# Existence and Algorithm for the Systems of Hierarchical Variational Inclusion Problems

Authors: Nopparat Wairojjana; Poom Kumam
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We study the existence and approximation of a solution for a system of hierarchical variational inclusion problems in Hilbert spaces. In this study, we use Maingé’s approach for finding the solutions of the system of hierarchical variational inclusion problems. Our result in this paper improves and generalizes some known corresponding results in the literature.


---

## Body

## 1. Introduction

LetH be a real Hilbert space with inner product and norm being 〈
·
,
·
〉 and ∥
·
∥, respectively, and let C be a nonempty closed convex subset of H. A mapping T
:
H
→
H is called nonexpansive if
(1)
T
x
-
T
y
≤
x
-
y
,
∀
x
,
y
∈
H
.
We use F
(
T
) to denote the set offixed points of T; that is, F
(
T
)
=
{
x
∈
H
:
T
x
=
x
}. It is well known that F
(
T
) is a closed convex set, if T is nonexpansive mappings.A variational inclusion problem [1–3] is the problem of finding a point u
∈
H such that
(2)
θ
∈
A
(
u
)
+
M
(
u
)
,
where A
:
H
→
H is a single-valued nonlinear mapping and M
:
H
→
2
H is a multivalued mapping. We use Ω to denote the set of solutions of the variational inclusion (2).On the other hand,a hierarchical fixed point problem [4–11] is the problem of finding a point x
*
∈
F
(
T
) such that
(3)
A
x
*
,
x
-
x
*
≥
0
,
∀
x
∈
F
(
T
)
.
If the set F
(
T
) is replaced by the solution set of the variational inequality, then the hierarchical fixed point problems are calledhierarchical variational inequality problems orhierarchical optimization problems. Many problems in mathematics, for example, the signal recovery [12], the power control problem [13], and the beamforming problem [14], can be considered in the framework of this kind of the hierarchical variational inequality problems.Recently, Chang et al. [15] introducedbilevel hierarchical variational inclusion problems; that is, find (
x
*
,
y
*
)
∈
Ω
1
×
Ω
2 such that, for given positive real numbers ρ and η, the following inequalities hold:
(4)
ρ
F
y
*
+
x
*
-
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
,
η
F
x
*
+
y
*
-
x
*
,
y
-
y
*
≥
0
,
∀
y
∈
Ω
2
,
where F
,
A
1
,
A
2
:
H
→
H are mappings, M
1
,
M
2
:
H
→
2
H are multivalued mappings, and Ω
i is the set of solutions to variational inclusion problem (2) with A
=
A
i, M
=
M
i for i
=
1,2. They solved the convex programming problems and quadratic minimization problems by using Maingés scheme.In this paper, we consider the followingsystem of hierarchical variational inclusion problem: find (
x
*
,
y
*
,
z
*
)
∈
Ω
1
×
Ω
2
×
Ω
3, such that, for given positive real numbers ρ
,
η, and ξ, the following inequalities hold:
(5)
ρ
F
y
*
+
x
*
-
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
,
η
F
z
*
+
y
*
-
z
*
,
y
-
y
*
≥
0
,
∀
y
∈
Ω
2
,
ξ
F
x
*
+
z
*
-
x
*
,
z
-
z
*
≥
0
,
∀
z
∈
Ω
3
.
Some special cases of the system of hierarchical variational inclusion problem (5) are as follows. (I)
IfM
i
=
0, A
i
=
I
-
T
i, where T
i
:
H
→
H is a nonlinear mapping for each i
=
1,2
,
3, in (5), then Ω
i
=
F
(
T
i
) and the system of hierarchical variational inclusion problem (5) reduces to the followingsystem of hierarchical optimization problem: find (
x
*
,
y
*
,
z
*
)
∈
F
(
T
1
)
×
F
(
T
2
)
×
F
(
T
3
), such that
(6)
ρ
F
y
*
+
x
*
-
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
F
(
T
1
)
,
η
F
z
*
+
y
*
-
z
*
,
y
-
y
*
≥
0
,
∀
y
∈
F
(
T
2
)
,
ξ
F
x
*
+
z
*
-
x
*
,
z
-
z
*
≥
0
,
∀
z
∈
F
(
T
3
)
,
which was studied by Li [16].
(II)
IfT
i
=
P
K
i for each i
=
1,2
,
3, where P
K
i is the metric projection from H onto a nonempty closed convex subset K
i in (6), then it is clear that the Ω
i
=
F
(
T
i
)
=
K
i and the system of hierarchical optimization problem (6) reduces to the followingsystem of optimization problem: find (
x
*
,
y
*
,
z
*
)
∈
K
1
×
K
2
×
K
3 such that
(7)
ρ
F
y
*
+
x
*
-
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
K
1
,
η
F
z
*
+
y
*
-
z
*
,
y
-
y
*
≥
0
,
∀
y
∈
K
2
,
ξ
F
x
*
+
z
*
-
x
*
,
z
-
z
*
≥
0
,
∀
z
∈
K
3
.
(III)
IfK
1
=
K
2
=
K
3, then the system of optimization problem (7) reduces to the followingsystem of variational inequality problem: find (
x
*
,
y
*
,
z
*
)
∈
K
1
×
K
1
×
K
1 such that
(8)
ρ
F
y
*
+
x
*
-
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
K
1
,
η
F
z
*
+
y
*
-
z
*
,
y
-
y
*
≥
0
,
∀
y
∈
K
1
,
ξ
F
x
*
+
z
*
-
x
*
,
z
-
z
*
≥
0
,
∀
z
∈
K
1
.
(IV)
Ifξ
=
0
,
ρ
,
η
>
0, Ω
1
=
Ω
3, and x
*
=
z
* in (5) then the system of hierarchical variational inclusion problem (5) reduces to the followingbilevel hierarchical variational inclusion problem: find (
x
*
,
y
*
)
∈
Ω
1
×
Ω
2 such that
(9)
ρ
F
y
*
+
x
*
-
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
,
η
F
x
*
+
y
*
-
x
*
,
y
-
y
*
≥
0
,
∀
y
∈
Ω
2
,
which was studied by Chang et al. [15].
(V)
In (9), if M
i
=
0, A
i
=
I
-
T
i, for each i
=
1,2, then bilevel hierarchical variational inclusion problem (9) reduces to the followingbilevel hierarchical optimization problem: find (
x
*
,
y
*
)
∈
F
(
T
1
)
×
F
(
T
2
) such that
(10)
ρ
F
y
*
+
x
*
-
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
F
(
T
1
)
,
η
F
x
*
+
y
*
-
x
*
,
y
-
y
*
≥
0
,
∀
y
∈
F
(
T
2
)
,
which was studied by Maingé [17] and Kraikaew and Saejung [18].
(VI)
In (10), if T
i
=
P
K
i for each i
=
1,2, then bilevel hierarchical optimization problem (10) reduces to the following problem [19–21]: find (
x
*
,
y
*
)
∈
K
1
×
K
2 such that
(11)
ρ
F
y
*
+
x
*
-
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
K
1
,
η
F
x
*
+
y
*
-
x
*
,
y
-
y
*
≥
0
,
∀
y
∈
K
2
.
(VII)
In (11), if K
1
=
K
2 then the problem (11) reduces to the following problem: find (
x
*
,
y
*
)
∈
K
1
×
K
1 such that
(12)
ρ
F
y
*
+
x
*
-
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
K
1
,
η
F
x
*
+
y
*
-
x
*
,
y
-
y
*
≥
0
,
∀
y
∈
K
1
.
(VIII)
In (5), if ξ
=
η
=
0, ρ
>
0, Ω
1
=
Ω
2
=
Ω
3, and x
*
=
y
*
=
z
* then the system of hierarchical variational inclusion problem (5) reduces to the followinghierarchical variational inclusion problem: find x
*
∈
Ω
1 such that
(13)
F
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
.
(IX)
In (13), if M
1
=
0, A
i
=
I
-
T
1 then the hierarchical variational inclusion problem (13) reduces to the followinghierarchical fixed point problem: find x
*
∈
F
(
T
1
) such that
(14)
F
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
F
(
T
1
)
.
(X)
In (15), if T
1
=
P
K
1 then the hierarchical fixed point problem (15) reduces to the followingclassic variational inequality problem: find x
*
∈
K
1 such that
(15)
F
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
K
1
.Motivated and inspired by Chang et al. [15], we introduce the system of a hierarchical variational inclusion problem (5) and investigate a more general variant of the scheme proposed by Chang et al. [15] to solve the system of a hierarchical variational inclusion problem. Our analysis and method allow us to prove the existence and approximation of solutions to the system of a hierarchical variational inclusion problem (5). The results presented in this paper extend and improve the results of Chang et al. [15], Maingé [17], Kraikaew and Saejung [18], and some authors.


