∈ C. 2.3
The following lemma was also given in 17.
Lemma 2.4 see 17. Assume that F : C × C → R satisfies (A1)–(A4). For r>0 and x ∈ H,
define a mapping T
r
: H → C as follows:
T
r
x
z ∈ C : F
z, y
1
r
y − z, z − x
≥ 0, ∀y ∈ C
2.4
for all z ∈ H. Then, the following holds:
1 T
r
is single-valued;
2 T
r
is firmly nonexpansive, that is, for any x, y ∈ H,
T
r
x − T
r
y
2
≤
T
r
x − T
r
y, x − y
; 2.5
8 Fixed Point Theory and Applications
3 FT
r
EPF;
4 EPF is closed and convex.
For each n, k ∈ N, let the mapping U
n,k
be defined by 1.23. Then we can have the
following crucial conclusions concerning W
n
. You can find them in 31. Now we only need
the following similar version in Hilbert spaces.
Lemma 2.5 see 31. Let C be a nonempty closed convex subset of a real Hilbert space H.Let
T
1
,T
2
, be nonexpansive mappings of C into itself such that ∩
∞
n1
FT
n
is nonempty, and let
μ
1
,μ
2
, be real numbers such that 0 ≤ μ
n
≤ b<1 for every n ≥ 1. Then, for every x ∈ C and
k ∈ N, the limit lim
n →∞
U
n,k
x exists.
Using Lemma 2.5, one can define a mapping W of C into itself as follows:
Wx lim
n →∞
W
n
x lim
n →∞
U
n,1
x 2.6
for every x ∈ C. Such a W is called the W-mapping generated by T
1
,T
2
, and μ
1
,μ
2
,
Throughout this paper, we will assume that 0 ≤ μ
n
≤ b<1 for every n ≥ 1. Then, we have the
following results.
Lemma 2.6 see 31. Let C be a nonempty closed convex subset of a real Hilbert space H.Let
T
1
,T
2
, be nonexpansive mappings of C into itself such that ∩
∞
n1
FT
n
is nonempty, and let
μ
1
,μ
2
, be real numbers such that 0 ≤ μ
n
≤ b<1 for every n ≥ 1. Then, FW∩
∞
n1
FT
n
.
Lemma 2.7 see 32. If {x
n
} is a b ounded sequence in C,thenlim
n →∞
Wx
n
− W
n
x
n
0.
Lemma 2.8 see 33. Let {x
n
} and {z
n
} be bounded sequences in a Banach space X, and let {β
n
} be
a sequence in 0, 1 with 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1. Suppose x
n1
1− β
n
z
n
β
n
x
n
for all integers n ≥ 0 and lim sup
n →∞
y
n1
− z
n
−x
n1
− x
n
≤ 0. Then, lim
n →∞
z
n
− x
n
0.
Lemma 2.9 see 34. Assume that {a
n
} is a sequence of nonnegative real numbers such that
a
n1
≤
1 − l
n
a
n
σ
n
,n≥ 0, 2.7
where {l
n
} is a sequence in 0, 1 and {σ
n
} is a sequence in R such that
1
∞
n1
l
n
∞;
2 lim sup
n →∞
σ
n
/l
n
≤ 0 or
∞
n1
|σ
n
| < ∞.
Then lim
n →∞
a
n
0.
Lemma 2.10. Let H be a real Hilbert space. Then for all x, y ∈ H,
1 x y
2
≤x
2
2y, x y;
2 x y
2
≥x
2
2y, x.
Fixed Point Theory and Applications 9
3. Main Results
In this section, we prove the strong convergence theorem for infinitely many nonexpansive
mappings in a real Hilbert space.
Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H,letF be a bifunction
from C × C to R satisfying (A1)–(A4), let {T
n
} be an infinitely many nonexpansive of C into itself,
and let B be an β-inverse-strongly monotone mapping of C into H such that Θ : ∩
∞
n1
FT
n
∩
EPF ∩ VIC, B
/
∅.Letf be a contraction of H into itself with α ∈ 0, 1, and let A be a strongly
positive linear bounded operator on H with coefficient
γ>0 and 0 <γ<γ/α.Let{x
n
}, {y
n
},
{k
n
}, and {u
n
} be sequences generated by 1.25,where{W
n
} is the sequence generated by 1.23,
{
n
}, {α
n
}, and {β
n
} are three sequences in 0, 1, and {r
n
} is a real sequence in 0, ∞ satisfying the
following conditions:
i lim
n →∞
n
0,
∞
n1
n
∞;
ii lim
n →∞
α
n
0 and
∞
n1
α
n
∞;
iii 0 < lim inf
n →∞
β
n
≤ lim sup
n →∞
β
n
< 1;
iv lim inf
n →∞
r
n
> 0 and lim
n →∞
|r
n1
− r
n
| 0;
v {λ
n
/β}⊂τ, 1 − δ for some τ, δ ∈ 0, 1 and lim
n →∞
λ
n
0.
