# Strong and Weak Convergence Criteria of Composite Iterative Algorithms for Systems of Generalized Equilibria

Authors: Lu-Chuan Ceng; Cheng-Wen Liao; Chin-Tzong Pang; Ching-Feng Wen; Zhao-Rong Kong
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We first introduce and analyze one iterative algorithm by using the composite shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. We prove a strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions. Our results improve and extend the corresponding results in the earlier and recent literature.


---

## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈
·
,
·
〉 and norm ∥
·
∥,
C a nonempty closed convex subset of H, and P
C the metric projection of H onto C. Let S
:
C
→
H be a nonlinear mapping on C. We denote by Fix
⁡
(
S
) the set of fixed points of S and by R the set of all real numbers. A mapping V is called strongly positive on H if there exists a constant γ
¯
>
0 such that
(1)
〈
V
x
,
x
〉
≥
γ
¯
∥
x
∥
2
,
∀
x
∈
H
.
A mapping S
:
C
→
H is called L-Lipschitz continuous if there exists a constant L
≥
0 such that
(2)
∥
S
x
-
S
y
∥
≤
L
∥
x
-
y
∥
,
∀
x
,
y
∈
C
.
In particular, if L
=
1 then S is called a nonexpansive mapping; if L
∈
[
0,1
) then A is called a contraction.LetA
:
C
→
H be a nonlinear mapping on C. We consider the following variational inequality problem (VIP): find a point x
∈
C such that
(3)
〈
A
x
,
y
-
x
〉
≥
0
,
∀
y
∈
C
.
The solution set of VIP (3) is denoted by VI
⁡
(
C
,
A
).The VIP (3) was first discussed by Lions [1] and now is well known; there are a lot of different approaches towards solving VIP (3) in finite-dimensional and infinite-dimensional spaces, and the research is intensively continued. The VIP (3) has many applications in computational mathematics, mathematical physics, operations research, mathematical economics, optimization theory, and other fields; see, for example, [2–5]. It is well known that if A is strongly monotone and Lipschitz-continuous mapping on C, then VIP (3) has a unique solution. Not only are the existence and uniqueness of solutions important topics in the study of VIP (3), but also how to actually find a solution of VIP (3) is important. Up to now, there have been many iterative algorithms in the literature, for finding approximate solutions of VIP (3) and its extended versions; see, for example, [6–11].In 1976, Korpelevič [12] proposed an iterative algorithm for solving the VIP (3) in Euclidean space R
n:
(4)
y
n
=
P
C
(
x
n
-
τ
A
x
n
)
,
x
n
+
1
=
P
C
(
x
n
-
τ
A
y
n
)
,
∀
n
≥
0
,
with τ
>
0 a given number, which is known as the extragradient method. The literature on the VIP is vast and Korpelevich's extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [10, 11, 13–23] and references therein, to name but a few.Letφ
:
C
→
R be a real-valued function,   A
:
H
→
H a nonlinear mapping, and Θ
:
C
×
C
→
R a bifunction. In 2008, Peng and Yao [18] introduced the following generalized mixed equilibrium problem (GMEP) of finding x
∈
C such that
(5)
Θ
(
x
,
y
)
+
φ
(
y
)
-
φ
(
x
)
+
〈
A
x
,
y
-
x
〉
≥
0
,
∀
y
∈
C
.
We denote the set of solutions of GMEP (5) by GMEP
⁡
(
Θ
,
φ
,
A
). The GMEP (5) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games, and others. The GMEP is further considered and studied; see, for example, [20, 23–28].Ifφ
=
0, then GMEP (5) reduces to the generalized equilibrium problem (GEP) which is to find x
∈
C such that
(6)
Θ
(
x
,
y
)
+
〈
A
x
,
y
-
x
〉
≥
0
,
∀
y
∈
C
.
It is introduced and studied by S. Takahashi and W. Takahashi [29]. The set of solutions of GEP is denoted by GEP
⁡
(
Θ
,
A
).IfA
=
0, then GMEP (5) reduces to the mixed equilibrium problem (MEP) which is to find x
∈
C such that
(7)
Θ
(
x
,
y
)
+
φ
(
y
)
-
φ
(
x
)
≥
0
,
∀
y
∈
C
.
It is considered and studied in [30–32]. The set of solutions of MEP is denoted by MEP
⁡
(
Θ
,
φ
).Ifφ
=
0, A
=
0, then GMEP (5) reduces to the equilibrium problem (EP) which is to find x
∈
C such that
(8)
Θ
(
x
,
y
)
≥
0
,
∀
y
∈
C
.
It is considered and studied in [33, 34]. The set of solutions of EP is denoted by EP
⁡
(
Θ
). It is worth mentioning that the EP is a unified model of several problems, namely, variational inequality problems, optimization problems, saddle point problems, complementarity problems, fixed point problems, Nash equilibrium problems, and so forth.Throughout this paper, we assume as in [18] that Θ
:
C
×
C
→
R is a bifunction satisfying conditions (H1)–(H4) and φ
:
C
→
R is a lower semicontinuous and convex function with restriction (H5), where(H1)
Θ
(
x
,
x
)
=
0 for all x
∈
C;
(H2)
Θ is monotone; that is, Θ
(
x
,
y
)
+
Θ
(
y
,
x
)
≤
0 for any x
,
y
∈
C;
(H3)
Θ is upper-hemicontinuous; that is, for each x
,
y
,
z
∈
C,
(9)
limsup
⁡
t
→
0
+
Θ
(
t
z
+
(
1
-
t
)
x
,
y
)
≤
Θ
(
x
,
y
)
;
(H4)
Θ
(
x
,
·
) is convex and lower semicontinuous for each x
∈
C;
(H5)
for eachx
∈
H and r
>
0 there exists a bounded subset D
x
⊂
C and y
x
∈
C such that, for any z
∈
C
∖
D
x,
(10)
Θ
(
z
,
y
x
)
+
φ
(
y
x
)
-
φ
(
z
)
+
1
r
〈
y
x
-
z
,
z
-
x
〉
<
0
.Given a positive numberr
>
0, let S
r
(
Θ
,
φ
)
:
H
→
C be the solution set of the auxiliary mixed equilibrium problem; that is, for each x
∈
H,
(11)
S
r
(
Θ
,
φ
)
(
x
)
≔
{
y
∈
C
:
Θ
(
y
,
z
)
+
φ
(
z
)
-
φ
(
y
)
1
r
hlhh
+
1
r
〈
K
′
(
y
)
-
K
′
(
x
)
,
z
-
y
〉
≥
0
,
∀
z
∈
C
}
.
In particular, whenever K
(
x
)
=
(
1
2
)
∥
x
∥
2, ∀
x
∈
H, S
r
(
Θ
,
φ
) is rewritten as T
r
(
Θ
,
φ
).LetΘ
1
,
Θ
2
:
C
×
C
→
R be two bifunctions and A
1
,
A
2
:
C
→
H two nonlinear mappings. Consider the following system of generalized equilibrium problems (SGEP): find (
x
*
,
y
*
)
∈
C
×
C such that
(12)
Θ
1
(
x
*
,
x
)
+
〈
A
1
y
*
,
x
-
x
*
〉
+
1
ν
1
〈
x
*
-
y
*
,
x
-
x
*
〉
≥
0
,
∀
x
∈
C
,
Θ
2
(
y
*
,
y
)
+
〈
A
2
x
*
,
y
-
y
*
〉
+
1
ν
2
〈
y
*
-
x
*
,
y
-
y
*
〉
≥
0
,
∀
y
∈
C
,
where ν
1
>
0 and ν
2
>
0 are two constants. It is introduced and studied in [19]. Whenever Θ
1
≡
Θ
2
≡
0, the SGEP reduces to a system of variational inequalities, which is considered and studied in [13]. It is worth mentioning that the system of variational inequalities is a tool to solve the Nash equilibrium problem for noncooperative games.In 2010, Ceng and Yao [19] transformed the SGEP into a fixed point problem in the following way.Proposition CY (see [19]).Let
Θ
1
,
Θ
2
:
C
×
C
→
R
be two bifunctions satisfying conditions (H1)–(H4) and let
A
k
:
C
→
H
be
ζ
k
-inverse strongly monotone for
k
=
1,2
. Let
ν
k
∈
(
0,2
ζ
k
)
for
k
=
1,2
. Then
(
x
*
,
y
*
)
∈
C
×
C
is a solution of SGEP (12) if and only if
x
*
is a fixed point of the mapping
G
:
C
→
C
defined by
G
=
T
ν
1
Θ
1
(
I
-
ν
1
A
1
)
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
, where
y
*
=
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
x
*
. Here, one denotes the fixed point set of
G
by
S
G
E
P
(
G
)
.Let{
T
n
}
n
=
1
∞ be an infinite family of nonexpansive mappings on H and {
λ
n
}
n
=
1
∞ a sequence of nonnegative numbers in [
0,1
]. For any n
≥
1, define a mapping W
n on H as follows:
(13)
U
n
,
n
+
1
=
I
,
U
n
,
n
=
λ
n
T
n
U
n
,
n
+
1
+
(
1
-
λ
n
)
I
,
U
n
,
n
-
1
=
λ
n
-
1
T
n
-
1
U
n
,
n
+
(
1
-
λ
n
-
1
)
I
,

⋮

U
n
,
k
=
λ
k
T
k
U
n
,
k
+
1
+
(
1
-
λ
k
)
I
,
U
n
,
k
-
1
=
λ
k
-
1
T
k
-
1
U
n
,
k
+
(
1
-
λ
k
-
1
)
I
,

⋮

U
n
,
2
=
λ
2
T
2
U
n
,
3
+
(
1
-
λ
2
)
I
,
W
n
=
U
n
,
1
=
λ
1
T
1
U
n
,
2
+
(
1
-
λ
1
)
I
.
Such a mapping W
n is called the W-mapping generated by T
n
,
T
n
-
1
,
…
,
T
1 and λ
n
,
λ
n
-
1
,
…
,
λ
1.In 2011, for the case whereC
=
H, Yao et al. [25] proposed the following hybrid iterative algorithm:
(14)
Θ
(
y
n
,
z
)
+
φ
(
z
)
-
φ
(
y
n
)
+
1
r
〈
K
′
(
y
n
)
-
K
′
(
x
n
)
,
z
-
y
n
〉
≥
0
,
z
∈
H
,
x
n
+
1
=
α
n
(
u
+
γ
f
(
x
n
)
)
+
β
n
x
n
+
(
(
1
-
β
n
)
I
-
α
n
(
I
+
μ
V
)
)
W
n
y
n
,
∀
n
≥
1
,
where f
:
H
→
H is a contraction, K
:
H
→
R is differentiable and strongly convex, {
α
n
}
,
{
β
n
}
⊂
(
0,1
), and x
0
,
u
∈
H are given, for finding a common element of the set MEP
⁡
(
Θ
,
φ
) and the fixed point set ∩
n
=
1
∞
Fix
⁡
(
T
n
) of an infinite family of nonexpansive mappings {
T
n
}
n
=
1
∞ on H. They proved the strong convergence of the sequence generated by the hybrid iterative algorithm (14) to a point x
*
∈
Ω
:
=
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
MEP
⁡
(
Θ
,
φ
) under some appropriate conditions. This point x
* also solves the following optimization problem:
(15)
min
⁡
x
∈
Ω
μ
2
〈
V
x
,
x
〉
+
1
2
∥
x
-
u
∥
2
-
h
(
x
)
,
where h
:
H
→
R is the potential function of γ
f.Letf
:
H
→
H be a contraction and V a strongly positive bounded linear operator on H. Assume that φ
:
H
→
R is a lower semicontinuous and convex functional, that Θ
,
Θ
1
,
Θ
2
:
H
×
H
→
R satisfy conditions (H1)–(H4), and that A
,
A
1
,
A
2
:
H
→
H are inverse strongly monotone. Let the mapping G be defined as in Proposition CY. Very recently, Ceng et al. [20] introduced the following hybrid extragradient-like iterative algorithm:
(16)
z
n
=
S
r
n
(
Θ
,
φ
)
(
x
n
-
r
n
A
x
n
)
,
x
n
+
1
=
α
n
(
u
+
γ
f
(
x
n
)
)
+
β
n
x
n
+
(
(
1
-
β
n
)
I
-
α
n
(
I
+
μ
V
)
)
W
n
G
z
n
,
∀
n
≥
0
,
for finding a common solution of GMEP (5), SGEP (12), and the fixed point problem of an infinite family of nonexpansive mappings {
T
n
}
n
=
1
∞ on H, where {
r
n
}
⊂
(
0
,
∞
), {
α
n
}
,
{
β
n
}
⊂
(
0,1
), ν
k
∈
(
0,2
ζ
k
), k
=
1,2, and x
0
,
u
∈
H are given. The authors proved the strong convergence of the sequence generated by the hybrid iterative algorithm (16) to a point x
*
∈
Ω
:
=
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
GMEP
⁡
(
Θ
,
φ
,
A
)
∩
SGEP
(
G
) under some suitable conditions. This point x
* also solves the following optimization problem:
(17)
min
⁡
x
∈
Ω
μ
2
〈
V
x
,
x
〉
+
1
2
∥
x
-
u
∥
2
-
h
(
x
)
,
where h
:
H
→
R is the potential function of γ
f.On the other hand, letC be a nonempty subset of a normed space X. A mapping S
:
C
→
C is called uniformly Lipschitzian if there exists a constant ℒ
>
0 such that
(18)
∥
S
n
x
-
S
n
y
∥
≤
ℒ
∥
x
-
y
∥
,
∀
n
≥
1
,
∀
x
,
y
∈
C
.
Recently, Kim and Xu [35] introduced the concept of asymptotically k-strict pseudocontractive mappings in a Hilbert space as below.Definition 1.
LetC be a nonempty subset of a Hilbert space H. A mapping S
:
C
→
C is said to be an asymptotically k-strict pseudocontractive mapping with sequence {
γ
n
} if there exist a constant k
∈
[
0,1
) and a sequence {
γ
n
} in [
0
,
∞
) with lim
⁡
n
→
∞
γ
n
=
0 such that
(19)
∥
S
n
x
-
S
n
y
∥
2
≤
(
1
+
γ
n
)
∥
x
-
y
∥
2
+
k
∥
x
-
S
n
x
-
(
y
-
S
n
y
)
∥
2
,
hhhhhlh
∀
n
≥
1
,
∀
x
,
y
∈
C
.
They studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptoticallyk-strict pseudocontractive mapping with sequence {
γ
n
} is a uniformly ℒ-Lipschitzian mapping with ℒ
=
sup
⁡
{
(
k
+
1
+
(
1
-
k
)
γ
n
)
/
(
1
+
k
)
:
n
≥
1
}. Subsequently, Sahu et al. [36] considered the concept of asymptotically k-strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.Definition 2.
LetC be a nonempty subset of a Hilbert space H. A mapping S
:
C
→
C is said to be an asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {
γ
n
} if there exist a constant k
∈
[
0,1
) and a sequence {
γ
n
} in [
0
,
∞
) with lim
⁡
n
→
∞
γ
n
=
0 such that
(20)
limsup
⁡
n
→
∞
sup
⁡
x
,
y
∈
C
(
∥
S
n
x
-
S
n
y
∥
2
-
(
1
+
γ
n
)
∥
x
-
y
∥
2
hhhhhhhlhhh
-
k
∥
x
-
S
n
x
-
(
y
-
S
n
y
)
∥
2
)
≤
0
.
Putc
n
:
=
max
⁡
{
0
,
sup
⁡
x
,
y
∈
C
(
∥
S
n
x
-
S
n
y
∥
2
-
(
1
+
γ
n
)
∥
x
-
y
∥
2
-
k
∥
x
-
S
n
x
-
(
y
-
S
n
y
)
∥
2
)
}. Then c
n
≥
0
(
∀
n
≥
1
), c
n
→
0
(
n
→
∞
), and (13) reduces to the relation
(21)
∥
S
n
x
-
S
n
y
∥
2
≤
(
1
+
γ
n
)
∥
x
-
y
∥
2
+
k
∥
x
-
S
n
x
-
(
y
-
S
n
y
)
∥
2
+
c
n
,
hhhhhhhhhl
∀
n
≥
1
,
∀
x
,
y
∈
C
.
Wheneverc
n
=
0 for all n
≥
1 in (21) then S is an asymptotically k-strict pseudocontractive mapping with sequence {
γ
n
}. In 2009, Sahu et al. [36] derived the weak and strong convergence of the modified Mann iteration processes for an asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {
γ
n
}. More precisely, they first established one weak convergence theorem for the following iterative scheme:
(22)
x
1
=
x
∈
C
chosen
arbitrarily
,
x
n
+
1
=
(
1
-
α
n
)
x
n
+
α
n
S
n
x
n
,
∀
n
≥
1
,
where 0
<
δ
≤
α
n
≤
1
-
k
-
δ, ∑
n
=
1
∞
‍
α
n
c
n
<
∞, and ∑
n
=
1
∞
‍
γ
n
<
∞, and then obtained another strong convergence theorem for the following iterative scheme:
(23)
x
1
=
x
∈
C
chosen
arbitrary
,
y
n
=
(
1
-
α
n
)
x
n
+
α
n
S
n
x
n
,
C
n
=
{
z
∈
C
:
∥
y
n
-
z
∥
2
≤
∥
x
n
-
z
∥
2
+
θ
n
}
,
Q
n
=
{
z
∈
C
:
〈
x
n
-
z
,
x
-
x
n
〉
≥
0
}
,
x
n
+
1
=
P
C
n
∩
Q
n
x
,
∀
n
≥
1
,
where 0
<
δ
≤
α
n
≤
1
-
k, θ
n
=
c
n
+
γ
n
Δ
n, and Δ
n
=
sup
⁡
{
∥
x
n
-
z
∥
2
:
z
∈
Fix
⁡
(
S
)
}
<
∞. Subsequently, the above iterative schemes are extended to develop new iterative algorithms for finding a common solution of the VIP and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense; see, for example, [10, 22].In 2009, Yao et al. [30] proposed and analyzed iterative algorithms for finding a common element of the set of fixed points of an asymptotically k-strict pseudocontraction and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Very recently, motivated by Yao et al. [30], Cai and Bu [26] introduced and analyzed the following iterative algorithm by the hybrid shrinking projection method:
(24)
pick
any
x
0
∈
H
,
set
C
1
=
C
,
x
1
=
P
C
1
x
0
,
u
n
=
T
r
M
,
n
(
Θ
M
,
φ
M
)
(
I
-
r
M
,
n
A
M
)
T
r
M
-
1
,
n
(
Θ
M
-
1
,
φ
M
-
1
)
×
(
I
-
r
M
-
1
,
n
A
M
-
1
)
⋯
T
r
1
,
n
(
Θ
1
,
φ
1
)
(
I
-
r
1
,
n
A
1
)
x
n
,
z
n
=
P
C
(
I
-
λ
N
,
n
B
N
)
×
P
C
(
I
-
λ
N
-
1
,
n
B
N
-
1
)
⋯
P
C
(
I
-
λ
2
,
n
B
2
)
×
P
C
(
I
-
λ
1
,
n
B
1
)
u
n
,
k
n
=
δ
n
z
n
+
(
1
-
δ
n
)
S
n
z
n
,
y
n
=
(
1
-
α
n
)
x
n
+
α
n
k
n
,
C
n
+
1
=
{
z
∈
C
n
:
∥
y
n
-
z
∥
2
≤
∥
x
n
-
z
∥
2
+
θ
n
}
,
x
n
+
1
=
P
C
n
+
1
x
0
,
∀
n
≥
0
,
for finding a common element of the set ∩
k
=
1
M
GMEP
⁡
(
Θ
k
,
φ
k
,
A
k
) of solutions of finitely many generalized mixed equilibrium problems, the set ∩
i
=
1
N
VI
⁡
(
C
,
B
i
) of solutions of finitely many variational inequalities for inverse strong monotone mappings {
B
i
}
i
=
1
N, and the set Fix
⁡
(
S
) of fixed points of an asymptotically k-strict pseudocontractive mapping S in the intermediate sense (provided that Ω
=
∩
k
=
1
M
GMEP
⁡
(
Θ
k
,
φ
k
,
A
k
)
∩
∩
i
=
1
N
VI
⁡
(
C
,
B
i
)
∩
Fix
⁡
(
S
) is nonempty and bounded), where θ
n
=
γ
n
Δ
n
2
+
c
n, Δ
n
=
sup
⁡
{
∥
x
n
-
p
∥
:
p
∈
Ω
}
<
∞, {
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
), {
r
k
,
n
}
⊂
[
e
k
,
f
k
]
⊂
(
0,2
μ
k
), i
∈
{
1,2
,
…
,
N
}, k
∈
{
1,2
,
…
,
M
}. It was proven in [26] that under appropriate conditions {
x
n
} converge strongly to P
Ω
x
0.Motivated and inspired by the above facts, we first introduce and analyze one iterative algorithm by using a composite shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another iterative algorithm involving no shrinking projection method and derive its weak convergence under mild assumptions. Our results improve and extend the corresponding results in the earlier and recent literature.


