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, [email protected]
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
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
≥
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 ∈ 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
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
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
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
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
≥ 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
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
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,
.
.
.
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
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
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
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
, ∀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
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
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
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
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
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
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.
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
Bx − By
2
x − y
2
λ
n
λ
n
− 2β
Bx − By
2
≤
x − y
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
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
y
≤
I − A γf
x
−
I − A γf
y
≤
I − A
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.
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 − λ
n
Bp
I − λ
n
A
u
n
−
I − λ
n
B
p
≤
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
−
λ
2
n
By
n
2
−
u
n
− v
n
2
2λ
n
By
n
,u
n
− v
n
− λ
n
,p− v
n
u
n
− p
2
−
u
n
− v
n
2
2λ
n
By
n
− Bp, p − y
n
2λ
v
n
− p
2
≤
u
n
− p
2
−
u
n
− v
n
2
2λ
n
By
n
,y
n
By
n
,y
n
− v
n
≤
u
n
− p
2
−
u
n
− y
n
2
−
y
u
n
− p
2
−
u
n
− y
n
2
−
y
n
− v
n
2
2
u
n
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
By
n
,v
n
− y
n
≤ λ
n
Bu
n
− By
n
v
n
− y
n
≤
λ
n
β
u
n
− p
2
−
u
n
− y
n
2
−
y
n
− v
n
2
2
u
n
−
y
n
− v
n
2
2
λ
n
β
u
n
− y
n
v
n
− y
n
λ
2
n
β
2
u
n
− y
n
2
v
n
− y
n
2
u
n
− p
u
n
− p
2
λ
2
n
β
2
− 1
u
n
− y
n
2
≤
u
n
− p
n
γf
x
n
− Ap
β
n
x
n
− p
1 − β
n
I −
n
n
− p
n
γf
x
n
− Ap
≤
1 − β
n
−
n
γ
α
n
u
n
x
n
− Ap
≤
1 − β
n
−
n
γ
α
n
x
n
− p
1 − α
n
1 − β
n
−
n
γ
x
n
− p
β
n
x
n
− p
n
γf
x
n
n
γf
p
− Ap
≤
1 −
n
γ
x
n
− p
n
γα
n
− p
γ − γα
n
γf
p
− Ap
γ − γα
.
3.15
By induction,
x
n
− p
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
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
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
n
,x
n1
− x
n
1 −
r
n
r
n1
u
n1
− x
n1
≤
u
n1
− u
,
3.22
and hence
u
n1
− u
n
≤
x
n1
− x
n
1
r
n1
|
r
n1
− r
n
|
u
n1
− x
v
n1
− v
n
≤
P
C
u
n1
− λ
n1
By
n1
− P
C
u
n
− λ
n
By
n
−
u
n
− λ
n1
Bu
n
λ
n1
Bu
n1
− By
n1
− Bu
n
λ
n
By
n
≤
u
n1
Bu
n
λ
n
By
n
≤
u
n1
− u
n
λ
n1
Bu
n1
By
n1
n1
− α
n
u
n
−
1 − α
n
v
n
α
n1
u
n1
− u
n
α
n1
− α
n
1 − α
n1
v
n1
− v
n
|
α
n
− α
n1
|
u
n
v
n
α
n1
u
n1
− u
Bu
n
λ
n
By
n
|
α
n
− α
n1
|
u
n
v
n
u
n1
− u
n
λ
n
By
n
|
α
n
− α
n1
|
u
n
v
n
≤
x
n1
− x
n
Bu
n
1 − α
n1
λ
n
By
n
|
α
n
− α
n1
|
u
n
v
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
I −
n
A
W
n
k
n
1 − β
n
n1
1 − β
n1
γf
x
n1
−
n
1 − β
n
γf
x
n
1 − β
n1
γf
x
n1
− AW
n1
k
n1
n
1 − β
n
AW
n
k
n
− γf
x
n
W
≤
n1
1 − β
n1
γf
x
n1
AW
n1
k
n1
n
1 − β
n
W
n1
k
n
− W
n
k
n
−
x
n1
− x
n
≤
n1
1 − β
n1
γf
x
n1
k
n1
− k
n
W
n1
k
n
− W
n
k
n
−
x
n1
− x
n
≤
n1
n
γf
x
n
M
1
c
|
r
n1
− r
n
|
1 − α
n1
λ
n1
|
