# The Hybrid Projection Methods for Pseudocontractive, Nonexpansive Semigroup, and Monotone Mapping

Authors: Phayap Katchang; Somyot Plubtieng
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We modify the three-step iterative schemes to prove the strong convergence theorems by using the hybrid projection methods for finding a common element of the set of solutions of fixed points for a pseudocontractive mapping and a nonexpansive semigroup mapping and the set of solutions of a variational inequality problem for a monotone mapping in a Hilbert space under some appropriate control conditions. Our theorems extend and unify most of the results that have been proved for this class of nonlinear mappings.


---

## Body

## 1. Introduction

LetC be a nonempty closed convex subset of a real Hilbert space H with inner product 〈
·
,
·
〉 and norm ∥
·
∥. Recall that a mapping T
:
C
→
C is said to be a k-strict pseudocontraction if there exists 0
≤
k
<
1 such that
(1)
∥
T
x
-
T
y
∥
2
≤
∥
x
-
y
∥
2
+
k
∥
(
I
-
T
)
x
-
(
I
-
T
)
y
∥
2
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
∀
x
,
y
∈
C
,
where I denotes the identity operator on C. When k
=
0, T
:
C
→
C is said to benonexpansive if
(2)
∥
T
x
-
T
y
∥
≤
∥
x
-
y
∥
,
∀
x
,
y
∈
C
.
And when k
=
1, T
:
C
→
C is said to bepseudocontraction if
(3)
∥
T
x
-
T
y
∥
2
≤
∥
x
-
y
∥
2
+
∥
(
I
-
T
)
x
-
(
I
-
T
)
y
∥
2
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
∀
x
,
y
∈
C
.
Clearly, the class of k-strict pseudocontraction falls into the one between classes of nonexpansive mappings and pseudocontraction mapping. We denote the set of fixed points of T by F
(
T
).A mappingA of C into H is calledmonotone if
(4)
〈
A
u
-
A
v
,
u
-
v
〉
≥
0
,
∀
u
,
v
∈
C
.
The classical variational inequality is used for finding u
∈
C such that
(5)
〈
A
u
,
v
-
u
〉
≥
0
,
∀
v
∈
C
.
The set of solutions of variational inequality problems is denoted by VI
⁡
(
C
,
A
). See, for example, [1–7] and the references therein.In 1953, Mann [8] introduced the iteration as follows:
(6)
x
n
+
1
=
α
n
x
n
+
(
1
-
α
n
)
S
x
n
,
where the initial guess element x
0
∈
C is arbitrary and {
α
n
} is a real sequence in [
0,1
]. The Mann iteration has been extensively investigated for nonexpansive mappings. In an infinite-dimensional Hilbert space, the Mann iteration can conclude only weak convergence [9]. Attempts to modify the Mann iteration method (6) so that strong convergence is guaranteed have recently been made. In 1974, The Ishikawa’s iteration process is defined by Ishikawa [10] as the following:
(7)
y
n
=
β
n
x
n
+
(
1
-
β
n
)
T
x
n
,
x
n
+
1
=
α
n
x
n
+
(
1
-
α
n
)
T
y
n
,
n
≥
0
,
where the initial guess x
0 is taken in C arbitrarily and the sequences α
n
,
β
n
∈
[
0,1
]. This is called Ishikawa Iteration. This has been studied in strong convergence theorem for lipschitzian pseudocontractive mapping in Hilbert spaces. Several years later, inspired by the idea of one and two step iterative scheme, Noor [11, 12] introduced a three-step iterative scheme and studied the approximate solution of variational inclusion in Hilbert spaces by using the techniques of updating the solution and the auxiliary principle. It has been shown in [13] by Goebel and Kirk that the three-step iterative scheme gives better numerical results than the two-step and one-step approximate iterations.A familyS
=
{
S
(
s
)
:
0
≤
s
<
∞
} of mappings of C into itself is called anonexpansive semigroup on C if it satisfies the following conditions:(i)
S
(
0
)
x
=
x for all x
∈
C;
(ii)
S
(
s
+
t
)
=
S
(
s
)
S
(
t
) for all s
,
t
≥
0;
(iii)
∥
S
(
s
)
x
-
S
(
s
)
y
∥
≤
∥
x
-
y
∥ for all x
,
y
∈
C and s
≥
0;
(iv)
for allx
∈
C
,
s
↦
S
(
s
)
x is continuous.We denote byF
(
S
) the set of all commonfixed points of
S; that is, F
(
S
)
=
{
x
∈
C
:
S
(
s
)
x
=
x
,
0
≤
s
<
∞
}. It is known that F
(
S
) is closed and convex. In the sense of nonexpansive semigroup mapping, we also see [14–24].In 2003, Nakajo and Takahashi [25] proposed the following modification of Mann iteration method for a nonexpansive mapping T from C into itself in a Hilbert space
(8)
x
0
∈
C
is  arbitrary
,
y
n
=
α
n
x
n
+
(
1
-
α
n
)
T
x
n
,
C
n
=
{
z
∈
C
:
∥
y
n
-
z
∥
≤
∥
x
n
-
z
∥
}
,
Q
n
=
{
z
∈
C
:
〈
x
n
-
z
,
x
0
-
x
n
〉
≥
0
}
,
x
n
+
1
=
P
C
n
∩
Q
n
x
0
,
n
≥
0
,
where P
K denotes the metric projection from a Hilbert space H onto a close convex subset K of H and proves that the sequence {
x
n
} converges strongly to P
F
(
T
)
x
0. A projection onto intersection of two halfspaces is computed by solving a linear system of two equations with two unknowns. In 2008, Takahashi et al. [26] proved the following strong convergence theorem by the newhybrid method in a Hilbert space. They assume x
0
∈
H, C
1
=
C, u
1
=
P
C
1
x
0, and defined the sequence by (8) where 0
≤
α
n
<
a
<
1
(9)
y
n
=
α
n
u
n
+
(
1
-
α
n
)
T
u
n
,
C
n
+
1
=
{
z
∈
C
n
:
∥
y
n
-
z
∥
≤
∥
u
n
-
z
∥
}
,
u
n
+
1
=
P
C
n
+
1
x
0
,
n
≥
0
,where 0
≤
α
n
<
a
<
1. Then {
u
n
} converges strongly to z
0
=
P
F
(
T
)
x
0.Recently, Zegeye and Shahzad [27] defined the mappings as follows:
(10)
F
r
n
(
x
)
=
{
z
∈
C
:
〈
y
-
z
,
A
z
〉
1
r
n
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
+
1
r
n
〈
y
-
z
,
z
-
x
〉
≥
0
,
∀
y
∈
C
}
,
(11)
T
r
n
(
x
)
=
{
z
∈
C
:
〈
y
-
z
,
T
z
〉
1
r
n
i
i
i
i
i
i
i
i
i
i
i
i
i
-
1
r
n
〈
y
-
z
,
(
1
+
r
n
)
z
-
x
〉
≤
0
,
∀
y
∈
C
}
,
for all x
∈
H and r
n
∈
(
0
,
∞
), where T
:
C
→
C is a continuous pseudocontractive mapping and A
:
C
→
H is a continuous monotone mapping. In the following year, Tang [28] introduced a viscosity iterative process, which converges strongly to a common element of the set of fixed points of a pseudocontractive mapping and the set of solutions of a monotone mapping as the following:
(12)
y
n
=
λ
n
x
n
+
(
1
-
λ
n
)
F
r
n
x
n
,
x
n
+
1
=
α
n
f
(
x
n
)
+
β
n
x
n
+
γ
n
T
r
n
y
n
,
where F
r
n and T
r
n are defined by (10) and (11), respectively.In this paper, we modify the three-step iterative schemes to prove the strong convergence theorems by using the hybrid projection methods for finding a common element of the set of solutions of fixed points for a pseudocontractive mapping and a nonexpansive semigroup mapping and the set of solutions of a variational inequality problem for a monotone mapping in a Hilbert space under some appropriate control conditions. The results that are presented in this paper extend and improve the corresponding ones announced by Nakajo and Takahashi [25], Takahashi et al. [26], Zegeye and Shahzad [27], Tang [28], and many authors.