## 2. Preliminaries

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. We use the notations x
n
⇀
x and x
n
→
x to indicate the weak convergence of {
x
n
} to x and the strong convergence of {
x
n
} to x, respectively. Moreover, we use ω
w
(
x
n
) to denote the weak ω-limit set of {
x
n
}; that is,
(25)
ω
w
(
x
n
)
≔
{
x
∈
H
:
x
n
i
⇀
x
for
some
h
h
subsequence
{
x
n
i
}
of
{
x
n
}
}
.Definition 3.
A mappingA
:
C
→
H is called(i)
monotone if(26)
〈
A
x
-
A
y
,
x
-
y
〉
≥
0
,
∀
x
,
y
∈
C
;
(ii)
η-strongly monotone if there exists a constant η
>
0 such that
(27)
〈
A
x
-
A
y
,
x
-
y
〉
≥
η
∥
x
-
y
∥
2
,
∀
x
,
y
∈
C
;
(iii)
ζ-inverse strongly monotone if there exists a constant ζ
>
0 such that
(28)
〈
A
x
-
A
y
,
x
-
y
〉
≥
ζ
∥
A
x
-
A
y
∥
2
,
∀
x
,
y
∈
C
.
It is easy to see that the projectionP
C is 1-inverse strongly monotone. The inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields.Definition 4.
A differentiable functionK
:
H
→
R is called(i)
convex if(29)
K
(
y
)
-
K
(
x
)
≥
〈
K
′
(
x
)
,
y
-
x
〉
,
∀
x
,
y
∈
H
,
where K
′
(
x
) is the Fréchet derivative of K at x;
(ii)
strongly convex if there exists a constantσ
>
0 such that
(30)
K
(
y
)
-
K
(
x
)
-
〈
K
′
(
x
)
,
y
-
x
〉
≥
σ
2
∥
x
-
y
∥
2
,
∀
x
,
y
∈
H
.
It is easy to see that ifK
:
H
→
R is a differentiable strongly convex function with constant σ
>
0 then K
′
:
H
→
H is strongly monotone with constant σ
>
0.The metric (or nearest point) projection fromH onto C is the mapping P
C
:
H
→
C which assigns to each point x
∈
H the unique point P
C
x
∈
C satisfying the property
(31)
∥
x
-
P
C
x
∥
=
inf
⁡
y
∈
C
∥
x
-
y
∥
=
:
d
(
x
,
C
)
.Some important properties of projections are gathered in the following proposition.Proposition 5.
For given
x
∈
H
and
z
∈
C,(i)
z
=
P
C
x
⇔
〈
x
-
z
,
y
-
z
〉
≤
0, ∀
y
∈
C;
(ii)
z
=
P
C
x
⇔
∥
x
-
z
∥
2
≤
∥
x
-
y
∥
2
-
∥
y
-
z
∥
2, ∀
y
∈
C;
(iii)
〈
P
C
x
-
P
C
y
,
x
-
y
〉
≥
∥
P
C
x
-
P
C
y
∥
2, ∀
y
∈
H.(This implies that
P
C
is nonexpansive and monotone.)By using the technique of [32], we can readily obtain the following elementary result.Proposition 6 (see [20, Lemma 1 and Proposition 1]).
LetC be a nonempty closed convex subset of a real Hilbert space H and let φ
:
C
→
R be a lower semicontinuous and convex function. Let Θ
:
C
×
C
→
R be a bifunction satisfying the conditions (H1)–(H4). Assume that(i)
K
:
H
→
R
is strongly convex with constant
σ
>
0
and the function
x
↦
〈
y
-
x
,
K
′
(
x
)
〉
is weakly upper semicontinuous for each
y
∈
H;
(ii)
for each
x
∈
H and r
>
0
there exists a bounded subset
D
x
⊂
C
and
y
x
∈
C
such that, for any
z
∈
C
∖
D
x,
(32)
Θ
(
z
,
y
x
)
+
φ
(
y
x
)
-
φ
(
z
)
+
1
r
〈
K
′
(
z
)
-
K
′
(
x
)
,
y
x
-
z
〉
<
0
.
Then the following hold:(a)
for each
x
∈
H
,
S
r
(
Θ
,
φ
)
(
x
)
≠
∅;
(b)
S
r
(
Θ
,
φ
)
is single valued;
(c)
S
r
(
Θ
,
φ
)
is nonexpansive if
K
′
is Lipschitz continuous with constant
ν
>
0
and
(33)
〈
K
′
(
x
1
)
-
K
′
(
x
2
)
,
u
1
-
u
2
〉
≤
〈
K
′
(
u
1
)
-
K
′
(
u
2
)
,
u
1
-
u
2
〉
,
hhhhhhhhlhhh
∀
(
x
1
,
x
2
)
∈
H
×
H
,
whereu
i
=
S
r
(
Θ
,
φ
)
(
x
i
) for i
=
1,2;
(d)
for alls
,
t
>
0 and x
∈
H,
(34)
〈
K
′
(
S
s
(
Θ
,
φ
)
x
)
-
K
′
(
S
t
(
Θ
,
φ
)
x
)
,
S
s
(
Θ
,
φ
)
x
-
S
t
(
Θ
,
φ
)
x
〉
≤
s
-
t
s
〈
K
′
(
S
s
(
Θ
,
φ
)
x
)
-
K
′
(
x
)
,
S
s
(
Θ
,
φ
)
x
-
S
t
(
Θ
,
φ
)
x
〉
;
(e)
Fix
⁡
(
S
r
(
Θ
,
φ
)
)
=
MEP
⁡
(
Θ
,
φ
);
(f)
MEP
⁡
(
Θ
,
φ
) is closed and convex.Remark 7.
In Proposition6, whenever Θ
:
C
×
C
→
R is a bifunction satisfying the conditions (H1)–(H4) and K
(
x
)
=
(
1
/
2
)
∥
x
∥
2, ∀
x
∈
H, we have, for any x
,
y
∈
H,
(35)
∥
S
r
(
Θ
,
φ
)
x
-
S
r
(
Θ
,
φ
)
y
∥
2
≤
〈
S
r
(
Θ
,
φ
)
x
-
S
r
(
Θ
,
φ
)
y
,
x
-
y
〉
(S
r
(
Θ
,
φ
) is firmly nonexpansive) and
(36)
∥
S
s
(
Θ
,
φ
)
x
-
S
t
(
Θ
,
φ
)
x
∥
≤
|
s
-
t
|
s
∥
S
s
(
Θ
,
φ
)
x
-
x
∥
,
hhh
∀
s
,
t
>
0
,
x
∈
H
.
In this case, S
r
(
Θ
,
φ
) is rewritten as T
r
(
Θ
,
φ
). If, in addition, φ
≡
0, then T
r
(
Θ
,
φ
) is rewritten as T
r
Θ; see [19, Lemma 2.1] for more details.We need some facts and tools in a real Hilbert spaceH which are listed as lemmas below.Lemma 8.
LetX be a real inner product space. Then the following inequality holds:
(37)
∥
x
+
y
∥
2
≤
∥
x
∥
2
+
2
〈
y
,
x
+
y
〉
,
∀
x
,
y
∈
X
.Lemma 9.
LetH be a real Hilbert space. Then the following hold:(a)
∥
x
-
y
∥
2
=
∥
x
∥
2
-
∥
y
∥
2
-
2
〈
x
-
y
,
y
〉 for all x
,
y
∈
H;
(b)
∥
λ
x
+
μ
y
∥
2
=
λ
∥
x
∥
2
+
μ
∥
y
∥
2
-
λ
μ
∥
x
-
y
∥
2 for all x
,
y
∈
H and λ
,
μ
∈
[
0,1
] with λ
+
μ
=
1;
(c)
if{
x
n
} is a sequence in H such that x
n
⇀
x, it follows that
(38)
limsup
⁡
n
→
∞
∥
x
n
-
y
∥
2
=
limsup
⁡
n
→
∞
∥
x
n
-
x
∥
2
+
∥
x
-
y
∥
2
,
∀
y
∈
H
.
We have the following crucial lemmas concerning theW-mappings defined by (13).Lemma 10 (see [37, Lemma 3.2]).
Let{
T
n
}
n
=
1
∞ be a sequence of nonexpansive self-mappings on H such that ∩
n
=
1
∞
Fix
⁡
(
T
n
)
≠
∅ and let {
λ
n
} be a sequence in (
0
,
b
] for some b
∈
(
0,1
). Then, for every x
∈
H and k
≥
1 the limit lim
⁡
n
→
∞
U
n
,
k
x exists, where U
n
,
k is defined by (13).Lemma 11 (see [37, Lemma 3.3]).
Let{
T
n
}
n
=
1
∞ be a sequence of nonexpansive self-mappings on H such that ∩
n
=
1
∞
Fix
⁡
(
T
n
)
≠
∅, and let {
λ
n
} be a sequence in (
0
,
b
] for some b
∈
(
0,1
). Then Fix
⁡
(
W
)
=
∩
n
=
1
∞
Fix
⁡
(
T
n
).Lemma 12 (see [38, Demiclosedness principle]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive self-mapping on C. Then I
-
T is demiclosed. That is, whenever {
x
n
} is a sequence in C weakly converging to some x
∈
C and the sequence {
(
I
-
T
)
x
n
} strongly converges to some y, it follows that (
I
-
T
)
x
=
y. Here I is the identity operator of H.Lemma 13.
LetA
:
C
→
H be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 5(i)) implies
(39)
u
∈
VI
⁡
(
C
,
A
)
⟺
u
=
P
C
(
u
-
λ
A
u
)
,
λ
>
0
.Lemma 14 (see [36, Lemma 2.5]).
LetH be a real Hilbert space. Given a nonempty closed convex subset of H and points x
,
y
,
z
∈
H and given also a real number a
∈
R, the set
(40)
{
v
∈
C
:
∥
y
-
v
∥
2
≤
∥
x
-
v
∥
2
+
〈
z
,
v
〉
+
a
}
is convex (and closed).Recall that a set-valued mappingT
:
D
(
T
)
⊂
H
→
2
H is called monotone if, for all x
,
y
∈
D
(
T
), f
∈
T
x and g
∈
T
y imply
(41)
〈
f
-
g
,
x
-
y
〉
≥
0
.
A set-valued mapping T is called maximal monotone if T is monotone and (
I
+
λ
T
)
D
(
T
)
=
H for each λ
>
0, where I is the identity mapping of H. We denote by G
(
T
) the graph of T. It is known that a monotone mapping T is maximal if and only if, for (
x
,
f
)
∈
H
×
H
,
〈
f
-
g
,
x
-
y
〉
≥
0 for every (
y
,
g
)
∈
G
(
T
) implies f
∈
T
x. Let A
:
C
→
H be a monotone, k-Lipschitz-continuous mapping, and let N
C
v be the normal cone to C at v
∈
C; that is,
(42)
N
C
v
=
{
w
∈
H
:
〈
v
-
u
,
w
〉
≥
0
,
∀
u
∈
C
}
.
Define
(43)
T
v
=
{
A
v
+
N
C
v
,
if
v
∈
C
,
∅
,
if
v
∉
C
.
Then, T is maximal monotone and 0
∈
T
v if and only if v
∈
VI
⁡
(
C
,
A
); see [39].Lemma 15 (see [36, Lemma 2.6]).
LetC be a nonempty subset of a Hilbert space H and S
:
C
→
C an asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {
γ
n
}. Then
(44)
∥
S
n
x
-
S
n
y
∥
≤
1
1
-
k
×
(
k
∥
x
-
y
∥
(
1
+
(
1
-
k
)
γ
n
)
∥
x
-
y
∥
2
+
(
1
-
k
)
c
n
h
h
h
+
(
1
+
(
1
-
k
)
γ
n
)
∥
x
-
y
∥
2
+
(
1
-
k
)
c
n
)
for all x
,
y
∈
C and n
≥
1.Lemma 16 (see [36, Lemma 2.7]).
LetC be a nonempty subset of a Hilbert space H and S
:
C
→
C a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {
γ
n
}. Let {
x
n
} be a sequence in C such that ∥
x
n
-
x
n
+
1
∥
→
0 and ∥
x
n
-
S
n
x
n
∥
→
0 as n
→
∞. Then ∥
x
n
-
S
x
n
∥
→
0 as n
→
∞.Lemma 17 (see Demiclosedness principle [36, Proposition 3.1]).
LetC be a nonempty closed convex subset of a Hilbert space H and S
:
C
→
C a continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {
γ
n
}. Then I
-
S is demiclosed at zero in the sense that if {
x
n
} is a sequence in C such that x
n
⇀
x
∈
C and limsup
⁡
m
→
∞
limsup
⁡
n
→
∞
∥
x
n
-
S
m
x
n
∥
=
0, then (
I
-
S
)
x
=
0.Lemma 18 (see [36, Proposition 3.2]).
LetC be a nonempty closed convex subset of a Hilbert space H and S
:
C
→
C a continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {
γ
n
} such that Fix
⁡
(
S
)
≠
∅. Then Fix
⁡
(
S
) is closed and convex.Remark 19.
Lemmas17 and 18 give some basic properties of an asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequence {
γ
n
}. Moreover, Lemma 17 extends the Demiclosedness principles studied for certain classes of nonlinear mappings in Kim and Xu [35], Górnicki [40], Xu [41], and Marino and Xu [42].Lemma 20 (see [43, page 80]).
Let{
a
n
}
n
=
1
∞, {
b
n
}
n
=
1
∞, and {
δ
n
}
n
=
1
∞ be sequences of nonnegative real numbers satisfying the inequality
(45)
a
n
+
1
≤
(
1
+
δ
n
)
a
n
+
b
n
,
∀
n
≥
1
.
If ∑
n
=
1
∞
‍
δ
n
<
∞ and ∑
n
=
1
∞
‍
b
n
<
∞, then lim
⁡
n
→
∞
a
n exists. If, in addition, {
a
n
}
n
=
1
∞ has a subsequence which converges to zero, then lim
⁡
n
→
∞
a
n
=
0.Recall that a Banach spaceX is said to satisfy the Opial condition [38] if, for any given sequence {
x
n
}
⊂
X which converges weakly to an element x
∈
X, there holds the inequality
(46)
limsup
⁡
n
→
∞
∥
x
n
-
x
∥
<
limsup
⁡
n
→
∞
∥
x
n
-
y
∥
,
∀
y
∈
X
,
y
≠
x
.
It is well known in [38] that every Hilbert space H satisfies the Opial condition.Lemma 21 (see [22, Proposition 3.1]).
LetC be a nonempty closed convex subset of a real Hilbert space H and let {
x
n
} be a sequence in H. Suppose that
(47)
∥
x
n
+
1
-
p
∥
2
≤
(
1
+
λ
n
)
∥
x
n
-
p
∥
2
+
δ
n
,
∀
p
∈
C
,
n
≥
1
,
where {
λ
n
} and {
δ
n
} are sequences of nonnegative real numbers such that ∑
n
=
1
∞
λ
n
‍
<
∞ and ∑
n
=
1
∞
δ
n
‍
<
∞. Then {
P
C
x
n
} converges strongly in C.Lemma 22 (see [44]).
LetC be a closed convex subset of a real Hilbert space H. Let {
x
n
} be a sequence in H and u
∈
H. Let q
=
P
C
u. If {
x
n
} is such that ω
w
(
x
n
)
⊂
C and satisfies the condition
(48)
∥
x
n
-
u
∥
≤
∥
u
-
q
∥
,
∀
n
,
then x
n
→
q as n
→
∞.