Then, {x
n
} and {u
n
} converge strongly to a point z ∈ Θ which is the unique solution of the variational
inequality
A − γf
z, z − x
≥ 0, ∀x ∈ Θ. 3.1
Equivalently, one has z P
Θ
I − A γfz.
Proof. Note that from the condition i, we may assume, without loss of generality, that
n
≤
1 − β
n
A
−1
for all n ∈ N.FromLemma 2.2, we know that if 0 ≤ ρ ≤A
−1
, then I − ρA≤
1−ρ
γ. We will assume that I −A≤1−γ.First,weshowthatI −λ
n
B is nonexpansive. Indeed,
from the β-inverse-strongly monotone mapping definition on B and condition v, we have
I − λ
n
Bx − I − λ
n
By
2
x − y − λ
n
Bx − By
2
x − y
2
− 2λ
n
x − y, Bx − By
λ
2
n
Bx − By
2
≤
x − y
2
− 2λ
n
β
Bx − By
2
λ
2
n
Bx − By
2
x − y
2
λ
n
λ
n
− 2β
Bx − By
2
≤
x − y
2
,
3.2
which implies that the mapping I − λ
n
B is nonexpansive. On the other hand, since A is a
strongly positive bounded linear operator on H, we have
A
sup
{|
Ax, x
|
: x ∈ H,
x
1
}
. 3.3
10 Fixed Point Theory and Applications
Observe that
1 − β
n
I −
n
A
x, x
1 − β
n
−
n
Ax, x
≥ 1 − β
n
−
n
A
≥ 0,
3.4
and this show that 1 − β
n
I −
n
A is positive. It follows that
1 − β
n
I −
n
A
sup
1 − β
n
I −
n
A
x, x
: x ∈ H,
x
1
sup
1 − β
n
−
n
Ax, x
: x ∈ H,
x
1
≤ 1 − β
n
−
n
γ.
3.5
Let Q P
Θ
, where Θ : ∩
∞
n1
FT
n
∩ EPF ∩ VIC, B.Notethatf is a contraction of H into
itself with α ∈ 0, 1. Then, we have
Q
I − A γf
x
− Q
I − A γf
y
P
Θ
I − A γf
x
− P
Θ
I − A γf
y
≤
I − A γf
x
−
I − A γf
y
≤
I − A
x − y
γ
f
x
− f
y
≤
1 −
γ
x − y
γα
x − y
1 −
γ γα
x − y
1 −
γ − γα
x − y
, ∀x, y ∈ H.
3.6
Since 0 < 1 −
γ − γα < 1, it follows that QI − A γf is a contraction of H into itself.
Therefore by the Banach Contraction Mapping Principle, which implies that there exists a
unique element z ∈ H such that z QI − A γfzP
Θ
I − A γfz.
We will divide the proof into five steps.
Step 1. We claim that {x
n
} is bounded. Indeed, pick any p ∈ Θ. From the definition of T
r
,we
note that u
n
T
r
n
x
n
. If follows that
u
n
− p
T
r
n
x
n
− T
r
n
p
≤
x
n
− p
. 3.7
Since I − λ
n
B is nonexpansive and p P
C
p − λ
n
Bp from 1.6, we have
y
n
− p
P
C
u
n
− λ
n
Bu
n
− P
C
p − λ
n
Bp
≤
u
n
− λ
n
Au
n
−
p − λ
n
Bp
I − λ
n
A
u
n
−
I − λ
n
B
p
≤
u
n
− p
≤
x
n
− p
.
3.8
Fixed Point Theory and Applications 11
Put v
n
P
C
u
n
− λ
n
By
n
. Since p ∈ VIC, B, we have p P
C
p − λ
n
Bp. Substituting x
u
n
− λ
n
Ay
n
and y p in 1.5, we can write
v
n
− p
2
≤
u
n
− λ
n
By
n
− p
2
−
u
n
− λ
n
By
n
− v
n
2
u
n
− p
2
− 2λ
n
By
n
,u
n
− p
λ
2
n
By
n
2
−
u
n
− v
n
2
2λ
n
By
n
,u
n
− v
n
− λ
2
n
By
n
2
u
n
− p
2
−
u
n
− v
n
2
2λ
n
By
n
,p− v
n
u
n
− p
2
−
u
n
− v
n
2
2λ
n
By
n
− Bp, p − y
n
2λ
n
Bp,p − y
n
2λ
n
By
n
,y
n
− v
n
.
