# Convergence Theorem for Equilibrium and Variational Inequality Problems and a Family of Infinitely Nonexpansive Mappings in Hilbert Space

Authors: Zhou Yinying; Cao Jiantao; Wang Yali
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

We introduce a hybrid iterative scheme for finding a common element of the set of common fixed points for a family of infinitely nonexpansive mappings, the set of solutions of the varitional inequality problem and the equilibrium problem in Hilbert space. Under suitable conditions, some strong convergence theorems are obtained. Our results improve and extend the corresponding results in (Chang et al. (2009), Min and Chang (2012), Plubtieng and Punpaeng (2007), S. Takahashi and W. Takahashi (2007), Tada and Takahashi (2007), Gang and Changsong (2009), Ying (2013), Y. Yao and J. C. Yao (2007), and Yong-Cho and Kang (2012)).


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## Body

## 1. Introduction

LetH be a real Hilbert space, whose inner product and norm are denoted by 〈
·
,
·
〉 and ∥
·
∥, respectively. Let C be a nonempty closed convex subset of H and P
C the metric projection of H onto C. Let ϕ
:
C
×
C
→
R be a bifunction. We consider the equilibrium problem EP which is to find z
∈
C such that
(1)
ϕ
(
z
,
y
)
≥
0
,
∀
y
∈
C
.
Let EP
⁡
(
ϕ
) be the set of solutions. Some methods have been proposed to solve the equilibrium problem.A mappingA is said to be α-inverse strongly monotone if there exists a real number α
>
0 such that 〈
A
x
-
A
y
,
x
-
y
〉
≥
α
∥
A
x
-
A
y
∥
2, for all x
,
y
∈
C.The classical variational inequality problem is to find an elementu
∈
C such that
(2)
〈
A
u
,
v
-
u
〉
≥
0
,
∀
v
∈
C
.
The solution set of inequality (2) is denoted by VI
⁡
(
C
,
A
). For given elements z
∈
H and u
∈
C, we have the following inequality:
(3)
〈
u
-
z
,
v
-
u
〉
≥
0
,
∀
v
∈
C
,
if and only if u
=
P
C
z. It is known that the projection operator P
C is nonexpansive. One can see that the variational inequality problem (2) is equivalent to a fixed point problem. Since an element u
∈
C is the solution of variational inequality (2) if and only if u
∈
C is a fixed point of the mapping P
C
(
I
-
λ
A
). Recently, many researchers studied various iterative algorithms for finding an element of VI
⁡
(
C
,
A
)
⋂
‍
F
(
S
). Takahashi and Toyoda [1] introduced the following iterative scheme:
(4)
x
n
+
1
=
α
n
x
n
+
(
1
-
α
n
)
S
P
C
(
I
-
λ
n
B
)
x
n
,
∀
n
≥
0
.
They proved that the sequence {
x
n
} converges weakly to a point q
∈
VI
⁡
(
C
,
B
)
⋂
‍
F
(
S
). Y. Yao and J. C. Yao [2] introduced the following iterative scheme:
(5)
x
1
=
u
∈
C
,
y
n
=
P
C
(
I
-
λ
n
A
)
x
n
,
x
n
+
1
=
α
n
u
+
β
n
x
n
+
γ
n
S
P
C
(
I
-
λ
n
A
)
y
n
.
Chang et al. [3] introduced the following iterative scheme:
(6)
ϕ
(
u
n
,
y
)
+
1
r
n
〈
y
-
u
n
,
u
n
-
x
n
〉
,
∀
y
∈
C
,
x
n
+
1
=
α
n
f
(
x
n
)
+
β
n
x
n
+
γ
n
W
n
k
n
,
k
n
=
P
C
(
I
-
λ
n
B
)
y
n
,
y
n
=
P
C
(
I
-
λ
n
B
)
u
n
,
and obtained some strong convergence theorems.In this paper, we will introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points for a family of infinitely nonexpansive mappings, the set of solutions of the variational inequality problem, and the equilibrium problem. Further, we obtain some strong convergence theorems and extend the results in [2–10].


## 2. Preliminaries

Letx
n
⇀
x and x
n
→
x be the weak convergence and strong convergence of the sequence {
x
n
} in H. Let C be a nonempty closed convex subset of a Hilbert space H. Let {
S
n
}
n
=
1
∞
:
C
→
C be a family of infinitely nonexpansive mappings and let {
λ
n
}
n
=
1
∞ be a sequence of positive numbers in [
0,1
]. For n
≥
1, we define a mapping W
n
:
C
→
C as follows:
(7)
U
n
,
n
+
1
=
I
U
n
,
n
=
λ
n
S
n
U
n
,
n
+
1
+
(
1
-
λ
n
)
I
U
n
,
n
-
1
=
λ
n
-
1
S
n
-
1
U
n
,
n
+
(
1
-
λ
n
-
1
)
I
⋮
U
n
,
k
=
λ
k
S
k
U
n
,
k
+
1
+
(
1
-
λ
k
)
I
U
n
,
k
-
1
=
λ
k
-
1
S
k
-
1
U
n
,
k
+
(
1
-
λ
k
-
1
)
I
⋮
U
n
,
2
=
λ
2
S
2
U
n
,
3
+
(
1
-
λ
2
)
I
W
n
=
U
n
,
1
=
λ
1
S
1
U
n
,
2
+
(
1
-
λ
1
)
I
.
W
n is the W-mapping of C into itself which is generated by S
n, S
n
-
1, …, S
1 and λ
n, λ
n
-
1, …, λ
1.In order to prove our main results, the following Lemmas are needed.Lemma 1 (see [11]).
LetC be a nonempty closed convex subset of a Banach space E, let {
S
n
}
n
=
1
∞
:
C
→
C be a family of infinitely nonexpansive mappings, such that ⋂
n
=
1
∞
‍
F
(
S
n
)
≠
∅, and let {
λ
n
}
n
=
1
∞ be a sequence of positive numbers in [
0
,
b
] for some b
∈
(
0,1
). For any n
≥
1, let {
W
n
} be the W-mapping of C into itself generated by S
n, S
n
-
1, …, S
1 and λ
n, λ
n
-
1, …, λ
1. Then W
n is asymptotically regular and nonexpansive. Further, if E is strictly convex, then F
(
W
n
)
=
⋂
i
=
1
n
F
(
S
i
).Lemma 2 (see [4]).