## 3. Strong Convergence Theorem

In this section, we will introduce and analyze one iterative algorithm by using a composite shrinking projection method for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. Under appropriate conditions we will prove strong convergence of the proposed algorithm.Theorem 23.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ, Θ
1, Θ
2 be three bifunctions from C
×
C to R satisfying (H1)–(H4) and let φ
:
C
→
R be a lower semicontinuous and convex functional. Let A
,
A
k
:
H
→
H and B
i
:
C
→
H be ζ-inverse strongly monotone, ζ
k-inverse strongly monotone, and η
i-inverse strongly monotone, respectively, where k
∈
{
1,2
} and i
∈
{
1,2
,
…
,
N
}. Let S
:
C
→
C be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense for some 0
≤
k
<
1 with sequence {
γ
n
}
⊂
[
0
,
∞
) such that lim
⁡
n
→
∞
γ
n
=
0 and {
c
n
}
⊂
[
0
,
∞
) such that lim
⁡
n
→
∞
c
n
=
0. Let {
T
n
}
n
=
1
∞ be a sequence of nonexpansive mappings on H and {
λ
n
} a sequence in (
0
,
b
] for some b
∈
(
0,1
). Let V be a γ
¯-strongly positive bounded linear operator with γ
¯
∈
(
1,2
]. Let W
n be the W-mapping defined by (13). Assume that Ω
:
=
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
G
M
E
P
(
Θ
,
φ
,
A
)
∩
S
G
E
P
(
G
)
∩
∩
i
=
1
N
VI
⁡
(
C
,
B
i
)
∩
Fix
⁡
(
S
) is nonempty and bounded where G is defined as in Proposition CY. Let {
r
n
} be a sequence in [
0,2
ζ
] and {
α
n
}, {
β
n
}, {
σ
n
}, and {
δ
n
} sequences in [
0,1
] such that lim
⁡
n
→
∞
σ
n
=
0,0
<
α
≤
α
n
≤
1, and k
≤
δ
n
≤
d
<
1. Pick any x
0
∈
H and set C
1
=
C, x
1
=
P
C
1
x
0. Let {
x
n
} be a sequence generated by the following algorithm:
(49)
u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
,
v
n
=
P
C
(
I
-
λ
N
,
n
B
N
)
×
P
C
(
I
-
λ
N
-
1
,
n
B
N
-
1
)
⋯
P
C
(
I
-
λ
1
,
n
B
1
)
u
n
,
z
n
=
β
n
x
n
+
σ
n
G
v
n
+
[
(
1
-
β
n
)
I
-
σ
n
V
]
W
n
G
v
n
,
k
n
=
δ
n
z
n
+
(
1
-
δ
n
)
S
n
z
n
,
y
n
=
(
1
-
α
n
)
x
n
+
α
n
k
n
,
C
n
+
1
=
{
z
∈
C
n
:
∥
y
n
-
z
∥
2
≤
∥
x
n
-
z
∥
2
+
θ
n
}
,
x
n
+
1
=
P
C
n
+
1
x
0
,
∀
n
≥
1
,
where θ
n
=
(
σ
n
+
γ
n
)
(
1
+
γ
n
)
Δ
n
+
c
n, Δ
n
=
sup
⁡
{
∥
x
n
-
p
∥
2
+
∥
(
I
-
V
)
p
∥
2
/
(
γ
¯
-
1
)
:
p
∈
Ω
}
<
∞, ν
k
∈
(
0,2
ζ
k
), k
∈
{
1,2
}, and {
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
), ∀
i
∈
{
1,2
,
…
,
N
}. Assume that the following conditions are satisfied:(i)
K
:
H
→
R is strongly convex with constant σ
>
0 and its derivative K
′ is Lipschitz continuous with constant ν
>
0 such that the function x
↦
〈
y
-
x
,
K
′
(
x
)
〉 is weakly upper semicontinuous for each y
∈
H;
(ii)
for eachx
∈
H, there exists a bounded subset D
x
⊂
C and z
x
∈
C such that, for any y
∉
D
x,
(50)
Θ
(
y
,
z
x
)
+
φ
(
z
x
)
-
φ
(
y
)
+
1
r
〈
K
′
(
y
)
-
K
′
(
x
)
,
z
x
-
y
〉
<
0
;
(iii)
0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
β
n
<
1 and 0
<
liminf
⁡
n
→
∞
r
n
≤
limsup
⁡
n
→
∞
r
n
<
2
ζ.Then {
x
n
} converges strongly to x
*
=
P
Ω
x
0 provided that S
r
(
Θ
,
φ
) is firmly nonexpansive.Proof.
Aslim
⁡
n
→
∞
σ
n
=
0
,
0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
β
n
<
1 and 0
<
liminf
⁡
n
→
∞
r
n
≤
limsup
⁡
n
→
∞
r
n
<
2
ζ, we may assume, without loss of generality, that {
β
n
}
⊂
[
a
,
a
^
]
⊂
[
0,1
]
,
{
r
n
}
⊂
[
c
,
c
^
]
⊂
(
0,2
ζ
) and β
n
+
σ
n
∥
V
∥
≤
1 for all n
≥
1. Since V is a γ
¯-strongly positive bounded linear operator on H, we know that
(51)
∥
V
∥
=
sup
⁡
{
〈
V
u
,
u
〉
:
u
∈
H
,
∥
u
∥
=
1
}
≥
γ
¯
>
1
.
Taking into account that β
n
+
σ
n
∥
V
∥
≤
1 for all n
≥
1, we have
(52)
〈
(
(
1
-
β
n
)
I
-
σ
n
V
)
u
,
u
〉
=
1
-
β
n
-
σ
n
〈
V
u
,
u
〉
≥
1
-
β
n
-
σ
n
∥
V
∥
≥
0
;
that is, (
1
-
β
n
)
I
-
σ
n
V is positive. It follows that
(53)
∥
(
1
-
β
n
)
I
-
σ
n
V
∥
=
sup
⁡
{
〈
(
(
1
-
β
n
)
I
-
σ
n
V
)
u
,
u
〉
:
u
∈
H
,
∥
u
∥
=
1
}
=
sup
⁡
⁡
{
1
-
β
n
-
σ
n
〈
V
u
,
u
〉
:
u
∈
H
,
∥
u
∥
=
1
}
≤
1
-
β
n
-
σ
n
γ
¯
.
Put
(54)
Λ
n
i
=
P
C
(
I
-
λ
i
,
n
B
i
)
P
C
(
I
-
λ
i
-
1
,
n
B
i
-
1
)
⋯
P
C
(
I
-
λ
1
,
n
B
1
)
for all i
∈
{
1,2
,
…
,
N
}, and Λ
n
0
=
I, where I is the identity mapping on H. Then we have v
n
=
Λ
n
N
u
n.
We divide the rest of the proof into several steps.
Step 1. We show that {
x
n
} is well defined. It is obvious that C
n is closed and convex. As the defining inequality in C
n is equivalent to the inequality
(55)
〈
2
(
x
n
-
y
n
)
,
z
〉
≤
∥
x
n
∥
2
-
∥
y
n
∥
2
+
θ
n
,
by Lemma 14 we know that C
n is convex for every n
≥
1.
First of all, let us show thatΩ
⊂
C
n for all n
≥
1. Suppose that Ω
⊂
C
n for some n
≥
1. Take p
∈
Ω arbitrarily. Since p
=
S
r
n
(
Θ
,
φ
)
(
p
-
r
n
A
p
), A is ζ-inverse strongly monotone, and 0
≤
r
n
≤
2
ζ, we have, for any n
≥
1,
(56)
∥
u
n
-
p
∥
2
=
∥
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
p
∥
2
≤
∥
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
p
∥
2
=
∥
(
x
n
-
p
)
-
r
n
(
A
x
n
-
A
p
)
∥
2
=
∥
x
n
-
p
∥
2
-
2
r
n
〈
x
n
-
p
,
A
x
n
-
A
p
〉
+
r
n
2
∥
A
x
n
-
A
p
∥
2
≤
∥
x
n
-
p
∥
2
-
2
r
n
ζ
∥
A
x
n
-
A
p
∥
2
+
r
n
2
∥
A
x
n
-
A
p
∥
2
=
∥
x
n
-
p
∥
2
+
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
≤
∥
x
n
-
p
∥
2
.
Since p
=
P
C
(
I
-
λ
i
,
n
B
i
)
p, Λ
n
i
p
=
p, and B
i is η
i-inverse strongly monotone, where λ
i
,
n
∈
(
0,2
η
i
), i
∈
{
1,2
,
…
,
N
}, by Proposition 5(iii) we deduce that for each n
≥
1
(57)
∥
v
n
-
p
∥
=
∥
P
C
(
I
-
λ
N
,
n
B
N
)
Λ
n
N
-
1
u
n
-
P
C
(
I
-
λ
N
,
n
B
N
)
Λ
n
N
-
1
p
∥
≤
∥
(
I
-
λ
N
,
n
B
N
)
Λ
n
N
-
1
u
n
-
(
I
-
λ
N
,
n
B
N
)
Λ
n
N
-
1
p
∥
≤
∥
Λ
n
N
-
1
u
n
-
Λ
n
N
-
1
p
∥
⋮
≤
∥
Λ
n
0
x
n
-
Λ
n
0
p
∥
=
∥
u
n
-
p
∥
.
Combining (56) and (57), we have
(58)
∥
v
n
-
p
∥
≤
∥
x
n
-
p
∥
.
Since p
=
G
p
=
T
ν
1
Θ
1
(
I
-
ν
1
A
1
)
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
p, A
k is ζ
k-inverse strongly monotone, for k
=
1,2, and 0
≤
ν
k
≤
2
ζ
k for k
=
1,2, we deduce that, for any n
≥
1,
(59)
∥
G
v
n
-
p
∥
2
=
∥
T
ν
1
Θ
1
(
I
-
ν
1
A
1
)
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
v
n
hh
-
T
ν
1
Θ
1
(
I
-
ν
1
A
1
)
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
p
∥
2
≤
∥
(
I
-
ν
1
A
1
)
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
v
n
hh
-
(
I
-
ν
1
A
1
)
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
p
∥
2
=
∥
[
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
v
n
-
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
p
]
hh
-
ν
1
[
A
1
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
v
n
-
A
1
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
p
]
∥
2
≤
∥
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
v
n
-
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
p
∥
2
+
ν
1
(
ν
1
-
2
ζ
1
)
×
∥
A
1
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
v
n
-
A
1
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
p
∥
2
≤
∥
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
v
n
-
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
p
∥
2
≤
∥
(
I
-
ν
2
A
2
)
v
n
-
(
I
-
ν
2
A
2
)
p
∥
2
=
∥
(
v
n
-
p
)
-
ν
2
(
A
2
v
n
-
A
2
p
)
∥
2
≤
∥
v
n
-
p
∥
2
+
ν
2
(
ν
2
-
2
ζ
2
)
∥
A
2
v
n
-
A
2
p
∥
2
≤
∥
v
n
-
p
∥
2
.
(This shows that G is nonexpansive.) Also, from (49), (53), (58), and (59) it follows that
(60)
∥
z
n
-
p
∥
=
∥
β
n
(
x
n
-
p
)
+
σ
n
(
G
v
n
-
p
)
+
[
(
1
-
β
n
)
I
-
σ
n
V
]
h
h
h
h
h
h
×
(
W
n
G
v
n
-
p
)
+
σ
n
(
I
-
V
)
p
∥
≤
β
n
∥
x
n
-
p
∥
+
σ
n
∥
G
v
n
-
p
∥
+
∥
[
(
1
-
β
n
)
I
-
σ
n
V
]
(
W
n
G
v
n
-
p
)
∥
+
σ
n
∥
(
I
-
V
)
p
∥
≤
β
n
∥
x
n
-
p
∥
+
σ
n
∥
G
v
n
-
p
∥
+
(
1
-
β
n
-
σ
n
γ
¯
)
∥
W
n
G
v
n
-
p
∥
+
σ
n
∥
(
I
-
V
)
p
∥
≤
β
n
∥
x
n
-
p
∥
+
γ
n
∥
G
v
n
-
p
∥
+
(
1
-
β
n
-
γ
n
γ
¯
)
∥
G
v
n
-
p
∥
+
γ
n
∥
(
I
-
V
)
p
∥
=
β
n
∥
x
n
-
p
∥
+
(
1
-
β
n
-
σ
n
(
γ
¯
-
1
)
)
×
∥
G
v
n
-
p
∥
+
σ
n
∥
(
I
-
V
)
p
∥
≤
β
n
∥
x
n
-
p
∥
+
(
1
-
β
n
-
σ
n
(
γ
¯
-
1
)
)
×
∥
v
n
-
p
∥
+
σ
n
∥
(
I
-
V
)
p
∥
≤
β
n
∥
x
n
-
p
∥
+
(
1
-
β
n
-
σ
n
(
γ
¯
-
1
)
)
×
∥
x
n
-
p
∥
+
σ
n
∥
(
I
-
V
)
p
∥
=
(
1
-
σ
n
(
γ
¯
-
1
)
)
∥
x
n
-
p
∥
+
σ
n
∥
(
I
-
V
)
p
∥
=
(
1
-
σ
n
(
γ
¯
-
1
)
)
∥
x
n
-
p
∥
+
σ
n
(
γ
¯
-
1
)
∥
(
I
-
V
)
p
∥
γ
¯
-
1
,
which hence yields
(61)
∥
z
n
-
p
∥
2
≤
(
1
-
σ
n
(
γ
¯
-
1
)
)
∥
x
n
-
p
∥
2
+
σ
n
(
γ
¯
-
1
)
∥
(
I
-
V
)
p
∥
2
(
γ
¯
-
1
)
2
≤
∥
x
n
-
p
∥
2
+
σ
n
∥
(
I
-
V
)
p
∥
2
γ
¯
-
1
.
By Lemma 9(b), we deduce from (49) and (61) that
(62)
∥
k
n
-
p
∥
2
=
∥
δ
n
(
z
n
-
p
)
+
(
1
-
δ
n
)
(
S
n
z
n
-
p
)
∥
2
=
δ
n
∥
z
n
-
p
∥
2
+
(
1
-
δ
n
)
∥
S
n
z
n
-
p
∥
2
-
δ
n
(
1
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
≤
δ
n
∥
z
n
-
p
∥
2
+
(
1
-
δ
n
)
×
[
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
k
∥
z
n
-
S
n
z
n
∥
2
+
c
n
]
-
δ
n
(
1
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
=
[
1
+
γ
n
(
1
-
δ
n
)
]
∥
z
n
-
p
∥
2
+
(
1
-
δ
n
)
×
(
k
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
+
(
1
-
δ
n
)
c
n
≤
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
(
1
-
δ
n
)
×
(
k
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
+
c
n
≤
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
c
n
≤
(
1
+
γ
n
)
(
∥
x
n
-
p
∥
2
+
σ
n
∥
(
I
-
V
)
p
∥
2
γ
¯
-
1
)
+
c
n
.
So, from (49) and (62) we get
(63)
∥
y
n
-
p
∥
2
=
∥
(
1
-
α
n
)
(
x
n
-
p
)
+
α
n
(
k
n
-
p
)
∥
2
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
∥
k
n
-
p
∥
2
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
(
∥
x
n
-
p
∥
2
+
σ
n
∥
(
I
-
V
)
p
∥
2
γ
¯
-
1
)
+
c
n
]
≤
(
1
+
γ
n
)
(
∥
x
n
-
p
∥
2
+
σ
n
∥
(
I
-
V
)
p
∥
2
γ
¯
-
1
)
+
c
n
=
∥
x
n
-
p
∥
2
+
γ
n
∥
x
n
-
p
∥
2
+
(
1
+
γ
n
)
σ
n
∥
(
I
-
V
)
p
∥
2
γ
¯
-
1
+
c
n
≤
∥
x
n
-
p
∥
2
+
γ
n
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
σ
n
(
1
+
γ
n
)
∥
(
I
-
V
)
p
∥
2
γ
¯
-
1
+
c
n
≤
∥
x
n
-
p
∥
2
+
(
γ
n
+
σ
n
)
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
(
σ
n
+
γ
n
)
(
1
+
γ
n
)
∥
(
I
-
V
)
p
∥
2
γ
¯
-
1
+
c
n
=
∥
x
n
-
p
∥
2
+
(
σ
n
+
γ
n
)
(
1
+
γ
n
)
×
(
∥
x
n
-
p
∥
2
+
∥
(
I
-
V
)
p
∥
2
γ
¯
-
1
)
+
c
n
≤
∥
x
n
-
p
∥
2
+
(
σ
n
+
γ
n
)
(
1
+
γ
n
)
Δ
n
+
c
n
=
∥
x
n
-
p
∥
2
+
θ
n
,
where θ
n
=
(
σ
n
+
γ
n
)
(
1
+
γ
n
)
Δ
n
+
c
n and Δ
n
=
sup
⁡
{
∥
x
n
-
p
∥
2
+
∥
(
I
-
V
)
p
∥
2
/
(
γ
¯
-
1
)
:
p
∈
Ω
}
<
∞. Hence p
∈
C
n
+
1. This implies that Ω
⊂
C
n for all n
≥
1. Therefore, {
x
n
} is well defined.
Step 2. We prove that ∥
x
n
-
k
n
∥
→
0, ∥
x
n
-
z
n
∥
→
0, and ∥
S
n
z
n
-
z
n
∥
→
0 as n
→
∞.
Indeed, letx
*
=
P
Ω
x
0. From x
n
=
P
C
n
x
0 and x
*
∈
Ω
⊂
C
n, we obtain
(64)
∥
x
n
-
x
0
∥
≤
∥
x
*
-
x
0
∥
.
This implies that {
x
n
} is bounded and hence {
u
n
}, {
v
n
}, {
z
n
}, {
k
n
}, and {
y
n
} are also bounded. Since x
n
+
1
∈
C
n
+
1
⊂
C
n and x
n
=
P
C
n
x
0, we have
(65)
∥
x
n
-
x
0
∥
≤
∥
x
n
+
1
-
x
0
∥
,
∀
n
≥
1
.
Therefore lim
⁡
n
→
∞
∥
x
n
-
x
0
∥ exists. From x
n
=
P
C
n
x
0
,
x
n
+
1
∈
C
n
+
1
⊂
C
n, by Proposition 5(ii) we obtain
(66)
∥
x
n
+
1
-
x
n
∥
2
≤
∥
x
0
-
x
n
+
1
∥
2
-
∥
x
0
-
x
n
∥
2
,
which implies
(67)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0