3.9
Using the fact that B is β-inverse-strongly monotone mapping, and p is a solution of the
variational inequality problem VIC, B, we also have
By
n
− Bp, p − y
n
≤ 0, Bp,p − y
n
≤0. 3.10
It follows from 3.9 and 3.10 that
v
n
− p
2
≤
u
n
− p
2
−
u
n
− v
n
2
2λ
n
By
n
,y
n
− v
n
u
n
− p
2
−
u
n
− y
n
y
n
− v
n
2
2λ
n
By
n
,y
n
− v
n
≤
u
n
− p
2
−
u
n
− y
n
2
−
y
n
− v
n
2
− 2
u
n
− y
n
,y
n
− v
n
2λ
n
By
n
,y
n
− v
n
u
n
− p
2
−
u
n
− y
n
2
−
y
n
− v
n
2
2
u
n
− λ
n
By
n
− y
n
,v
n
− y
n
.
3.11
Substituting x by u
n
− λ
n
Bu
n
and y v
n
in 1.4,weobtain
u
n
− λ
n
Bu
n
− y
n
,v
n
− y
n
≤ 0. 3.12
It follows that
u
n
− λ
n
By
n
− y
n
,v
n
− y
n
u
n
− λ
n
Bu
n
− y
n
,v
n
− y
n
λ
n
Bu
n
− λ
n
By
n
,v
n
− y
n
≤
λ
n
Bu
n
− λ
n
By
n
,v
n
− y
n
≤ λ
n
Bu
n
− By
n
v
n
− y
n
≤
λ
n
β
u
n
− y
n
v
n
− y
n
.
3.13
12 Fixed Point Theory and Applications
Substituting 3.13 into 3.11, we have
v
n
− p
2
≤
u
n
− p
2
−
u
n
− y
n
2
−
y
n
− v
n
2
2
u
n
− λ
n
By
n
− y
n
,v
n
− y
n
≤
u
n
− p
2
−
u
n
− y
n
2
−
y
n
− v
n
2
2
λ
n
β
u
n
− y
n
v
n
− y
n
≤
u
n
− p
2
−
u
n
− y
n
2
−
y
n
− v
n
2
λ
2
n
β
2
u
n
− y
n
2
v
n
− y
n
2
u
n
− p
2
−
u
n
− y
n
2
λ
2
n
β
2
u
n
− y
n
2
u
n
− p
2
λ
2
n
β
2
− 1
u
n
− y
n
2
≤
u
n
− p
2
≤
x
n
− p
2
.
3.14
Setting k
n
α
n
u
n
1 − α
n
v
n
, we can calculate
x
n1
− p
n
γf
x
n
− Ap
β
n
x
n
− p
1 − β
n
I −
n
A
W
n
k
n
− p
≤
1 − β
n
−
n
γ
k
n
− p
β
n
x
n
− p
n
γf
x
n
− Ap
≤
1 − β
n
−
n
γ
α
n
u
n
− p
1 − α
n
v
n
− p
β
n
x
n
− p
n
γf
x
n
− Ap
≤
1 − β
n
−
n
γ
α
n
x
n
− p
1 − α
n
x
n
− p
β
n
x
n
− p
n
γf
x
n
− Ap
1 − β
n
−
n
γ
x
n
− p
β
n
x
n
− p
n
γf
x
n
− Ap
1 −
n
γ
x
n
− p
n
γ
f
x
n
− f
p
n
γf
p
− Ap
≤
1 −
n
γ
x
n
− p
n
γα
x
n
− p
n
γf
p
− Ap
1 −
γ − γα
n
x
n
− p
γ − γα
n
γf
p
− Ap
γ − γα
.
3.15
By induction,
x
n
− p
≤ max
x
1
− p
,
γf
p
− Ap
γ − γα
,n∈ N. 3.16
Hence, {x
n
} is bounded, so are {u
n
}, {v
n
}, {W
n
k
n
}, {fx
n
}, {Bu
n
}, {y
n
}, and {By
n
}.
Step 2. We claim that lim
n →∞
x
n1
− x
n
0.
Fixed Point Theory and Applications 13
Observing that u
n
T
r
n
x
n
and u
n1
T
r
n1
x
n1
, we get
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0 ∀y ∈ H 3.17
F
u
n1
,y
1
r
n1
y − u
n1
,u
n1
− x
n1
≥ 0 ∀y ∈ H. 3.18
Putting y u
n1
in 3.17 and y u
n
in 3.18, we have
F
u
n
,u
n1
1
r
n
u
n1
− u
n
,u
n
− x
n
≥ 0
F
u
n1
,u
n
1
r
n1
u
n
− u
n1
,u
n1
− x
n1
≥0.