## 2. Preliminaries

This section collects some definitions and lemmas which can be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive. We use→ for strong convergence and ⇀ for weak convergence.Definition 1.
LetA
,
T
,
F
:
H
→
H be a mapping and let M
:
H
→
2
H be a multivalued mapping. (1)
A mappingT is callednonexpansive if
(16)
T
x
-
T
y
≤
x
-
y
,
∀
x
,
y
∈
H
.
(2)
A mappingT is calledquasinonexpansive if F
(
T
)
≠
∅ and
(17)
T
x
-
p
≤
x
-
p
,
∀
x
∈
H
,
p
∈
F
(
T
)
.
It should be noted thatT is quasinonexpansive if and only if for all x
∈
H, p
∈
F
(
T
)
(18)
x
-
T
x
,
x
-
p
≥
1
2
x
-
T
x
2
.
(3)
A mappingT is calledstrongly quasinonexpansiveif T is quasinonexpansive and x
n
-
T
x
n
→
0, whenever {
x
n
} is a bounded sequence in H and ∥
x
n
-
p
∥
-
∥
T
x
n
-
p
∥
→
0 for some p
∈
F
(
T
).
(4)
A mappingF is called μ
-Lipschitzian if there exists α
>
0 such that
(19)
F
x
-
F
y
≤
μ
x
-
y
,
∀
x
,
y
∈
H
.
(5)
A mappingF is called r
-strongly monotoneif there exists r
>
0 such that
(20)
F
x
-
F
y
,
x
-
y
≥
r
x
-
y
2
,
∀
x
,
y
∈
H
.
It is easy to prove that ifF
:
H
→
H is a μ-Lipschitzian and r-strongly monotone mapping and if ρ
∈
(
0
,
2
r
/
μ
2
), then the mapping I
-
ρ
F is a contraction.
(6)
A mappingA is called α
-inverse-strongly monotoneif there exists μ
>
0 such that
(21)
A
x
-
A
y
,
x
-
y
≥
α
A
x
-
A
y
2
,
∀
x
,
y
∈
H
.
(7)
A multivalued mappingM is calledmonotone if for all x
,
y
∈
H, u
∈
M
x and v
∈
M
y imply that
(22)
u
-
v
,
x
-
y
≥
0
.
(8)
A multivalued mappingM is calledmaximal monotone if it is monotone and for any (
x
,
u
)
∈
H
×
H,
(23)
u
-
v
,
x
-
y
≥
0
for every(
y
,
v
)
∈
Graph
⁡
(
M
) (the graph of mapping M) implies that u
∈
M
x.Lemma 2 (see [22]).
LetA
:
H
→
H be an α-inverse-strongly monotone mapping. Then (1)
A is an 1
/
α-Lipschitz continuous and monotone mapping;
(2)
for any constantλ
>
0, one has
(24)
(
I
-
λ
A
)
x
-
(
I
-
λ
A
)
y
2
≤
x
-
y
2
+
λ
(
λ
-
2
α
)
A
x
-
A
y
2
;
(3)
ifλ
∈
(
0,2
α
], then I
-
λ
A is a nonexpansive mapping, where I is the identity mapping on H.Lemma 3.
Letx
∈
H and z
∈
C be any points. Then one has the following. (1)
Thatz
=
P
C
[
x
] if and only if there holds the relation:
(25)
x
-
z
,
y
-
z
≤
0
,
∀
y
∈
C
.
(2)
Thatz
=
P
C
[
x
] if and only if there holds the relation:
(26)
x
-
z
2
≤
x
-
y
2
-
y
-
z
2
,
∀
y
∈
C
.
(3)
There holds the relation:(27)
P
C
x
-
P
C
y
,
x
-
y
≥
P
C
x
-
P
C
y
2
,
∀
x
,
y
∈
H
.
Consequently,P
C is nonexpansive and monotone.Definition 4.
LetM
:
H
→
2
H be a multivalued maximal monotone mapping. Then the mapping J
M
,
λ
:
H
→
H defined by
(28)
J
M
,
λ
(
u
)
=
I
+
λ
M
-
1
(
u
)
,
u
∈
H
is calledthe resolvent operator associated with
M, where λ is any positive number and I is the identity mapping.Proposition 5 (see [22]).
LetM
:
H
→
2
H be a multivalued maximal monotone mapping, and let A
:
H
→
H be an α-inverse-strongly monotone mapping. Then the following conclusions hold. (1)
The resolvent operatorJ
M
,
λ associated with M is single-valued and nonexpansive for all λ
>
0.
(2)
The resolvent operatorJ
M
,
λ is 1-inverse-strongly monotone; that is,
(29)
J
M
,
λ
(
x
)
-
J
M
,
λ
(
y
)
2
≤
x
-
y
,
J
M
,
λ
x
-
J
M
,
λ
y
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
∀
x
,
y
∈
H
.
(3)
u
∈
H is a solution of the variational inclusion (2) if and only if u
=
J
M
,
λ
(
u
-
λ
A
u
), for all λ
>
0; that is, u is a fixed point of the mapping J
M
,
λ
(
I
-
λ
A
). Therefore one has
(30)
Ω
=
F
J
M
,
λ
I
-
λ
A
,
∀
λ
>
0
,
whereΩ is the set of solutions of variational inclusion problem (2).
(4)
Ifλ
∈
(
0,2
α
], then Ω is a closed convex subset in H.Lemma 6 (see [23]).
Forx
,
y
∈
H and ω
∈
(
0,1
), the following statements hold: (1)
∥
x
+
y
∥
2
≤
∥
x
∥
2
+
2
〈
y
,
x
+
y
〉;
(2)
∥
(
1
-
ω
)
x
+
ω
y
∥
2
=
(
1
-
ω
)
∥
x
∥
2
+
ω
∥
y
∥
2
-
ω
(
1
-
ω
)
∥
x
-
y
∥
2.Lemma 7 (see [24]).
Let{
a
n
} be a sequence of real numbers, and there exists a subsequence {
a
m
j
} of {
a
n
} such that a
m
j
<
a
m
j
+
1 for all j
∈
N, where N is the set of all positive integers. Then there exists a nondecreasing sequence {
n
k
} of N such that lim
⁡
k
→
∞
n
k
=
∞ and the following properties are satisfied by all (sufficiently large) number k
∈
N:
(31)
a
n
k
≤
a
n
k
+
1
,
a
k
≤
a
n
k
+
1
.
In fact, n
k is the largest number n in the set {
1,2
,
…
,
k
} such that a
n
<
a
n
+
1 holds.Lemma 8 (see [18]).
Let{
a
n
}
⊂
[
0
,
∞
), {
α
n
}
⊂
[
0,1
), {
b
n
}
⊂
(
-
∞
,
+
∞
), and h
∈
[
0,1
) be such that (1)
{
a
n
} is a bounded sequence;
(2)
a
n
+
1
≤
(
1
-
α
n
)
2
a
n
+
2
α
n
h
a
n
a
n
+
1
+
α
n
b
n, for all n
≥
1;
(3)
whenever{
a
n
k
} is a subsequence of {
a
n
} satisfying
(32)
lim
⁡
inf
⁡
k
→
∞
(
a
n
k
+
1
-
a
n
k
)
≥
0
,
it follows thatlimsup
⁡
k
→
∞
b
n
k
≤
0;
(4)
lim
⁡
n
→
∞
α
n
=
0 and ∑
n
=
1
∞
‍
α
n
=
∞.
Thenlim
⁡
n
→
∞
a
n
=
0.Lemma 9 (see [15]).
LetM
:
H
→
2
H be a multivalued maximal monotone mapping, let A
:
H
→
H be an α-inverse-strongly monotone mapping, and let Ω be the set of solutions of variational inclusion problem (2) and Ω
≠
∅. Then the following statements hold. (1)
Ifλ
∈
(
0,2
α
], then the mapping K
:
H
→
H defined by
(33)
K
∶
=
J
M
,
λ
(
I
-
λ
A
)
is quasinonexpansive, whereI is the identity mapping and J
M
,
λ is the resolvent operator associated with M.
(2)
The mappingI
-
K
:
H
→
H is demiclosed at zero; that is, for any sequence {
x
n
}
⊂
H, if x
n
⇀
x and (
I
-
K
)
x
n
→
0, then x
=
K
x.
(3)
For anyβ
∈
(
0,1
), the mapping K
β defined by
(34)
K
β
=
(
1
-
β
)
I
+
β
K
is a strongly quasinonexpansive mapping andF
(
K
β
)
=
F
(
K
).
(4)
I
-
K
β
,
β
∈
(
0,1
) is demiclosed at zero.