## 2. Preliminaries

LetH be a real Hilbert space with norm ∥
·
∥ and inner product 〈
·
,
·
〉 and let C be a closed convex subset of H. Then
(13)
∥
λ
x
+
(
1
-
λ
)
y
∥
2
=
λ
∥
x
∥
2
+
(
1
-
λ
)
∥
y
∥
2
-
λ
(
1
-
λ
)
∥
x
-
y
∥
2
,
for all x
,
y
∈
H and λ
∈
[
0,1
].Recall that, the metric projectionP
C  from a Hilbert space  H  to a closed convex subset  C  of  H  is defined as the following: given x
∈
H
,
P
C
x is the only point in C with the property
(14)
∥
x
-
P
C
x
∥
=
inf
⁡
⁡
{
∥
x
-
y
∥
:
y
∈
C
}
.
P
C
x is characterized as follows:
(15)
∥
x
-
P
C
x
∥
≤
∥
x
-
y
∥
,
〈
x
-
P
C
x
,
y
-
P
C
x
〉
≤
0
,
∥
x
-
y
∥
2
≥
∥
x
-
P
C
x
∥
2
+
∥
y
-
P
C
x
∥
2
,
for all x
∈
H
,
y
∈
C.Hilbert spaceH satisfies theKadec-Klee property [28, 29]; that is, for any sequence {
x
n
}. x
n
⇀
x and ∥
x
n
∥
→
∥
x
∥ together imply ∥
x
n
-
x
∥
→
0.A normed spaceX is said to satisfy Opial’s condition [30], if for each sequence {
x
n
} in X which converges weakly to a point x
∈
X, we have
(16)
liminf
⁡
n
→
∞
∥
x
n
-
x
∥
<
liminf
⁡
n
→
∞
∥
x
n
-
y
∥
,
∀
y
∈
X
,
y
≠
x
.Lemma 1 (see [27]).
LetC be a nonempty closed convex subset of a Hilbert space H. Let T
:
C
→
C be a continuous pseudocontractive mapping and let A
:
C
→
H be a continuous monotone mapping and define mappings F
r and T
r as follows: x
∈
H
,
r
∈
(
0
,
∞
)
(17)
F
r
(
x
)
=
{
z
∈
C
:
〈
y
-
z
,
A
z
〉
1
r
iiiiiiiiiiiiiii
+
1
r
〈
y
-
z
,
z
-
x
〉
≥
0
,
∀
y
∈
C
}
,
T
r
(
x
)
=
{
z
∈
C
:
〈
y
-
z
,
T
z
〉
1
r
iiiiiiiiiiiiiiii
-
1
r
〈
y
-
z
,
(
1
+
r
)
z
-
x
〉
≤
0
,
∀
y
∈
C
}
.
Then, the following hold: (1)
F
r and T
r are single-valued;
(2)
F
r and T
r are firmly nonexpansive; that is, for any x
,
y
∈
H, ∥
F
r
x
-
F
r
y
∥
2
≤
〈
F
r
x
-
F
r
y
,
x
-
y
〉,  ∥
T
r
x
-
T
r
y
∥
2
≤
〈
T
r
x
-
T
r
y
,
x
-
y
〉;
(3)
F
(
T
r
)
=
F
(
T
), F
(
F
r
)
=
V
I
(
C
,
A
);
(4)
F
(
T
) and V
I
(
C
,
A
) are closed and convex.


## 3. Main Results

### 3.1. The Hybrid Method

Theorem 2.
LetC be a nonempty bounded closed and convex subset of a real Hilbert space H. Let T
:
C
→
C be a continuous pseudocontractive mapping and let A
:
C
→
H be a continuous monotone mapping. Let S
=
{
S
(
s
)
:
0
≤
s
<
∞
} be a nonexpansive semigroup on C such that liminf
⁡
n
→
0
μ
n
=
0, limsup
⁡
n
→
0
μ
n
>
0, and lim
⁡
n
→
0
(
μ
n
+
1
-
μ
n
)
=
0. Let {
α
n
}, {
β
n
}, and {
γ
n
} be the sequences in [
0
,
a
) for some a
∈
[
0,1
), {
r
n
}
⊂
(
0
,
∞
) such that liminf
⁡
n
→
∞
r
n
>
0 and suppose F
=
F
(
S
)
∩
F
(
T
)
∩
V
I
(
C
,
A
)
≠
∅. The mappings T
r
n and F
r
n are defined by (10) and (11). Let {
x
n
} be sequences generated by x
0
∈
C and
(18)
w
n
=
γ
n
x
n
+
(
1
-
γ
n
)
F
r
n
x
n
,
z
n
=
β
n
w
n
+
(
1
-
β
n
)
T
r
n
w
n
,
y
n
=
α
n
z
n
+
(
1
-
α
n
)
S
(
μ
n
)
z
n
,
C
n
=
{
z
∈
C
∣
∥
y
n
-
z
∥
≤
∥
x
n
-
z
∥
}
,
Q
n
=
{
z
∈
C
∣
〈
x
n
-
z
,
x
0
-
x
n
〉
≥
0
}
,
x
n
+
1
=
P
C
n
∩
Q
n
x
0
,
∀
n
≥
0
.
Then the sequence {
x
n
} converges strongly to P
F
x
0.Proof.
Consider that(19)
∥
y
n
-
x
n
∥
2
=
∥
(
y
n
-
z
)
-
(
x
n
-
z
)
∥
2
=
∥
y
n
-
z
∥
2
-
2
〈
y
n
-
z
,
x
n
-
z
〉
+
∥
x
n
-
z
∥
2
=
∥
y
n
-
z
∥
2
-
2
〈
y
n
-
x
n
+
x
n
-
z
,
x
n
-
z
〉
+
∥
x
n
-
z
∥
2
=
∥
y
n
-
z
∥
2
-
2
〈
y
n
-
x
n
,
x
n
-
z
〉
-
2
〈
x
n
-
z
,
x
n
-
z
〉
+
∥
x
n
-
z
∥
2
=
∥
y
n
-
z
∥
2
-
2
〈
y
n
-
x
n
,
x
n
-
z
〉
-
∥
x
n
-
z
∥
2
.
On the other hand, we get that C
n is closed and Q
n is closed and convex for all n
≥
0. From (19), ∥
y
n
-
z
∥
≤
∥
x
n
-
z
∥ is equivalent to ∥
y
n
-
x
n
∥
2
+
2
〈
y
n
-
x
n
,
x
n
-
z
〉
≤
0 for z
∈
C. Thus, we have C
n is convex for all n
≥
0. Therefore, C
n
∩
Q
n is closed and convex for all n
≥
0. Let x
*
∈
F, we have
(20)
∥
w
n
-
x
*
∥
≤
γ
n
∥
x
n
-
x
*
∥
+
(
1
-
γ
n
)
∥
F
r
n
x
n
-
x
*
∥
≤
γ
n
∥
x
n
-
x
*
∥
+
∥
F
r
n
x
n
-
x
*
∥
-
γ
n
∥
x
n
-
x
*
∥
=
∥
F
r
n
x
n
-
x
*
∥
≤
∥
x
n
-
x
*
∥
,
∥
z
n
-
x
*
∥
≤
β
n
∥
w
n
-
x
*
∥
+
(
1
-
β
n
)
∥
T
r
n
w
n
-
x
*
∥
≤
β
n
∥
x
n
-
x
*
∥
+
∥
T
r
n
w
n
-
x
*
∥
-
β
n
∥
x
n
-
x
*
∥
=
∥
T
r
n
w
n
-
x
*
∥
≤
∥
w
n
-
x
*
∥
≤
∥
F
r
n
x
n
-
x
*
∥
≤
∥
x
n
-
x
*
∥
.
It follows that
(21)
∥
y
n
-
x
*
∥
≤
α
n
∥
z
n
-
x
*
∥
+
(
1
-
α
n
)
∥
S
(
μ
n
)
z
n
-
x
*
∥
≤
α
n
∥
x
n
-
x
*
∥
+
∥
z
n
-
x
*
∥
-
α
n
∥
x
n
-
x
*
∥
=
∥
z
n
-
x
*
∥
≤
∥
w
n
-
x
*
∥
≤
∥
x
n
-
x
*
∥
.
Therefore, x
*
∈
C
n for all n
≥
0. Thus, we have F
⊂
C
n for all n
≥
0.
Next, we use mathematical induction. Start withn
=
0, we have x
0
∈
C and Q
0
=
C then F
⊂
C
0
∩
Q
0. Assume that x
k is given and F
⊂
C
k
∩
Q
k for some k
≥
0. There exists a unique x
k
+
1
∈
C
k
∩
Q
k such that x
k
+
1
=
P
C
k
∩
Q
k
x
0, then we get 〈
x
k
+
1
-
z
,
x
0
-
x
k
+
1
〉
≥
0 for z
∈
C
k
∩
Q
k. From F
⊂
C
k
∩
Q
k, we have F
⊂
Q
k
+
1. Therefore, F
⊂
C
k
+
1
∩
Q
k
+
1. Thus {
x
n
} is well defined and F
⊂
C
n
∩
Q
n for all n
≥
0.
SinceF is a nonempty closed convex subset of C, there exists a unique x
¯
∈
F such that x
¯
=
P
F
x
0. From x
n
+
1
=
P
C
n
∩
Q
n
x
0 and the metric projection property, we have
(22)
0
≤
〈
x
n
+
1
-
x
0
,
x
¯
-
x
n
+
1
〉
=
〈
x
n
+
1
-
x
0
,
x
¯
-
x
0
+
x
0
-
x
n
+
1
〉
=
〈
x
n
+
1
-
x
0
,
x
¯
-
x
0
〉
+
〈
x
n
+
1
-
x
0
,
x
0
-
x
n
+
1
〉
=
∥
x
n
+
1
-
x
0
∥
∥
x
¯
-
x
0
∥
-
∥
x
n
+
1
-
x
0
∥
2
.
It follows that ∥
x
n
+
1
-
x
0
∥
≤
∥
x
¯
-
x
0
∥ for all x
¯
∈
F
⊂
C
n
∩
Q
n and n
≥
0. This implied {
x
n
} is bounded. So, {
F
r
n
x
n
}, {
T
r
n
w
n
}, {
S
(
μ
n
)
z
n
}, {
y
n
}, {
z
n
}, and {
w
n
} are bounded.
Next, we show that(23)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0
.
Since x
n
+
1
∈
C
n
∩
Q
n
⊂
Q
n, x
n
=
P
Q
n
x
0, and 〈
x
n
-
x
0
,
x
n
+
1
-
x
n
〉
≥
0, as same as the prove of (22), we get ∥
x
n
+
1
-
x
0
∥
≥
∥
x
n
-
x
0
∥ for all n
≥
0. Thus, {
∥
x
n
-
x
0
∥
} is nondecreasing. By {
∥
x
n
-
x
0
∥
} is bounded and nondecreasing, there exists the limit of {
∥
x
n
-
x
0
∥
}. Since x
n
+
1
∈
Q
n, we have
(24)
〈
x
n
-
x
n
+
1
,
x
0
-
x
n
〉
≥
0
,
∀
n
≥
0
,
(25)
∥
x
n
-
x
n
+
1
∥
2