3.19
So, from A2 we have
u
n1
− u
n
,
u
n
− x
n
r
n
−
u
n1
− x
n1
r
n1
≥ 0, 3.20
and hence
u
n1
− u
n
,u
n
− u
n1
u
n1
− x
n
−
r
n
r
n1
u
n1
− x
n1
≥ 0. 3.21
Without loss of generality, let us assume that there exists a real number c such that r
n
>c>0
for all n ∈ N. Then, we have
u
n1
− u
n
2
≤
u
n1
− u
n
,x
n1
− x
n
1 −
r
n
r
n1
u
n1
− x
n1
≤
u
n1
− u
n
x
n1
− x
n
1 −
r
n
r
n1
u
n1
− x
n1
,
3.22
and hence
u
n1
− u
n
≤
x
n1
− x
n
1
r
n1
|
r
n1
− r
n
|
u
n1
− x
n1
≤
x
n1
− x
n
M
1
c
|
r
n1
− r
n
|
,
3.23
14 Fixed Point Theory and Applications
where M
1
sup{u
n
− x
n
: n ∈ N}.Notethat
v
n1
− v
n
≤
P
C
u
n1
− λ
n1
By
n1
− P
C
u
n
− λ
n
By
n
≤
u
n1
− λ
n1
By
n1
−
u
n
− λ
n
By
n
u
n1
− λ
n1
Bu
n1
−
u
n
− λ
n1
Bu
n
λ
n1
Bu
n1
− By
n1
− Bu
n
λ
n
By
n
≤
u
n1
− λ
n1
Bu
n1
−
u
n
− λ
n1
Bu
n
λ
n1
Bu
n1
By
n1
Bu
n
λ
n
By
n
≤
u
n1
− u
n
λ
n1
Bu
n1
By
n1
Bu
n
λ
n
By
n
,
k
n1
− k
n
α
n1
u
n1
1 − α
n1
v
n1
− α
n
u
n
−
1 − α
n
v
n
α
n1
u
n1
− u
n
α
n1
− α
n
u
n
1 − α
n1
v
n1
− v
n
α
n
− α
n1
v
n
≤ α
n1
u
n1
− u
n
1 − α
n1
v
n1
− v
n
|
α
n
− α
n1
|
u
n
v
n
α
n1
u
n1
− u
n
1 − α
n1
×
u
n1
− u
n
λ
n1
Bu
n1
By
n1
Bu
n
λ
n
By
n
|
α
n
− α
n1
|
u
n
v
n
u
n1
− u
n
1 − α
n1
λ
n1
Bu
n1
By
n1
Bu
n
1 − α
n1
λ
n
By
n
|
α
n
− α
n1
|
u
n
v
n
≤
x
n1
− x
n
M
1
c
|
r
n1
− r
n
|
1 − α
n1
λ
n1
×
Bu
n1
By
n1
Bu
n
1 − α
n1
λ
n
By
n
|
α
n
− α
n1
|
u
n
v
n
.
3.24
Setting
z
n
x
n1
− β
n
x
n
1 − β
n
n
γf
x
n
1 − β
n
I −
n
A
W
n
k
n
1 − β
n
, 3.25
Fixed Point Theory and Applications 15
we have x
n1
1 − β
n
z
n
β
n
x
n
,n≥ 1. It follows that
z
n1
− z
n
n1
γf
x
n1
1 − β
n1
I −
n1
A
W
n1
k
n1
1 − β
n1
−
n
γf
x
n
1 − β
n
I −
n
A
W
n
k
n
1 − β
n
n1
1 − β
n1
γf
x
n1
−
n
1 − β
n
γf
x
n
W
n1
k
n1
− W
n
k
n
n
1 − β
n
AW
n
k
n
−
n1
1 − β
n1
AW
n1
k
n1
n1
1 − β
n1
γf
x
n1
− AW
n1
k
n1
n
1 − β
n
AW
n
k
n
− γf
x
n
W
n1
k
n1
− W
n1
k
n
W
n1
k
n
− W
n
k
n
.
3.26
It follows from 3.24 and 3.26 that
z
n1
− z
n
−
x
n1
− x
n
≤
n1
1 − β
n1
γf
x
n1
AW
n1
k
n1
n
1 − β
n
AW
n
k
n
γf
x
n
W
n1
k
n1
− W
n1
k
n
W
n1
k
n
− W
n
k
n
−
x
n1
− x
n
≤
n1
1 − β
n1
γf
x
n1
AW
n1
k
n1
n
1 − β
n
AW
n
k
n
γf
x
n
k
n1
− k
n
W
n1
k
n
− W
n
k
n
−
x
n1
− x
n
≤
n1
1 − β
n1
γf
x
n1
AW
n1
k
n1
n
1 − β
n
AW
n
k
n
γf
x
n
M
1
c
|
r
n1
− r
n
|
1 − α
n1
λ
n1
Bu
n1
By
n1
Bu
n
1 − α
n1
λ
n
By
n
|
α
n
− α
n1
|
u
n
v
n
W
n1
k
n
− W
n
k
n
.
3.27
16 Fixed Point Theory and Applications
Since T
i
and U
n,i
are nonexpansive, we have
W
n1
k
n
− W
n
k
n
μ
1
T
1
U
n1,2
k
n
− μ
1
T
1
U
n,2
k
n
≤ μ
1
U
n1,2
k
n
− U
n,2
k
n
μ
1
μ
2
T
2
U
n1,3
k
n
− μ
2
T
2
U
n,3
k
n
≤ μ
1
μ
2
U
n1,3
k
n
− U
n,3
k
n
.