## 3. Main Results

Throughout this section, we always assume that the following conditions are satisfied:(C1)
M
i
:
H
→
2
H is a multivalued maximal monotone mapping, A
i
:
H
→
H is an α
i-inverse-strongly monotone mapping, and Ω
i is the set of solutions to variational inclusion problem (2) with A
=
A
i, M
=
M
i, and Ω
i
≠
∅, for all i
=
1,2
,
3;
(C2)
K
i and K
i
,
β
,
β
∈
(
0,1
), i
=
1,2
,
3, are the mappings defined by
(35)
K
i
∶
=
J
M
i
,
λ
(
I
-
λ
A
i
)
,
λ
∈
(
0,2
α
i
]
,
K
i
,
β
∶
=
(
1
-
β
)
I
+
β
K
i
,
β
∈
(
0,1
)
,
respectively.Next, there are our main results.


### 3.1. An Existence Theorem

Theorem 10.
LetA
i, M
i, Ω
i, K
i, and K
i
,
β satisfy conditions (C1) and (C2), and let f
i
:
H
→
H be contractions with a contractive constant h
i
∈
(
0,1
), for all i
=
1,2
,
3. Then there exists a unique element (
x
*
,
y
*
,
z
*
)
∈
Ω
1
×
Ω
2
×
Ω
3 such that the following three inequalities are satisfied:
(36)
x
*
-
f
1
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
,
y
*
-
f
2
z
*
,
y
-
y
*
≥
0
,
∀
y
∈
Ω
2
,
z
*
-
f
3
x
*
,
z
-
z
*
≥
0
,
∀
z
∈
Ω
3
.Proof.
The proof is a consequence of Banach’s contraction principle but it is given here for the sake of completeness. By Proposition5 and Lemma 9, Ω
1, Ω
2, and Ω
3 are nonempty closed and convex. Therefore the metric projection P
Ω
i is well defined for each i
=
1,2
,
3.
Sincef
i is a contraction mapping for each i
=
1,2
,
3, then we have P
Ω
i
f
i which is a contraction and also have
(37)
P
Ω
1
f
1
∘
P
Ω
2
f
2
∘
P
Ω
3
f
3
which is a contraction. Hence there exists a unique element x
*
∈
H such that
(38)
x
*
=
(
P
Ω
1
f
1
∘
P
Ω
2
f
2
∘
P
Ω
3
f
3
)
x
*
.
Putting z
*
=
P
Ω
3
f
3
(
x
*
) and y
*
=
P
Ω
2
f
2
(
z
*
), then z
*
∈
Ω
3, y
*
∈
Ω
2, and x
*
=
P
Ω
1
f
1
(
y
*
).
Suppose that there is an element(
x
^
,
y
^
,
z
^
)
∈
Ω
1
×
Ω
2
×
Ω
3 such that the following three inequalities are satisfied:
(39)
x
^
-
f
1
y
^
,
x
-
x
^
≥
0
,
∀
x
∈
Ω
1
,
y
^
-
f
2
z
^
,
y
-
y
^
≥
0
,
∀
y
∈
Ω
2
,
z
^
-
f
3
x
^
,
z
-
z
^
≥
0
,
∀
z
∈
Ω
3
.
Then
(40)
x
^
=
P
Ω
1
f
1
(
y
^
)
,
y
^
=
P
Ω
2
f
2
(
z
^
)
,
z
^
=
P
Ω
3
f
3
(
x
^
)
.
Therefore
(41)
x
^
=
P
Ω
1
f
1
∘
P
Ω
2
f
2
∘
P
Ω
3
f
3
x
^
.
This implies that x
^
=
x
*, y
^
=
y
*, and z
^
=
z
*. This completes the proof.