.
It follows from x
n
+
1
∈
C
n
+
1 that ∥
y
n
-
x
n
+
1
∥
2
≤
∥
x
n
-
x
n
+
1
∥
2
+
θ
n and hence
(68)
∥
x
n
-
y
n
∥
2
≤
2
(
∥
x
n
-
x
n
+
1
∥
2
+
∥
x
n
+
1
-
y
n
∥
2
)
≤
2
(
∥
x
n
-
x
n
+
1
∥
2
+
∥
x
n
-
x
n
+
1
∥
2
+
θ
n
)
=
2
(
2
∥
x
n
-
x
n
+
1
∥
2
+
θ
n
)
.
From (67) and lim
⁡
n
→
∞
θ
n
=
0, we have
(69)
lim
⁡
n
→
∞
∥
x
n
-
y
n
∥
=
0
.
Since y
n
-
x
n
=
α
n
(
k
n
-
x
n
) and 0
<
α
≤
α
n
≤
1, we have
(70)
α
∥
k
n
-
x
n
∥
≤
α
n
∥
k
n
-
x
n
∥
=
∥
y
n
-
x
n
∥
,
which immediately leads to
(71)
lim
⁡
n
→
∞
∥
k
n
-
x
n
∥
=
0
.
Also, utilizing Lemmas 8 and 9(b) we obtain from (49), (58), (59), and (62) that
(72)
∥
z
n
-
p
∥
2
=
∥
β
n
x
n
+
σ
n
G
v
n
+
[
(
1
-
β
n
)
I
-
σ
n
V
]
W
n
G
v
n
-
p
∥
2
=
∥
β
n
(
x
n
-
p
)
+
(
1
-
β
n
)
(
W
n
G
v
n
-
p
)
h
h
h
h
h
h
+
σ
n
(
G
v
n
-
V
W
n
G
v
n
)
∥
2
≤
∥
β
n
(
x
n
-
p
)
+
(
1
-
β
n
)
(
W
n
G
v
n
-
p
)
∥
2
+
2
σ
n
〈
(
G
v
n
-
V
W
n
G
v
n
)
,
z
n
-
p
〉
=
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
W
n
G
v
n
-
p
∥
2
-
β
n
(
1
-
β
n
)
∥
x
n
-
W
n
G
v
n
∥
2
+
2
σ
n
〈
(
G
v
n
-
V
W
n
G
v
n
)
,
z
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
G
v
n
-
p
∥
2
-
β
n
(
1
-
β
n
)
∥
x
n
-
W
n
G
v
n
∥
2
+
2
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
v
n
-
p
∥
2
-
β
n
(
1
-
β
n
)
∥
x
n
-
W
n
G
v
n
∥
2
+
2
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
x
n
-
p
∥
2
-
β
n
(
1
-
β
n
)
∥
x
n
-
W
n
G
v
n
∥
2
+
2
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
=
∥
x
n
-
p
∥
2
-
β
n
(
1
-
β
n
)
∥
x
n
-
W
n
G
v
n
∥
2
+
2
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
,
and hence
(73)
∥
y
n
-
p
∥
2
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
∥
k
n
-
p
∥
2
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
c
n
]
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
hhhhh
×
(
∥
x
n
-
p
∥
2
-
β
n
(
1
-
β
n
)
∥
x
n
-
p
∥
2
hhhhhhhh
×
∥
x
n
-
W
n
G
v
n
∥
2
hhhhhhhh
∥
x
n
-
p
∥
2
+
2
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
)
+
c
n
]
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
(
1
+
γ
n
)
×
(
∥
x
n
-
p
∥
2
-
β
n
(
1
-
β
n
)
∥
x
n
-
W
n
G
v
n
∥
2
hhh
∥
x
n
-
p
∥
2
+
2
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
)
+
c
n
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
-
α
n
(
1
+
γ
n
)
β
n
(
1
-
β
n
)
∥
x
n
-
W
n
G
v
n
∥
2
+
(
1
+
γ
n
)
2
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
+
c
n
≤
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
-
α
n
(
1
+
γ
n
)
×
β
n
(
1
-
β
n
)
∥
x
n
-
W
n
G
v
n
∥
2
+
2
σ
n
(
1
+
γ
n
)
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
+
c
n
.
So, it follows that
(74)
α
(
1
+
γ
n
)
a
(
1
-
a
^
)
∥
x
n
-
W
n
G
v
n
∥
2
≤
α
n
(
1
+
γ
n
)
β
n
(
1
-
β
n
)
∥
x
n
-
W
n
G
v
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
y
n
-
p
∥
2
+
γ
n
∥
x
n
-
p
∥
2
+
2
σ
n
(
1
+
γ
n
)
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
+
c
n
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
p
∥
+
∥
y
n
-
p
∥
)
+
γ
n
∥
x
n
-
p
∥
2
+
2
σ
n
(
1
+
γ
n
)
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
+
c
n
.
Since lim
⁡
n
→
∞
σ
n
=
0
,
lim
⁡
n
→
∞
γ
n
=
0, and lim
⁡
n
→
∞
c
n
=
0, it follows from (69) and the boundedness of {
x
n
}
,
{
y
n
}
,
{
z
n
}, and {
v
n
} that
(75)
lim
⁡
n
→
∞
∥
x
n
-
W
n
G
v
n
∥
=
0
.
Note that
(76)
∥
z
n
-
x
n
∥
=
∥
(
1
-
β
n
)
(
W
n
G
v
n
-
x
n
)
+
σ
n
(
G
v
n
-
V
W
n
G
v
n
)
∥
≤
(
1
-
β
n
)
∥
W
n
G
v
n
-
x
n
∥
+
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
≤
∥
W
n
G
v
n
-
x
n
∥
+
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
.
Hence, it follows from (75) and lim
⁡
n
→
∞
σ
n
=
0 that
(77)
lim
⁡
n
→
∞
∥
x
n
-
z
n
∥
=
0
.
Note that
(78)
∥
k
n
-
z
n
∥
≤
∥
k
n
-
x
n
∥
+
∥
x
n
-
z
n
∥
.
Thus, we deduce from (71) and (77) that
(79)
lim
⁡
n
→
∞
∥
k
n
-
z
n
∥
=
0
.
Since k
n
-
z
n
=
(
1
-
δ
n
)
(
S
n
z
n
-
z
n
) and k
≤
δ
n
≤
d
<
1, we have
(80)
(
1
-
d
)
∥
S
n
z
n
-
z
n
∥
≤
(
1
-
δ
n
)
∥
S
n
z
n
-
z
n
∥
=
∥
k
n
-
z
n
∥
,
which, together with (79), yields
(81)
lim
⁡
n
→
∞
∥
S
n
z
n
-
z
n
∥
=
0
.
Step 3. We prove that ∥
x
n
-
u
n
∥
→
0, ∥
x
n
-
v
n
∥
→
0, ∥
v
n
-
G
v
n
∥
→
0, ∥
x
n
-
W
x
n
∥
→
0, and ∥
z
n
-
S
z
n
∥
→
0 as n
→
∞.
Indeed, from (57), (59), and γ
¯
∈
(
1,2
] it follows that
(82)
∥
z
n
-
p
∥
2
=
∥
β
n
(
x
n
-
p
)
+
σ
n
(
G
v
n
-
p
)
hl
+
[
(
1
-
β
n
)
I
-
σ
n
V
]
(
W
n
G
v
n
-
p
)
hl
+
σ
n
(
I
-
V
)
p
∥
2
≤
∥
β
n
(
x
n
-
p
)
+
σ
n
(
G
v
n
-
p
)
hl
+
[
(
1
-
β
n
)
I
-
σ
n
V
]
(
W
n
G
v
n
-
p
)
∥
2
+
2
σ
n
〈
(
I
-
V
)
p
,
z
n
-
p
〉
≤
[
β
n
∥
x
n
-
p
∥
+
σ
n
∥
G
v
n
-
p
∥
hl
+
(
1
-
β
n
-
σ
n
γ
¯
)
∥
W
n
G
v
n
-
p
∥
]
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
[
β
n
∥
x
n
-
p
∥
+
σ
n
∥
G
v
n
-
p
∥
hl
+
(
1
-
β
n
-
σ
n
γ
¯
)
∥
G
v
n
-
p
∥
]
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
=
[
β
n
∥
x
n
-
p
∥
+
(
1
-
β
n
-
σ
n
(
γ
¯
-
1
)
)
hl
×
∥
G
v
n
-
p
∥
]
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
[
β
n
∥
x
n
-
p
∥
+
(
1
-
β
n
)
∥
G
v
n
-
p
∥
]
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
G
v
n
-
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
v
n
-
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
u
n
-
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
.
Next let us show that
(83)
lim
⁡
n
→
∞
∥
x
n
-
u
n
∥
=
0
.
For p
∈
Ω, we find that
(84)
∥
u
n
-
p
∥
2
=
∥
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
p
∥
2
≤
∥
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
p
∥
2
=
∥
x
n
-
p
-
r
n
(
A
x
n
-
A
p
)
∥
2
≤
∥
x
n
-
p
∥
2
+
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
.
Combining (82) and (84), we obtain
(85)
∥
z
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
u
n
-
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
×
[
∥
x
n
-
p
∥
2
+
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
]
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
=
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
r
n
(
r
n
-
2
ζ
)
×
∥
A
x
n
-
A
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
,
which immediately implies that
(86)
(
1
-
a
^
)
c
(
2
ζ
-
c
^
)
∥
A
x
n
-
A
p
∥
2
≤
(
1
-
β
n
)
r
n
(
2
ζ
-
r
n
)
∥
A
x
n
-
A
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
z
n
-
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
∥
x
n
-
z
n
∥
(
∥
x
n
-
p
∥
+
∥
z
n
-
p
∥
)
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
.
Since lim
⁡
n
→
∞
σ
n
=
0 and {
x
n
} and {
z
n
} are bounded sequences, it follows from (77) that
(87)
lim
⁡
n
→
∞
∥
A
x
n
-
A
p
∥
=
0
.
Furthermore, from the firm nonexpansivity of S
r
n
(
Θ
,
φ
), we have
(88)
∥
u
n
-
p
∥
2
=
∥
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
p
∥
2
≤
〈
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
p
,
u
n
-
p
〉
=
1
2
[
∥
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
p
∥
2
+
∥
u
n
-
p
∥
2
hhh
-
∥
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
p
-
(
u
n
-
p
)
∥
2
]
≤
1
2
[
∥
x
n
-
p
∥
2
+
∥
u
n
-
p
∥
2
hhh
-
∥
x
n
-
u
n
-
r
n
(
A
x
n
-
A
p
)
∥
2
]
=
1
2
[
∥
x
n
-
p
∥
2
+
∥
u
n
-
p
∥
2
-
∥
x
n
-
u
n
∥
2
hhh
+
2
r
n
〈
A
x
n
-
A
p
,
x
n
-
u
n
〉
-
r
n
2
∥
A
x
n
-
A
p
∥
2
]
,
which leads to
(89)
∥
u
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
x
n
-
u
n
∥
2
+
2
r
n
∥
A
x
n
-
A
p
∥
∥
x
n
-
u
n
∥
.
From (82) and (89), we have
(90)
∥
z
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
u
n
-
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
×
[
∥
x
n
-
p
∥
2
-
∥
x
n
-
u
n
∥
2
hhl
∥
x
n
-
u
n
∥
2
+
2
r
n
∥
A
x
n
-
A
p
∥
∥
x
n
-
u
n
∥
]
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
∥
x
n
-
p
∥
2
-
(
1
-
β
n
)
∥
x
n
-
u
n
∥
2
+
2
r
n
∥
A
x
n
-
A
p
∥
∥
x
n
-
u
n
∥
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
,
which hence implies that
(91)
∥
x
n
-
u
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
z
n
-
p
∥
2
+
2
r
n
∥
A
x
n
-
A
p
∥
×
∥
x
n
-
u
n
∥
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
∥
x
n
-
z
n
∥
(
∥
x
n
-
p
∥
+
∥
z
n
-
p
∥
)
+
2
r
n
∥
A
x
n
-
A
p
∥
∥
x
n
-
u
n
∥
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
.
Since lim
⁡
n
→
∞
σ
n
=
0 and {
x
n
}, {
u
n
}, and {
z
n
} are bounded sequences, it follows from (77) and (87) that (83) holds.
Next we show thatlim
⁡
n
→
∞
∥
B
i
Λ
n
i
u
n
-
B
i
p
∥
=
0, i
=
1,2
,
…
,
N. As a matter of fact, observe that
(92)
∥
Λ
n
i
u
n
-
p
∥
2
=
∥
P
C
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
P
C
(
I
-
λ
i
,
n
B
i
)
p
∥
2
≤
∥
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
(
I
-
λ
i
,
n
B
i
)
p
∥
2
≤
∥
Λ
n
i
-
1
u
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
u
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
x
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
.
Combining (59), (82), and (92), we have
(93)
∥
z
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
v
n
-
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
Λ
n
i
u
n
-
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
×
[
∥
x
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
hhh
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
]
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
=
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
λ
i
,
n
×
(
λ
i
,
n
-
2
η
i
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
,
which together with {
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
), ∀
i
∈
{
1,2
,
…
,
N
}, implies that
(94)
(
1
-
a
^
)
a
i
(
2
η
i
-
b
i
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
≤
(
1
-
β
n
)
λ
i
,
n
(
2
η
i
-
λ
i
,
n
)
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
z
n
-
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
×
∥
z
n
-
p
∥
≤
∥
x
n
-
z
n
∥
(
∥
x
n
-
p
∥
+
∥
z
n
-
p
∥
)
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
.
Since lim
⁡
n
→
∞
σ
n
=
0 and {
x
n
} and {
z
n
} are bounded sequences, it follows from (77) that
(95)
lim
⁡
n
→
∞
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
=
0
,
i
=
1,2
,
…
,
N
.
By Proposition 5(iii) and Lemma 9(a), we obtain
(96)
∥
Λ
n
i
u
n
-
p
∥
2
=
∥
P
C
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
P
C
(
I
-
λ
i
,
n
B
i
)
p
∥
2
≤
〈
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
(
I
-
λ
i
,
n
B
i
)
p
,
Λ
n
i
u
n
-
p
〉
=
1
2
(
∥
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
-
(
I
-
λ
i
,
n
B
i
)
p
∥
2
hhl
+
∥
Λ
n
i
u
n
-
p
∥
2
-
∥
(
I
-
λ
i
,
n
B
i
)
Λ
n
i
-
1
u
n
hhhhhhhhlhhhhhhhl
-
(
I
-
λ
i
,
n
B
i
)
p
-
(
Λ
n
i
u
n
-
p
)
∥
2
)
≤
1
2
(
∥
Λ
n
i
-
1
u
n
-
p
∥
2
+
∥
Λ
n
i
u
n
-
p
∥
2
hhh
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
-
λ
i
,
n
(
B
i
Λ
n
i
-
1
u
n
-
B
i
p
)
∥
2
)
≤
1
2
(
∥
u
n
-
p
∥
2
+
∥
Λ
n
i
u
n
-
p
∥
2
hhh
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
-
λ
i
,
n
(
B
i
Λ
n
i
-
1
u
n
-
B
i
p
)
∥
2
)
≤
1
2
(
∥
x
n
-
p
∥
2
+
∥
Λ
n
i
u
n
-
p
∥
2
hhh
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
-
λ
i
,
n
(
B
i
Λ
n
i
-
1
u
n
-
B
i
p
)
∥
2
)
,
which implies
(97)
∥
Λ
n
i
u