.
.
≤ μ
1
μ
2
···μ
n
U
n1,n1
k
n
− U
n,n1
k
n
≤ M
2
n
i1
μ
i
,
3.28
where M
2
≥ 0 is a constant such that U
n1,n1
k
n
− U
n,n1
k
n
≤M
2
for all n ≥ 0.
Combining 3.27 and 3.28, we have
z
n1
− z
n
−
x
n1
− x
n
≤
n1
1 − β
n1
γf
x
n1
AW
n1
k
n1
n
1 − β
n
AW
n
k
n
γf
x
n
M
1
c
|
r
n1
− r
n
|
1 − α
n1
λ
n1
Bu
n1
By
n1
Bu
n
1 − α
n1
λ
n
By
n
|
α
n
− α
n1
|
u
n
v
n
M
2
n
i1
μ
i
,
3.29
which implies that noting that i, ii, iii, iv, v,and0<μ
i
≤ b<1, for all i ≥ 1
lim sup
n →∞
z
n1
− z
n
−
x
n1
− x
n
≤ 0. 3.30
Hence, by Lemma 2.8,weobtain
lim
n →∞
z
n
− x
n
0. 3.31
It follows that
lim
n →∞
x
n1
− x
n
lim
n →∞
1 − β
n
z
n
− x
n
0. 3.32
Fixed Point Theory and Applications 17
Applying 3.32 and ii, iv,andv to 3.23 and 3.24,weobtainthat
lim
n →∞
u
n1
− u
n
lim
n →∞
k
n1
− k
n
0. 3.33
Since x
n1
n
γfx
n
β
n
x
n
1 − β
n
I −
n
AW
n
k
n
, we have
x
n
− W
n
k
n
≤
x
n
− x
n1
x
n1
− W
n
k
n
x
n
− x
n1
n
γf
x
n
β
n
x
n
1 − β
n
I −
n
A
W
n
k
n
− W
n
k
n
x
n
− x
n1
n
γf
x
n
− AW
n
k
n
β
n
x
n
− W
n
k
n
≤
x
n
− x
n1
n
γf
x
n
AW
n
k
n
β
n
x
n
− W
n
k
n
,
3.34
that is
x
n
− W
n
k
n
≤
1
1 − β
n
x
n
− x
n1
n
1 − β
n
γf
x
n
AW
n
k
n
. 3.35
By i, iii,and3.32 it follows that
lim
n →∞
W
n
k
n
− x
n
0. 3.36
Step 3. We claim that the following statements hold:
i lim
n →∞
u
n
− k
n
0;
ii lim
n →∞
x
n
− u
n
0.
For any p ∈ Θ : ∩
∞
n1
FT
n
∩ EPF ∩ VIC, B and 3.14, we have
k
n
− p
2
α
n
u
n
− p1 − α
n
v
n
− p
2
≤ α
n
u
n
− p
2
1 − α
n
v
n
− p
2
≤ α
n
u
n
− p
2
1 − α
n
u
n
− p
2
λ
2
n
β
2
− 1
u
n
− y
n
2
u
n
− p
2
1 − α
n
λ
2
n
β
2
− 1
u
n
− y
n
2
≤
x
n
− p
2
1 − α
n
λ
2
n
β
2
− 1
u
n
− y
n
2
.
3.37
18 Fixed Point Theory and Applications
Observe that
x
n1
− p
2
1 − β
n
I −
n
AW
n
k
n
− pβ
n
x
n
− p
n
γfx
n
− Ap
2
1 − β
n
I −
n
AW
n
k
n
− pβ
n
x
n
− p
2
2
n
γfx
n
− Ap
2
2β
n
n
x
n
− p, γf
x
n
− Ap
2
n
1 − β
n
I −
n
A
W
n
k
n
− p
,γf
x
n
− Ap
≤
1 − β
n
−
n
γ
W
n
k
n
− p
β
n
x
n
− p
2
2
n
γfx
n
− Ap
2
2β
n
n
x
n
− p, γf
x
n
− Ap
2
n
1 − β
n
I −
n
A
W
n
k
n
− p
,γf
x
n
− Ap
≤
1 − β
n
−
n
γ
k
n
− p
β
n
x
n
− p
2
c
n
≤
1 − β
n
−
n
γ
2
k
n
− p
2
β
2
n
x
n
− p
2
2
1 − β
n
−
n
γ
β
n
k
n
− p
x
n
− p
c
n
≤
1 − β
n
−
n
γ
2
k
n
− p
2
β
2
n
x
n
− p
2
1 − β
n
−
n
γ
β
n
k
n
− p
2
x
n
− p
2
c
n
1 −
n
γ
2
− 2
1 −
n
γ
β
n
β
2
n
k
n
− p
2
β
2
n
x
n
− p
2
1 −
n
γ
β
n
− β
2
n
k
n
− p
2
x
n
− p
2
c
n
1 −
n
γ
2
k
n
− p
2
−
1 −
n
γ
β
n
k
n
− p
2
1 −
n
γ
β
n
x
n
− p
2
c
n
1 −
n
γ
1 − β
n
−
n
γ
k
n
− p
2
1 −
n
γ
β
n
x
n
− p
2
c
n
,
3.38
where
c
n
2
n
γfx
n
− Ap
2
2β
n
n
x
n
− p, γf
x
n
− Ap
2
n
1 − β
n
I −
n
A
W
n
k
n
− p
,γf
x
n
− Ap
.