### 3.2. A Convergence Theorem

Theorem 11.
LetA
i, M
i, Ω
i, K
i, and K
i
,
β satisfy conditions (C1) and (C2), and let f
i
:
H
→
H be contractions with a contractive constant h
i
∈
(
0,1
), for all i
=
1,2
,
3. Let {
x
n
}, {
y
n
}, and {
z
n
} be three sequences defined by
(42)
x
0
,
y
0
,
z
0
∈
H
,
x
n
+
1
=
1
-
α
n
K
1
,
β
x
n
+
α
n
f
1
K
2
,
β
y
n
,
y
n
+
1
=
1
-
α
n
K
2
,
β
y
n
+
α
n
f
2
K
3
,
β
z
n
,
z
n
+
1
=
(
1
-
α
n
)
K
3
,
β
z
n
+
α
n
f
3
(
K
1
,
β
x
n
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
=
0,1
,
2
,
…
,
where {
α
n
} is a sequence in (
0,1
) satisfying α
n
→
0 and ∑
n
=
0
∞
‍
α
n
=
∞. Then the sequences {
x
n
}, {
y
n
}, and {
z
n
} generated to be (42) converge to x
*
,
y
*, and z
*, respectively, where (
x
*
,
y
*
,
z
*
) is the unique element in Ω
1
×
Ω
2
×
Ω
3 verifying (36).Proof.
(i) First we prove that sequences{
x
n
}, {
y
n
}, and {
z
n
} are bounded.
From Lemma9, it follow that K
i
,
β is strongly quasinonexpansive and F
(
K
i
,
β
)
=
F
(
K
i
)
=
Ω
i for each i
=
1,2
,
3. Since f
i is contraction with the coefficient h
i for each i
=
1,2
,
3 and x
*
∈
F
(
K
1
,
β
), y
*
∈
F
(
K
2
,
β
), and z
*
∈
F
(
K
3
,
β
), it follows that
(43)
x
n
+
1
-
x
*
≤
(
1
-
α
n
)
K
1
,
β
x
n
-
x
*
+
α
n
f
1
(
K
2
,
β
y
n
)
-
x
*
≤
(
1
-
α
n
)
x
n
-
x
*
+
α
n
f
1
(
K
2
,
β
y
n
)
-
f
1
(
y
*
)
+
α
n
f
1
(
y
*
)
-
x
*
≤
(
1
-
α
n
)
x
n
-
x
*
+
α
n
h
1
K
2
,
β
y
n
-
y
*
+
α
n
f
1
(
y
*
)
-
x
*
≤
(
1
-
α
n
)
x
n
-
x
*
+
α
n
h
1
y
n
-
y
*
+
α
n
f
1
(
y
*
)
-
x
*
≤
(
1
-
α
n
)
x
n
-
x
*
+
α
n
h
y
n
-
y
*
+
α
n
f
1
(
y
*
)
-
x
*
,
where h
=
max
⁡
{
h
1
,
h
2
,
h
3
}. Similarly, we can also compute that
(44)
y
n
+
1
-
y
*
≤
(
1
-
α
n
)
y
n
-
y
*
+
α
n
h
z
n
-
z
*
+
α
n
f
2
z
*
-
y
*
,
z
n
+
1
-
z
*
≤
(
1
-
α
n
)
z
n
-
z
*
+
α
n
h
x
n
-
x
*
+
α
n
f
3
(
x
*
)
-
z
*
.
This implies that
(45)
x
n
+
1
-
x
*
+
y
n
+
1
-
y
*
+
z
n
+
1
-
z
*
≤
1
-
α
n
1
-
h
x
n
-
x
*
+
y
n
-
y
*
+
z
n
-
z
*
+
α
n
(
1
-
h
)
×
f
1
(
y
*
)
-
x
*
+
f
2
(
z
*
)
-
y
*
+
f
3
(
x
*
)
-
z
*
1
-
h
≤
max
⁡
⁡
x
n
-
x
*
+
y
n
-
y
*
+
z
n
-
z
*
,
1
-
h
-
1
f
1
y
*
-
x
*
+
f
2
z
*
-
y
*
+
f
3
x
*
-
z
*
×
1
-
h
-
1
.
By induction, we have
(46)
x
n
+
1
-
x
*
+
y
n
+
1
-
y
*
+
z
n
+
1
-
z
*
≤
max
⁡
⁡
x
0
-
x
*
+
y
0
-
y
*
+
z
0
-
z
*
,
1
-
h
-
1
f
1
y
*
-
x
*
+
f
2
z
*
-
y
*
+
f
3
(
x
*
)
-
z
*
×
1
-
h
-
1
,
for all n
≥
1.
Hence{
x
n
}, {
y
n
}, and {
z
n
} are bounded. Consequently, {
K
1
,
β
x
n
}, {
K
2
,
β
y
n
}, and {
K
3
,
β
z
n
} are bounded.
(ii) Next we prove that for eachn
≥
1 the following inequality holds:
(47)
x
n
+
1
-
x
*
2
+
y
n
+
1
-
y
*
2
+
z
n
+
1
-
z
*
2
≤
1
-
α
n
2
(
x
n
-
x
*
2
+
y
n
-
y
*
2
+
z
n
-
z
*
2
)
+
2
α
n
h
x
n
+
1
-
x
*
y
n
-
y
*
+
y
n
+
1
-
y
*
×
z
n
-
z
*
+
z
n
+
1
-
z
*
x
n
-
x
*
+
2
α
n
f
1
y
*
-
x
*
,
x
n
+
1
-
x
*
+
f
2
z
*
-
y
*
,
y
n
+
1
-
y
*
+
f
3
x
*
-
z
*
,
z
n
+
1
-
z
*
.
From (42) and Lemma 6, we have
(48)
x
n
+
1
-
x
*
2
=
(
1
-
α
n
)
(
K
1
,
β
x
n
-
x
*
)
+
α
n
(
f
1
(
K
2
,
β
y
n
)
-
x
*
)
2
≤
1
-
α
n
K
1
,
β
x
n
-
x
*
2
+
2
α
n
f
1
K
2
,
β
y
n
-
x
*
,
x
n
+
1
-
x
*
=
1
-
α
n
2
K
1
,
β
x
n
-
x
*
2
+
2
α
n
f
1
K
2
,
β
y
n
-
f
1
y
*
,
x
n
+
1
-
x
*
+
2
α
n
f
1
y
*
-
x
*
,
x
n
+
1
-
x
*
≤
1
-
α
n
2
x
n
-
x
*
2
+
2
α
n
f
1
(
K
2
,
β
y
n
)
-
f
1
(
y
*
)
×
x
n
+
1
-
x
*
+
2
α
n
f
1
y
*
-
x
*
,
x
n
+
1
-
x
*
≤
1
-
α
n
2
x
n
-
x
*
2
+
2
α
n
h
1
K
2
,
β
y
n
-
y
*
×
x
n
+
1
-
x
*
+
2
α
n
f
1
y
*
-
x
*
,
x
n
+
1
-
x
*
≤
1
-
α
n
2
x
n
-
x
*
2
+
2
α
n
h
y
n
-
y
*
x
n
+
1
-
x
*
+
2
α
n
f
1
y
*
-
x
*
,
x
n
+
1
-
x
*
.
Similarly, we can also prove that
(49)
y
n
+
1
-
y
*
2
≤
1
-
α
n
2
y
n
-
y
*
2
+
2
α
n
h
z
n
-
z
*
y
n
+
1
-
y
*
+
2
α
n
f
2
z
*
-
y
*
,
y
n
+
1
-
y
*
,
z
n
+
1
-
z
*
2
≤
1
-
α
n
2
z
n
-
z
*
2
+
2
α
n
h
x
n
-
x
*
z
n
+
1
-
z
*
+
2
α
n
f
3
x
*
-
z
*
,
z
n
+
1
-
z
*
.
Adding up inequalities (48) and (49), inequality (47) is proved.
(iii) Next, we prove that if there exists a subsequence{
n
k
}
⊂
{
n
} such that
(50)
lim
⁡
inf
⁡
k
→
∞
x
n
k
+
1
-
x
*
2
+
y
n
k
+
1
-
y
*
2
+
z
n
k
+
1
-
z
*
2
-
x
n
k
-
x
*
2
+
y
n
k
-
y
*
2
+
z
n
k
-
z
*
2
≥
0
,
then
(51)
lim
⁡
sup
⁡
k
→
∞
f
1
y
*
-
x
*
,
x
n
k
+
1
-
x
*
+
f
2
z
*
-
y
*
,
y
n
k
+
1
-
y
*
f
2
z
*
-
y
*
-
y
*
+
f
3
x
*
-
z
*
,
z
n
k
+
1
-
z
*
≤
0
.
Since the norm ∥
·
∥
2 is convex and lim
⁡
n
→
∞
α
n
=
0, by (42), we have
(52)
0
≤
lim
⁡
inf
⁡
k
→
∞
x
n
k
+
1
-
x
*
2
+
y
n
k
+
1
-
y
*
2
+
z
n
k
+
1
-
z
*
2
-
x
n
k
-
x
*
2
+
y
n
k
-
y
*
2
+
z
n
k
-
z
*
2
≤
lim
⁡
inf
⁡
k
→
∞
(
1
-
α
n
k
)
K
1
,
β
x
n
k
-
x
*
2
+
α
n
k
f
1
(
K
2
,
β
y
n
k
)
-
x
*
2
+
1
-
α
n
k
K
2
,
β
y
n
k
-
y
*
2
+
α
n
k
f
2
(
K
3
,
β
z
n
k
)
-
y
*
2
+
(
1
-
α
n
k
)
K
3
,
β
z
n
k
-
z
*
2
+
α
n
k
f
3
(
K
1
,
β
x
n
k
)
-
z
*
2
-
x
n
k
-
x
*
2
+
y
n
k
-
y
*
2
+
z
n
k
-
z
*
2
=
lim
⁡
inf
⁡
k
→
∞
K
1
,
β
x
n
k
-
x
*
2
-
x
n
k
-
x
*
2
+
(
K
2
,
β
y
n
k
-
y
*
2
-
y
n
k
-
y
*
2
)
+
K
3
,
β
z
n
k
-
z
*
2
-
z
n
k
-
z
*
2
≤
lim
⁡
sup
⁡
k
→
∞
K
1
,
β
x
n
k
-
x
*
2
-
x
n
k
-
x
*
2
+
K
2
,
β
y
n
k
-
y
*
2
-
y
n
k
-
y
*
2
+
K
3
,
β
z
n
k
-
z
*
2
-
z
n
k
-
z
*
2
≤
0
.
This implies that
(53)
lim
⁡
k
→
∞
(
K
1
,
β
x
n
k
-
x
*
2
-
x
n
k
-
x
*
2
)
=
lim
⁡
k
→
∞
(
K
2
,
β
y
n
k
-
y
*
2
-
y
n
k
-
y
*
2
)
=
lim
⁡
k
→
∞
(
K
3
,
β
z
n
k
-
z
*
2
-
z
n
k
-
z
*
2
)
=
0
.
Since the sequences {
∥
K
1
,
β
x
n
k
-
x
*
∥
+
∥
x
n
k
-
x
*
∥
}, {
∥
K
2
,
β
y
n
k
-
y
*
∥
+
∥
y
n
k
-
y
*
∥
}, and {
∥
K
3
,
β
z
n
k
-
z
*
∥
+
∥
z
n
k
-
z
*
∥
} are bounded, we have
(54)
lim
⁡
k
→
∞
K
1
,
β
x
n
k
-
x
*
-
x
n
k
-
x
*
=
lim
⁡
k
→
∞
(
K
2
,
β
y
n
k
-
y
*
-
y
n
k
-
y
*
)
=
lim
⁡
k
→
∞
K
3
,
β
z
n
k
-
z
*
-
z
n
k
-
z
*
=
0
.
By Lemma 9, K
1
,
β, K
2
,
β, and K
3
,
β are strongly quasinonexpansive. We have
(55)
K
1
,
β
x
n
k
-
x
n
k
⟶
0
,
K
2
,
β
y
n
k
-
y
n
k
⟶
0
,
K
3
,
β
z
n
k
-
z
n
k
⟶
0
.
Consequently, we obtain that
(56)
x
n
k
-
x
n
k
+
1
⟶
0
,
y
n
k
-
y
n
k
+
1
⟶
0
,
z
n
k
-
z
n
k
+
1
⟶
0
.
It follows from the boundedness of {
x
n
k
} and H which is reflexive that there exists a subsequence {
x
n
k
l
} of {
x
n
k
} such that x
n
k
l
⇀
p and
(57)
lim
⁡
l
→
∞
f
1
y
*
-
x
*
,
x
n
k
l
-
x
*
=
lim
⁡
sup
⁡
k
→
∞
f
1
y
*
-
x
*
,
x
n
k
-
x
*
=
lim
⁡
sup
⁡
k
→
∞
f
1
y
*
-
x
*
,
x
n
k
+
1
-
x
*
.
By Lemma 9, I
-
K
1
,
β is demiclosed at zero, and so p
∈
F
(
K
1
,
β
)
=
Ω
1. Hence from (36) we have
(58)
lim
⁡
l
→
∞
f
1
y
*
-
x
*
,
x
n
k
l
-
x
*
=
f
1
y
*
-
x
*
,
p
-
x
*
≤
0
.
Therefore
(59)
lim
⁡
sup
⁡
k
→
∞
f
1
y
*
-
x
*
,
x
n
k
+
1
-
x
*
=
lim
⁡
l
→
∞
f
1
y
*
-
x
*
,
x
n
k
l
-
x
*
≤
0
.
Similarly, we can also prove that
(60)
lim
⁡
sup
⁡
k
→
∞
f
2
z
*
-
y
*
,
y
n
k
+
1
-
y
*
≤
0
,
lim
⁡
sup
⁡
k
→
∞
f
3
x
*
-
z
*
,
z
n
k
+
1
-
z
*
≤
0
.
Hence, we have the desired inequality.
(iv) Finally, we prove that the sequences{
x
n
}, {
y
n
}, and {
z
n
} generated to be (42) converge to x
*
,
y
*, and z
*, respectively.
It is clear that(61)
x
n
+
1
-
x
*
y
n
-
y
*
+
y
n
+
1
-
y
*
z
n
-
z
*
+
z
n
+
1
-
z
*
x
n
-
x
*
≤
x
n
-
x
*
2
+
y
n
-
y
*
2
+
z
n
-
z
*
2
1
/
2
×
x
n
+
1
-
x
*
2
+
y
n
+
1
-
y
*
2
+
z
n
+
1
-
z
*
2
1
/
2
.
Substituting (61) into (47), we have
(62)
x
n
+
1
-
x
*
2
+
y
n
+
1
-
y
*
2
+
z
n
+
1
-
z
*
2
≤
1
-
α
n
2
x
n
-
x
*
2
+
y
n
-
y
*
2
+
z
n
-
z
*
2
+
2
α
n
h
x
n
-
x
*
2
+
y
n
-
y
*
2
+
z
n
-
z
*
2
1
/
2
+
z
n
-
z
*
2
1
/
2
×
x
n
+
1
-
x
*
2
+
y
n
+
1
-
y
*
2
+
z
n
+
1
-
z
*
2
1
/
2
+
2
α
n
f
1
y
*
-
x
*
,
x
n
+
1
-
x
*
+
f
2
z
*
-
y
*
,
y
n
+
1
-
y
*
+
f
3
x
*
-
z
*
,
z
n
+
1
-
z
*
.
Set
(63)
a
n
∶
=
x
n
-
x
*
2
+
y
n
-
y
*
2
+
z
n
-
z
*
2
,
b
n
∶
=
2
f
1
y
*
-
x
*
,
x
n
+
1
-
x
*
+
f
2
z
*
-
y
*
,
y
n
+
1
-
y
*
+
f
3
x
*
-
z
*
,
z
n
+
1
-
z
*
.
Then, we have the following statements. (i)
From (i),{
a
n
} is bounded sequence.
(ii)
From (62), a
n
+
1
≤
(
1
-
α
n
)
2
a
n
+
2
α
n
h
a
n
a
n
+
1
+
α
n
b
n, for all n
≥
1.
(iii)
From (iii), whenever{
a
n
k
} is a subsequence of {
a
n
} satisfying
(64)
lim
⁡
inf
⁡
k
→
∞
(
a
n
k
+
1
-
a
n
k
)
≥
0
,
it follows thatlimsup
⁡
k
→
∞
b
n
k
≤
0.
By Lemma8, we have
(65)
lim
⁡
n
→
∞
x
n
-
x
*
2
+
y
n
-
y
*
2
+
z
n
-
z
*
2
=
0
.
Hence, we obtain that
(66)
lim
⁡
n
→
∞
x
n
-
x
*
=
lim
⁡
n
→
∞
y
n
-
y
*
=
lim
⁡
n
→
∞
z
n
-
z
*
=
0
.
This completes the proof.