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
-
λ
i
,
n
(
B
i
Λ
n
i
-
1
u
n
-
B
i
p
)
∥
2
=
∥
x
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
-
λ
i
,
n
2
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
+
2
λ
i
,
n
〈
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
,
B
i
Λ
n
i
-
1
u
n
-
B
i
p
〉
≤
∥
x
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
.
Combining (59), (82), and (97), we have
(98)
∥
z
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
v
n
-
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
Λ
n
i
u
n
-
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
×
[
∥
x
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
hhh
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
]
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
∥
x
n
-
p
∥
2
-
(
1
-
β
n
)
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
.
So, we conclude that
(99)
(
1
-
a
^
)
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
≤
(
1
-
β
n
)
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
z
n
-
p
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
∥
x
n
-
z
n
∥
(
∥
x
n
-
p
∥
+
∥
z
n
-
p
∥
)
+
2
b
i
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
.
Since lim
⁡
n
→
∞
σ
n
=
0 and {
x
n
}
,
{
z
n
}, and {
u
n
} are bounded, from (77) and (95) we get
(100)
lim
⁡
n
→
∞
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
=
0
.
From (100) we get
(101)
∥
u
n
-
v
n
∥
=
∥
Λ
n
0
u
n
-
Λ
n
N
u
n
∥
≤
∥
Λ
n
0
u
n
-
Λ
n
1
u
n
∥
+
∥
Λ
n
1
u
n
-
Λ
n
2
u
n
∥
+
⋯
+
∥
Λ
n
N
-
1
u
n
-
Λ
n
N
u
n
∥
⟶
0
hhhhhhhhhhhhhhh
as
n
⟶
∞
.
Taking into account that ∥
x
n
-
v
n
∥
≤
∥
x
n
-
u
n
∥
+
∥
u
n
-
v
n
∥, we conclude from (83) and (101) that
(102)
lim
⁡
n
→
∞
∥
x
n
-
v
n
∥
=
0
.
On the other hand, for simplicity, we writep
~
=
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
p, v
~
n
=
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
v
n, and w
n
=
G
v
n
=
T
ν
1
Θ
1
(
I
-
ν
1
A
1
)
v
~
n for all n
≥
1. Then
(103)
p
=
G
p
=
T
ν
1
Θ
1
(
I
-
ν
1
A
1
)
p
~
=
T
ν
1
Θ
1
(
I
-
ν
1
A
1
)
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
p
.
We now show thatlim
⁡
n
→
∞
∥
G
v
n
-
v
n
∥
=
0; that is, lim
⁡
n
→
∞
∥
w
n
-
v
n
∥
=
0. As a matter of fact, for p
∈
Ω, it follows from (58), (59), and (82) that
(104)
∥
z
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
G
v
n
-
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
=
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
w
n
-
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
×
[
∥
v
~
n
-
p
~
∥
2
+
ν
1
(
ν
1
-
2
ζ
1
)
∥
A
1
v
~
n
-
A
1
p
~
∥
2
]
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
×
[
∥
v
n
-
p
∥
2
+
ν
2
(
ν
2
-
2
ζ
2
)
∥
A
2
v
n
-
A
2
p
∥
2
hhh
+
ν
1
(
ν
1
-
2
ζ
1
)
∥
A
1
v
~
n
-
A
1
p
~
∥
2
]
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
×
[
∥
x
n
-
p
∥
2
+
ν
2
(
ν
2
-
2
ζ
2
)
∥
A
2
v
n
-
A
2
p
∥
2
hhh
+
ν
1
(
ν
1
-
2
ζ
1
)
∥
A
1
v
~
n
-
A
1
p
~
∥
2
]
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
=
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
×
[
ν
2
(
ν
2
-
2
ζ
2
)
∥
A
2
v
n
-
A
2
p
∥
2
hhh
+
ν
1
(
ν
1
-
2
ζ
1
)
∥
A
1
v
~
n
-
A
1
p
~
∥
2
]
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
,
which immediately yields
(105)
(
1
-
a
^
)
[
ν
2
(
2
ζ
2
-
ν
2
)
∥
A
2
v
n
-
A
2
p
∥
2
hhh
+
ν
1
(
2
ζ
1
-
ν
1
)
∥
A
1
v
~
n
-
A
1
p
~
∥
2
]
≤
(
1
-
β
n
)
[
ν
2
(
2
ζ
2
-
ν
2
)
∥
A
2
v
n
-
A
2
p
∥
2
hhhhhhhh
+
ν
1
(
2
ζ
1
-
ν
1
)
∥
A
1
v
~
n
-
A
1
p
~
∥
2
]
≤
∥
x
n
-
p
∥
2
-
∥
z
n
-
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
∥
x
n
-
z
n
∥
(
∥
x
n
-
p
∥
+
∥
z
n
-
p
∥
)
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
.
Since lim
⁡
n
→
∞
σ
n
=
0 and {
x
n
} and {
z
n
} are bounded, from (77) we get
(106)
lim
⁡
n
→
∞
∥
A
2
v
n
-
A
2
p
∥
=
0
,
lim
⁡
n
→
∞
∥
A
1
v
~
n
-
A
1
p
~
∥
=
0
.
Also, in terms of the firm nonexpansivity of T
ν
k
Θ
k and the ζ
k-inverse strong monotonicity of A
k for k
=
1,2, we obtain from ν
k
∈
(
0,2
ζ
k
)
,
k
∈
{
1,2
}, and (59) that
(107)
∥
v
~
n
-
p
~
∥
2
=
∥
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
v
n
-
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
p
∥
2
≤
〈
(
I
-
ν
2
A
2
)
v
n
-
(
I
-
ν
2
A
2
)
p
,
v
~
n
-
p
~
〉
=
1
2
[
∥
(
I
-
ν
2
A
2
)
v
n
-
(
I
-
ν
2
A
2
)
p
∥
2
+
∥
v
~
n
-
p
~
∥
2
hhh
-
∥
(
I
-
ν
2
A
2
)
v
n
-
(
I
-
ν
2
A
2
)
p
-
(
v
~
n
-
p
~
)
∥
2
]
≤
1
2
[
∥
v
n
-
p
∥
2
+
∥
v
~
n
-
p
~
∥
2
hhh
-
∥
(
v
n
-
v
~
n
)
-
ν
2
(
A
2
v
n
-
A
2
p
)
-
(
p
-
p
~
)
∥
2
]
=
1
2
[
∥
v
n
-
p
∥
2
+
∥
v
~
n
-
p
~
∥
2
-
∥
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
∥
2
hhh
+
2
ν
2
〈
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
,
A
2
v
n
-
A
2
p
〉
hhh
-
ν
2
2
∥
A
2
v
n
-
A
2
p
∥
2
]
,
∥
w
n
-
p
∥
2
=
∥
T
ν
1
Θ
1
(
I
-
ν
1
A
1
)
v
~
n
-
T
ν
1
Θ
1
(
I
-
ν
1
A
1
)
p
~
∥
2
≤
〈
(
I
-
ν
1
A
1
)
v
~
n
-
(
I
-
ν
1
A
1
)
p
~
,
w
n
-
p
〉
=
1
2
[
∥
(
I
-
ν
1
A
1
)
v
~
n
-
(
I
-
ν
1
A
1
)
p
~
∥
2
+
∥
w
n
-
p
∥
2
hlh
-
∥
(
I
-
ν
1
A
1
)
v
~
n
-
(
I
-
ν
1
A
1
)
p
~
-
(
w
n
-
p
)
∥
2
]
≤
1
2
[
∥
v
~
n
-
p
~
∥
2
+
∥
w
n
-
p
∥
2
-
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
2
hlh
+
2
ν
1
〈
A
1
v
~
n
-
A
1
p
~
,
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
〉
hlh
-
ν
1
2
∥
A
1
v
~
n
-
A
1
p
~
∥
2
]
≤
1
2
[
∥
v
n
-
p
∥
2
+
∥
w
n
-
p
∥
2
-
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
2
hlh
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
2
+
2
ν
1
〈
A
1
v
~
n
-
A
1
p
~
,
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
〉
]
.
Thus, we have
(108)
∥
v
~
n
-
p
~
∥
2
≤
∥
v
n
-
p
∥
2
-
∥
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
∥
2
+
2
ν
2
〈
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
,
A
2
v
n
-
A
2
p
〉
-
ν
2
2
∥
A
2
v
n
-
A
2
p
∥
2
,
(109)
∥
w
n
-
p
∥
2
≤
∥
v
n
-
p
∥
2
-
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
2
+
2
ν
1
∥
A
1
v
~
n
-
A
1
p
~
∥
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
.
Consequently, from (58), (104), and (108) it follows that
(110)
∥
z
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
×
[
∥
v
~
n
-
p
~
∥
2
+
ν
1
(
ν
1
-
2
ζ
1
)
∥
A
1
v
~
n
-
A
1
p
~
∥
2
]
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
v
~
n
-
p
~
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
×
[
∥
v
n
-
p
∥
2
-
∥
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
∥
2
∥
A
2
v
n
-
A
2
p
∥
2
hlh
+
2
ν
2
〈
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
,
A
2
v
n
-
A
2
p
〉
hlh
-
ν
2
2
∥
A
2
v
n
-
A
2
p
∥
2
]
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
×
[
∥
x
n
-
p
∥
2
-
∥
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
∥
2
hlh
∥
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
∥
2
+
2
ν
2
∥
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
∥
∥
A
2
v
n
-
A
2
p
∥
]
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
∥
x
n
-
p
∥
2
-
(
1
-
β
n
)
∥
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
∥
2
+
2
ν
2
∥
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
∥
∥
A
2
v
n
-
A
2
p
∥
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
,
which hence leads to
(111)
(
1
-
a
^
)
∥
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
∥
2
≤
(
1
-
β
n
)
∥
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
z
n
-
p
∥
2
+
2
ν
2
∥
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
∥
∥
A
2
v
n
-
A
2
p
∥
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
∥
x
n
-
z
n
∥
(
∥
x
n
-
p
∥
+
∥
z
n
-
p
∥
)
+
2
ν
2
∥
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
∥
∥
A
2
v
n
-
A
2
p
∥
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
.
Since lim
⁡
n
→
∞
σ
n
=
0 and {
x
n
}
,
{
z
n
}
,
{
v
n
}, and {
v
~
n
} are bounded sequences, we conclude from (77) and (106) that
(112)
lim
⁡
n
→
∞
∥
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
∥
=
0
.
Furthermore, from (58), (104), and (109) it follows that
(113)
∥
z
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
w
n
-
p
∥
2
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
×
[
∥
v
n
-
p
∥
2
-
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
2
hh
∥
v
n
-
p
∥
2
+
2
ν
1
∥
A
1
v
~
n
-
A
1
p
~
∥
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
]
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
×
[
∥
x
n
-
p
∥
2
-
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
2
hh
∥
v
n
-
p
∥
2
+
2
ν
1
∥
A
1
v
~
n
-
A
1
p
~
∥
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
]
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
∥
x
n
-
p
∥
2
-
(
1
-
β
n
)
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
2
+
2
ν
1
∥
A
1
v
~
n
-
A
1
p
~
∥
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
,
which hence yields
(114)
(
1
-
a
^
)
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
2
≤
(
1
-
β
n
)
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
z
n
-
p
∥
2
+
2
ν
1
∥
A
1
v
~
n
-
A
1
p
~
∥
×
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
≤
∥
x
n
-
z
n
∥
(
∥
x
n
-
p
∥
+
∥
z
n
-
p
∥
)
+
2
ν
1
∥
A
1
v
~
n
-
A
1
p
~
∥
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
+
2
σ
n
∥
(
I
-
V
)
p
∥
∥
z
n
-
p
∥
.
Since lim
⁡
n
→
∞
σ
n
=
0 and {
x
n
}
,
{
z
n
}
,
{
w
n
}, and {
v
~
n
} are bounded sequences, we conclude from (77) and (106) that
(115)
lim
⁡
n
→
∞
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
=
0
.
Note that
(116)
∥
v
n
-
w
n
∥
≤
∥
(
v
n
-
v
~
n
)
-
(
p
-
p
~
)
∥
+
∥
(
v
~
n
-
w
n
)
+
(
p
-
p
~
)
∥
.
Hence from (112) and (115) we get
(117)
lim
⁡
n
→
∞
∥
v
n
-
G
v
n
∥
=
lim
⁡
n
→
∞
∥
v
n
-
w
n
∥
=
0
,
then by (75), (102), and (117), we have
(118)
∥
x
n
-
W
n
x
n
∥
≤
∥
x
n
-
W
n
G
v
n
∥
+
∥
W
n
G
v
n
-
W
n
x
n
∥
≤
∥
x
n
-
W
n
G
v
n
∥
+
∥
G
v
n
-
x
n
∥
≤
∥
x
n
-
W
n
G
v
n
∥
+
∥
G
v
n
-
v
n
∥
+
∥
v
n
-
x
n
∥
⟶
0
hhhhhhhhhhhhhhhhhhhhhhhhh
.
as
n
⟶
∞
.
Also, observe that
(119)
∥
x
n
-
W
x
n
∥
≤
∥
x
n
-
W
n
x
n
∥
+
∥
W
n
x
n
-
W
x
n
∥
.
From (118), [45, Remark 3.2], and the boundedness of {
x
n
} we immediately obtain
(120)
lim
⁡
n
→
∞
∥
x
n
-
W
x
n
∥
=
0
.
In addition, from (67) and (77), we have
(121)
∥
z
n
+
1
-
z
n
∥
≤
∥
z
n
+
1
-
x
n
+
1
∥
+
∥
x
n
+
1
-
x
n
∥
+
∥
x
n
-
z
n
∥
⟶
0
hhhhhhhhhhhhhhhhhhhhhhhhh
.
as
n
⟶
∞
.
We note that
(122)
∥
S
n
z
n
-
S
n
+
1
z
n
∥
≤
∥
S
n
z
n
-
z
n
∥
+
∥
z
n
-
z
n
+
1
∥
+
∥
z
n
+
1
-
S
n
+
1
z
n
+
1
∥
+
∥
S
n
+
1
z
n
+
1
-
S
n
+
1
z
n
∥
.
From (81), (121), and Lemma 15, we obtain
(123)
lim
⁡
n
→
∞
∥
S
n
z
n
-
S
n
+
1
z
n
∥
=
0
.
In the meantime, we note that
(124)
∥
z
n
-
S
z
n
∥
≤
∥
z
n
-
S
n
z
n
∥
+
∥
S
n
z
n
-
S
n
+
1
z
n
∥
+
∥
S
n
+
1
z
n
-
S
z
n
∥
.
From (81), (123), and the uniform continuity of S, we have
(125)
lim
⁡
n
→
∞
∥
z
n
-
S
z
n
∥
=
0
.
Step 4. We prove that x
n
→
x
*
=
P
Ω
x
0 as n
→
∞.
Indeed, since{
x
n
} is bounded, there exists a subsequence {
x
n
i
} which converges weakly to some w. From (102), (83), (100), and (77), we have that v
n
i
⇀
w
,
u
n
i
⇀