=
∥
x
n
-
x
0
+
x
0
-
x
n
+
1
∥
2

=
∥
x
n
-
x
0
∥
2
+
2
〈
x
n
-
x
0
,
x
0
-
x
n
+
1
〉

+
∥
x
n
+
1
-
x
0
∥
2

=
∥
x
n
-
x
0
∥
2
+
2
〈
x
n
-
x
0
,
x
0
-
x
n
+
x
n
-
x
n
+
1
〉

+
∥
x
n
+
1
-
x
0
∥
2

=
∥
x
n
-
x
0
∥
2
+
2
〈
x
n
-
x
0
,
x
0
-
x
n
〉

+
2
〈
x
n
-
x
0
,
x
n
-
x
n
+
1
〉
+
∥
x
n
+
1
-
x
0
∥
2

≤
∥
x
n
-
x
0
∥
2
+
2
〈
x
n
-
x
0
,
x
0
-
x
n
〉

+
∥
x
n
+
1
-
x
0
∥
2

=
∥
x
n
-
x
0
∥
2
-
2
∥
x
n
-
x
0
∥
2
+
∥
x
n
+
1
-
x
0
∥
2

=
∥
x
n
+
1
-
x
0
∥
2
-
∥
x
n
-
x
0
∥
2
.
Since lim
⁡
n
→
∞
∥
x
n
-
x
0
∥ exists, therefore (23) holds.
Next, we show that(26)
lim
⁡
n
→
∞
∥
S
(
μ
n
)
x
n
-
x
n
∥
=
0
.
Since x
n
+
1
∈
C
n, we have
(27)
∥
x
n
-
y
n
∥
≤
∥
x
n
-
x
n
+
1
∥
+
∥
x
n
+
1
-
y
n
∥
≤
2
∥
x
n
+
1
-
x
n
∥
⟶
0
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
as
n
⟶
∞
.
Let x
*
∈
F, v
n
=
F
r
n
x
n, and u
n
=
T
r
n
w
n; it follows from Lemma 1, we get
(28)
∥
v
n
-
x
*
∥
2
=
∥
F
r
n
x
n
-
F
r
n
x
*
∥
2
≤
〈
F
r
n
x
n
-
F
r
n
x
*
,
x
n
-
x
*
〉
=
〈
v
n
-
x
*
,
x
n
-
x
*
〉
=
1
2
(
∥
v
n
-
x
*
∥
2
+
∥
x
n
-
x
*
∥
2
-
∥
v
n
-
x
n
∥
2
)
,
∥
u
n
-
x
*
∥
2
=
∥
T
r
n
w
n
-
T
r
n
x
*
∥
2
≤
〈
T
r
n
w
n
-
T
r
n
x
*
,
w
n
-
x
*
〉
=
〈
u
n
-
x
*
,
w
n
-
x
*
〉
=
1
2
(
∥
u
n
-
x
*
∥
2
+
∥
w
n
-
x
*
∥
2
-
∥
u
n
-
w
n
∥
2
)
.
Hence
(29)
∥
v
n
-
x
*
∥
2
≤
∥
x
n
-
x
*
∥
2
-
∥
v
n
-
x
n
∥
2
,
∥
u
n
-
x
*
∥
2
≤
∥
w
n
-
x
*
∥
2
-
∥
u
n
-
w
n
∥
2
.
It follows that
(30)
∥
y
n
-
x
*
∥
2
≤
α
n
∥
z
n
-
x
*
∥
2
+
(
1
-
α
n
)
∥
S
(
μ
n
)
z
n
-
x
*
∥
2
≤
α
n
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
∥
v
n
-
x
*
∥
2
≤
α
n
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
(
∥
x
n
-
x
*
∥
2
-
∥
v
n
-
x
n
∥
2
)
≤
α
n
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
∥
x
n
-
x
*
∥
2
-
∥
v
n
-
x
n
∥
2
=
∥
x
n
-
x
*
∥
2
-
∥
v
n
-
x
n
∥
2
,
∥
y
n
-
x
*
∥
2
≤
α
n
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
∥
u
n
-
x
*
∥
2
≤
α
n
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
(
∥
w
n
-
x
*
∥
2
-
∥
u
n
-
w
n
∥
2
)
≤
α
n
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
∥
x
n
-
x
*
∥
2
-
∥
u
n
-
w
n
∥
2
=
∥
x
n
-
x
*
∥
2
-
∥
u
n
-
w
n
∥
2
.
Consequently, we have that
(31)
∥
v
n
-
x
n
∥
2
≤
∥
x
n
-
x
*
∥
2
-
∥
y
n
-
x
*
∥
2
≤
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
∥
x
n
-
y
n
∥
,
∥
u
n
-
w
n
∥
2
≤
∥
x
n
-
x
*
∥
2
-
∥
y
n
-
x
*
∥
2
≤
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
∥
x
n
-
y
n
∥
.
Equation (27) implies that
(32)
lim
⁡
n
→
∞
∥
v
n
-
x
n
∥
=
0
,
lim
⁡
n
→
∞
∥
u
n
-
w
n
∥
=
0
.
On the other hand, from (18) and (32), we also have
(33)
∥
w
n
-
v
n
∥
=
γ
n
∥
x
n
-
v
n
∥
⟶
0
as
n
⟶
∞
,
∥
z
n
-
u
n
∥
=
β
n
∥
w
n
-
u
n
∥
⟶
0
as
n
⟶
∞
.
It follows from (32)-(33) we get
(34)
lim
⁡
n
→
∞
∥
z
n
-
x
n
∥
=
0
.
Since y
n
-
x
n
=
α
n
(
z
n
-
x
n
)
+
(
1
-
α
n
)
(
S
(
μ
n
)
z
n
-
x
n
) and from (27) and (34), we get
(35)
∥
x
n
-
S
(
μ
n
)
z
n
∥
≤
α
n
1
-
α
n
∥
z
n
-
x
n
∥
+
1
1
-
α
n
∥
x
n
-
y
n
∥
⟶
0
ppppppppppppppppppppppppp
as
n
⟶
∞
.
It follows that
(36)
∥
x
n
-
S
(
μ
n
)
x
n
∥
≤
∥
x
n
-
S
(
μ
n
)
z
n
∥
+
∥
S
(
μ
n
)
z
n
-
S
(
μ
n
)
x
n
∥
≤
∥
x
n
-
S
(
μ
n
)
z
n
∥
+
∥
z
n
-
x
n
∥
.
Therefore (26) holds.
Next, we show thatx
′
∈
F
=
F
(
S
)
∩
F
(
T
)
∩
VI
⁡
(
C
,
A
). First, we show that x
′ is the unique solution in F
(
S
). Since {
x
n
} is bounded, we choose subsequence {
x
n
i
} of {
x
n
} and assume that x
n
i
⇀
x
′. Suppose that x
′
∉
F
(
S
)
=
∩
0
≤
s
<
∞
F
(
T
(
s
)
); that is, x
′
≠
S
(
s
)
x
′. From Opial’s condition and (26), we have
(37)
liminf
⁡
i
→
∞
∥
x
n
i
-
x
′
∥
<
liminf
⁡
i
→
∞
∥
x
n
i
-
S
(
s
)
x
′
∥
≤
liminf
⁡
i
→
∞
(
∥
x
n
i
-
S
(
s
)
x
n
i
∥
+
∥
S
(
s
)
x
n
i
-
S
(
s
)
x
′
∥
)
≤
liminf
⁡
i
→
∞
∥
x
n
i
-
x
′
∥
.
This is a contradiction. Thus, we obtain x
′
∈
F
(
S
).
Since{
x
n
i
} is bounded, there exists a subsequence {
x
n
i
j
} of {
x
n
i
} such that x
n
i
j
⇀
x
′, x
′
∈
C. Without loss of generality, we may assume that x
n
i
⇀
x
′. From the setting v
n
=
F
r
n
x
n and (10), we have
(38)
〈
y
-
v
n
i
,
A
v
n
i
〉
+
1
r
n
i
〈
y
-
v
n
i
,
v
n
i
-
x
n
i
〉
≥
0
,
∀
y
∈
C
.
For t
∈
(
0,1
) and v
∈
C, let v
t
=
t
v
+
(
1
-
t
)
x
′. Since v
∈
C and x
′
∈
C, we have v
t
∈
C and
(39)
〈
v
t
-
v
n
i
,
A
v
t
〉
≥
〈
v
t
-
v
n
i
,
A
v
t
-
A
v
n
i
〉
-
〈
v
t
-
v
n
i
,
v
n
i
-
x
n
i
r
n
i
〉
.
Since A is a monotone and v
n
i
-
x
n
i
→
0, we obtain
(40)
〈
v
t
-
x
′
,
A
v
t
〉
=
lim
⁡
i
→
∞
〈
v
t
-
v
n
i
,
A
v
t
〉
≥
0
.
By the continuity of A, if t
→
0 then 〈
v
-
x
′
,
A
v
〉
≥
0
,
∀
v
∈
C. Therefore, x
′
∈
VI
⁡
(
C
,
A
).
On the other hand, sinceu
n
=
T
r
n
w
n, from (11) we have
(41)
〈
y
-
u
n
i
,
T
u
n
i
〉
-
1
r
n
i
〈
y
-
u
n
i
,
(
1
+
r
n
i
)
u
n
i
-
w
n
i
〉
≤
0
,
iiiiiiiiiiiiiiiiiiiippppiiiiiiiiiiiiiiiiii
∀
y
∈
C
.
For t
∈
(
0,1
) and v
∈
C, let v
t
=
t
v
+
(
1
-
t
)
x
′. Since v
∈
C and x
′
∈
C, we have v
t
∈
C and
(42)
〈
u
n
i
-
v
t
,
T
v
t
〉
≥
〈
u
n
i
-
v
t
,
T
v
t
〉
+
〈
v
t
-
u
n
i
,
T
u
n
i
〉
-
1
r
n
i
〈
v
t
-
u
n
i
,
(
1
+
r
n
i
)
u
n
i
-
w
n
i
〉
=
-
〈
v
t
-
u
n
i
,
T
v
t
-
T
u
n
i
〉
-
1
r
n
i
〈
v
t
-
u
n
i
,
u
n
i
-
w
n
i
〉
-
〈
v
t
-
u
n
i
,
u
n
i
〉
≥
-
∥
v
t
-
u
n
i
∥
2
-
1
r
n
i
〈
v
t
-
u
n
i
,
u
n
i
-
w
n
i
〉
-
〈
v
t
-
u
n
i
,
u
n
i
〉
=
-
1
r
n
i
〈
v
t
-
u
n
i
,
u
n
i
-
w
n
i
〉
-
〈
v
t
-
u
n
i
,
v
t
〉
.
It follows from u
n
-
w
n
→
0 as n
→
∞, we get 〈
x
′
-
v
t
,
T
v
t
〉
≥
-
〈
v
t
-
x
′
,
v
t
〉 and hence -
〈
v
-
x
′
,
T
v
t
〉
≥
-
〈
v
-
x
′
,
v
t
〉
,
∀
v
∈
C. By the continuity of T, if t
→
0 then -
〈
v
-
x
′
,
T
x
′
〉
≥
-
〈
v
-
x
′
,
x
′
〉
,
∀
v
∈
C. Let v
=
T
x
′, we have x
′
=
T
x
′; therefore, x
′
∈
F
(
T
). Consequently, we conclude that x
′
∈
F
=
F
(
S
)
∩
F
(
T
)
∩
VI
⁡
(
C
,
A
).
Finally, we show thatx
n
→
x
′, where x
′
=
P
F
x
0. Since x
n
+
1
=
P
C
n
∩
Q
n
x
0 and x
′
∈
F
⊂
C
n
∩
Q
n, we get
(43)
∥
x
n
+
1
-
x
0
∥
≤
∥
x
′
-
x
0
∥
,
∀
n
≥
0
.
If  x
¯
=
P
F
x
0, it follows from (43), and the lower semicontinuity of the norm that
(44)
∥
x
′
-
x
0
∥
≤
∥
x
¯
-
x
0
∥
≤
liminf
⁡
i
→
∞
∥
x
n
i
-
x
0
∥
≤
limsup
⁡
i
→
∞
∥
x
n
i
-
x
0
∥
≤
∥
x
′
-
x
0
∥
.
Thus, we obtain that lim
⁡
i
→
∞
∥
x
n
i
-
x
0
∥
=
∥
x
′
-
x
0
∥
=
∥
x
¯
-
x
0
∥. Using the Kadec-Klee property of H, we obtain that
(45)
lim
⁡
i
→
∞
x
n
i
=
x
′
=
x
¯
.
Since {
x
n
i
} is an arbitrary weakly convergent subsequence of {
x
n
}, we can conclude that {
x
n
} converges strongly to x
′, where x
′
=
P
F
x
0. This completes the proof.Corollary 3.
LetC be a nonempty bounded closed and convex subset of a real Hilbert space H. Let T
:
C
→
C be a continuous pseudocontractive mapping and let A
:
C
→
H be a continuous monotone mapping. Let S
=
{
T
(
s
)
:
0
≤
s
<
∞
} be a nonexpansive semigroup on C such that liminf
⁡
n
→
0
μ
n
=
0, limsup
⁡
n
→
0
μ
n
>
0, and lim
⁡
n
→
0
(
μ
n
+
1
-
μ
n
)
=
0. Let {
r
n
}
⊂
(
0
,
∞
) such that liminf
⁡
n
→
∞
r
n
>
0 and suppose F
=
F
(
S
)
∩
F
(
T
)
∩
V
I
(
C
,
A
)
≠
∅. The mappings T
r
n and F
r
n are defined by (10) and (11). Let {
x
n
} be a sequences generated by x
0
∈
C and
(46)
y
n
=
S
(
μ
n
)
T
r
n
F
r
n
,
C
n
=
{
z
∈
C
∣
∥
y
n
-
z
∥
≤
∥
x
n
-
z
∥
}
,
Q
n
=
{
z
∈
C
∣
〈
x
n
-
z
,
x
0
-
x
n
〉
≥
0
}
,
x
n
+
1
=
P
C
n
∩
Q
n
x
0
,
∀
n
≥
0
.
Then the sequence {
x
n
} converges strongly to P
F
x
0.Proof.
Puttingα
n
=
β
n
=
γ
n
=
0
,
∀
n
≥
0 in Theorem 2, we can obtain the result.