3.39
It follows from condition i that
lim
n →∞
c
n
0. 3.40
Fixed Point Theory and Applications 19
Substituting 3.37 into 3.38,andusingv, we have
x
n1
− p
2
≤
1 −
n
γ
1 − β
n
−
n
γ
x
n
− p
2
1 − α
n
λ
2
n
β
2
− 1
u
n
− y
n
2
1 −
n
γ
β
n
x
n
− p
2
c
n
1 −
n
γ
2
x
n
− p
2
1 −
n
γ
1 − β
n
−
n
γ
1 − α
n
λ
2
n
β
2
− 1
u
n
− y
n
2
c
n
≤
x
n
− p
2
1 − α
n
λ
2
n
β
2
− 1
u
n
− y
n
2
c
n
.
3.41
It follows that
1 − α
n
δ
u
n
− y
n
2
≤
1 − α
n
1 −
λ
2
n
β
2
u
n
− y
n
2
≤
x
n
− p
2
−
x
n1
− p
2
c
n
x
n
− p
−
x
n1
− p
x
n
− p
x
n1
− p
c
n
≤
x
n
− x
n1
x
n
− p
x
n1
− p
c
n
.
3.42
Since lim
n →∞
c
n
0andfrom3.32,weobtain
lim
n →∞
u
n
− y
n
0. 3.43
Note that
k
n
− v
n
α
n
u
n
− v
n
. 3.44
Since lim
n →∞
α
n
0, we have
lim
n →∞
k
n
− v
n
0. 3.45
20 Fixed Point Theory and Applications
As B is 1/β-Lipschitz continuous, we obtain
v
n
− y
n
P
C
u
n
− λ
n
By
n
− P
C
u
n
− λ
n
Bu
n
≤
u
n
− λ
n
By
n
−
u
n
− λ
n
Bu
n
λ
n
Bu
n
− By
n
≤
λ
n
β
u
n
− y
n
,
3.46
then, we get
lim
n →∞
v
n
− y
n
0. 3.47
from
u
n
− k
n
≤
u
n
− y
n
y
n
− v
n
v
n
− k
n
. 3.48
Applying 3.43, 3.45,and3.47, we have
lim
n →∞
u
n
− k
n
0. 3.49
For any p ∈ Θ,notethatT
r
is firmly nonexpansive Lemma 2.4, then we have
u
n
− p
2
T
r
n
x
n
− T
r
n
p
2
≤
T
r
n
x
n
− T
r
n
p, x
n
− p
u
n
− p, x
n
− p
1
2
u
n
− p
2
x
n
− p
2
−
x
n
− u
n
2
,
3.50
and hence
u
n
− p
2
≤
x
n
− p
2
−
x
n
− u
n
2
, 3.51
Fixed Point Theory and Applications 21
which together with 3.38 gives
x
n1
− p
2
≤
1 −
n
γ
1 − β
n
−
n
γ
k
n
− p
2
1 −
n
γ
β
n
x
n
− p
2
c
n
1 −
n
γ
1 −
n
γ − β
n
×
k
n
− u
n
2
u
n
− p
2
2k
n
− u
n
,u
n
− p
1 −
n
γ
β
n
x
n
− p
2
c
n
≤
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
2
1 −
n
γ
1 −
n
γ − β
n
u
n
− p
2
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
u
n
− p
1 −
n
γ
β
n
x
n
− p
2
c
n
≤
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
2
1 −
n
γ
1 −
n
γ − β
n
x
n
− p
2
−
x
n
− u
n
2
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
u
n
− p
1 −
n
γ
β
n
x
n
− p
2
c
n
≤
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
2
1 −
n
γ
1 −
n
γ − β
n
x
n
− p
2
−
1 −
n
γ
1 −
n
γ − β
n
x
n
− u
n
2
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
u
n
− p
1 −
n
γ
β
n
x
n
− p
2
c
n
1 −
n
γ
2
x
n
− p
2
−
1 −
n
γ
1 −
n
γ − β
n
x
n
− u
n
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
2
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
u
n
− p
c
n
1 − 2
n
γ
n
γ
2
x
n
− p
2
−
1 −
n
γ
1 −
n
γ − β
n
x
n
− u
n
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
2
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
u
n
− p
c
n
≤
x
n
− p
2
n
γ
2
x
n
− p
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
2
22 Fixed Point Theory and Applications
−
1 −
n
γ
1 −
n
γ − β
n
x
n
− u
n
2
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
u
n
− p
c
n
.