w
,
Λ
n
i
m
u
n
i
⇀
w, and z
n
i
⇀
w, where m
∈
{
1,2
,
…
,
N
}. Since S is uniformly continuous, by (125) we get lim
⁡
n
→
∞
∥
z
n
-
S
m
z
n
∥
=
0 for any m
≥
1. Hence from Lemma 17, we obtain w
∈
Fix
⁡
(
S
). In the meantime, utilizing Lemma 12, we deduce from v
n
i
⇀
w
,
x
n
i
⇀
w, (117), and (120) that w
∈
SGEP
(
G
) and w
∈
Fix
⁡
(
W
)
=
∩
n
=
1
∞
Fix
⁡
(
T
n
) (due to Lemma 11). Next, we prove that w
∈
∩
m
=
1
N
VI
⁡
(
C
,
B
m
). As a matter of fact, let
(126)
T
~
m
v
=
{
B
m
v
+
N
C
v
,
v
∈
C
,
∅
,
v
∉
C
,
where m
∈
{
1,2
,
…
,
N
}. Let (
v
,
u
)
∈
G
(
T
~
m
). Since u
-
B
m
v
∈
N
C
v and Λ
n
m
u
n
∈
C, we have
(127)
〈
v
-
Λ
n
m
u
n
,
u
-
B
m
v
〉
≥
0
.
On the other hand, from Λ
n
m
u
n
=
P
C
(
I
-
λ
m
,
n
B
m
)
Λ
n
m
-
1
u
n and v
∈
C, we have
(128)
〈
v
-
Λ
n
m
u
n
,
Λ
n
m
u
n
-
(
Λ
n
m
-
1
u
n
-
λ
m
,
n
B
m
Λ
n
m
-
1
u
n
)
〉
≥
0
,
and hence
(129)
〈
v
-
Λ
n
m
u
n
,
Λ
n
m
u
n
-
Λ
n
m
-
1
u
n
λ
m
,
n
+
B
m
Λ
n
m
-
1
u
n
〉
≥
0
.
Therefore we have
(130)
〈
v
-
Λ
n
i
m
u
n
i
,
u
〉
≥
〈
v
-
Λ
n
i
m
u
n
i
,
B
m
v
〉
≥
〈
v
-
Λ
n
i
m
u
n
i
,
B
m
v
〉
-
〈
v
-
Λ
n
i
m
u
n
i
,
Λ
n
i
m
u
n
i
-
Λ
n
i
m
-
1
u
n
i
λ
m
,
n
i
+
B
m
Λ
n
i
m
-
1
u
n
i
〉
=
〈
v
-
Λ
n
i
m
u
n
i
,
B
m
v
-
B
m
Λ
n
i
m
u
n
i
〉
+
〈
v
-
Λ
n
i
m
u
n
i
,
B
m
Λ
n
i
m
u
n
i
-
B
m
Λ
n
i
m
-
1
u
n
i
〉
-
〈
v
-
Λ
n
i
m
u
n
i
,
Λ
n
i
m
u
n
i
-
Λ
n
i
m
-
1
u
n
i
λ
m
,
n
i
〉
≥
〈
v
-
Λ
n
i
m
u
n
i
,
B
m
Λ
n
i
m
u
n
i
-
B
m
Λ
n
i
m
-
1
u
n
i
〉
-
〈
v
-
Λ
n
i
m
u
n
i
,
Λ
n
i
m
u
n
i
-
Λ
n
i
m
-
1
u
n
i
λ
m
,
n
i
〉
.
From (100) and since B
m is uniformly continuous, we obtain that lim
⁡
n
→
∞
∥
B
m
Λ
n
m
u
n
-
B
m
Λ
n
m
-
1
u
n
∥
=
0. From Λ
n
i
m
u
n
i
⇀
w
,
{
λ
m
,
n
}
⊂
[
a
m
,
b
m
]
⊂
(
0,2
η
m
), ∀
m
∈
{
1,2
,
…
,
N
}, and (100), we have
(131)
〈
v
-
w
,
u
〉
≥
0
.
Since T
~
m is maximal monotone, we have w
∈
T
~
m
-
1
0 and hence w
∈
VI
⁡
(
C
,
B
m
), m
=
1,2
,
…
,
N, which implies w
∈
∩
m
=
1
N
VI
⁡
(
C
,
B
m
).
Next, we show thatw
∈
GMEP
⁡
(
Θ
,
φ
,
A
). In fact, from u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n, we know that
(132)
Θ
(
u
n
,
y
)
+
φ
(
y
)
-
φ
(
u
n
)
+
〈
A
x
n
,
y
-
u
n
〉
+
1
r
n
〈
K
′
(
u
n
)
-
K
′
(
x
n
)
,
y
-
u
n
〉
≥
0
,
∀
y
∈
C
.
From (H2) it follows that
(133)
φ
(
y
)
-
φ
(
u
n
)
+
〈
A
x
n
,
y
-
u
n
〉
+
1
r
n
〈
K
′
(
u
n
)
-
K
′
(
x
n
)
,
y
-
u
n
〉
≥
Θ
(
y
,
u
n
)
,
hhhhhhhhhhhhhhhhhhhhhhhlhhh
∀
y
∈
C
.
Replacing n by n
i, we have
(134)
φ
(
y
)
-
φ
(
u
n
i
)
+
〈
A
x
n
i
,
y
-
u
n
i
〉
+
〈
K
′
(
u
n
i
)
-
K
′
(
x
n
i
)
r
n
i
,
y
-
u
n
i
〉
≥
Θ
(
y
,
u
n
i
)
,
hhhhhhhhhhhhhhhhhhhhhhhhhlhhh
∀
y
∈
C
.
Put u
t
=
t
y
+
(
1
-
t
)
w for all t
∈
(
0,1
] and y
∈
C. Then from (134) we have
(135)
〈
u
t
-
u
n
i
,
A
u
t
〉
≥
〈
u
t
-
u
n
i
,
A
u
t
〉
-
φ
(
u
t
)
+
φ
(
u
n
i
)
-
〈
u
t
-
u
n
i
,
A
x
n
i
〉
-
〈
K
′
(
u
n
i
)
-
K
′
(
x
n
i
)
r
n
i
,
u
t
-
u
n
i
〉
+
Θ
(
u
t
,
u
n
i
)
≥
〈
u
t
-
u
n
i
,
A
u
t
-
A
u
n
i
〉
+
〈
u
t
-
u
n
i
,
A
u
n
i
-
A
x
n
i
〉
-
φ
(
u
t
)
+
φ
(
u
n
i
)
-
〈
K
′
(
u
n
i
)
-
K
′
(
x
n
i
)
r
n
i
,
u
t
-
u
n
i
〉
+
Θ
(
u
t
,
u
n
i
)
.
Since ∥
u
n
i
-
x
n
i
∥
→
0 as i
→
∞, we deduce from the Lipschitz continuity of A and K
′ that ∥
A
u
n
i
-
A
x
n
i
∥
→
0 and ∥
K
′
(
u
n
i
)
-
K
′
(
x
n
i
)
∥
→
0 as i
→
∞. Further, from the monotonicity of A, we have 〈
u
t
-
u
n
i
,
A
u
t
-
A
u
n
i
〉
≥
0. So, from (H4), the weakly lower semicontinuity of φ, (
K
′
(
u
n
i
)
-
K
′
(
x
n
i
)
)
/
r
n
i
→
0 and u
n
i
⇀
w, we have
(136)
〈
u
t
-
w
,
A
u
t
〉
≥
-
φ
(
u
t
)
+
φ
(
w
)
+
Θ
(
u
t
,
w
)
,
as
i
⟶
∞
.
From (H1), (H4), and (136) we also have
(137)
0
=
Θ
(
u
t
,
u
t
)
+
φ
(
u
t
)
-
φ
(
u
t
)
≤
t
Θ
(
u
t
,
y
)
+
(
1
-
t
)
Θ
(
u
t
,
w
)
+
t
φ
(
y
)
+
(
1
-
t
)
φ
(
w
)
-
φ
(
u
t
)
=
t
[
Θ
(
u
t
,
y
)
+
φ
(
y
)
-
φ
(
u
t
)
]
+
(
1
-
t
)
[
Θ
(
u
t
,
w
)
+
φ
(
w
)
-
φ
(
w
)
-
φ
(
u
t
)
]
≤
t
[
Θ
(
u
t
,
y
)
+
φ
(
y
)
-
φ
(
u
t
)
]
+
(
1
-
t
)
〈
u
t
-
w
,
A
u
t
〉
=
t
[
Θ
(
u
t
,
y
)
+
φ
(
y
)
-
φ
(
u
t
)
]
+
(
1
-
t
)
t
〈
y
-
w
,
A
u
t
〉
,
and hence
(138)
0
≤
Θ
(
u
t
,
y
)
+
φ
(
y
)
-
φ
(
u
t
)
+
(
1
-
t
)
〈
y
-
w
,
A
u
t
〉
.
Letting t
→
0
+, we have, for each y
∈
C,
(139)
0
≤
Θ
(
w
,
y
)
+
φ
(
y
)
-
φ
(
w
)
+
〈
A
w
,
y
-
w
〉
.
This implies that w
∈
GMEP
⁡
(
Θ
,
φ
,
A
). Consequently, w
∈
Ω
=
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
GMEP
⁡
(
Θ
,
φ
,
A
)
∩
SGEP
(
G
)
∩
⋂
i
=
1
N
VI
⁡
(
C
,
B
i
)
∩
Fix
⁡
(
S
). This shows that ω
w
(
x
n
)
⊂
Ω. From (64) and Lemma 22 we infer that x
n
→
x
*
=
P
Ω
x
0 as n
→
∞. This completes the proof.Corollary 24.
ChooseN
=
2 in Theorem 23. For any x
0
∈
H, C
1
=
C, and x
1
=
P
C
1
x
0, the iterative scheme (49) reduces to the following iterative one:
(140)
u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
,
v
n
=
P
C
(
I
-
λ
2
,
n
B
2
)
P
C
(
I
-
λ
1
,
n
B
1
)
u
n
,
z
n
=
β
n
x
n
+
σ
n
G
v
n
+
[
(
1
-
β
n
)
I
-
σ
n
V
]
W
n
G
v
n
,
k
n
=
δ
n
z
n
+
(
1
-
δ
n
)
S
n
z
n
,
y
n
=
(
1
-
α
n
)
x
n
+
α
n
k
n
,
C
n
+
1
=
{
z
∈
C
n
:
∥
y
n
-
z
∥
2
≤
∥
x
n
-
z
∥
2
+
θ
n
}
,
x
n
+
1
=
P
C
n
+
1
x
0
,
∀
n
≥
1
,
where θ
n
=
(
σ
n
+
γ
n
)
(
1
+
γ
n
)
Δ
n
+
c
n, Δ
n
=
sup
⁡
{
∥
x
n
-
p
∥
2
+
∥
(
I
-
V
)
p
∥
2
/
(
γ
¯
-
1
)
:
p
∈
Ω
}
<
∞, ν
k
∈
(
0,2
ζ
k
), k
=
1,2, and {
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
)
,
i
=
1,2. Then {
x
n
} converges strongly to x
*
=
P
Ω
x
0 provided that S
r
(
Θ
,
φ
) is firmly nonexpansive.Corollary 25.
ChooseN
=
1 and T
n
≡
I the identity operator of H in Theorem 23. For any x
0
∈
H, C
1
=
C, and x
1
=
P
C
1
x
0, the iterative scheme (49) reduces to the following iterative one:
(141)
u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
,
v
n
=
P
C
(
I
-
λ
1
,
n
B
1
)
,
z
n
=
β
n
x
n
+
(
1
-
β
n
)
G
v
n
+
σ
n
(
I
-
V
)
G
v
n
,
k
n
=
δ
n
z
n
+
(
1
-
δ
n
)
S
n
z
n
,
y
n
=
(
1
-
α
n
)
x
n
+
α
n
k
n
,
C
n
+
1
=
{
z
∈
C
n
:
∥
y
n
-
z
∥
2
≤
∥
x
n
-
z
∥
2
+
θ
n
}
,
x
n
+
1
=
P
C
n
+
1
x
0
,
∀
n
≥
1
,
where θ
n
=
(
σ
n
+
γ
n
)
(
1
+
γ
n
)
Δ
n
+
c
n, Δ
n
=
sup
⁡
{
∥
x
n
-
p
∥
2
+
∥
(
I
-
V
)
p
∥
2
/
(
γ
¯
-
1
)
:
p
∈
Ω
}
<
∞, ν
k
∈
(
0,2
ζ
k
), k
=
1,2, and {
λ
1
,
n
}
⊂
[
a
1
,
b
1
]
⊂
(
0,2
η
1
). Then {
x
n
} converges strongly to x
*
=
P
Ω
x
0 provided that S
r
(
Θ
,
φ
) is firmly nonexpansive.Proof.
In Theorem23, putting N
=
1 and T
n
≡
I the identity operator of H, we have W
n
≡
I. In this case, we get
(142)
z
n
=
β
n
x
n
+
σ
n
G
v
n
+
[
(
1
-
β
n
)
I
-
σ
n
V
]
W
n
G
v
n
=
β
n
x
n
+
σ
n
G
v
n
+
[
(
1
-
β
n
)
I
-
σ
n
V
]
G
v
n
=
β
n
x
n
+
(
1
-
β
n
)
G
v
n
+
σ
n
(
I
-
V
)
G
v
n
.
So, the iterative scheme (49) reduces to the iterative one (141). Utilizing Theorem 23, we derive the desired result.Remark 26.
Theorem23 extends, improves, supplements, and develops Ceng et al.’s [20, Theorem 1] in the following aspects.
(i) The problem of finding a point(143)
x
*
∈
⋂
i
=
1
N
Fix
⁡
(
T
n
)
∩
GMEP
⁡
(
Θ
,
φ
,
A
)
∩
SGEP
(
G
)
∩
⋂
i
=
1
N
VI
⁡
(
C
,
B
i
)
∩
Fix
⁡
(
S
)
in Theorem 23 is very different from the problem of finding a point
(144)
x
*
∈
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
GMEP
⁡
(
Θ
,
φ
,
A
)
∩
SGEP
(
G
)
in Ceng et al.’s [20, Theorem 1]. There is no doubt that our problem of finding a point x
*
∈
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
GMEP
⁡
(
Θ
,
φ
,
A
)
∩
SGEP
(
G
)
∩
⋂
i
=
1
N
VI
⁡
(
C
,
B
i
)
∩
Fix
⁡
(
S
) is more general and more subtle than the problem of finding a point x
*
∈
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
GMEP
⁡
(
Θ
,
φ
,
A
)
∩
SGEP
(
G
) in [20, Theorem 1].
(ii) The iterative scheme in [20, Theorem 1] is extended to develop the iterative scheme in Theorem 23 by the virtue of Mann-type iterative method and the shrinking projection method. The iterative scheme in Theorem 23 is more advantageous and more flexible than the iterative scheme in [20, Theorem 1] because it involves solving four problems: the GMEP (5), the SGEP (12), finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings.
(iii) The iterative scheme in Theorem23 is very different from the iterative scheme in [20, Theorem 1] because the iterative scheme in Theorem 23 involves Mann-type iterative method and the shrinking projection method. The proof of [20, Theorem 1] makes use of Lemma 12 (i.e., Demiclosedness principle for a nonexpansive mapping) but no use of Lemma 17 (i.e., Demiclosedness principle for an asymptotically strict pseudocontractive mapping in the intermediate sense). However, the proof of Theorem 23 depends on not only Lemma 12 but also Lemma 17 because there is an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings appearing in the problem of Theorem 23.
(iv) The proof of Theorem23 combines Cai and Bu convergence analysis for Mann-type iterative method and the shrinking projection method to solve finitely many GMEPs, finitely many VIPs, and the fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense (see [26, Theorem 3.1]) and Ceng et al.'s convergence analysis for hybrid extragradient-like iterative algorithm (see [20, Theorem 3.1]), where γ
¯
∈
(
0,1
] for a γ
¯-strongly positive bounded linear operator V. Because in iterative scheme (49) the composite shrinking projection method involves a γ
¯-strongly positive bounded linear operator V with γ
¯
∈
(
1,2
] and infinitely many nonexpansive mappings, the properties of the W-mappings W
n and W and the operator V play a key role in the proof of Theorem 23.
(v) Theorem23 extends Ceng et al.’s [20, Theorem 1] from the fixed point problem of infinitely many nonexpansive mappings to the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings and generalizes Ceng et al.’s [20, Theorem 1] to the setting of finitely many variational inequalities. The proof of Theorem 23 depends on the properties of the γ
¯-strongly positive bounded linear operator V with γ
¯
∈
(
1,2
], the result on the W-mappings W
n and W (i.e., lim
⁡
n
→
∞
∥
W
n
x
n
-
W
x
n
∥
=
0 for any bounded sequence {
x
n
}
⊂
C) (see [45, Remark 3.2]), and the properties of asymptotically strict pseudocontractive mapping in the intermediate sense (see Lemmas 15–18).Remark 27.
Theorem23 extends, improves, supplements, and develops Yao et al.’s [30, Theorem 3.1] in the following aspects.(i)
Theorem23 generalizes and extends [30, Theorem 3.1] from the asymptotically k-strict pseudocontractive mapping to the asymptotically k-strict pseudocontractive mapping in the intermediate sense and from the MEP to the GMEP and generalizes [30, Theorem 3.1] to the setting of SGEP.
(ii)
We add finitely many variational inequalities and infinitely many nonexpansive mappings{
T
n
}
n
=
1
∞ in our algorithm such that it can be applied to find a common solution of the GMEP (5), the SGEP (12), finitely many variational inequalities for inverse strongly monotone mappings, and the common fixed point problem of an asymptotically k-strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings {
T
n
}
n
=
1
∞.