### 3.2. The Shrinking Projection Method

Theorem 4.
LetC be a nonempty bounded closed and convex subset of a real Hilbert space H. Let T
:
C
→
C be a continuous pseudocontractive mapping and let A
:
C
→
H be a continuous monotone mapping. Let S
=
{
S
(
s
)
:
0
≤
s
<
∞
} be a nonexpansive semigroup on C such that liminf
⁡
n
→
0
μ
n
=
0, limsup
⁡
n
→
0
μ
n
>
0, and lim
⁡
n
→
0
(
μ
n
+
1
-
μ
n
)
=
0. Let {
α
n
}, {
β
n
}, and {
γ
n
} be the sequences in [
0
,
a
) for some a
∈
[
0,1
), {
r
n
}
⊂
(
0
,
∞
) such that liminf
⁡
n
→
∞
r
n
>
0 and suppose F
=
F
(
S
)
∩
F
(
T
)
∩
V
I
(
C
,
A
)
≠
∅. The mappings T
r
n and F
r
n are defined by (10) and (11). Let {
x
n
} be sequences generated by x
0
∈
C and
(47)
w
n
=
γ
n
x
n
+
(
1
-
γ
n
)
F
r
n
x
n
,
z
n
=
β
n
w
n
+
(
1
-
β
n
)
T
r
n
w
n
,
y
n
=
α
n
z
n
+
(
1
-
α
n
)
S
(
μ
n
)
z
n
,
C
n
+
1
=
{
z
∈
C
n
∣
∥
y
n
-
z
∥
≤
∥
x
n
-
z
∥
}
,
x
n
+
1
=
P
C
n
+
1
x
0
,
∀
n
≥
0
.
Then the sequence {
x
n
} converges strongly to P
F
x
0.Proof.
First, we show thatF
⊂
C
n. By induction, it is obvious that F
⊂
C
1. Suppose that F
⊂
C
k for some k
≥
0, so we have x
*
∈
F
⊂
C
k such that ∥
y
k
-
x
*
∥
≤
∥
x
k
-
x
*
∥. Then, we get x
*
∈
C
n
+
1. Therefore F
⊂
C
n for all n
≥
0.
On the other hand, we show thatC
n is closed and convex for all n
≥
0. By mathematical induction, it is obvious that C
1
=
C is closed and convex. Suppose that C
k is closed and convex for some k
≥
0. For z
∈
C
k, we have that ∥
y
k
-
z
∥
≤
∥
x
k
-
z
∥ is equivalent to ∥
y
k
-
x
k
∥
2
+
2
〈
y
k
-
x
k
,
x
k
-
z
〉
≤
0. Thus, we have C
k
+
1 is closed and convex for all n
≥
0. Therefore, C
n is closed and convex for all n
≥
0. This implies that {
x
n
} is well defined.
Next, we show that{
x
n
} is bounded. From the metric projection property and (47), we have x
n
=
P
C
n
x
0 and 〈
x
n
-
x
0
,
x
¯
-
x
n
〉
≥
0 for all x
¯
∈
F
⊂
C
n and n
≥
0. Consider
(48)
0
≤
〈
x
n
-
x
0
,
x
¯
-
x
n
〉
=
〈
x
n
-
x
0
,
x
¯
-
x
0
+
x
0
-
x
n
〉
=
〈
x
n
-
x
0
,
x
¯
-
x
0
〉
+
〈
x
n
-
x
0
,
x
0
-
x
n
〉
=
∥
x
n
-
x
0
∥
∥
x
¯
-
x
0
∥
-
∥
x
n
-
x
0
∥
2
.
It follows that ∥
x
n
-
x
0
∥
≤
∥
x
¯
-
x
0
∥ for all x
¯
∈
F and n
≥
0. This implied {
x
n
} is bounded. So, {
F
r
n
x
n
}, {
T
r
n
w
n
}, {
S
(
μ
n
)
z
n
}, {
y
n
}, {
z
n
}, and {
w
n
} are bounded.
Next, we show that(49)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0
.
From the metric projection property and (47), we have x
n
=
P
C
n
x
0, x
n
+
1
=
P
C
n
+
1
x
0
∈
C
n
+
1
⊂
C
n, and 〈
x
n
-
x
0
,
x
n
+
1
-
x
n
〉
≥
0. As same as the prove of (48), we get ∥
x
n
-
x
0
∥
≤
∥
x
n
+
1
-
x
0
∥ for all n
≥
0. Thus, {
∥
x
n
-
x
0
∥
} is nondecreasing. Since {
∥
x
n
-
x
0
∥
} is bounded and nondecreasing, there exists the limit of {
∥
x
n
-
x
0
∥
}. Similar to the proved of (25), we get (49) is hold.
Since sequence{
x
n
} is bounded, we can choose subsequence {
x
n
i
} of {
x
n
} and assume that x
n
i
⇀
x
′. Similar to the proof of Theorem 2, we also have x
′
∈
F.
Finally, we show thatx
n
→
x
′, where x
′
=
P
F
x
0. Since x
n
=
P
C
n
x
0 and x
′
∈
F
⊂
C
n, we have
(50)
∥
x
n
-
x
0
∥
≤
∥
x
¯
-
x
0
∥
.
It follows from (50), if x
¯
=
P
F
x
0 and the lower semicontinuity of the norm
(51)
∥
x
′
-
x
0
∥
≤
∥
x
¯
-
x
0
∥
≤
liminf
⁡
i
→
∞
∥
x
n
i
-
x
0
∥
≤
limsup
⁡
i
→
∞
∥
x
n
i
-
x
0
∥
≤
∥
x
′
-
x
0
∥
,
thus, we obtain that lim
⁡
i
→
∞
∥
x
n
i
-
x
0
∥
=
∥
x
¯
-
x
0
∥
=
∥
x
′
-
x
0
∥. Using the Kadec-Klee property of H, we obtain that
(52)
lim
⁡
i
→
∞
x
n
i
=
x
¯
=
x
′
.
Since {
x
n
i
} is an arbitrary weakly convergent subsequence of {
x
n
}, we can conclude that {
x
n
} converges strongly to x
′, where x
′
=
P
F
x
0.Corollary 5.
LetC be a nonempty bounded closed and convex subset of a real Hilbert space H. Let T
:
C
→
C be a continuous pseudocontractive mapping and let A
:
C
→
H be a continuous monotone mapping. Let S
=
{
T
(
s
)
:
0
≤
s
<
∞
} be a nonexpansive semigroup on C such that liminf
⁡
n
→
0
μ
n
=
0, limsup
⁡
n
→
0
μ
n
>
0, and lim
⁡
n
→
0
(
μ
n
+
1
-
μ
n
)
=
0. Let {
r
n
}
⊂
(
0
,
∞
) such that liminf
⁡
n
→
∞
r
n
>
0 and suppose F
=
F
(
S
)
∩
F
(
T
)
∩
V
I
(
C
,
A
)
≠
∅. The mappings T
r
n and F
r
n are defined by (10) and (11). Let {
x
n
} be a sequences generated by x
0
∈
C and
(53)
y
n
=
S
(
μ
n
)
T
r
n
F
r
n
,
C
n
+
1
=
{
z
∈
C
n
∣
∥
y
n
-
z
∥
≤
∥
x
n
-
z
∥
}
,
x
n
+
1
=
P
C
n
+
1
x
0
,
∀
n
≥
0
.
Then the sequence {
x
n
} converges strongly to P
F
x
0.Proof.
Puttingα
n
=
β
n
=
γ
n
=
0
,
∀
n
≥
0 in Theorem 4, we can obtain the result.Remark 6.
According to nonexpansive semigroup mapping, it will be interesting if we replace the semigroupS
=
(
N
,
+
) by an additive positive real numbers of a commutative semigroup or a left amenable semigroup or a left reversible semigroup with using an asymptotically invariant sequence. See [31, 32].