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
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
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
AW
n1
k
n1
n
1 − β
n
AW
n
k
n
γf
x
n
Bu
n
1 − α
n1
λ
n
By
n
|
α
n
− α
n1
|
u
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
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
1 − β
n
I −
n
A
W
n
k
n
− W
n
k
n
x
n
− x
n1
n
n
γf
x
n
AW
n
k
n
β
n
x
n
− W
n
k
n
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 →∞
n
u
n
− p1 − α
n
v
n
− p
2
≤ α
n
u
n
− p
2
1 − α
n
v
n
λ
2
n
β
2
− 1
u
n
− y
n
2
u
n
− p
2
1 − α
n
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
n
AW
n
k
n
− pβ
n
x
n
− p
2
2
n
γfx
n
− Ap
2
2β
n
n
x
≤
1 − β
n
−
n
γ
W
n
k
n
− p
β
n
x
n
− p
2
2
n
A
W
n
k
n
− p
,γf
x
n
− Ap
≤
1 − β
n
−
n
γ
k
n
− p
β
2
n
x
n
− p
2
2
1 − β
n
−
n
γ
β
n
k
n
− p
n
− p
2
1 − β
n
−
n
γ
β
n
k
n
− p
2
x
n
− p
− p
2
β
2
n
x
n
− p
2
1 −
n
γ
β
n
− β
2
n
2
−
1 −
n
γ
β
n
k
n
− p
2
1 −
n
γ
β
n
x
n
γ
β
n
x
n
− p
2
c
n
,
3.38
where
c
n
2
n
γfx
n
− Ap
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
u
n
− y
n
2
1 −
n
γ
β
n
x
n
− p
2
c
n
1 −
n
n
β
2
− 1
u
n
− y
n
2
c
n
≤
x
n
− p
2
1 − α
n
n
− y
n
2
≤
1 − α
n
1 −
λ
2
n
β
2
u
n
− y
n
2
≤
x
n1
− p
x
n
− p
x
n1
− p
c
n
≤
x
n
− x
n1
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 →∞
C
u
n
− λ
n
Bu
n
≤
u
n
− λ
n
By
n
−
u
n
− λ
n
Bu
n
v
n
− y
n
0. 3.47
from
u
n
− k
n
≤
u
n
− y
n
y
n
2
T
r
n
x
n
− T
r
n
p
2
≤
T
r
n
x
n
− T
r
n
p, x
n
2
−
x
n
− u
n
2
,
3.50
and hence
u
n
− p
2
≤
x
n
− p
2
k
n
− p
2
1 −
n
γ
β
n
x
n
− p
2
c
n
1 −
,u
n
− p
1 −
n
γ
β
n
x
n
− p
2
c
n
≤
1 −
n
γ
1 −
n
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
u
n
− p
1 −
n
γ
β
1 −
n
γ
1 −
n
γ − β
n
x
n
− p
2
−
x
n
− u
n
2
2
x
n
− p
2
c
n
≤
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
2
n
− u
n
2
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
u
n
− p
2
−
1 −
n
γ
1 −
n
γ − β
n
x
n
− u
n
2
1 −
n
γ
1 −
n
γ − β
n
c
n
1 − 2
n
γ
n
γ
2
x
n
− p
2
−
1 −
n
γ
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
u
n
− p
c
n
≤
x
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 −
1 −
n
γ − β
n
x
n
− u
n
2
≤
x
n
− p
2
−
x
n1
− p
− u
n
2
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
u
n
− p
c
n
− p
n
γ
2
x
n
− p
2
1 −
n
γ
1 −
n
γ − β
n
c
n
≤
x
n
− x
n1
x
n
− p
x
n1
− p
2
1 −
n
γ
1 −
n
γ − β
n
k
n
− u
n
u
n
− p
c
n
.
3.53
Using
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
− 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
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
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
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
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
}
≤ 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
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
x
n
− z
n
γfx
n
− Az
2
1 − β
n
I −
n
AW
n
k
n
− zβ
n
x
n
− z
2
n
A
W
n
k
n
− z
,γf
x
n
− Az
≤
1 − β
n
−
n
γ
W
n
k
n
− z
− 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
− z
,γf
z
− Az
≤
1 − β
n
−
n
γ
W
n
k
n
− z
β
n
x
n
− z
z
2β
n
n
x
n
− z, γf
z
− Az
2
1 − β
n
γ
n
W
n
k
n
− z
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
n
n
x
n
− z, γf
z
− Az
2
1 − β
n
γ
n
α
x
n
− z
2
2
1 − β
n
1 −
n
γ
2
2β
n
n
γα 2
1 − β
n
γ
n
α
x
n
− z
2
2
n
− Az
− 2
2
n
A
W
n
k
n
− z
,γf
z
− Az
≤
1 − 2
γ − αγ
n
x
n
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
γ
2
x
n
− z
2
γfx
n
− Az
2
2
A
W
n
k
n
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
− z
2
≤
1 − 2
γ − αγ
n
x
n
− z
2
n
σ
n
, 3.69
where
σ
n
2β
n
x
n
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.
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