### 3.3. Consequence Results

Using Theorem11, we can prove the following results.Theorem 12.
LetA
i, M
i, Ω
i, K
i, and K
i
,
β satisfy conditions (C1) and (C2), and let F
:
H
→
H be a μ-Lipschitzian and r-strongly monotone mapping. Let {
x
n
}, {
y
n
}, and {
z
n
} be three sequences defined by
(67)
x
0
,
y
0
,
z
0
∈
H
,
x
n
+
1
=
(
1
-
α
n
)
K
1
,
β
x
n
+
α
n
f
1
(
K
2
,
β
y
n
)
,
y
n
+
1
=
(
1
-
α
n
)
K
2
,
β
y
n
+
α
n
f
2
(
K
3
,
β
z
n
)
,
z
n
+
1
=
(
1
-
α
n
)
K
3
,
β
z
n
+
α
n
f
3
(
K
1
,
β
x
n
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
=
0,1
,
2
,
…
,
where f
1
∶
=
I
-
ρ
F, f
2
∶
=
I
-
η
F, f
3
∶
=
I
-
ξ
F with ρ
,
η
,
ξ
∈
(
0
,
2
r
/
μ
2
), and {
α
n
} is a sequence in (
0,1
) satisfying α
n
→
0 and ∑
n
=
0
∞
‍
α
n
=
∞. Then the sequences {
x
n
}, {
y
n
}, and {
z
n
} converge to x
*
,
y
*, and z
*, respectively, where (
x
*
,
y
*
,
z
*
) is the unique element in Ω
1
×
Ω
2
×
Ω
3 such that the following three inequalities are satisfied:
(68)
ρ
F
y
*
+
x
*
-
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
,
η
F
z
*
+
y
*
-
z
*
,
y
-
y
*
≥
0
,
∀
y
∈
Ω
2
,
ξ
F
x
*
+
z
*
-
x
*
,
z
-
z
*
≥
0
,
∀
z
∈
Ω
3
.Proof.
It is easy to see thatf
1, f
2, and f
3 are contraction mappings and all the conditions in Theorem 11 are satisfied. By Theorem 11, we have the sequences {
x
n
}, {
y
n
}, and {
z
n
} which converge to (
x
*
,
y
*
,
z
*
)
∈
Ω
1
×
Ω
2
×
Ω
3 such that the following three inequalities are satisfied:
(69)
x
*
-
f
1
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
,
y
*
-
f
2
z
*
,
y
-
y
*
≥
0
,
∀
y
∈
Ω
2
,
z
*
-
f
3
x
*
,
z
-
z
*
≥
0
,
∀
z
∈
Ω
3
.
Substituting f
1
∶
=
I
-
ρ
F, f
2
∶
=
I
-
η
F, and f
3
∶
=
I
-
ξ
F into (69), we obtain that the sequences {
x
n
}, {
y
n
}, and {
z
n
} converge to (
x
*
,
y
*
,
z
*
)
∈
Ω
1
×
Ω
2
×
Ω
3 such that the following three inequalities are satisfied:
(70)
ρ
F
y
*
+
x
*
-
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
,
η
F
z
*
+
y
*
-
z
*
,
y
-
y
*
≥
0
,
∀
y
∈
Ω
2
,
ξ
F
x
*
+
z
*
-
x
*
,
z
-
z
*
≥
0
,
∀
z
∈
Ω
3
.
This completes the proofSettingA
1
=
A
2
=
A
3 in Theorem 11, we obtain the following corollary.Corollary 13.
LetA
1, M
1, Ω
1, K
1, and K
1
,
β satisfy conditions (C1) and (C2), and let f
i
:
H
→
H be contractions with a contractive constant h
i
∈
(
0,1
), for all i
=
1,2
,
3. Let {
x
n
}, {
y
n
}, and {
z
n
} be three sequences defined by
(71)
x
0
,
y
0
,
z
0
∈
H
,
x
n
+
1
=
(
1
-
α
n
)
K
1
,
β
x
n
+
α
n
f
1
(
K
1
,
β
y
n
)
,
y
n
+
1
=
1
-
α
n
K
1
,
β
y
n
+
α
n
f
2
K
1
,
β
z
n
,
z
n
+
1
=
(
1
-
α
n
)
K
1
,
β
z
n
+
α
n
f
3
(
K
1
,
β
x
n
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
=
0,1
,
2
,
…
,
where {
α
n
} is a sequence in (
0,1
) satisfying α
n
→
0 and ∑
n
=
0
∞
‍
α
n
=
∞. Then the sequences {
x
n
}, {
y
n
}, and {
z
n
} generated to be (42) converge to x
*, y
*, and z
*, respectively, where (
x
*
,
y
*
,
z
*
) is the unique element in Ω
1
×
Ω
1
×
Ω
1 such that the following three inequalities are satisfied:
(72)
x
*
-
f
1
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
,
y
*
-
f
2
z
*
,
x
-
y
*
≥
0
,
∀
x
∈
Ω
1
,
z
*
-
f
3
x
*
,
x
-
z
*
≥
0
,
∀
x
∈
Ω
1
.Corollary 14.
LetA
1, M
1, Ω, K
1, and K
1
,
β satisfy conditions (C1) and (C2), and let F
:
H
→
H be μ-Lipschitzian and r-strongly monotone mapping. Let {
x
n
}, {
y
n
}, and {
z
n
} be three sequences defined by
(73)
x
0
,
y
0
,
z
0
∈
H
,
x
n
+
1
=
(
1
-
α
n
)
K
1
,
β
x
n
+
α
n
f
1
(
K
1
,
β
y
n
)
,
y
n
+
1
=
(
1
-
α
n
)
K
1
,
β
y
n
+
α
n
f
2
(
K
1
,
β
z
n
)
,
z
n
+
1
=
1
-
α
n
K
1
,
β
z
n
+
α
n
f
3
K
1
,
β
x
n
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
=
0,1
,
2
,
…
,
where f
1
∶
=
I
-
ρ
F, f
2
∶
=
I
-
η
F, f
3
∶
=
I
-
ξ
F with ρ
,
η
,
ξ
∈
(
0
,
2
r
/
μ
2
), and {
α
n
} is a sequence in (
0,1
) satisfying α
n
→
0 and ∑
n
=
0
∞
‍
α
n
=
∞. Then the sequences {
x
n
}, {
y
n
}, and {
z
n
} converge to x
*, y
*, and z
*, respectively, where (
x
*
,
y
*
,
z
*
) is the unique element in Ω
1
×
Ω
1
×
Ω
1 such that the following three inequalities are satisfied:
(74)
ρ
F
y
*
+
x
*
-
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
,
η
F
z
*
+
y
*
-
z
*
,
x
-
y
*
≥
0
,
∀
x
∈
Ω
1
,
ξ
F
x
*
+
z
*
-
x
*
,
x
-
z
*
≥
0
,
∀
x
∈
Ω
1
.SettingA
1
=
A
2
=
A
3, f
1
=
f
2
=
f
3, and x
0
=
y
0
=
z
0 in Theorem 11, we obtain the following corollary.Corollary 15.
LetA
1, M
1, Ω
1, K
1, and K
1
,
β satisfy conditions (C1) and (C2), and let f
:
H
→
H be contractions with a contractive constant h
∈
(
0,1
). Let {
x
n
} be the sequences defined by
(75)
x
0
∈
H
,
x
n
+
1
=
(
1
-
α
n
)
K
1
,
β
x
n
+
α
n
f
(
K
1
,
β
x
n
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
=
0,1
,
2
,
…
,
where {
α
n
} is a sequence in (
0,1
) satisfying α
n
→
0 and ∑
n
=
0
∞
‍
α
n
=
∞. Then the sequences {
x
n
} converge to x
*
∈
Ω
1 such that the following three inequalities are satisfied:
(76)
x
*
-
f
1
x
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
.Corollary 16.
LetA
1, M
1, Ω
1, K
1, and K
1
,
β satisfy conditions (C1) and (C2), and let F
:
H
→
H be μ-Lipschitzian and r-strongly monotone mapping. Let {
x
n
} be the sequences defined by
(77)
x
0
∈
H
,
x
n
+
1
=
(
1
-
α
n
)
K
1
,
β
x
n
+
α
n
(
I
-
ρ
F
)
(
K
1
,
β
x
n
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
=
0,1
,
2
,
…
,
where ρ
∈
(
0
,
2
r
/
μ
2
) and {
α
n
} is a sequence in (
0,1
) satisfying α
n
→
0 and ∑
n
=
0
∞
‍
α
n
=
∞. Then the sequences {
x
n
} converge to x
*
∈
Ω
1 such that the following three inequalities are satisfied:
(78)
F
x
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
.