## 3.1. The Hybrid Method

Theorem 2.
LetC be a nonempty bounded closed and convex subset of a real Hilbert space H. Let T
:
C
→
C be a continuous pseudocontractive mapping and let A
:
C
→
H be a continuous monotone mapping. Let S
=
{
S
(
s
)
:
0
≤
s
<
∞
} be a nonexpansive semigroup on C such that liminf
⁡
n
→
0
μ
n
=
0, limsup
⁡
n
→
0
μ
n
>
0, and lim
⁡
n
→
0
(
μ
n
+
1
-
μ
n
)
=
0. Let {
α
n
}, {
β
n
}, and {
γ
n
} be the sequences in [
0
,
a
) for some a
∈
[
0,1
), {
r
n
}
⊂
(
0
,
∞
) such that liminf
⁡
n
→
∞
r
n
>
0 and suppose F
=
F
(
S
)
∩
F
(
T
)
∩
V
I
(
C
,
A
)
≠
∅. The mappings T
r
n and F
r
n are defined by (10) and (11). Let {
x
n
} be sequences generated by x
0
∈
C and
(18)
w
n
=
γ
n
x
n
+
(
1
-
γ
n
)
F
r
n
x
n
,
z
n
=
β
n
w
n
+
(
1
-
β
n
)
T
r
n
w
n
,
y
n
=
α
n
z
n
+
(
1
-
α
n
)
S
(
μ
n
)
z
n
,
C
n
=
{
z
∈
C
∣
∥
y
n
-
z
∥
≤
∥
x
n
-
z
∥
}
,
Q
n
=
{
z
∈
C
∣
〈
x
n
-
z
,
x
0
-
x
n
〉
≥
0
}
,
x
n
+
1
=
P
C
n
∩
Q
n
x
0
,
∀
n
≥
0
.
Then the sequence {
x
n
} converges strongly to P
F
x
0.Proof.
Consider that(19)
∥
y
n
-
x
n
∥
2
=
∥
(
y
n
-
z
)
-
(
x
n
-
z
)
∥
2
=
∥
y
n
-
z
∥
2
-
2
〈
y
n
-
z
,
x
n
-
z
〉
+
∥
x
n
-
z
∥
2
=
∥
y
n
-
z
∥
2
-
2
〈
y
n
-
x
n
+
x
n
-
z
,
x
n
-
z
〉
+
∥
x
n
-
z
∥
2
=
∥
y
n
-
z
∥
2
-
2
〈
y
n
-
x
n
,
x
n
-
z
〉
-
2
〈
x
n
-
z
,
x
n
-
z
〉
+
∥
x
n
-
z
∥
2
=
∥
y
n
-
z
∥
2
-
2
〈
y
n
-
x
n
,
x
n
-
z
〉
-
∥
x
n
-
z
∥
2
.
On the other hand, we get that C
n is closed and Q
n is closed and convex for all n
≥
0. From (19), ∥
y
n
-
z
∥
≤
∥
x
n
-
z
∥ is equivalent to ∥
y
n
-
x
n
∥
2
+
2
〈
y
n
-
x
n
,
x
n
-
z
〉
≤
0 for z
∈
C. Thus, we have C
n is convex for all n
≥
0. Therefore, C
n
∩
Q
n is closed and convex for all n
≥
0. Let x
*
∈
F, we have
(20)
∥
w
n
-
x
*
∥
≤
γ
n
∥
x
n
-
x
*
∥
+
(
1
-
γ
n
)
∥
F
r
n
x
n
-
x
*
∥
≤
γ
n
∥
x
n
-
x
*
∥
+
∥
F
r
n
x
n
-
x
*
∥
-
γ
n
∥
x
n
-
x
*
∥
=
∥
F
r
n
x
n
-
x
*
∥
≤
∥
x
n
-
x
*
∥
,
∥
z
n
-
x
*
∥
≤
β
n
∥
w
n
-
x
*
∥
+
(
1
-
β
n
)
∥
T
r
n
w
n
-
x
*
∥
≤
β
n
∥
x
n
-
x
*
∥
+
∥
T
r
n
w
n
-
x
*
∥
-
β
n
∥
x
n
-
x
*
∥
=
∥
T
r
n
w
n
-
x
*
∥
≤
∥
w
n
-
x
*
∥
≤
∥
F
r
n
x
n
-
x
*
∥
≤
∥
x
n
-
x
*
∥
.
It follows that
(21)
∥
y
n
-
x
*
∥
≤
α
n
∥
z
n
-
x
*
∥
+
(
1
-
α
n
)
∥
S
(
μ
n
)
z
n
-
x
*
∥
≤
α
n
∥
x
n
-
x
*
∥
+
∥
z
n
-
x
*
∥
-
α
n
∥
x
n
-
x
*
∥
=
∥
z
n
-
x
*
∥
≤
∥
w
n
-
x
*
∥
≤
∥
x
n
-
x
*
∥
.
Therefore, x
*
∈
C
n for all n
≥
0. Thus, we have F
⊂
C
n for all n
≥
0.
Next, we use mathematical induction. Start withn
=
0, we have x
0
∈
C and Q
0
=
C then F
⊂
C
0
∩
Q
0. Assume that x
k is given and F
⊂
C
k
∩
Q
k for some k
≥
0. There exists a unique x
k
+
1
∈
C
k
∩
Q
k such that x
k
+
1
=
P
C
k
∩
Q
k
x
0, then we get 〈
x
k
+
1
-
z
,
x
0
-
x
k
+
1
〉
≥
0 for z
∈
C
k
∩
Q
k. From F
⊂
C
k
∩
Q
k, we have F
⊂
Q
k
+
1. Therefore, F
⊂
C
k
+
1
∩
Q
k
+
1. Thus {
x
n
} is well defined and F
⊂
C
n
∩
Q
n for all n
≥
0.
SinceF is a nonempty closed convex subset of C, there exists a unique x
¯
∈
F such that x
¯
=
P
F
x
0. From x
n
+
1
=
P
C
n
∩
Q
n
x
0 and the metric projection property, we have
(22)
0
≤
〈
x
n
+
1
-
x
0
,
x
¯
-
x
n
+
1
〉
=
〈
x
n
+
1
-
x
0
,
x
¯
-
x
0
+
x
0
-
x
n
+
1
〉
=
〈
x
n
+
1
-
x
0
,
x
¯
-
x
0
〉
+
〈
x
n
+
1
-
x
0
,
x
0
-
x
n
+
1
〉
=
∥
x
n
+
1
-
x
0
∥
∥
x
¯
-
x
0
∥
-
∥
x
n
+
1
-
x
0
∥
2
.
It follows that ∥
x
n
+
1
-
x
0
∥
≤
∥
x
¯
-
x
0
∥ for all x
¯
∈
F
⊂
C
n
∩
Q
n and n
≥
0. This implied {
x
n
} is bounded. So, {
F
r
n
x
n
}, {
T
r
n
w
n
}, {
S
(
μ
n
)
z
n
}, {
y
n
}, {
z
n
}, and {
w
n
} are bounded.
Next, we show that(23)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0
.
Since x
n
+
1
∈
C
n
∩
Q
n
⊂
Q
n, x
n
=
P
Q
n
x
0, and 〈
x
n
-
x
0
,
x
n
+
1
-
x
n
〉
≥
0, as same as the prove of (22), we get ∥
x
n
+
1
-
x
0
∥
≥
∥
x
n
-
x
0
∥ for all n
≥
0. Thus, {
∥
x
n
-
x
0
∥
} is nondecreasing. By {
∥
x
n
-
x
0
∥
} is bounded and nondecreasing, there exists the limit of {
∥
x
n
-
x
0
∥
}. Since x
n
+
1
∈
Q
n, we have
(24)
〈
x
n
-
x
n
+
1
,
x
0
-
x
n
〉
≥
0
,
∀
n
≥
0
,
(25)
∥
x
n
-
x
n
+
1
∥
2