## 4. Weak Convergence Theorem

In this section, we will propose and analyze another iterative algorithm (involving no shrinking projection method) for finding a solution of the system of generalized equilibria with constraints of several problems: a generalized mixed equilibrium problem, finitely many variational inequalities, and the common fixed point problem of an asymptotically strict pseudocontractive mapping in the intermediate sense and infinitely many nonexpansive mappings in a real Hilbert space. Moreover, under mild conditions we will prove weak convergence of the proposed algorithm.Theorem 28.
LetC
,
N
,
Θ
,
Θ
1
,
Θ
2
,
φ
,
A
,
A
k
,
B
i
,
V
,
{
T
n
}
n
=
1
∞, {
λ
n
}, and W
n be the same notations as in Theorem 23, where k
∈
{
1,2
} and i
∈
{
1,2
,
…
,
N
}. Let S
:
C
→
C be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense for some 0
≤
k
<
1 with the sequence {
γ
n
}
⊂
[
0
,
∞
) such that ∑
n
=
1
∞
‍
γ
n
<
∞ and {
c
n
}
⊂
[
0
,
∞
) such that ∑
n
=
1
∞
‍
c
n
<
∞. Assume that Ω
⋂
i
=
1
N
Fix
⁡
(
T
n
)
∩
GMEP
⁡
(
Θ
,
φ
,
A
)
∩
S
G
E
P
(
G
)
∩
⋂
i
=
1
N
VI
⁡
(
C
,
B
i
)
∩
Fix
⁡
(
S
) is nonempty where G is defined as in Proposition CY. Let {
r
n
} be a sequence in [
0,2
ζ
] and let {
α
n
}
,
{
β
n
}
,
{
σ
n
}, and {
δ
n
} be sequences in [
0,1
] such that ∑
n
=
1
∞
‍
σ
n
<
∞
,
0
<
α
≤
α
n
≤
1, and 0
<
k
+
ϵ
≤
δ
n
≤
d
<
1. Pick any x
1
∈
H and let {
x
n
} be a sequence generated by the following algorithm:
(145)
u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
,
v
n
=
P
C
(
I
-
λ
N
,
n
B
N
)
×
P
C
(
I
-
λ
N
-
1
,
n
B
N
-
1
)
⋯
P
C
(
I
-
λ
1
,
n
B
1
)
u
n
,
z
n
=
β
n
x
n
+
σ
n
G
v
n
+
[
(
1
-
β
n
)
I
-
σ
n
V
]
W
n
G
v
n
,
k
n
=
δ
n
z
n
+
(
1
-
δ
n
)
S
n
z
n
,
x
n
+
1
=
(
1
-
α
n
)
x
n
+
α
n
k
n
,
∀
n
≥
1
,
where ν
k
∈
(
0,2
ζ
k
), k
∈
{
1,2
}, and {
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
)
,
∀
i
∈
{
1,2
,
…
,
N
}. Assume that the conditions (i)–(iii) are satisfied. Then {
x
n
} converges weakly to x
*
=
lim
⁡
n
→
∞
P
Ω
x
n provided that S
r
(
Θ
,
φ
) is firmly nonexpansive.Proof.
Aslim
⁡
n
→
∞
σ
n
=
0, 0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
β
n
<
1 and 0
<
liminf
⁡
n
→
∞
r
n
≤
limsup
⁡
n
→
∞
r
n
<
2
ζ, we may assume, without loss of generality, that {
β
n
}
⊂
[
a
,
a
^
]
⊂
[
0,1
]
,
{
r
n
}
⊂
[
c
,
c
^
]
⊂
(
0,2
ζ
), and β
n
+
σ
n
∥
V
∥
≤
1 for all n
≥
1. First, let us show that lim
⁡
n
→
∞
∥
x
n
-
p
∥ exists for any p
∈
Ω. Put
(146)
Λ
n
i
=
P
C
(
I
-
λ
i
,
n
B
i
)
P
C
(
I
-
λ
i
-
1
,
n
B
i
-
1
)
⋯
P
C
(
I
-
λ
1
,
n
B
1
)
for all i
∈
{
1,2
,
…
,
N
}
,
n
≥
1, and Λ
n
0
=
I, where I is the identity mapping on H. Then we get v
n
=
Λ
n
N
u
n. Take p
∈
Ω arbitrarily. Repeating the same arguments as in the proof of Theorem 23, we can obtain that
(147)
∥
(
1
-
β
n
)
I
-
σ
n
V
∥
≤
1
-
β
n
-
σ
n
γ
¯
,
(148)
∥
u
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
+
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
≤
∥
x
n
-
p
∥
2
,
(149)
∥
v
n
-
p
∥
≤
∥
u
n
-
p
∥
,
(150)
∥
G
v
n
-
p
∥
2
≤
∥
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
v
n
-
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
p
∥
2
+
ν
1
(
ν
1
-
2
ζ
1
)
×
∥
A
1
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
v
n
-
A
1
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
p
∥
2
≤
∥
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
v
n
-
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
p
∥
2
≤
∥
v
n
-
p
∥
2
+
ν
2
(
ν
2
-
2
ζ
2
)
∥
A
2
v
n
-
A
2
p
∥
2
≤
∥
v
n
-
p
∥
2
,
(151)
∥
Λ
n
i
u
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
2
,
i
∈
{
1,2
,
…
,
N
}
,
(152)
∥
k
n
-
p
∥
2
≤
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
(
1
-
δ
n
)
×
(
k
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
+
c
n
≤
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
c
n
≤
(
1
+
γ
n
)
(
∥
x
n
-
p
∥
2
+
σ
n
∥
(
I
-
V
)
p
∥
2
γ
¯
-
1
)
+
c
n
,
(153)
∥
Λ
n
i
u
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
×
∥
B
i
Λ
n
i
-
1
u
n
-
B
i
p
∥
,
i
∈
{
1,2
,
…
,
N
}
.
Utilizing (145) and (152), we obtain
(154)
∥
x
n
+
1
-
p
∥
2
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
∥
k
n
-
p
∥
2
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
(
∥
x
n
-
p
∥
2
+
σ
n
∥
(
I
-
V
)
p
∥
2
γ
¯
-
1
)
+
c
n
]
≤
(
1
+
γ
n
)
(
∥
x
n
-
p
∥
2
+
σ
n
∥
(
I
-
V
)
p
∥
2
γ
¯
-
1
)
+
c
n
=
∥
x
n
-
p
∥
2
+
γ
n
∥
x
n
-
p
∥
2
+
σ
n
(
1
+
γ
n
)
∥
(
I
-
V
)
p
∥
2
γ
¯
-
1
+
c
n
.
Since ∑
n
=
1
∞
‍
σ
n
<
∞
,
∑
n
=
1
∞
‍
γ
n
<
∞, and ∑
n
=
1
∞
‍
c
n
<
∞, by Lemma 21 we have that lim
⁡
n
→
∞
∥
x
n
-
p
∥ exists. Thus {
x
n
} is bounded and so are the sequences {
u
n
}
,
{
v
n
}
,
{
z
n
}, and {
k
n
}.
Also, utilizing Lemmas8 and 9(b), we obtain from (145), (148), (149), and (152) that
(155)
∥
z
n
-
p
∥
2
=
∥
β
n
(
x
n
-
p
)
+
(
1
-
β
n
)
(
W
n
G
v
n
-
p
)
hh
+
σ
n
(
G
v
n
-
V
W
n
G
v
n
)
∥
2
≤
∥
β
n
(
x
n
-
p
)
+
(
1
-
β
n
)
(
W
n
G
v
n
-
p
)
∥
2
+
2
σ
n
〈
G
v
n
-
V
W
n
G
v
n
,
z
n
-
p
〉
=
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
W
n
G
v
n
-
p
∥
2
-
β
n
(
1
-
β
n
)
∥
x
n
-
W
n
G
v
n
∥
2
+
2
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
=
∥
x
n
-
p
∥
2
-
β
n
(
1
-
β
n
)
∥
x
n
-
W
n
G
v
n
∥
2
+
2
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
≤
∥
x
n
-
p
∥
2
+
2
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
,
and hence
(156)
∥
x
n
+
1
-
p
∥
2
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
∥
k
n
-
p
∥
2
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
c
n
]
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
hhhhh
×
(
∥
x
n
-
p
∥
2
hhhhhhhh
-
β
n
(
1
-
β
n
)
∥
x
n
-
W
n
G
v
n
∥
2
hhhhhhhh
+
2
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
hhhhhhhh
∥
x
n
-
p
∥
2
×
∥
z
n
-
p
∥
)
+
c
n
]
≤
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
-
α
n
(
1
+
γ
n
)
β
n
×
(
1
-
β
n
)
∥
x
n
-
W
n
G
v
n
∥
2
+
2
σ
n
(
1
+
γ
n
)
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
+
c
n
.
So, it follows that
(157)
α
(
1
+
γ
n
)
a
(
1
-
a
^
)
∥
x
n
-
W
n
G
v
n
∥
2
≤
α
n
(
1
+
γ
n
)
β
n
(
1
-
β
n
)
∥
x
n
-
W
n
G
v
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
x
n
+
1
-
p
∥
2
+
γ
n
∥
x
n
-
p
∥
2
+
2
σ
n
(
1
+
γ
n
)
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
+
c
n
.
Since lim
⁡
n
→
∞
σ
n
=
0, lim
⁡
n
→
∞
γ
n
=
0, and lim
⁡
n
→
∞
c
n
=
0, it follows from the existence of lim
⁡
n
→
∞
∥
x
n
-
p
∥ and the boundedness of {
x
n
}
,
{
z
n
}, and {
v
n
} that
(158)
lim
⁡
n
→
∞
∥
x
n
-
W
n
G
v
n
∥
=
0
.
Note that
(159)
∥
z
n
-
x
n
∥
=
∥
(
1
-
β
n
)
(
W
n
G
v
n
-
x
n
)
+
σ
n
(
G
v
n
-
V
W
n
G
v
n
)
∥
≤
∥
W
n
G
v
n
-
x
n
∥
+
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
.
Hence, it follows from (158) and lim
⁡
n
→
∞
σ
n
=
0 that
(160)
lim
⁡
n
→
∞
∥
x
n
-
z
n
∥
=
0
.
In the meantime, from (152) and (155) it follows that
(161)
∥
x
n
+
1
-
p
∥
2
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
∥
k
n
-
p
∥
2
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
(
1
-
δ
n
)
hhhhh
×
(
k
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
+
c
n
]
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
hhhhh
×
(
∥
x
n
-
p
∥
2
hhhhhhhh
∥
x
n
-
p
∥
2
+
2
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
)
hhhhh
∥
x
n
-
p
∥
2
+
(
1
-
δ
n
)
(
k
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
+
c
n
]
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
(
1
+
γ
n
)
×
(
∥
x
n
-
p
∥
2
+
2
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
)
+
α
n
(
1
-
δ
n
)
(
k
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
+
c
n
≤
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
2
(
1
+
γ
n
)
σ
n
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
+
α
n
(
1
-
δ
n
)
(
k
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
+
c
n
≤
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
2
σ
n
(
1
+
γ
n
)
×
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
+
α
n
(
1
-
δ
n
)
(
k
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
+
c
n
,
which, together with 0
<
k
+
ϵ
≤
δ
n
≤
d
<
1, leads to
(162)
α
(
1
-
d
)
ϵ
∥
z
n
-
S
n
z
n
∥
2
≤
α
n
(
1
-
δ
n
)
(
δ
n
-
k
)