LetC be a nonempty closed convex subset of a strictly convex Banach space E. Let {
S
n
}
n
=
1
∞
:
C
→
C be a family of infinitely nonexpansive mappings, such that ⋂
n
=
1
∞
‍
F
(
S
n
)
≠
∅, and let {
λ
n
}
n
=
1
∞ be a sequence of positive numbers in [
0
,
b
] for some b
∈
(
0,1
). Then for every x
∈
C and k
≥
1
lim
⁡
k
→
∞
U
n
,
k
x exists.
Using Lemma2, we can define a mapping W
:
C
→
C as follows:
(8)
W
x
=
lim
⁡
n
→
∞
⁡
W
n
x
=
lim
⁡
n
→
∞
⁡
U
n
,
1
x
,
∀
x
∈
C
.
Such a W is called the W-mapping generated by the sequence {
S
n
}
n
=
1
∞ and {
λ
n
}
n
=
1
∞. Throughout this paper, we always assume that {
λ
n
}
n
=
1
∞ is a sequence of positive numbers in [
0
,
b
] for an element b
∈
(
0,1
).Lemma 3 (see [4]).
LetC be a nonempty closed convex subset of a strictly convex Banach space E. Let {
S
n
}
n
=
1
∞
:
C
→
C be a family of infinitely nonexpansive mappings such that ⋂
n
=
1
∞
‍
F
(
S
n
)
≠
∅ and let {
λ
n
}
n
=
1
∞ be a sequence of positive numbers in [
0
,
b
] for some b
∈
(
0,1
). Then, W is a nonexpansive mapping and F
(
W
)
=
⋂
n
=
1
∞
‍
F
(
S
n
).Lemma 4 (see [4]).
LetC be a nonempty closed convex subset of a Hilbert space H, let {
S
n
}
n
=
1
∞
:
C
→
C be a family of infinitely nonexpansive mappings, such that ⋂
n
=
1
∞
‍
F
(
S
n
)
≠
∅, and let {
λ
n
}
n
=
1
∞ be a sequence of positive numbers in [
0
,
b
] for some b
∈
(
0,1
). If K is any bounded subset of C, then lim
⁡
n
→
∞
sup
⁡
x
∈
K
∥
W
x
-
W
n
x
∥
=
0.Lemma 5 (see [10]).
Let{
x
n
} and {
z
n
} be bounded sequences in a Banach space E and let {
β
n
} be a sequence in [
0,1
] with 0
<
liminf
⁡
n
→
∞
⁡
β
n
≤
limsup
⁡
n
→
∞
⁡
β
n
<
1. Suppose that x
n
+
1
=
(
1
-
β
n
)
z
n
+
β
n
x
n for all integers n
≥
1 and limsup
⁡
n
→
∞
⁡
(
∥
z
n
+
1
-
z
n
∥
-
∥
x
n
+
1
-
x
n
∥
)
≤
0. Then, lim
⁡
n
→
∞
⁡
∥
z
n
-
x
n
∥
=
0.Lemma 6 (see [10]).
Assume thata
n is a sequence of nonnegative real numbers, such that
(9)
a
n
+
1
≤
(
1
-
γ
n
)
a
n
+
δ
n
,
∀
n
≥
n
0
,
where n
0 is some nonnegative integer, γ
n
∈
(
0,1
), and δ
n are sequences satisfying (1)
∑
n
=
1
∞
‍
γ
n
=
∞;
(2)
limsup
⁡
n
→
∞
(
δ
n
/
γ
n
)
≤
0 or ∑
n
=
1
∞
‍
|
δ
n
|
=
∞; then, lim
⁡
n
→
∞
a
n
=
0.Lemma 7 (see [10]).
LetC be a nonempty closed convex subset of a Hilbert space H. Let {
T
m
:
1
≤
m
≤
r
} be a sequence of nonexpansive mappings on C. Suppose that ⋂
m
=
1
r
‍
F
(
T
m
) is nonempty. Let {
λ
m
} be a sequence of positive numbers with ∑
m
=
1
r
‍
λ
m
=
1. Then, a mapping S on C defined by S
x
=
∑
m
=
1
r
‍
λ
m
T
m
x for all x
∈
C is well defined and nonexpansive and F
(
S
)
=
⋂
m
=
1
r
‍
F
(
T
m
) holds.For solving the equilibrium problem for bifunctionF
:
C
×
C
→
R, assume that F satisfies the following conditions:(
A
1
)
F
(
x
,
x
)
=
0 for all x
∈
C;
(
A
2
)
F is monotone; that is, F
(
x
,
y
)
+
F
(
y
,
x
)
≤
0 for all x
,
y
∈
C;
(
A
3
)
lim
⁡
t
→
0
⁡
F
(
t
z
+
(
1
-
t
)
x
,
y
)
≤
F
(
x
,
y
), for any x
,
y
,
z
∈
C;
(
A
4
)
for eachx
∈
C
,
y
↦
F
(
x
,
y
) is convex and lower semicontinuous.If an equilibrium bifunctionF
:
C
×
C
→
R satisfies conditions (
A
1
)
–
(
A
4
), then we have the following two important results.Lemma 8 (see [4]).
LetC be a nonempty closed convex subset of a Hilbert space H and let F be an equilibrium bifunction F
:
C
×
C
→
R that satisfies conditions (
A
1
)
–
(
A
4
). Let r
>
0 and x
∈
C; then, there exists y
∈
C such that F
(
y
,
z
)
+
(
1
/
r
)
〈
z
-
y
,
y
-
x
〉
≥
0, for all z
∈
C.Lemma 9 (see [4]).
LetF be an equilibrium bifunction F
:
C
×
C
→
Rthat satisfies conditions (
A
1
)
–
(
A
4
). For given r
>
0 and x
∈
C, define a mapping V
r
:
H
→
C as follows:
(10)
V
r
(
x
)
=
{
y
∈
C
:
F
(
y
,
z
)
+
1
r
〈
z
-
y
,
y
-
x
〉
≥
0
,
∀
z
∈
C
}
.
Then, the following conclusions hold: (1)
V
r is single-valued;
(2)
V
r is firmly nonexpansive; that is, for any x
,
y
∈
H, ∥
V
r
x
-
V
r
y
∥
2
≤
〈
V
r
x
-
V
r
y
,
x
-
y
〉;
(3)
F
(
V
r
)
=
EP
⁡
(
F
)
(4)
EP
⁡
(
F
) is a closed and convex set.