=
∥
x
n
-
x
0
+
x
0
-
x
n
+
1
∥
2

=
∥
x
n
-
x
0
∥
2
+
2
〈
x
n
-
x
0
,
x
0
-
x
n
+
1
〉

+
∥
x
n
+
1
-
x
0
∥
2

=
∥
x
n
-
x
0
∥
2
+
2
〈
x
n
-
x
0
,
x
0
-
x
n
+
x
n
-
x
n
+
1
〉

+
∥
x
n
+
1
-
x
0
∥
2

=
∥
x
n
-
x
0
∥
2
+
2
〈
x
n
-
x
0
,
x
0
-
x
n
〉

+
2
〈
x
n
-
x
0
,
x
n
-
x
n
+
1
〉
+
∥
x
n
+
1
-
x
0
∥
2

≤
∥
x
n
-
x
0
∥
2
+
2
〈
x
n
-
x
0
,
x
0
-
x
n
〉

+
∥
x
n
+
1
-
x
0
∥
2

=
∥
x
n
-
x
0
∥
2
-
2
∥
x
n
-
x
0
∥
2
+
∥
x
n
+
1
-
x
0
∥
2

=
∥
x
n
+
1
-
x
0
∥
2
-
∥
x
n
-
x
0
∥
2
.
Since lim
⁡
n
→
∞
∥
x
n
-
x
0
∥ exists, therefore (23) holds.
Next, we show that(26)
lim
⁡
n
→
∞
∥
S
(
μ
n
)
x
n
-
x
n
∥
=
0
.
Since x
n
+
1
∈
C
n, we have
(27)
∥
x
n
-
y
n
∥
≤
∥
x
n
-
x
n
+
1
∥
+
∥
x
n
+
1
-
y
n
∥
≤
2
∥
x
n
+
1
-
x
n
∥
⟶
0
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
as
n
⟶
∞
.
Let x
*
∈
F, v
n
=
F
r
n
x
n, and u
n
=
T
r
n
w
n; it follows from Lemma 1, we get
(28)
∥
v
n
-
x
*
∥
2
=
∥
F
r
n
x
n
-
F
r
n
x
*
∥
2
≤
〈
F
r
n
x
n
-
F
r
n
x
*
,
x
n
-
x
*
〉
=
〈
v
n
-
x
*
,
x
n
-
x
*
〉
=
1
2
(
∥
v
n
-
x
*
∥
2
+
∥
x
n
-
x
*
∥
2
-
∥
v
n
-
x
n
∥
2
)
,
∥
u
n
-
x
*
∥
2
=
∥
T
r
n
w
n
-
T
r
n
x
*
∥
2
≤
〈
T
r
n
w
n
-
T
r
n
x
*
,
w
n
-
x
*
〉
=
〈
u
n
-
x
*
,
w
n
-
x
*
〉
=
1
2
(
∥
u
n
-
x
*
∥
2
+
∥
w
n
-
x
*
∥
2
-
∥
u
n
-
w
n
∥
2
)
.
Hence
(29)
∥
v
n
-
x
*
∥
2
≤
∥
x
n
-
x
*
∥
2
-
∥
v
n
-
x
n
∥
2
,
∥
u
n
-
x
*
∥
2
≤
∥
w
n
-
x
*
∥
2
-
∥
u
n
-
w
n
∥
2
.
It follows that
(30)
∥
y
n
-
x
*
∥
2
≤
α
n
∥
z
n
-
x
*
∥
2
+
(
1
-
α
n
)
∥
S
(
μ
n
)
z
n
-
x
*
∥
2
≤
α
n
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
∥
v
n
-
x
*
∥
2
≤
α
n
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
(
∥
x
n
-
x
*
∥
2
-
∥
v
n
-
x
n
∥
2
)
≤
α
n
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
∥
x
n
-
x
*
∥
2
-
∥
v
n
-
x
n
∥
2
=
∥
x
n
-
x
*
∥
2
-
∥
v
n
-
x
n
∥
2
,
∥
y
n
-
x
*
∥
2
≤
α
n
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
∥
u
n
-
x
*
∥
2
≤
α
n
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
(
∥
w
n
-
x
*
∥
2
-
∥
u
n
-
w
n
∥
2
)
≤
α
n
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
∥
x
n
-
x
*
∥
2
-
∥
u
n
-
w
n
∥
2
=
∥
x
n
-
x
*
∥
2
-
∥
u
n
-
w
n
∥
2
.
Consequently, we have that
(31)
∥
v
n
-
x
n
∥
2
≤
∥
x
n
-
x
*
∥
2
-
∥
y
n
-
x
*
∥
2
≤
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
∥
x
n
-
y
n
∥
,
∥
u
n
-
w
n
∥
2
≤
∥
x
n
-
x
*
∥
2
-
∥
y
n
-
x
*
∥
2
≤
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
∥
x
n
-
y
n
∥
.
Equation (27) implies that
(32)
lim
⁡
n
→
∞
∥
v
n
-
x
n
∥
=
0
,
lim
⁡
n
→
∞
∥
u
n
-
w
n
∥
=
0
.
On the other hand, from (18) and (32), we also have
(33)
∥
w
n
-
v
n
∥
=
γ
n
∥
x
n
-
v
n
∥
⟶
0
as
n
⟶
∞
,
∥
z
n
-
u
n
∥
=
β
n
∥
w
n
-
u
n
∥
⟶
0
as
n
⟶
∞
.
It follows from (32)-(33) we get
(34)
lim
⁡
n
→
∞
∥
z
n
-
x
n
∥
=
0
.
Since y
n
-
x
n
=
α
n
(
z
n
-
x
n
)
+
(
1
-
α
n
)
(
S
(
μ
n
)
z
n
-
x
n
) and from (27) and (34), we get
(35)
∥
x
n
-
S
(
μ
n
)
z
n
∥
≤
α
n
1
-
α
n
∥
z
n
-
x
n
∥
+
1
1
-
α
n
∥
x
n
-
y
n
∥
⟶
0
ppppppppppppppppppppppppp
as
n
⟶
∞
.
It follows that
(36)
∥
x
n
-
S
(
μ
n
)
x
n
∥
≤
∥
x
n
-
S
(
μ
n
)
z
n
∥
+
∥
S
(
μ
n
)
z
n
-
S
(
μ
n
)
x
n
∥
≤
∥
x
n
-
S
(
μ
n
)
z
n
∥
+
∥
z
n
-
x
n
∥
.
Therefore (26) holds.
Next, we show thatx
′
∈
F
=
F
(
S
)
∩
F
(
T
)
∩
VI
⁡
(
C
,
A
). First, we show that x
′ is the unique solution in F
(
S
). Since {
x
n
} is bounded, we choose subsequence {
x
n
i
} of {
x
n
} and assume that x
n
i
⇀
x
′. Suppose that x
′
∉
F
(
S
)
=
∩
0
≤
s
<
∞
F
(
T
(
s
)
); that is, x
′
≠
S
(
s
)
x
′. From Opial’s condition and (26), we have
(37)
liminf
⁡
i
→
∞
∥
x
n
i
-
x
′
∥
<
liminf
⁡
i
→
∞
∥
x
n
i
-
S
(
s
)
x
′
∥
≤
liminf
⁡
i
→
∞
(
∥
x
n
i
-
S
(
s
)
x
n
i
∥
+
∥
S
(
s
)
x
n
i
-