∥
z
n
-
S
n
z
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
x
n
+
1
-
p
∥
2
+
γ
n
∥
x
n
-
p
∥
2
+
2
σ
n
(
1
+
γ
n
)
∥
G
v
n
-
V
W
n
G
v
n
∥
∥
z
n
-
p
∥
+
c
n
.
Consequently, from lim
⁡
n
→
∞
σ
n
=
0, lim
⁡
n
→
∞
γ
n
=
0, lim
⁡
n
→
∞
c
n
=
0, and the existence of lim
⁡
n
→
∞
∥
x
n
-
p
∥, we get
(163)
lim
⁡
n
→
∞
∥
z
n
-
S
n
z
n
∥
=
0
.
Since k
n
-
z
n
=
(
1
-
δ
n
)
(
S
n
z
n
-
z
n
), from (163) we have
(164)
lim
⁡
n
→
∞
∥
k
n
-
z
n
∥
=
0
.
Note that
(165)
∥
x
n
+
1
-
x
n
∥
=
α
n
∥
k
n
-
x
n
∥
≤
∥
k
n
-
z
n
∥
+
∥
z
n
-
x
n
∥
.
Hence from (160) and (164) we have
(166)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0
.
Repeating the same arguments as those of Step 3 in the proof of Theorem 23, we can obtain that ∥
x
n
-
u
n
∥
→
0, ∥
x
n
-
v
n
∥
→
0, ∥
v
n
-
G
v
n
∥
→
0, ∥
x
n
-
W
x
n
∥
→
0
,
∥
z
n
-
S
z
n
∥
→
0, and ∥
Λ
n
i
-
1
u
n
-
Λ
n
i
u
n
∥
→
0, i
∈
{
1,2
,
…
,
N
} as n
→
∞.
Since{
x
n
} is bounded, there exists a subsequence {
x
n
i
} of {
x
n
} which converges weakly to w. It is easy to see that v
n
i
⇀
w
,
u
n
i
⇀
w
,
Λ
n
i
m
u
n
i
⇀
w, and z
n
i
⇀
w, where m
∈
{
1,2
,
…
,
N
}. Since S is uniformly continuous and ∥
z
n
-
S
z
n
∥
→
0 as n
→
∞, we get lim
⁡
n
→
∞
∥
z
n
-
S
m
z
n
∥
=
0 for any m
≥
1. Hence from Lemma 17, we obtain w
∈
Fix
⁡
(
S
). In the meantime, utilizing Lemma 12, we deduce from v
n
i
⇀
w
,
x
n
i
⇀
w,
∥
v
n
-
G
v
n
∥
→
0, and ∥
x
n
-
W
x
n
∥
→
0 that w
∈
SGEP
(
G
) and w
∈
Fix
⁡
(
W
)
=
∩
n
=
1
∞
Fix
⁡
(
T
n
) (due to Lemma 11). Repeating the same arguments as those of Step 4 in the proof of Theorem 23, we can conclude that w
∈
⋂
m
=
1
N
VI
⁡
(
C
,
B
m
) and w
∈
GMEP
⁡
(
Θ
,
φ
,
A
). Consequently, w
∈
Ω. This shows that ω
w
(
x
n
)
⊂
Ω.
Next let us show thatω
w
(
x
n
) is a single-point set. As a matter of fact, let {
x
n
j
} be another subsequence of {
x
n
} such that x
n
j
⇀
w
′. Then we get w
′
∈
Ω. If w
≠
w
′, from the Opial condition, we have
(167)
lim
⁡
n
→
∞
∥
x
n
-
w
∥
=
lim
⁡
i
→
∞
∥
x
n
i
-
w
∥
<
lim
⁡
i
→
∞
∥
x
n
i
-
w
′
∥
=
lim
⁡
n
→
∞
∥
x
n
-
w
′
∥
=
lim
⁡
j
→
∞
∥
x
n
j
-
w
′
∥
<
lim
⁡
j
→
∞
∥
x
n
j
-
w
∥
=
lim
⁡
n
→
∞
∥
x
n
-
w
∥
.
This attains a contradiction. So we have w
=
w
′. Put w
n
=
P
Ω
x
n. Since w
∈
Ω, we have 〈
x
n
-
w
n
,
w
n
-
w
〉
≥
0. By Lemma 21, we have that {
w
n
} converges strongly to some w
~
∈
Ω. Since {
x
n
} converges weakly to w, we have
(168)
〈
w
-
w
~
,
w
~
-
w
〉
≥
0
.
Therefore we obtain w
=
w
~
=
lim
⁡
n
→
∞
P
Ω
x
n. This completes the proof.Corollary 29.
ChooseN
=
2 in Theorem 28. For any x
1
∈
H the iterative scheme (145) reduces to the following iterative one:
(169)
u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
,
v
n
=
P
C
(
I
-
λ
2
,
n
B
2
)
P
C
(
I
-
λ
1
,
n
B
1
)
u
n
,
z
n
=
β
n
x
n
+
(
1
-
β
n
)
G
v
n
+
σ
n
(
I
-
V
)
G
v
n
,
k
n
=
δ
n
z
n
+
(
1
-
δ
n
)
S
n
z
n
,
x
n
+
1
=
(
1
-
α
n
)
x
n
+
α
n
k
n
,
∀
n
≥
1
,
where ν
k
∈
(
0,2
ζ
k
) and {
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
) for k
=
1,2 and i
=
1,2. Then {
x
n
} converges weakly to x
*
=
lim
⁡
n
→
∞
P
Ω
x
n provided that S
r
(
Θ
,
φ
) is firmly nonexpansive.Corollary 30.
ChooseN
=
1 and T
n
≡
I the identity operator of H in Theorem 28. For any x
1
∈
H the iterative scheme (145) reduces to the following iterative one:
(170)
u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
,
v
n
=
P
C
(
I
-
λ
1
,
n
B
1
)
u
n
,
z
n
=
β
n
x
n
+
σ
n
G
v
n
+
[
(
1
-
β
n
)
I
-
σ
n
V
]
G
v
n
,
k
n
=
δ
n
z
n
+
(
1
-
δ
n
)
S
n
z
n
,
x
n
+
1
=
(
1
-
α
n
)
x
n
+
α
n
k
n
,
∀
n
≥
1
,
where ν
k
∈
(
0,2
ζ
k
) and {
λ
1
,
n
}
⊂
[
a
1
,
b
1
]
⊂
(
0,2
η
1
) for k
=
1,2. Then {
x
n
} converges weakly to x
*
=
lim
⁡
n
→
∞
P
Ω
x
n provided that S
r
(
Θ
,
φ
) is firmly nonexpansive.In the following, we provide a numerical example to illustrate how Corollary30 works.Example 31.
LetH
=
R
2 with inner product 〈
·
,
·
〉 and norm ∥
·
∥ which are defined by
(171)
〈
x
,
y
〉
=
a
c
+
b
d
,
∥
x
∥
=
a
2
+
b
2
,
for all x
,
y
∈
R
2 with x
=
(
a
,
b
) and y
=
(
c
,
d
). Let C
=
{
(
a
,
a
)
:
a
∈
R
}. Clearly, C is a nonempty closed convex subset of a real Hilbert space H
=
R
2. Let K
(
x
)
=
(
1
/
2
)
∥
x
∥
2,  ∀
x
∈
H, Θ
(
x
,
y
)
=
Θ
1
(
x
,
y
)
=
Θ
2
(
x
,
y
)
=
0, ∀
(
x
,
y
)
∈
C
×
C, and φ
=
0, ∀
x
∈
C. Then Θ, Θ
1, and Θ
2 are three bifunctions from C
×
C to R satisfying (H1)–(H4) and φ
:
C
→
R is a lower semicontinuous and convex functional. Let V be a γ
¯-strongly positive bounded linear operator with γ
¯
∈
(
1,2
], let A
,
A
k
:
H
→
H and B
1
:
C
→
H be ζ-inverse strongly monotone, ζ
k-inverse strongly monotone, and η
1-inverse strongly monotone, respectively, for k
=
1,2, and let S
:
C
→
C be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense for some 0
≤
k
<
1 with sequence {
γ
n
}
⊂
[
0
,
∞
) such that ∑
n
=
1
∞
‍
γ
n
<
∞ and {
c
n
}
⊂
[
0
,
∞
) such that ∑
n
=
1
∞
‍
c
n
<
∞ such that Ω
:
=
GMEP
⁡
(
Θ
,
φ
,
A
)
∩
SGEP
(
G
)
∩
VI
⁡
(
C
,
B
)
∩
Fix
⁡
(
S
) is nonempty, for instance, putting
(172)
A
=
{
3
5
2
5
2
5
3
5
}
,
B
1
=
S
=
{
2
3
1
3
1
3
2
3
}
,
V
=
5
4
A
,
A
1
=
I
-
A
=
{
2
5
-
2
5
-
2
5
2
5
}
,
A
2
=
I
-
B
1
=
{
1
3
-
1
3
-
1
3
1
3
}
.
It is easy to see that ∥
A
∥
=
∥
B
1
∥
=
∥
S
∥
=
1, that A is ζ-inverse strongly monotone with ζ
=
1
/
2, that V is a 5
/
4-strongly positive bounded linear operator, that B
1
,
A
1, and A
2 are 1
/
2-inverse strongly monotone, and that S is a nonexpansive mapping, that is, a uniformly continuous asymptotically 0-strict pseudocontractive mapping in the intermediate sense with sequences {
γ
n
} (γ
n
≡
0) and {
c
n
} (c
n
≡
0). Moreover, it is clear that Fix
⁡
(
S
)
=
C
,
GMEP
⁡
(
Θ
,
φ
,
A
)
=
{
0
}
,
VI
⁡
(
C
,
B
1
)
=
{
0
}, and SGEP
(
G
)
=
C. Hence, Ω
:
=
GMEP
⁡
(
Θ
,
φ
,
A
)
∩
SGEP
(
G
)
∩
VI
⁡
(
C
,
B
1
)
∩
Fix
⁡
(
S
)
=
{
0
}. In this case, from iterative scheme (170) in Corollary 30, we obtain that, for any given x
1
∈
C,
(173)
u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
=
P
C
(
I
-
r
n
A
)
x
n
=
(
1
-
r
n
)
x
n
,
v
n
=
P
C
(
I
-
λ
1
,
n
B
1
)
u
n
=
(
1
-
λ
1
,
n
)
u
n
=
(
1
-
λ
1
,
n
)
(
1
-
r
n
)
x
n
,
z
n
=
β
n
x
n
+
(
1
-
β
n
)
G
v
n
+
σ
n
(
I
-
V
)
G
v
n
=
β
n
x
n
+
(
1
-
β
n
)
T
ν
1
Θ
1
(
I
-
ν
1
A
1
)
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
v
n
+
σ
n
(
I
-
V
)
T
ν
1
Θ
1
(
I
-
ν
1
A
1
)
T
ν
2
Θ
2
(
I
-
ν
2
A
2
)
v
n
=
β
n
x
n
+
(
1
-
β
n
)
v
n
+
σ
n
(
I
-
5
4
A
)
v
n
=
β
n
x
n
+
(
1
-
β
n
)
v
n
-
1
4
σ
n
v
n
=
β
n
x
n
+
(
1
-
β
n
-
1
4
σ
n
)
v
n
=
β
n
x
n
+
(
1
-
β
n
-
1
4
σ
n
)
(
1
-
λ
1
,
n
)
(
1
-
r
n
)
x
n
=
[
β
n
+
(
1
-
β
n
-
1
4
σ
n
)
(
1
-
λ
1
,
n
)
(
1
-
r
n
)
]
x
n
,
k
n
=
δ
n
z
n
+
(
1
-
δ
n
)
S
n
z
n
=
δ
n
z
n
+
(
1
-
δ
n
)
z
n
=
z
n
=
[
β
n
+
(
1
-
β
n
-
1
4
σ
n
)
(
1
-
λ
1
,
n
)
(
1
-
r
n
)
]
x
n
,
x
n
+
1
=
(
1
-
α
n
)
x
n
+
α
n
k
n
=
(
1
-
α
n
)
x
n
+
α
n
[
β
n
+
(
1
-
β
n
-
1
4
σ
n
)
(
1
-
λ
1
,
n
)
(
1
-
r
n
)
]
x
n
=
{
1
-
α
n
1
4
g
g
h
+
α
n
[
β
n
+
(
1
-
β
n
-
1
4
σ
n
)
(
1
-
λ
1
,
n
)
(
1
-
r
n
)
]
}
x
n
.
Whenever 0
<
α
≤
α
n
≤
1
,
{
λ
1
,
n
}
⊂
[
a
1
,
b
1
]
⊂
(
0,1
)
,
{
β
n
}
⊂
[
c
,
c
^
]
⊂
(
0,1
)
,
{
r
n
}
⊂
[
e
,
e
^
]
⊂
(
0,1
) and {
σ
n
}
⊂
[
0,1
] satisfying (
1
/
4
)
σ
n
≤
1
-
c
^, we have
(174)
∥
x
n
+
1
∥
=
{
1
-
α
n
+
α
n
[
β
n
+
(
1
-
β
n
-
1
4
σ
n
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
1
4
×
(
1
-
λ
1
,
n
)
(
1
-
r
n
)
]
}
∥
x
n
∥
≤
{
1
-
α
n
+
α
n
[
β
n
+
(
1
-
β
n
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
(
1
-
λ
1
,
n
)
(
1
-
r
n
)
]
}
∥
x
n
∥
≤
{
1
-
α
n
+
α
n
[
β
n
+
(
1
-
β
n
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
(
1
-
a
1
)
(
1
-
e
)
]
}
∥
x
n
∥
=
[
1
-
α
n
(
1
-
β
n
)
+
α
n
(
1
-
β
n
)
h
h
h
h
h
×
(
1
-
a
1
)
(
1
-
e
)
]
∥
x
n
∥
=
[
1
-
α
n
(
1
-
β
n
)
(
1
-
(
1
-
a
1
)
(
1
-
e
)
)
]
∥
x
n
∥
≤
[
1
-
α
(
1
-
c
^
)
(
1
-
(
1
-
a
1
)
(
1
-
e
)
)
]
∥
x
n
∥
⋮
≤
[
1
-
α
(
1
-
c
^
)
(
1
-
(
1
-
a
1
)
(
1
-
e
)
)
]
n
∥
x
1
∥
.
Since 0
<
α
(
1
-
c
^
)
(
1
-
(
1
-
a
1
)
(
1
-
e
)
)
<
1, we immediately get
(175)
lim
⁡
n
→
∞
∥
x
n
∥
=
0
.
This shows that {
x
n
} converges to the unique element 0 of Ω.



---

*Source: 513678-2014-03-25.xml*