## 3. The Main Results

Theorem 10.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let B
m
:
C
→
H be a l
m-inverse strongly monotone mapping for each 1
≤
m
≤
r, where r is some positive integer. Let D
:
C
→
H be a α-inverse strongly monotone mapping. Let F be an equilibrium bifunction F
:
C
×
C
→
Rthat satisfies conditions (
A
1
)
–
(
A
4
). Let {
T
n
}
n
=
1
∞
:
C
→
C be a family of infinite k
n-strict pseudocontractive mappings with 0
≤
k
n
<
1 and let {
λ
n
}
n
=
1
∞ be a sequence of positive numbers in [
0
,
b
] for some b
∈
(
0,1
). {
S
n
}
n
=
1
∞
:
C
→
C is a family of infinitely nonexpansive mappings such that ℱ
=
F
(
W
)
⋂
‍
VI
⁡
(
C
,
B
m
)
⋂
‍
EP
⁡
≠
∅, where F
(
W
)
:
=
⋂
n
=
1
∞
‍
F
(
S
n
). Let A be a strongly positive linear bounded operator with coefficient γ
¯
>
0 and let f
:
H
→
H be a contraction with contraction constant h
(
0
<
h
<
1
) and 0
<
γ
<
(
γ
¯
/
h
). Let {
x
n
}
,
{
y
n
}
,
{
ρ
n
} be sequence generated by x
1
∈
H and
(11)
F
(
y
n
,
η
)
+
〈
D
y
n
,
η
-
y
n
〉
+
1
r
n
〈
η
-
y
n
,
y
n
-
x
n
〉
≥
0
,
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
∀
η
∈
C
,
ρ
n
=
∑
m
=
1
r
‍
η
n
m
P
C
(
I
-
μ
m
B
m
)
y
n
,
x
n
+
1
=
α
n
γ
f
(
x
n
)
+
β
n
x
n
+
(
(
1
-
β
n
)
I
-
α
n
A
)
W
n
ρ
n
,
where μ
m
∈
(
0,2
l
m
)
,
{
α
n
}
,
{
β
n
}
⊂
[
0,1
], and {
r
n
}
⊂
[
0
,
∞
]. If the following conditions are satisfied: (
C
1
)
lim
⁡
n
→
∞
α
n
=
0 and ∑
n
=
1
∞
‍
α
n
=
∞;
(
C
2
)
lim
⁡
n
→
∞
η
n
m
=
η
m
∈
(
0,1
);
(
C
3
)
∑
n
=
1
∞
‍
|
r
n
+
1
-
r
n
|
<
∞;
(
C
4
)
liminf
⁡
n
→
∞
⁡
r
n
>
0, 0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
β
n
<
1;
(
C
5
)
∑
n
=
1
∞
‍
η
n
m
=
1, for all n
≥
1,
then{
x
n
} converges strongly to q
∈
ℱ, where q
=
P
ℱ
(
γ
f
+
(
I
-
A
)
)
q.Proof.
We define a bifunctionϕ
:
C
×
C
→
R by ϕ
(
z
,
y
)
=
F
(
z
,
y
)
+
〈
D
z
,
y
-
z
〉, for all y
,
z
∈
C, so the equilibrium problem is equivalent to the following equilibrium problem: find an element z
∈
C such that ϕ
(
z
,
y
)
≥
0, for all y
∈
C and (11) can be written as
(12)
ϕ
(
y
n
,
η
)
+
1
r
n
〈
η
-
y
n
,
y
n
-
x
n
〉
≥
0
,
∀
η
∈
C
,
ρ
n
=
∑
m
=
1
r
η
n
m
P
C
(
I
-
μ
m
B
m
)
y
n
,
x
n
+
1
=
α
n
γ
f
(
x
n
)
+
β
n
x
n
+
(
(
1
-
β
n
)
I
-
α
n
A
)
W
n
ρ
n
.Step 1.
First, we prove the sequences{
x
n
}, {
y
n
}, {
ρ
n
} are bounded.
Letp
∈
ℱ; as y
n
=
V
r
n
x
n, we have ∥
y
n
-
p
∥
=
∥
V
r
n
x
n
-
p
∥
≤
∥
x
n
-
p
∥. Next we show that the mapping I
-
μ
m
B
m is nonexpansive for each m. Consider
(13)
∥
(
I
-
μ
m
B
m
)
x
-
(
I
-
μ
m
B
m
)
y
∥
2
=
∥
(
x
-
y
)
-
μ
m
(
B
m
x
-
B
m
y
)
∥
2
=
∥
x
-
y
∥
2
+
μ
m
2
∥
B
m
x
-
B
m
y
∥
2
-
2
μ
m
〈
B
m
x
-
B
m
y
,
x
-
y
〉
≤
∥
x
-
y
∥
2
+
μ
m
2
∥
B
m
x
-
B
m
y
∥
2
-
2
μ
m
l
m
∥
B
m
x
-
B
m
y
∥
2
=
∥
x
-
y
∥
2
-
μ
m
(
2
l
m
-
μ
m
)
∥
B
m
x
-
B
m
y
∥
2
≤
∥
x
-
y
∥
2
,
∥
ρ
n
-
p
∥
=
∥
∑
m
=
1
r
η
n
m
P
C
(
I
-
μ
m
B
m
)
y
n
-
p
∥
≤
∑
m
=
1
r
η
n
m
∥
y
n
-
p
∥
≤
∥
x
n
-
p
∥
.