S
(
s
)
x
′
∥
)
≤
liminf
⁡
i
→
∞
∥
x
n
i
-
x
′
∥
.
This is a contradiction. Thus, we obtain x
′
∈
F
(
S
).
Since{
x
n
i
} is bounded, there exists a subsequence {
x
n
i
j
} of {
x
n
i
} such that x
n
i
j
⇀
x
′, x
′
∈
C. Without loss of generality, we may assume that x
n
i
⇀
x
′. From the setting v
n
=
F
r
n
x
n and (10), we have
(38)
〈
y
-
v
n
i
,
A
v
n
i
〉
+
1
r
n
i
〈
y
-
v
n
i
,
v
n
i
-
x
n
i
〉
≥
0
,
∀
y
∈
C
.
For t
∈
(
0,1
) and v
∈
C, let v
t
=
t
v
+
(
1
-
t
)
x
′. Since v
∈
C and x
′
∈
C, we have v
t
∈
C and
(39)
〈
v
t
-
v
n
i
,
A
v
t
〉
≥
〈
v
t
-
v
n
i
,
A
v
t
-
A
v
n
i
〉
-
〈
v
t
-
v
n
i
,
v
n
i
-
x
n
i
r
n
i
〉
.
Since A is a monotone and v
n
i
-
x
n
i
→
0, we obtain
(40)
〈
v
t
-
x
′
,
A
v
t
〉
=
lim
⁡
i
→
∞
〈
v
t
-
v
n
i
,
A
v
t
〉
≥
0
.
By the continuity of A, if t
→
0 then 〈
v
-
x
′
,
A
v
〉
≥
0
,
∀
v
∈
C. Therefore, x
′
∈
VI
⁡
(
C
,
A
).
On the other hand, sinceu
n
=
T
r
n
w
n, from (11) we have
(41)
〈
y
-
u
n
i
,
T
u
n
i
〉
-
1
r
n
i
〈
y
-
u
n
i
,
(
1
+
r
n
i
)
u
n
i
-
w
n
i
〉
≤
0
,
iiiiiiiiiiiiiiiiiiiippppiiiiiiiiiiiiiiiiii
∀
y
∈
C
.
For t
∈
(
0,1
) and v
∈
C, let v
t
=
t
v
+
(
1
-
t
)
x
′. Since v
∈
C and x
′
∈
C, we have v
t
∈
C and
(42)
〈
u
n
i
-
v
t
,
T
v
t
〉
≥
〈
u
n
i
-
v
t
,
T
v
t
〉
+
〈
v
t
-
u
n
i
,
T
u
n
i
〉
-
1
r
n
i
〈
v
t
-
u
n
i
,
(
1
+
r
n
i
)
u
n
i
-
w
n
i
〉
=
-
〈
v
t
-
u
n
i
,
T
v
t
-
T
u
n
i
〉
-
1
r
n
i
〈
v
t
-
u
n
i
,
u
n
i
-
w
n
i
〉
-
〈
v
t
-
u
n
i
,
u
n
i
〉
≥
-
∥
v
t
-
u
n
i
∥
2
-
1
r
n
i
〈
v
t
-
u
n
i
,
u
n
i
-
w
n
i
〉
-
〈
v
t
-
u
n
i
,
u
n
i
〉
=
-
1
r
n
i
〈
v
t
-
u
n
i
,
u
n
i
-
w
n
i
〉
-
〈
v
t
-
u
n
i
,
v
t
〉
.
It follows from u
n
-
w
n
→
0 as n
→
∞, we get 〈
x
′
-
v
t
,
T
v
t
〉
≥
-
〈
v
t
-
x
′
,
v
t
〉 and hence -
〈
v
-
x
′
,
T
v
t
〉
≥
-
〈
v
-
x
′
,
v
t
〉
,
∀
v
∈
C. By the continuity of T, if t
→
0 then -
〈
v
-
x
′
,
T
x
′
〉
≥
-
〈
v
-
x
′
,
x
′
〉
,
∀
v
∈
C. Let v
=
T
x
′, we have x
′
=
T
x
′; therefore, x
′
∈
F
(
T
). Consequently, we conclude that x
′
∈
F
=
F
(
S
)
∩
F
(
T
)
∩
VI
⁡
(
C
,
A
).
Finally, we show thatx
n
→
x
′, where x
′
=
P
F
x
0. Since x
n
+
1
=
P
C
n
∩
Q
n
x
0 and x
′
∈
F
⊂
C
n
∩
Q
n, we get
(43)
∥
x
n
+
1
-
x
0
∥
≤
∥
x
′
-
x
0
∥
,
∀
n
≥
0
.
If  x
¯
=
P
F
x
0, it follows from (43), and the lower semicontinuity of the norm that
(44)
∥
x
′
-
x
0
∥
≤
∥
x
¯
-
x
0
∥
≤
liminf
⁡
i
→
∞
∥
x
n
i
-
x
0
∥
≤
limsup
⁡
i
→
∞
∥
x
n
i
-
x
0
∥
≤
∥
x
′
-
x
0
∥
.
Thus, we obtain that lim
⁡
i
→
∞
∥
x
n
i
-
x
0
∥
=
∥
x
′
-
x
0
∥
=
∥
x
¯
-
x
0
∥. Using the Kadec-Klee property of H, we obtain that
(45)
lim
⁡
i
→
∞
x
n
i
=
x
′
=
x
¯
.
Since {
x
n
i
} is an arbitrary weakly convergent subsequence of {
x
n
}, we can conclude that {
x
n
} converges strongly to x
′, where x
′
=
P
F
x
0. This completes the proof.Corollary 3.
LetC be a nonempty bounded closed and convex subset of a real Hilbert space H. Let T
:
C
→
C be a continuous pseudocontractive mapping and let A
:
C
→
H be a continuous monotone mapping. Let S
=
{
T
(
s
)
:
0
≤
s
<
∞
} be a nonexpansive semigroup on C such that liminf
⁡
n
→
0
μ
n
=
0, limsup
⁡
n
→
0
μ
n
>
0, and lim
⁡
n
→
0
(
μ
n
+
1
-
μ
n
)
=
0. Let {
r
n
}
⊂
(
0
,
∞
) such that liminf
⁡
n
→
∞
r
n
>
0 and suppose F
=
F
(
S
)
∩
F
(
T
)
∩
V
I
(
C
,
A
)
≠
∅. The mappings T
r
n and F
r
n are defined by (10) and (11). Let {
x
n
} be a sequences generated by x
0
∈
C and
(46)
y
n
=
S
(
μ
n
)
T
r
n
F
r
n
,
C
n
=
{
z
∈
C
∣
∥
y
n
-
z
∥
≤
∥
x
n
-
z
∥
}
,
Q
n
=
{
z
∈
C
∣
〈
x
n
-
z
,
x
0
-
x
n
〉
≥
0
}
,
x
n
+
1
=
P
C
n
∩
Q
n
x
0
,
∀
n
≥
0
.
Then the sequence {
x
n
} converges strongly to P
F
x
0.Proof.
Puttingα
n
=
β
n
=
γ
n
=
0
,
∀
n
≥
0 in Theorem 2, we can obtain the result.