## 3.1. An Existence Theorem

Theorem 10.
LetA
i, M
i, Ω
i, K
i, and K
i
,
β satisfy conditions (C1) and (C2), and let f
i
:
H
→
H be contractions with a contractive constant h
i
∈
(
0,1
), for all i
=
1,2
,
3. Then there exists a unique element (
x
*
,
y
*
,
z
*
)
∈
Ω
1
×
Ω
2
×
Ω
3 such that the following three inequalities are satisfied:
(36)
x
*
-
f
1
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
,
y
*
-
f
2
z
*
,
y
-
y
*
≥
0
,
∀
y
∈
Ω
2
,
z
*
-
f
3
x
*
,
z
-
z
*
≥
0
,
∀
z
∈
Ω
3
.Proof.
The proof is a consequence of Banach’s contraction principle but it is given here for the sake of completeness. By Proposition5 and Lemma 9, Ω
1, Ω
2, and Ω
3 are nonempty closed and convex. Therefore the metric projection P
Ω
i is well defined for each i
=
1,2
,
3.
Sincef
i is a contraction mapping for each i
=
1,2
,
3, then we have P
Ω
i
f
i which is a contraction and also have
(37)
P
Ω
1
f
1
∘
P
Ω
2
f
2
∘
P
Ω
3
f
3
which is a contraction. Hence there exists a unique element x
*
∈
H such that
(38)
x
*
=
(
P
Ω
1
f
1
∘
P
Ω
2
f
2
∘
P
Ω
3
f
3
)
x
*
.
Putting z
*
=
P
Ω
3
f
3
(
x
*
) and y
*
=
P
Ω
2
f
2
(
z
*
), then z
*
∈
Ω
3, y
*
∈
Ω
2, and x
*
=
P
Ω
1
f
1
(
y
*
).
Suppose that there is an element(
x
^
,
y
^
,
z
^
)
∈
Ω
1
×
Ω
2
×
Ω
3 such that the following three inequalities are satisfied:
(39)
x
^
-
f
1
y
^
,
x
-
x
^
≥
0
,
∀
x
∈
Ω
1
,
y
^
-
f
2
z
^
,
y
-
y
^
≥
0
,
∀
y
∈
Ω
2
,
z
^
-
f
3
x
^
,
z
-
z
^
≥
0
,
∀
z
∈
Ω
3
.
Then
(40)
x
^
=
P
Ω
1
f
1
(
y
^
)
,
y
^
=
P
Ω
2
f
2
(
z
^
)
,
z
^
=
P
Ω
3
f
3
(
x
^
)
.
Therefore
(41)
x
^
=
P
Ω
1
f
1
∘
P
Ω
2
f
2
∘
P
Ω
3
f
3
x
^
.
This implies that x
^
=
x
*, y
^
=
y
*, and z
^
=
z
*. This completes the proof.