Since A is a strongly positive linear bounded operator, then ∥
A
∥
=
sup
⁡
{
|
〈
A
u
,
u
〉
|
:
u
∈
H
,
∥
u
∥
=
1
}, 〈
(
(
1
-
β
n
)
I
-
α
n
A
)
u
,
u
〉
=
1
-
β
n
-
α
n
〈
A
u
,
u
〉
≥
1
-
β
n
-
α
n
∥
A
∥
≥
0, so
(14)
∥
(
1
-
β
n
)
I
-
α
n
A
∥
=
sup
⁡
{
|
〈
(
(
1
-
β
n
)
I
-
α
n
A
)
u
,
u
〉
|
:
u
∈
H
,
∥
u
∥
=
1
}
≤
1
-
β
n
-
α
n
γ
¯
,
(15)
∥
x
n
+
1
-
p
∥
=
∥
α
n
γ
f
(
x
n
)
+
β
n
x
n
+
(
(
1
-
β
n
)
I
-
α
n
A
)
W
n
ρ
n
-
p
∥
=
∥
α
n
γ
(
f
(
x
n
)
-
f
(
p
)
)
+
β
n
(
x
n
-
p
)
hhhhhh
+
(
(
1
-
β
n
)
I
-
α
n
A
)
(
W
n
ρ
n
-
p
)
h
h
h
h
h
h
+
α
n
(
γ
f
(
p
)
-
A
p
)
∥
≤
α
n
γ
h
∥
x
n
-
p
∥
+
β
n
∥
x
n
-
p
∥
+
(
(
1
-
β
n
)
I
-
α
n
A
)
∥
x
n
-
p
∥
+
α
n
∥
γ
f
(
p
)
-
A
p
∥
≤
α
n
γ
h
∥
x
n
-
p
∥
+
β
n
∥
x
n
-
p
∥
+
(
1
-
β
n
)
∥
x
n
-
p
∥
-
α
n
γ
¯
∥
x
n
-
p
∥
+
α
n
∥
γ
f
(
p
)
-
A
p
∥
=
(
1
-
α
n
(
γ
¯
-
γ
h
)
)
∥
x
n
-
p
∥
+
α
n
∥
γ
f
(
p
)
-
A
p
∥
≤
max
⁡
{
∥
x
n
-
p
∥
,
1
γ
¯
-
γ
h
∥
γ
f
(
p
)
-
A
p
∥
}
⋮
≤
max
⁡
{
∥
x
1
-
p
∥
,
1
γ
¯
-
γ
h
∥
γ
f
(
p
)
-
A
p
∥
}
.
This implies that {
x
n
} is bounded sequence in H. Therefore {
y
n
}, {
ρ
n
}, {
γ
f
(
x
n
)
}, {
W
n
ρ
n
} are all bounded.Step 2.
Next, we prove thatlim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0 and lim
⁡
n
→
∞
∥
ρ
n
+
1
-
ρ
n
∥
=
0.
In fact, let us define a sequence{
z
n
} by x
n
+
1
=
(
1
-
β
n
)
z
n
+
β
n
x
n, for all n
≥
1; then, we have
(16)
z
n
+
1
-
z
n
=
x
n
+
2
-
β
n
+
1
x
n
+
1
1
-
β
n
+
1
-
x
n
+
1
-
β
n
x
n
1
-
β
n
=
α
n
+
1
γ
f
(
x
n
+
1
)
+
(
(
1
-
β
n
+
1
)
I
-
α
n
+
1
A
)
W
n
+
1
ρ
n
+
1
1
-
β
n
+
1
-
α
n
γ
f
(
x
n
)
+
(
(
1
-
β
n
)
I
-
α
n
A
)
W
n
ρ
n
1
-
β
n
=
α
n
+
1
1
-
β
n
+
1
[
γ
f
(
x
n
+
1
)
-
A
W
n
+
1
ρ
n
+
1
]
-
α
n
1
-
β
n
[
γ
f
(
x
n
)
-
A
W
n
ρ
n
]
+
W
n
+
1
ρ
n
+
1
-
W
n
ρ
n
=
α
n
+
1
1
-
β
n
+
1
[
γ
f
(
x
n
+
1
)
-
A
W
n
+
1
ρ
n
+
1
]
-
α
n
1
-
β
n
[
γ
f
(
x
n
)
-
A
W
n
ρ
n
]
+
(
W
n
+
1
ρ
n
+
1
-
W
n
+
1
ρ
n
)
+
(
W
n
+
1
ρ
n
-
W
n
ρ
n
)
,
∥
z
n
+
1
-
z
n
∥
-
∥
x
n
+
1
-
x
n
∥
≤
α
n
+
1
1
-
β
n
+
1
(
∥
γ
f
(
x
n
+
1
)
∥
+
∥
A
W
n
+
1
ρ
n
+
1
∥
)
+
α
n
1
-
β
n
(
∥
γ
f
(
x
n
)
∥
+
∥
A
W
n
ρ
n
∥
)
+
∥
ρ
n
+
1
-
ρ
n
∥
+
∥
W
n
+
1
ρ
n
-
W
n
ρ
n
∥
-
∥
x
n
+
1
-
x
n
∥
.
Because ρ
n
=
∑
m
=
1
r
‍
η
n
m
P
C
(
I
-
μ
m
B
m
)
y
n, we have
(17)
∥
ρ
n
+
1
-
ρ
n
∥
=
∥
∑
m
=
1
r
‍
η
n
+
1
m
P
C
(
I
-
μ
m
B
m
)
y
n
+
1
-
∑
m
=
1
r
‍
η
n
m
P
C
(
I
-
μ
m
B
m
)
y
n
∥
=
∥
∑
m
=
1
r
‍
η
n
+
1
m
P
C
(
I
-
μ
m
B
m
)
y
n
+
1
-
∑
m
=
1
r
‍
η
n
+
1
m
P
C
(
I
-
μ
m
B
m
)
y
n
h
h
h
h
+
∑
m
=
1
r
‍
η
n
+
1
m
P
C
(
I
-
μ
m
B
m
)
y
n
-
∑
m
=
1
r
‍
η
n
m
P
C
(
I
-
μ
m
B
m
)
y
n
∥
≤
∥
y
n
+
1
-
y
n
∥
+
M
∑
m
=
1
r
‍
|
η
n
+
1
m
-
η
n
m
|
,
where M
=
max
⁡
{
sup
⁡
{
∥
P
C
(
I
-
μ
m
B
m
)
y
n
∥
:
n
≥
1
}
:
1
≤
m
≤
r
}, so
(18)
∥
z
n
+
1
-
z
n
∥
-
∥
x
n
+
1
-
x
n
∥
≤
α
n
+
1
1
-
β
n
+
1
(
∥
γ
f
(
x
n
+
1
)
∥
+
∥
A
W
n
+
1
ρ
n
+
1
∥
)
+
α
n
1
-
β
n
(
∥
γ
f
(
x
n
)
∥
+
∥
A
W
n
ρ
n
∥
)
+
∥
y
n
+
1
-
y
n
∥
+
M
∑
m
=
1
r
|
η
n
+
1
m
-
η
n
m
|
+
∥
W
n
+
1
ρ
n
-
W
n
ρ
n
∥
-
∥
x
n
+
1
-
x
n
∥
.
Observing y
n
=
V
r
n
x
n, y
n
+
1
=
V
r
n
+
1
x
n
+
1, we have
(19)
ϕ
(
y
n
,
η
)
+
1
r
n
〈
η
-
y
n
,
y
n
-
x
n
〉
≥
0
,
∀
η
∈
C
,
(20)
ϕ
(
y
n
+
1
,
η
)
+
1
r
n
+
1
〈
η
-
y
n
+
1
,
y
n
+
1
-
x
n
+
1
〉
≥
0
,
∀
η
∈
C
.