3.52
So
1 −
n
γ
1 −
n
γ − β
n
x
n
− u
n
2
≤
x
n
− p
2
−
x
n1
− p
2
n
γ
2
x
n
− p
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
2
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
u
n
− p
c
n
x
n
− p
−
x
n1
− p
x
n
− p
x
n1
− p
n
γ
2
x
n
− p
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
2
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
u
n
− p
c
n
≤
x
n
− x
n1
x
n
− p
x
n1
− p
n
γ
2
x
n
− p
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
2
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
u
n
− p
c
n
.
3.53
Using
n
→ 0, c
n
→ 0asn →∞, 3.32,and3.49,weobtain
lim
n →∞
x
n
− u
n
0. 3.54
Since lim inf
n →∞
r
n
> 0, we obtain
lim
n →∞
x
n
− u
n
r
n
lim
n →∞
1
r
n
x
n
− u
n
0. 3.55
Observe that
W
n
u
n
− u
n
≤
W
n
u
n
− W
n
k
n
W
n
k
n
− x
n
x
n
− u
n
≤
u
n
− k
n
W
n
k
n
− x
n
x
n
− u
n
.
3.56
Applying 3.36, 3.49,and3.54 to the last inequality, we obtain
lim
n →∞
W
n
u
n
− u
n
0. 3.57
Fixed Point Theory and Applications 23
Let W be the mapping defined by 2.6. Since {u
n
} is bounded, applying Lemma 2.7 and
3.57, we have
Wu
n
− u
n
≤
Wu
n
− W
n
u
n
W
n
u
n
− u
n
→ 0asn →∞. 3.58
Step 4. We claim that lim sup
n →∞
A − γfz, z − x
n
≤0, where z is the unique solution of
the variational inequality A − γfz, z − x≥0, for all x ∈ Θ.
Since z P
Θ
I − A γfz is a unique solution of the variational inequality 3.1,to
show this inequality, we choose a subsequence {u
n
i
} of {u
n
} such that
lim
i →∞
A − γf
z, z − u
n
i
lim sup
n →∞
A − γf
z, z − u
n
. 3.59
Since {u
n
i
} is bounded, there exists a subsequence {u
n
i
j
} of {u
n
i
} which converges weakly to
w ∈ C. Without loss of generality, we can assume that u
n
i
w.From Wu
n
− u
n
→0, we
obtain Wu
n
i
w. Next, We show that w ∈ Θ, where Θ : ∩
∞
n1
FT
n
∩ EPF∩ VIC, B.First,
we show that w ∈ EPF. Since u
n
T
r
n
x
n
, we have
F
u
n
,y
1
r
n
y − u
n
,u
n
− x
n
≥ 0, ∀y ∈ C. 3.60
If follows from A2 that
1
r
n
y − u
n
,u
n
− x
n
≥−F
u
n
,y
≥ F
y, u
n
, 3.61
and hence
y − u
n
i
,
u
n
i
− x
n
i
r
n
i
≥ F
y, u
n
i
. 3.62
Since u
n
i
− x
n
i
/r
n
i
→ 0andu
n
i
w,it follows by A4 that Fy, w ≤ 0 for all y ∈ H. For t
with 0 <t≤ 1andy ∈ H, let y
t
ty 1 − tw. Since y ∈ H and w ∈ H, we have y
t
∈ H and
hence Fy
t
,w ≤ 0. So, from A1 and A4 we have
0 F
y
t
,y
t
≤ tF
y
t
,y
1 − t
F
y
t
,w
≤ tF
y
t
,y
, 3.63
and hence Fy
t
,y ≥ 0. From A3, we have Fw, y ≥ 0 for all y ∈ H and hence w ∈ EPF.
Next, we show that w ∈∩
∞
n1
FT
n
. By Lemma 2.6, we have FW∩
∞
n1
FT
n
.
Assume w
/
∈ FW. Since u
n
i
wand w
/
Ww, it follows by the Opial’s condition that
lim inf
i →∞
u
n
i
− w
< lim inf
i →∞
u
n
i
− Ww
≤ lim inf
i →∞
{
u
n
i
− Wu
n
i
Wu
n
i
− Ww
}
≤ lim inf
i →∞
u
n
i
− w
,
3.64
24 Fixed Point Theory and Applications
which derives a contradiction. Thus, we have w ∈ FW∩
∞
n1
FT
n
. By the same argument
as that in the proof of 35, Theorem 2.1, Pages 10–11, we can show that w ∈ VIC, B. Hence
w ∈ Θ. Since z P
Θ
I − A γfz, it follows that
lim sup
n →∞
A − γf
z, z − x
n
lim sup
n →∞
A − γf
z, z − u
n
lim
i →∞
A − γf
z, z − u
n
i
A − γf
z, z − w
≤ 0.