## 3.2. A Convergence Theorem

Theorem 11.
LetA
i, M
i, Ω
i, K
i, and K
i
,
β satisfy conditions (C1) and (C2), and let f
i
:
H
→
H be contractions with a contractive constant h
i
∈
(
0,1
), for all i
=
1,2
,
3. Let {
x
n
}, {
y
n
}, and {
z
n
} be three sequences defined by
(42)
x
0
,
y
0
,
z
0
∈
H
,
x
n
+
1
=
1
-
α
n
K
1
,
β
x
n
+
α
n
f
1
K
2
,
β
y
n
,
y
n
+
1
=
1
-
α
n
K
2
,
β
y
n
+
α
n
f
2
K
3
,
β
z
n
,
z
n
+
1
=
(
1
-
α
n
)
K
3
,
β
z
n
+
α
n
f
3
(
K
1
,
β
x
n
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
=
0,1
,
2
,
…
,
where {
α
n
} is a sequence in (
0,1
) satisfying α
n
→
0 and ∑
n
=
0
∞
‍
α
n
=
∞. Then the sequences {
x
n
}, {
y
n
}, and {
z
n
} generated to be (42) converge to x
*
,
y
*, and z
*, respectively, where (
x
*
,
y
*
,
z
*
) is the unique element in Ω
1
×
Ω
2
×
Ω
3 verifying (36).Proof.
(i) First we prove that sequences{
x
n
}, {
y
n
}, and {
z
n
} are bounded.
From Lemma9, it follow that K
i
,
β is strongly quasinonexpansive and F
(
K
i
,
β
)
=
F
(
K
i
)
=
Ω
i for each i
=
1,2
,
3. Since f
i is contraction with the coefficient h
i for each i
=
1,2
,
3 and x
*
∈
F
(
K
1
,
β
), y
*
∈
F
(
K
2
,
β
), and z
*
∈
F
(
K
3
,
β
), it follows that
(43)
x
n
+
1
-
x
*
≤
(
1
-
α
n
)
K
1
,
β
x
n
-
x
*
+
α
n
f
1
(
K
2
,
β
y
n
)
-
x
*
≤
(
1
-
α
n
)
x
n
-
x
*
+
α
n
f
1
(
K
2
,
β
y
n
)
-
f
1
(
y
*
)
+
α
n
f
1
(
y
*
)
-
x
*
≤
(
1
-
α
n
)
x
n
-
x
*
+
α
n
h
1
K
2
,
β
y
n
-
y
*
+
α
n
f
1
(
y
*
)
-
x
*
≤
(
1
-
α
n
)
x
n
-
x
*
+
α
n
h
1
y
n
-
y
*
+
α
n
f
1
(
y
*
)
-
x
*
≤
(
1
-
α
n
)
x
n
-
x
*
+
α
n
h
y
n
-
y
*
+
α
n
f
1
(
y
*
)
-
x
*
,
where h
=
max
⁡
{
h
1
,
h
2
,
h
3
}. Similarly, we can also compute that
(44)
y
n
+
1
-
y
*
≤
(
1
-
α
n
)
y
n
-
y
*
+
α
n
h
z
n
-
z
*
+
α
n
f
2
z
*
-
y
*
,
z
n
+
1
-
z
*
≤
(
1
-
α
n
)
z
n
-
z
*
+
α
n
h
x
n
-
x
*
+
α
n
f
3
(
x
*
)
-
z
*
.
This implies that
(45)
x
n
+
1
-
x
*
+
y
n
+
1
-
y
*
+
z
n
+
1
-
z
*
≤
1
-
α
n
1
-
h
x
n
-
x
*
+
y
n
-
y
*
+
z
n
-
z
*
+
α
n
(
1
-
h
)
×
f
1
(
y
*
)
-
x
*
+
f
2
(
z
*
)
-
y
*
+
f
3
(
x
*
)
-
z
*
1
-
h
≤
max
⁡
⁡
x
n
-
x
*
+
y
n
-
y
*
+
z
n
-
z
*
,
1
-
h
-
1
f
1
y
*
-
x
*
+
f
2
z
*
-
y
*
+
f
3
x
*
-
z
*
×
1
-
h
-
1
.
By induction, we have
(46)
x
n
+
1
-
x
*
+
y
n
+
1
-
y
*
+
z
n
+
1
-
z
*
≤
max
⁡
⁡
x
0
-
x
*
+
y
0
-
y
*
+
z
0
-
z
*
,
1
-
h
-
1
f
1
y
*
-
x
*
+
f
2
z
*
-
y
*
+
f
3
(
x
*
)
-
z
*
×
1
-
h
-
1
,
for all n
≥
1.
Hence{
x
n
}, {
y
n
}, and {
z
n
} are bounded. Consequently, {
K
1
,
β
x
n
}, {
K
2
,
β
y
n
}, and {
K
3
,
β
z
n
} are bounded.
(ii) Next we prove that for eachn
≥
1 the following inequality holds:
(47)
x
n
+
1
-
x
*
2
+
y
n
+
1
-
y
*
2
+
z
n
+
1
-
z
*
2
≤
1
-
α
n
2
(
x
n
-
x
*
2
+
y
n
-
y
*
2
+
z
n
-
z
*
2
)
+
2
α
n
h
x
n
+
1
-
x
*
y
n
-
y
*
+
y
n
+
1
-
y
*
×
z
n
-
z
*
+
z
n
+
1
-
z
*
x
n
-
x
*
+
2
α
n
f
1
y
*
-
x
*
,
x
n
+
1
-
x
*
+
f
2
z
*
-
y
*
,
y
n
+
1
-
y
*
+
f
3
x
*
-
z
*
,
z
n
+
1
-
z
*
.
From (42) and Lemma 6, we have
(48)
x
n
+
1
-
x
*
2
=
(
1
-
α
n
)
(
K
1
,
β
x
n
-
x
*
)
+
α
n
(
f
1
(
K
2
,
β
y
n
)
-
x
*
)
2
≤
1
-
α
n
K
1
,
β
x
n
-
x
*
2
+
2
α
n
f
1
K
2
,
β
y
n
-
x
*
,
x
n
+
1
-
x
*
=
1
-
α
n
2
K
1
,
β
x
n
-
x
*
2
+
2
α
n
f
1
K
2
,
β
y
n
-
f
1
y
*
,
x
n
+
1
-
x
*
+
2
α
n
f
1
y
*
-
x
*
,
x
n
+
1
-
x
*
≤
1
-
α
n
2
x
n
-
x
*
2
+
2
α
n
f
1
(
K
2
,
β
y
n
)
-
f
1
(
y
*
)
×
x
n
+
1
-
x
*
+
2
α
n
f
1
y
*
-
x
*
,
x
n
+
1
-
x
*
≤
1
-
α
n
2
x
n
-
x
*
2
+
2
α
n
h
1
K
2
,
β
y
n
-
y
*
×
x
n
+
1
-
x
*
+
2
α
n
f
1
y
*
-
x
*
,
x
n
+
1
-
x
*
≤
1
-
α
n
2
x
n
-
x
*
2
+
2
α
n
h
y
n
-
y
*
x
n
+
1
-
x
*
+
2
α
n
f
1
y
*
-
x
*
,
x
n
+
1
-
x
*
.
Similarly, we can also prove that
(49)
y
n
+
1
-
y
*
2
≤
1
-
α
n
2
y
n
-
y
*
2
+
2
α
n
h
z
n
-
z
*
y
n
+
1
-
y
*
+
2
α
n
f
2
z
*
-
y
*
,
y
n
+
1
-
y
*
,
z
n
+
1
-
z
*
2
≤
1
-
α
n
2
z
n
-
z
*
2
+
2
α
n
h
x
n
-
x
*
z
n
+
1
-
z
*
+
2
α
n
f
3
x
*
-
z
*
,
z
n
+
1
-
z
*
.
Adding up inequalities (48) and (49), inequality (47) is proved.
(iii) Next, we prove that if there exists a subsequence{
n
k
}
⊂
{
n
} such that
(50)
lim
⁡
inf
⁡
k
→
∞
x
n
k
+
1
-
x
*
2
+
y
n
k
+
1
-
y
*
2
+
z
n
k
+
1
-
z
*
2
-
x
n
k
-
x
*
2
+
y
n
k
-
y
*
2
+
z
n
k
-
z
*
2
≥
0
,
then
(51)
lim
⁡
sup
⁡
k
→
∞
f
1
y
*
-
x
*
,
x
n
k
+
1
-
x
*
+
f
2
z
*
-
y
*
,
y
n
k
+
1
-
y
*
f
2
z
*
-
y
*
-
y
*
+
f
3
x
*
-
z
*
,
z
n
k
+
1
-
z
*
≤
0
.
Since the norm ∥
·
∥
2 is convex and lim
⁡
n
→
∞
α
n
=
0, by (42), we have
(52)
0
≤
lim
⁡
inf
⁡
k
→
∞
x
n
k
+
1
-
x
*
2
+
y
n
k
+
1
-
y
*
2
+
z
n
k
+
1
-
z
*
2
-
x
n
k
-
x
*
2
+
y
n
k
-
y
*
2
+
z
n
k
-
z
*
2
≤
lim
⁡
inf
⁡
k
→
∞
(
1
-
α
n
k
)
K
1
,
β
x
n
k
-
x
*
2
+
α
n
k
f
1
(
K
2
,
β
y
n
k
)
-
x
*
2
+
1
-
α
n
k
K
2
,
β
y
n
k
-
y
*
2
+
α
n
k
f
2
(
K
3
,
β
z
n
k
)
-
y
*
2
+
(
1
-
α
n
k
)
K
3
,
β
z
n
k
-
z
*
2
+
α
n
k
f
3
(
K
1
,
β
x
n
k
)
-
z
*
2
-
x
n
k
-
x
*
2
+
y
n
k
-
y
*
2
+
z
n
k
-
z
*
2
=
lim
⁡
inf
⁡
k
→
∞
K
1
,
β
x
n
k
-
x
*
2
-
x
n
k
-
x
*
2
+
(
K
2
,
β
y
n
k
-
y
*
2
-
y
n
k
-
y
*
2
)
+
K
3
,
β
z
n
k
-
z
*
2
-
z
n
k
-
z
*
2
≤
lim
⁡
sup
⁡
k
→
∞
K
1
,
β
x
n
k
-
x
*
2
-
x
n
k
-
x
*
2
+
K
2
,
β
y
n
k
-
y
*
2
-
y
n
k
-
y
*
2
+
K
3
,
β
z
n
k
-
z
*
2
-
z
n
k
-
z
*
2
≤
0
.
This implies that
(53)
lim
⁡
k
→
∞
(
K
1
,
β
x
n
k
-
x
*
2
-
x
n
k
-
x
*
2
)
=
lim
⁡
k
→
∞
(
K
2
,
β
y
n
k
-
y
*
2
-
y
n
k
-
y
*
2
)
=
lim
⁡
k
→
∞
(
K
3
,
β
z
n
k
-
z
*
2
-
z
n
k
-
z
*
2
)
=
0
.
Since the sequences {
∥
K
1
,
β
x
n
k
-
x
*
∥
+
∥
x
n
k
-
x
*
∥
}, {
∥
K
2
,
β
y
n
k
-
y
*
∥
+
∥
y
n
k
-
y
*
∥
}, and {
∥
K
3
,
β
z
n
k
-
z
*
∥
+
∥
z
n
k
-
z
*
∥
} are bounded, we have
(54)
lim
⁡
k
→
∞
K
1
,
β
x
n
k
-
x
*
-
x
n
k
-
x
*
=
lim
⁡
k
→
∞
(
K
2
,
β
y
n
k
-
y
*
-
y
n
k
-
y
*
)
=
lim
⁡
k
→
∞
K
3
,
β
z
n
k
-
z
*
-
z
n
k
-
z
*
=
0
.
By Lemma 9, K
1
,
β, K
2
,
β, and K
3
,
β are strongly quasinonexpansive. We have
(55)
K
1
,
β
x
n
k
-
x
n
k
⟶
0
,
K
2
,
β
y
n
k
-
y
n
k
⟶
0
,
K
3
,
β
z
n
k
-
z
n
k
⟶
0
.
Consequently, we obtain that
(56)
x
n
k
-
x
n
k
+
1
⟶
0
,
y
n
k
-
y
n
k
+
1
⟶
0
,
z
n
k
-
z
n
k
+
1
⟶
0
.
It follows from the boundedness of {
x
n
k
} and H which is reflexive that there exists a subsequence {
x
n
k
l
} of {
x
n
k
} such that x
n
k
l
⇀
p and
(57)
lim
⁡
l
→
∞
f
1
y
*
-
x
*
,
x
n
k
l
-
x
*
=
lim
⁡
sup
⁡
k
→
∞
f
1
y
*
-
x
*
,
x
n
k
-
x
*
=
lim
⁡
sup
⁡
k
→
∞
f
1
y
*
-
x
*
,
x
n
k
+
1
-
x
*
.
By Lemma 9, I
-
K
1
,
β is demiclosed at zero, and so p
∈
F
(
K
1
,
β
)
=
Ω
1. Hence from (36) we have
(58)
lim
⁡
l
→
∞
f
1
y
*
-
x
*
,
x
n
k
l
-
x
*
=
f
1
y
*
-
x
*
,
p
-
x
*
≤
0
.
Therefore
(59)
lim
⁡
sup
⁡
k
→
∞
f
1
y
*
-
x
*
,
x
n
k
+
1
-
x
*
=
lim
⁡
l
→
∞
f
1
y
*
-
x
*
,
x
n
k
l
-
x
*
≤
0
.
Similarly, we can also prove that
(60)
lim
⁡
sup
⁡
k
→
∞
f
2
z
*
-
y
*
,
y
n
k
+
1
-
y
*
≤
0
,
lim
⁡
sup
⁡
k
→
∞
f
3
x
*
-
z
*
,
z
n
k
+
1
-
z
*
≤
0
.
Hence, we have the desired inequality.
(iv) Finally, we prove that the sequences{
x
n
}, {
y
n
}, and {
z
n
} generated to be (42) converge to x
*
,
y
*, and z
*, respectively.
It is clear that(61)
x
n
+
1
-
x
*
y
n
-
y
*
+
y
n
+
1
-
y
*
z
n
-
z
*
+
z
n
+
1
-
z
*
x
n
-
x
*
≤
x
n
-
x
*
2
+
y
n
-
y
*
2
+
z
n
-
z
*
2
1
/
2
×
x
n
+
1
-
x
*
2
+
y
n
+
1
-
y
*
2
+
z
n
+
1
-
z
*
2
1
/
2
.
Substituting (61) into (47), we have
(62)
x
n
+
1
-
x
*
2
+
y
n
+
1
-
y
*
2
+
z
n
+
1
-
z
*
2
≤
1
-
α
n
2
x
n
-
x
*
2
+
y
n
-
y
*
2
+
z
n
-
z
*
2
+
2
α
n
h
x
n
-
x
*
2
+
y
n
-
y
*
2
+
z
n
-
z
*
2
1
/
2
+
z
n
-
z
*
2
1
/
2
×
x
n
+
1
-
x
*
2
+
y
n
+
1
-
y
*
2
+
z
n
+
1
-
z
*
2
1
/
2
+
2
α
n
f
1
y
*
-
x
*
,
x
n
+
1
-
x
*
+
f
2
z
*
-
y
*
,
y
n
+
1
-
y
*
+
f
3
x
*
-
z
*
,
z
n
+
1
-
z
*
.
Set
(63)
a
n
∶
=
x
n
-
x
*
2
+
y
n
-
y
*
2
+
z
n
-
z
*
2
,
b
n
∶
=
2
f
1
y
*
-
x
*
,
x
n
+
1
-
x
*
+
f
2
z
*
-
y
*
,
y
n
+
1
-
y
*
+
f
3
x
*
-
z
*
,
z
n
+
1
-
z
*
.
Then, we have the following statements. (i)
From (i),{
a
n
} is bounded sequence.
(ii)
From (62), a
n
+
1
≤
(
1
-
α
n
)
2
a
n
+
2
α
n
h
a
n
a
n
+
1
+
α
n
b
n, for all n
≥
1.
(iii)
From (iii), whenever{
a
n
k
} is a subsequence of {
a
n
} satisfying
(64)
lim
⁡
inf
⁡
k
→
∞
(
a
n
k
+
1
-
a
n
k
)
≥
0
,
it follows thatlimsup
⁡
k
→
∞
b
n
k
≤
0.
By Lemma8, we have
(65)
lim
⁡
n
→
∞
x
n
-
x
*
2
+
y
n
-
y
*
2
+
z
n
-
z
*
2
=
0
.
Hence, we obtain that
(66)
lim
⁡
n
→
∞
x
n
-
x
*
=
lim
⁡
n
→
∞
y
n
-
y
*
=
lim
⁡
n
→
∞
z
n
-
z
*
=
0
.
This completes the proof.