Putting η
=
y
n
+
1 in (19), η
=
y
n in (20), adding up these two inequalities, and using condition (
A
2
) to simplify, we have
(21)
〈
r
n
r
n
+
1
y
n
+
1
-
y
n
,
y
n
-
y
n
+
1
+
y
n
+
1
hh
-
x
n
-
r
n
r
n
+
1
(
y
n
+
1
-
x
n
+
1
)
〉
≥
0
.
By condition (
C
4
), without loss of generality, we can assume that there exists a real number m such that r
n
>
m
>
0, so
(22)
∥
y
n
+
1
-
y
n
∥
2
≤
∥
y
n
+
1
-
y
n
∥
×
(
∥
x
n
+
1
-
x
n
∥
+
|
1
-
r
n
r
n
+
1
|
(
y
n
+
1
-
x
n
+
1
)
)
∥
y
n
+
1
-
y
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
M
1
m
|
r
n
+
1
-
r
n
|
,
where M
1
=
sup
⁡
n
≥
1
∥
y
n
-
x
n
∥.
SinceS
i
,
U
n
,
i are nonexpansive, so ∥
W
n
+
1
ρ
n
-
W
n
ρ
n
∥
≤
∏
i
=
1
n
+
1
‍
λ
i
∥
S
n
+
1
ρ
n
-
ρ
n
∥
≤
L
(
∏
i
=
1
n
+
1
‍
λ
i
), where L
=
sup
⁡
n
≥
1
∥
S
n
+
1
ρ
n
-
ρ
n
∥. Consider
(23)
∥
z
n
+
1
-
z
n
∥
-
∥
x
n
+
1
-
x
n
∥
≤
α
n
+
1
1
-
β
n
+
1
(
∥
γ
f
(
x
n
+
1
)
∥
+
∥
A
W
n
+
1
ρ
n
+
1
∥
)
+
α
n
1
-
β
n
(
∥
γ
f
(
x
n
)
∥
+
∥
A
W
n
ρ
n
∥
)
+
M
1
m
|
r
n
+
1
-
r
n
|
+
L
(
∏
i
=
1
n
+
1
λ
i
)
.
Using 0
<
λ
i
≤
b
<
1
(
i
≥
1
) and the conditions (
C
1
)
–
(
C
3
), limsup
⁡
n
→
∞
(
∥
z
n
+
1
-
z
n
∥
-
∥
x
n
+
1
-
x
n
∥
)
≤
0. By Lemma 5, we conclude that lim
⁡
n
→
∞
∥
z
n
-
x
n
∥
=
0, lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
lim
⁡
n
→
∞
(
1
-
β
n
)
∥
z
n
-
x
n
∥
=
0. Consider
(24)
∥
ρ
n
+
1
-
ρ
n
∥
≤
∥
y
n
+
1
-
y
n
∥
+
M
∑
m
=
1
r
|
η
n
+
1
m
-
η
n
m
|
≤
∥
x
n
+
1
-
x
n
∥
M
1
m
|
r
n
+
1
-
r
n
|
+
M
∑
m
=
1
r
|
η
n
+
1
m
-
η
n
m
|
.
So lim
⁡
n
→
∞
∥
ρ
n
+
1
-
ρ
n
∥
=
0.Step 3.
Consider(25)
∥
x
n
-
W
n
ρ
n
∥
=
∥
x
n
-
x
n
+
1
+
x
n
+
1
-
W
n
ρ
n
∥
≤
∥
x
n
-
x
n
+
1
∥
+
∥
α
n
γ
f
(
x
n
)
+
β
n
x
n
h
h
h
h
+
(
(
1
-
β
n
)
I
-
α
n
A
)
W
n
ρ
n
-
W
n
ρ
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
α
n
∥
γ
f
(
x
n
)
-
A
W
n
ρ
n
∥
+
β
n
∥
x
n
-
W
n
ρ
n
∥
∥
x
n
-
W
n
ρ
n
∥
≤
1
1
-
β
n
∥
x
n
+
1
-
x
n
∥
+
α
n
1
-
β
n
∥
γ
f
(
x
n
)
-
A
W
n
ρ
n
∥
.
So we have lim
⁡
n
→
∞
∥
x
n
-
W
n
ρ
n
∥
=
0.Step 4.
For any givenp
∈
ℱ,
(26)
∥
y
n
-
p
∥
2
=
∥
V
r
n
x
n
-
p
∥
2
=
〈
V
r
n
x
n
-
V
r
n
p
,
x
n
-
p
〉
=
〈
y
n
-
p
,
x
n
-
p
〉
=
1
2
(
∥
y
n
-
p
∥
2
+
∥
x
n
-
p
∥
2
-
∥
x
n
-
y
n
∥
2
)
,
(27)
∥
x
n
+
1
-
p
∥
2
=
∥
α
n
γ
f
(
x
n
)
+
β
n
x
n
h
h
h
h
h
h
h
+
(
(
1
-
β
n
)
I
-
α
n
A
)
W
n
ρ
n
-
p
∥
2
=
∥
[
(
I
-
α
n
A
)
(
W
n
ρ
n
-
p
)
+
β
n
(
x
n
-
W
n
ρ
n
)
]
h
h
h
h
h
h
h
+
α
n
(
γ
f
(
x
n
)
-
A
p
)
∥
≤
∥
(
I
-
α
n
A
)
(
W
n
ρ
n
-
p
)
+
β
n
(
x
n
-
W
n
ρ
n
)
∥
2
+
2
α
n
〈
γ
f
(
x
n
)
-
A
p
,
x
n
+
1
-
p
〉
≤
[
(
1
-
α
n
γ
¯
)
∥
ρ
n
-
p
∥
+
β
n
∥
x
n
-
W
n
ρ
n
∥
]
2
+
2
α
n
∥
γ
f
(
x
n
)
-
A
p
∥
·
∥
x
n
+
1
-
p
∥
=
(
1
-
α
n
γ
¯
)
2
∥
ρ
n
-
p
∥
2
+
β
n
2
∥
x
n
-
W
n
ρ
n
∥
2
+
2
(
1
-
α
n
γ
¯
)
β
n
∥
ρ
n
-
p
∥
·
∥
x
n
-
W
n
ρ
n
∥
+
2
α
n
∥
γ
f
(
x
n
)
-
A
p
∥
·
∥
x
n
+
1
-
p
∥
≤
(
1
-
α
n
γ
¯
)
2
[
∥
x
n
-
p
∥
2
-
∥
x
n
-
y
n
∥
2
]
+
β
n
2
∥
x
n
-
W
n
ρ
n
∥
2
+
2
(
1
-
α
n
γ
¯
)
β
n
∥
ρ
n
-
p
∥
·
∥
x
n
-
W
n
ρ
n
∥
+
2
α
n
∥
γ
f
(
x
n
)
-
A
p
∥
·
∥
x
n
+
1
-
p
∥
.