## 3.2. The Shrinking Projection Method

Theorem 4.
LetC be a nonempty bounded closed and convex subset of a real Hilbert space H. Let T
:
C
→
C be a continuous pseudocontractive mapping and let A
:
C
→
H be a continuous monotone mapping. Let S
=
{
S
(
s
)
:
0
≤
s
<
∞
} be a nonexpansive semigroup on C such that liminf
⁡
n
→
0
μ
n
=
0, limsup
⁡
n
→
0
μ
n
>
0, and lim
⁡
n
→
0
(
μ
n
+
1
-
μ
n
)
=
0. Let {
α
n
}, {
β
n
}, and {
γ
n
} be the sequences in [
0
,
a
) for some a
∈
[
0,1
), {
r
n
}
⊂
(
0
,
∞
) such that liminf
⁡
n
→
∞
r
n
>
0 and suppose F
=
F
(
S
)
∩
F
(
T
)
∩
V
I
(
C
,
A
)
≠
∅. The mappings T
r
n and F
r
n are defined by (10) and (11). Let {
x
n
} be sequences generated by x
0
∈
C and
(47)
w
n
=
γ
n
x
n
+
(
1
-
γ
n
)
F
r
n
x
n
,
z
n
=
β
n
w
n
+
(
1
-
β
n
)
T
r
n
w
n
,
y
n
=
α
n
z
n
+
(
1
-
α
n
)
S
(
μ
n
)
z
n
,
C
n
+
1
=
{
z
∈
C
n
∣
∥
y
n
-
z
∥
≤
∥
x
n
-
z
∥
}
,
x
n
+
1
=
P
C
n
+
1
x
0
,
∀
n
≥
0
.
Then the sequence {
x
n
} converges strongly to P
F
x
0.Proof.
First, we show thatF
⊂
C
n. By induction, it is obvious that F
⊂
C
1. Suppose that F
⊂
C
k for some k
≥
0, so we have x
*
∈
F
⊂
C
k such that ∥
y
k
-
x
*
∥
≤
∥
x
k
-
x
*
∥. Then, we get x
*
∈
C
n
+
1. Therefore F
⊂
C
n for all n
≥
0.
On the other hand, we show thatC
n is closed and convex for all n
≥
0. By mathematical induction, it is obvious that C
1
=
C is closed and convex. Suppose that C
k is closed and convex for some k
≥
0. For z
∈
C
k, we have that ∥
y
k
-
z
∥
≤
∥
x
k
-
z
∥ is equivalent to ∥
y
k
-
x
k
∥
2
+
2
〈
y
k
-
x
k
,
x
k
-
z
〉
≤
0. Thus, we have C
k
+
1 is closed and convex for all n
≥
0. Therefore, C
n is closed and convex for all n
≥
0. This implies that {
x
n
} is well defined.
Next, we show that{
x
n
} is bounded. From the metric projection property and (47), we have x
n
=
P
C
n
x
0 and 〈
x
n
-
x
0
,
x
¯
-
x
n
〉
≥
0 for all x
¯
∈
F
⊂
C
n and n
≥
0. Consider
(48)
0
≤
〈
x
n
-
x
0
,
x
¯
-
x
n
〉
=
〈
x
n
-
x
0
,
x
¯
-
x
0
+
x
0
-
x
n
〉
=
〈
x
n
-
x
0
,
x
¯
-
x
0
〉
+
〈
x
n
-
x
0
,
x
0
-
x
n
〉
=
∥
x
n
-
x
0
∥
∥
x
¯
-
x
0
∥
-
∥
x
n
-
x
0
∥
2
.
It follows that ∥
x
n
-
x
0
∥
≤
∥
x
¯
-
x
0
∥ for all x
¯
∈
F and n
≥
0. This implied {
x
n
} is bounded. So, {
F
r
n
x
n
}, {
T
r
n
w
n
}, {
S
(
μ
n
)
z
n
}, {
y
n
}, {
z
n
}, and {
w
n
} are bounded.
Next, we show that(49)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0
.
From the metric projection property and (47), we have x
n
=
P
C
n
x
0, x
n
+
1
=
P
C
n
+
1
x
0
∈
C
n
+
1
⊂
C
n, and 〈
x
n
-
x
0
,
x
n
+
1
-
x
n
〉
≥
0. As same as the prove of (48), we get ∥
x
n
-
x
0
∥
≤
∥
x
n
+
1
-
x
0
∥ for all n
≥
0. Thus, {
∥
x
n
-
x
0
∥
} is nondecreasing. Since {
∥
x
n
-
x
0
∥
} is bounded and nondecreasing, there exists the limit of {
∥
x
n
-
x
0
∥
}. Similar to the proved of (25), we get (49) is hold.
Since sequence{
x
n
} is bounded, we can choose subsequence {
x
n
i
} of {
x
n
} and assume that x
n
i
⇀
x
′. Similar to the proof of Theorem 2, we also have x
′
∈
F.
Finally, we show thatx
n
→
x
′, where x
′
=
P
F
x
0. Since x
n
=
P
C
n
x
0 and x
′
∈
F
⊂
C
n, we have
(50)
∥
x
n
-
x
0
∥
≤
∥
x
¯
-
x
0
∥
.
It follows from (50), if x
¯
=
P
F
x
0 and the lower semicontinuity of the norm
(51)
∥
x
′
-
x
0
∥
≤
∥
x
¯
-
x
0
∥
≤
liminf
⁡
i
→
∞
∥
x
n
i
-
x
0
∥
≤
limsup
⁡
i
→
∞
∥
x
n
i
-
x
0
∥
≤
∥
x
′
-
x
0
∥
,
thus, we obtain that lim
⁡
i
→
∞
∥
x
n
i
-
x
0
∥
=
∥
x
¯
-
x
0
∥
=
∥
x
′
-
x
0
∥. Using the Kadec-Klee property of H, we obtain that
(52)
lim
⁡
i
→
∞
x
n
i
=
x
¯
=
x
′
.
Since {
x
n
i
} is an arbitrary weakly convergent subsequence of {
x
n
}, we can conclude that {
x
n
} converges strongly to x
′, where x
′
=
P
F
x
0.Corollary 5.
LetC be a nonempty bounded closed and convex subset of a real Hilbert space H. Let T
:
C
→
C be a continuous pseudocontractive mapping and let A
:
C
→
H be a continuous monotone mapping. Let S
=
{
T
(
s
)
:
0
≤
s
<
∞
} be a nonexpansive semigroup on C such that liminf
⁡
n
→
0
μ
n
=
0, limsup
⁡
n
→
0
μ
n
>
0, and lim
⁡
n
→
0
(
μ
n
+
1
-
μ
n
)
=
0. Let {
r
n
}
⊂
(
0
,
∞
) such that liminf
⁡
n
→
∞
r
n
>
0 and suppose F
=
F
(
S
)
∩
F
(
T
)
∩
V
I
(
C
,
A
)
≠
∅. The mappings T
r
n and F
r
n are defined by (10) and (11). Let {
x
n
} be a sequences generated by x
0
∈
C and
(53)
y
n
=
S
(
μ
n
)
T
r
n
F
r
n
,
C
n
+
1
=
{
z
∈
C
n
∣
∥
y
n
-
z
∥
≤
∥
x
n
-
z
∥
}
,
x
n
+
1
=
P
C
n
+
1
x
0
,
∀
n
≥
0
.
Then the sequence {
x
n
} converges strongly to P
F
x
0.Proof.
Puttingα
n
=
β
n
=
γ
n
=
0
,
∀
n
≥
0 in Theorem 4, we can obtain the result.Remark 6.
According to nonexpansive semigroup mapping, it will be interesting if we replace the semigroupS
=
(
N
,
+
) by an additive positive real numbers of a commutative semigroup or a left amenable semigroup or a left reversible semigroup with using an asymptotically invariant sequence. See [31, 32].



---

*Source: 813701-2014-05-21.xml*