3.65
It follows from the last inequality, 3.36,and3.54 that
lim sup
n →∞
γf
z
− Az, W
n
k
n
− z
≤ 0. 3.66
Step 5. Finally, we show that {x
n
}and {u
n
} converge strongly to z P
Θ
I −Aγfz. Indeed,
from 1.25 , we have
x
n1
− z
2
n
γfx
n
β
n
x
n
1 − β
n
I −
n
AW
n
k
n
− z
2
1 − β
n
I −
n
AW
n
k
n
− zβ
n
x
n
− z
n
γfx
n
− Az
2
1 − β
n
I −
n
AW
n
k
n
− zβ
n
x
n
− z
2
2
n
γfx
n
− Az
2
2β
n
n
x
n
− z, γf
x
n
− Az
2
n
1 − β
n
I −
n
A
W
n
k
n
− z
,γf
x
n
− Az
≤
1 − β
n
−
n
γ
W
n
k
n
− z
β
n
x
n
− z
2
2
n
γfx
n
− Az
2
2β
n
n
γx
n
− z, f
x
n
− f
z
2β
n
n
x
n
− z, γf
z
− Az
2
1 − β
n
γ
n
W
n
k
n
− z, f
x
n
− f
z
2
1 − β
n
n
W
n
k
n
− z, γf
z
− Az
− 2
2
n
A
W
n
k
n
− z
,γf
z
− Az
≤
1 − β
n
−
n
γ
W
n
k
n
− z
β
n
x
n
− z
2
2
n
γfx
n
− Az
2
2β
n
n
γ
x
n
− z
f
x
n
− f
z
2β
n
n
x
n
− z, γf
z
− Az
2
1 − β
n
γ
n
W
n
k
n
− z
f
x
n
− f
z
2
1 − β
n
n
W
n
k
n
− z, γf
z
− Az
− 2
2
n
A
W
n
k
n
− z
,γf
z
− Az
Fixed Point Theory and Applications 25
≤
1 − β
n
−
n
γ
x
n
− z
β
n
x
n
− z
2
2
n
γfx
n
− Az
2
2β
n
n
γα
x
n
− z
2
2β
n
n
x
n
− z, γf
z
− Az
2
1 − β
n
γ
n
α
x
n
− z
2
2
1 − β
n
n
W
n
k
n
− z, γf
z
− Az
− 2
2
n
A
W
n
k
n
− z
,γf
z
− Az
1 −
n
γ
2
2β
n
n
γα 2
1 − β
n
γ
n
α
x
n
− z
2
2
n
γfx
n
− Az
2
2β
n
n
x
n
− z, γf
z
− Az 2
1 − β
n
n
W
n
k
n
− z, γf
z
− Az
− 2
2
n
A
W
n
k
n
− z
,γf
z
− Az
≤
1 − 2
γ − αγ
n
x
n
− z
2
γ
2
2
n
x
n
− z
2
2
n
γfx
n
− Az
2
2β
n
n
x
n
− z, γf
z
− Az 2
1 − β
n
n
W
n
k
n
− z, γf
z
− Az
2
2
n
A
W
n
k
n
− z
γf
z
− Az
1 − 2
γ − αγ
n
x
n
− z
2
n
×
n
γ
2
x
n
− z
2
γfx
n
− Az
2
2
A
W
n
k
n
− z
γf
z
− Az
2β
n
x
n
− z, γf
z
− Az
2
1 − β
n
W
n
k
n
− z, γf
z
− Az
.
3.67
Since {x
n
}, {fx
n
}, and {W
n
k
n
} are bounded, we can take a constant M>0 such that
γ
2
x
n
− z
2
γfx
n
− Az
2
2
A
W
n
k
n
− z
γf
z
− Az
≤ M 3.68
for all n ≥ 0. It then follows that
x
n1
− z
2
≤
1 − 2
γ − αγ
n
x
n
− z
2
n
σ
n
, 3.69
where
σ
n
2β
n
x
n
− z, γf
z
− Az 2
1 − β
n
W
n
k
n
− z, γf
z
− Az
n
M. 3.70
Using i, 3.65,and3.66, we get lim sup
n →∞
σ
n
≤ 0. Applying Lemma 2.9 to 3.69,we
conclude that x
n
→ z in norm. Finally, noticing
u
n
− z
T
r
n
x
n
− T
r
n
z
≤
x
n
− z
, 3.71
we also conclude that u
n
→ z in norm. This completes the proof.