Simplifying it, we have
(28)
(
1
-
α
n
γ
¯
)
2
∥
x
n
-
y
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
x
n
+
1
-
p
∥
2
+
α
n
2
γ
¯
2
∥
x
n
-
p
∥
2
+
β
n
2
∥
x
n
-
W
n
ρ
n
∥
2
+
2
(
1
-
α
n
γ
¯
)
β
n
∥
ρ
n
-
p
∥
·
∥
x
n
-
W
n
ρ
n
∥
+
2
α
n
∥
γ
f
(
x
n
)
-
A
p
∥
·
∥
x
n
+
1
-
p
∥
≤
(
∥
x
n
-
p
∥
+
∥
x
n
+
1
-
p
∥
)
∥
x
n
-
x
n
+
1
∥
+
α
n
2
γ
¯
2
∥
x
n
-
p
∥
2
+
β
n
2
∥
x
n
-
W
n
ρ
n
∥
2
+
2
(
1
-
α
n
γ
¯
)
β
n
∥
ρ
n
-
p
∥
·
∥
x
n
-
W
n
ρ
n
∥
+
2
α
n
∥
γ
f
(
x
n
)
-
A
p
∥
·
∥
x
n
+
1
-
p
∥
.
So lim
⁡
n
→
∞
∥
x
n
-
y
n
∥
=
0.Step 5.
Consider(29)
∥
ρ
n
-
p
∥
2
=
∥
∑
m
=
1
r
η
n
m
P
C
(
I
-
μ
m
B
m
)
y
n
-
p
∥
2
≤
∥
(
y
n
-
p
)
-
μ
m
(
B
m
y
n
-
B
m
p
)
∥
2
=
∥
y
n
-
p
∥
2
+
μ
m
2
∥
B
m
y
n
-
B
m
p
∥
2
-
2
μ
m
〈
y
n
-
p
,
B
m
y
n
-
B
m
p
〉
≤
∥
y
n
-
p
∥
2
+
μ
m
2
∥
B
m
y
n
-
B
m
p
∥
2
-
2
μ
m
l
m
∥
B
m
y
n
-
B
m
p
∥
2
≤
∥
x
n
-
p
∥
2
+
μ
m
(
μ
m
-
2
l
m
)
∥
B
m
y
n
-
B
m
p
∥
2
∥
x
n
+
1
-
p
∥
2
≤
(
1
-
α
n
γ
¯
)
2
∥
x
n
-
p
∥
2
+
(
1
-
α
n
γ
¯
)
2
μ
m
(
μ
m
-
2
l
m
)
∥
B
m
y
n
-
B
m
p
∥
2
+
2
(
1
-
α
n
γ
¯
)
β
n
∥
ρ
n
-
p
∥
·
∥
x
n
-
W
n
ρ
n
∥
+
2
α
n
∥
γ
f
(
x
n
)
-
A
p
∥
·
∥
x
n
+
1
-
p
∥
(
2
l
m
-
μ
m
)
(
1
-
α
n
γ
¯
)
2
μ
m
∥
B
m
y
n
-
B
m
p
∥
2
≤
(
∥
x
n
-
p
∥
+
∥
x
n
+
1
-
p
∥
)
∥
x
n
-
x
n
+
1
∥
+
α
n
2
γ
¯
2
∥
x
n
-
p
∥
2
+
2
(
1
-
α
n
γ
¯
)
β
n
∥
ρ
n
-
p
∥
·
∥
x
n
-
W
n
ρ
n
∥
+
2
α
n
∥
γ
f
(
x
n
)
-
A
p
∥
·
∥
x
n
+
1
-
p
∥
.
So lim
⁡
n
→
∞
∥
B
m
y
n
-
B
m
p
∥
=
0.Step 6.
Consider(30)
∥
ρ
n
-
p
∥
2
=
∥
∑
m
=
1
r
η
n
m
P
C
(
I
-
μ
m
B
m
)
y
n
-
p
∥
2
≤
〈
(
y
n
-
μ
m
B
m
y
n
)
-
(
p
-
μ
m
B
m
p
)
,
ρ
n
-
p
〉
=
1
2
{
∥
I
-
μ
m
B
m
y
n
-
I
-
μ
m
B
m
p
∥
2
+
∥
ρ
n
-
p
∥
2
h
h
h
h
h
-
∥
(
y
n
-
ρ
n
)
-
μ
m
(
B
m
y
n
-
B
m
p
)
∥
2
}
≤
1
2
{
∥
y
n
-
p
∥
2
+
∥
ρ
n
-
p
∥
2
-
∥
y
n
-
ρ
n
∥
2
hhhh
-
μ
m
2
∥
B
m
y
n
-
B
m
p
∥
2
h
h
h
h
+
2
μ
m
〈
y
n
-
ρ
n
,
B
m
y
n
-
B
m
p
〉
∥
y
n
-
p
∥
2
}
∥
ρ
n
-
p
∥
2
≤
∥
y
n
-
p
∥
2
-
∥
y
n
-
ρ
n
∥
2
-
μ
m
2
∥
B
m
y
n
-
B
m
p
∥
2
+
2
μ
m
〈
y
n
-
ρ
n
,
B
m
y
n
-
B
m
p
〉
∥
x
n
+
1
-
p
∥
2
≤
(
1
-
α
n
γ
¯
)
2
∥
ρ
n
-
p
∥
2
+
β
n
2
∥
x
n
-
W
n
ρ
n
∥
2
+
2
(
1
-
α
n
γ
¯
)
β
n
∥
ρ
n
-
p
∥
·
∥
x
n
-
W
n
ρ
n
∥
+
2
α
n
∥
γ
f
(
x
n
)
-
A
p
∥
·
∥
x
n
+
1
-
p
∥
≤
(
1
-
α
n
γ
¯
)
2
[
∥
y
n
-
p
∥
2
-
∥
y
n
-
ρ
n
∥
2
hhhhhhhhhhhhhh
-
μ
m
2
∥
B
m
y
n
-
B
m
p
∥
2
h
h
h
h
h
h
h
h
h
h
h
h
h
h
+
2
μ
m
〈
y
n
-
ρ
n
,
B
m
y
n
-
B
m
p
〉
∥
y
n
-
p
∥
2
]
+
β
n
2
∥
x
n
-
W
n
ρ
n
∥
2
+
2
(
1
-
α
n
γ
¯
)
β
n
∥
ρ
n
-
p
∥
·
∥
x
n
-
W
n
ρ
n
∥
+
2
α
n
∥
γ
f
(
x
n
)
-
A
p
∥
·
∥
x
n
+
1
-
p
∥
≤
∥
x
n
-
p
∥
2
+
α
n
γ
¯
2
∥
x
n
-
p
∥
2
+
2
μ
m
(
1
-
α
n
γ
¯
)
2
∥
y
n
-
ρ
n
∥
·
∥
B
m
y
n
-
B
m
p
∥
-
(
1
-
α
n
γ
¯
)
2
∥
y
n
-
ρ
n
∥
2
+
β
n
2
∥
x
n
-
W
n
ρ
n
∥
2
+
2
(
1
-
α
n
γ
¯
)
β
n
∥
ρ
n
-
p
∥
·
∥
x
n
-
W
n
ρ
n
∥
+
2
α
n
∥
γ
f
(
x
n
)
-
A
p
∥
·
∥
x
n
+
1
-
p
∥
(
1
-
α
n
γ
¯
)
2
∥
y
n
-
ρ
n
∥
2
≤
(
∥
x
n
-
p
∥
+
∥
x
n
+
1
-
p
∥
)
∥
x
n
-
x
n
+
1
∥
+
2
μ
m
(
1
-
α
n
γ
¯
)
2
∥
y
n
-
ρ
n
∥
·
∥
B
m
y
n
-
B
m
p
∥
+
β
n
2
∥
x
n
-
W
n
ρ
n
∥
2
+
2
(
1
-
α
n
γ
¯
)
β
n
∥
ρ
n
-
p
∥
·
∥
x
n
-
W
n
ρ
n
∥
+
2
α
n
∥
γ
f
(
x
n
)
-
A
p
∥
·
∥
x
n
+
1
-
p
∥
.
So lim
⁡
n
→
∞
∥
y
n
-
ρ
n
∥
=
0.Step 7.
Consider(31)
∥
y
n
-
W
n
y
n
∥
≤
∥
W
n
y
n
-
W
n
ρ
n
∥
+
∥
W
n
ρ
n
-
x
n
∥
+
∥
x
n
-
y
n
∥
≤
∥
y
n
-
ρ
n
∥
+
∥
W
n
ρ
n
-
x
n
∥
+
∥
x
n
-
y
n
∥
.
So lim
⁡
n
→
∞
∥
y
n
-
W
n
y
n
∥
=
0.Step 8.
Next, we prove thatlimsup
⁡
n
→
∞
〈
γ
f
(
q
)
-
A
q
,
x
n
-
q
〉
≤
0, where q
=
P
ℱ
(
γ
f
+
(
I
-
A
q
)
).
To show it, we can choose a subsequence{
x
n
i
} of {
x
n
} such that
(32)
limsup
⁡
n
→
∞
⁡
〈
γ
f
(
q
)
-
A
q
,
x
n
-
q
〉
=
lim
⁡
i
→
∞
⁡
〈
γ
f
(
q
)
-
A
q
,
x
n
i
-
q
〉
.
Since {
y
n
i
} is bounded, so there exists a subsequence {
y
n
i
j
} of {
y
n
i
} which converges weakly to ω. Without loss of generality, we can assume that y
n
i
⇀
ω; then, ϕ
(
y
n
,
η
)
+
(
1
/
r
n
)
〈
η
-
y
n
,
y
n
-
x
n
〉
≥
0, for all η
∈
C,〈
η
-
y
n
,
(
y
n
-
x
n
)
/
r
n
〉
≥
ϕ
(
η
,
y
n
), 〈
η
-
y
n
i
,
(
y
n
i
-
x
n
i
)
/
r
n
i
〉
≥
ϕ
(
η
,
y
n
i
). So ϕ
(
η
,
ω
)
≤
0.
For anyt with 0
≤
t
≤
1 and η
∈
C, let η
t
=
t
η
+
(
1
-
t
)
ω. Since η
∈
C and ϕ
(
η
t
,
ω
)
≤
0, from conditions (
A
1
) and (
A
4
), we have
(33)
0
=
ϕ
(
η
t
,
η
t
)
≤
t
ϕ
(
η
t
,
η
)
+
(
1
-
t
)
ϕ
(
η
t
,
ω
)
≤
t
ϕ
(
η
t
,
η
)
.
This implies that ϕ
(
η
t
,
η
)
≥
0. From condition (
A
3
), we have ϕ
(
ω
,
η
)
≥
0. So ω
∈
EP
⁡
(
ϕ
).
Define a mappingJ
:
C
→
C by J
x
=
∑
m
=
1
r
‍
η
n
m
P
C
(
I
-
μ
m
B
m
)
x, for all x
∈
C, where lim
⁡
n
→
∞
η
n
m
=
η
m. From Lemma 7 we see that J is nonexpansive such that
(34)
F
(
J
)
=
⋂
m
=
1
r
F
(
P
C
(
I
-
μ
m
B
m
)
)
=
⋂
m
=
1
r
VI
⁡
(
C
,
B
m
)
.
∥
y
n
-
J
y
n
∥
≤
∥
y
n
-
ρ
n
∥
+
∥
ρ
n
-
J
y
n
∥
=
∥
y
n
-
ρ
n
∥
+
∥
∑
m
=
1
r
η
n
m
P
C
(
I
-
μ
m
B
m
)
y
n
h
h
h
h
h
h
h
h
h
h
h
h
h
-
∑
m
=
1
r
η
m
P
C
(
I
-
μ
m
B
m
)
y
n
∥
≤
∥
y
n
-
ρ
n
∥
+
M
∑
m
=
1
r
|
η
n
m
-
η
m
|
⟶
0
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
(
n
⟶
∞
)
.
Since every nonexpansive mapping is strictly pseudocontractive, so ω
∈
F
(
J
)
=
⋂
m
=
1
r
‍
VI
⁡
(
C
,
B
m
).
Now we prove thatω
∈
F
(
ω
), and if not, we have ω
≠
W
(
ω
). From Opial’s condition, we have
(35)
liminf
⁡
i
→
∞
⁡
∥
y
n
i
-
ω
∥
<
liminf
⁡
i
→
∞
⁡
∥
y
n
i
-
W
ω
∥
≤
liminf
⁡
i
→
∞
⁡
(
∥
y
n
i
-
W
y
n
i
∥
+
∥
W
y
n
i
-
W
ω
∥
)
≤
liminf
⁡
i
→
∞
⁡
(
∥
y
n
i
-
W
y
n
i
∥
+
∥
y
n
i
-
ω
∥
)
,
lim
⁡
i
→
∞
⁡
∥
W
y
n
i
-
y
n
i
∥
≤
lim
⁡
i
→
∞
(
∥
W
y
n
i
-
W
n
i
y
n
i
∥
+
∥
W
n
i
y
n
i
-
y
n
i
∥
)
≤
lim
⁡
i
→
∞
{
sup
⁡
x
∈
C
∥
W
ω
-
W
n
i
ω
∥
}
+
lim
⁡
i
→
∞
∥
W
n
i
y
n
i
-
y
n
i
∥
=
0
.
Therefore, liminf
⁡
i
→
∞
∥
y
n
i
-
ω
∥
<
liminf
⁡
i
→
∞
∥
y
n
i
-
ω
∥, so ω
∈
F
(
ω
).Step 9.
Finally, we prove thatx
n
→
q
=
P
ℱ
(
γ
f
+
(
I
-
A
)
q
). Since q
=
P
ℱ
(
γ
f
+
(
I
-
A
)
q
), so
(36)
limsup
⁡
n
→
∞
⁡
〈
γ
f
(
q
)
-
A
q
,
x
n
-
q
〉
=
lim
⁡
i
→
∞
⁡
〈
γ
f
(
q
)
-
A
q
,
x
n
i
-
q
〉
=
lim
⁡
i
→
∞
⁡
〈
γ
f
(
q
)
-
A
q
,
x
n
i
-
y
n
i
+
y
n
i
-
q
〉
=
〈
γ
f
(
q
)
-
A
q
,
ω
-
q
〉
≤
0
,
∥
x
n
+
1
-
q
∥
2
=
∥
α
n
(
γ
f
(
x
n
)
-
A
q
)
h
h
h
h
h
+
[
(
1
-
β
n
I
-
α
n
A
)
(
W
n
ρ
n
-
q
)
+
β
n
(
x
n
-
q
)
]
∥
2
≤
∥
(
1
-
β
n
I
-
α
n
A
)
(
W
n
ρ
n
-
q
)
+
β
n
(
x
n
-
q
)
]
∥
2
+
2
α
n
γ
〈
f
(
x
n
)
-
f
(
q
)
,
x
n
+
1
-
q
〉
+
2
α
n
〈
γ
f
(
q
)
-
A
q
,
x
n
+
1
-
q
〉
≤
[
(
1
-
β
n
-
α
n
γ
¯
)
∥
ρ
n
-
p
∥
+
β
n
∥
x
n
-
q
∥
]
2
+
2
α
n
γ
h
∥
x
n
-
q
∥
·
∥
x
n
+
1
-
q
∥
+
2
α
n
〈
γ
f
(
q
)
-
A
q
,
x
n
+
1
-
q
〉
≤
(
1
-
α
n
γ
¯
)
2
∥
x
n
-
q
∥
2
+
α
n
γ
h
(
∥
x
n
-
q
∥
2
+
∥
x
n
+
1
-
q
∥
2
)
+
2
α
n
〈
γ
f
(
q
)
-
A
q
,
x
n
+
1
-
q
〉
.
This implies that
(37)
∥
x
n
+
1
-
q
∥
2
≤
[
1
-
2
(
γ
¯
-
γ
h
)
α
n
1
-
α
n
γ
h
]
∥
x
n
-
q
∥
2
+
2
(
γ
¯
-
γ
h
)
α
n
1
-
α
n
γ
h
{
α
n
γ
2
2
(
γ
¯
-
γ
h
)
∥
x
n
-
q
∥
2
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
+
1
γ
¯
-
γ
h
〈
γ
f
(
q
)
-
A
q
,
x
n
+
1
-
q
〉
α
n
γ
2
2
(
γ
¯
-
γ
h
)
}
.
From Lemma 6, {
x
n
} converges strongly to q.
TakingF
(
x
,
y
)
=
0
,
D
=
0 for all x
,
y
∈
H, r
n
=
1
,
m
=
1 in Theorem 10, then Theorem 10 is reduced to the following.Corollary 11.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let B
:
C
→
H be a λ-inverse strongly monotone mapping and D
:
C
→
H a α-inverse strongly monotone mapping. Let {
T
n
}
n
=
1
∞
:
C
→
C be a family of infinite k
n-strict pseudocontractive mappings with 0
≤
k
n
<
1 and let {
λ
n
}
n
=
1
∞ be a sequence of positive numbers in [
0
,
b
] for some b
∈
(
0,1
). {
S
n
}
n
=
1
∞
:
C
→
C is a family of infinitely nonexpansive mappings such that ℱ
=
F
(
W
)
⋂
‍
VI
⁡
(
C
,
B
)
≠
∅, where F
(
W
)
:
=
⋂
n
=
1
∞
‍
F
(
S
n
). Let A be a strongly positive linear bounded operator with coefficient γ
¯
>
0 and let f
:
H
→
H be a contraction with contraction constant h
(
0
<
h
<
1
) and 0
<
γ
<
(
γ
¯
/
h
). Let {
x
n
} and {
y
n
} be generated in sequence by x
1
∈
H and
(38)
y
n
=
P
C
(
I
-
λ
B
)
x
n
,
x
n
+
1
=
α
n
γ
f
(
x
n
)
+
β
n
x
n
+
(
(
1
-
β
n
)
I
-
α
n
A
)
W
n
y
n
,
where λ
∈
(
0,2
α
]
,
{
α
n
}
,
{
β
n
}
⊂
[
0,1
]. If the following conditions are satisfied: (
C
1
)
lim
⁡
n
→
∞
α
n
=
0, ∑
n
=
1
∞
‍
α
n
=
∞;
(
C
2
)
0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
β
n
<
1,
then{
x
n
} converges strongly to q
∈
ℱ, where q
=
P
ℱ
(
γ
f
+
(
I
-
A
)
)
q.



---

*Source: 232541-2014-04-07.xml*