## 3.3. Consequence Results

Using Theorem11, we can prove the following results.Theorem 12.
LetA
i, M
i, Ω
i, K
i, and K
i
,
β satisfy conditions (C1) and (C2), and let F
:
H
→
H be a μ-Lipschitzian and r-strongly monotone mapping. Let {
x
n
}, {
y
n
}, and {
z
n
} be three sequences defined by
(67)
x
0
,
y
0
,
z
0
∈
H
,
x
n
+
1
=
(
1
-
α
n
)
K
1
,
β
x
n
+
α
n
f
1
(
K
2
,
β
y
n
)
,
y
n
+
1
=
(
1
-
α
n
)
K
2
,
β
y
n
+
α
n
f
2
(
K
3
,
β
z
n
)
,
z
n
+
1
=
(
1
-
α
n
)
K
3
,
β
z
n
+
α
n
f
3
(
K
1
,
β
x
n
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
=
0,1
,
2
,
…
,
where f
1
∶
=
I
-
ρ
F, f
2
∶
=
I
-
η
F, f
3
∶
=
I
-
ξ
F with ρ
,
η
,
ξ
∈
(
0
,
2
r
/
μ
2
), and {
α
n
} is a sequence in (
0,1
) satisfying α
n
→
0 and ∑
n
=
0
∞
‍
α
n
=
∞. Then the sequences {
x
n
}, {
y
n
}, and {
z
n
} converge to x
*
,
y
*, and z
*, respectively, where (
x
*
,
y
*
,
z
*
) is the unique element in Ω
1
×
Ω
2
×
Ω
3 such that the following three inequalities are satisfied:
(68)
ρ
F
y
*
+
x
*
-
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
,
η
F
z
*
+
y
*
-
z
*
,
y
-
y
*
≥
0
,
∀
y
∈
Ω
2
,
ξ
F
x
*
+
z
*
-
x
*
,
z
-
z
*
≥
0
,
∀
z
∈
Ω
3
.Proof.
It is easy to see thatf
1, f
2, and f
3 are contraction mappings and all the conditions in Theorem 11 are satisfied. By Theorem 11, we have the sequences {
x
n
}, {
y
n
}, and {
z
n
} which converge to (
x
*
,
y
*
,
z
*
)
∈
Ω
1
×
Ω
2
×
Ω
3 such that the following three inequalities are satisfied:
(69)
x
*
-
f
1
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
,
y
*
-
f
2
z
*
,
y
-
y
*
≥
0
,
∀
y
∈
Ω
2
,
z
*
-
f
3
x
*
,
z
-
z
*
≥
0
,
∀
z
∈
Ω
3
.
Substituting f
1
∶
=
I
-
ρ
F, f
2
∶
=
I
-
η
F, and f
3
∶
=
I
-
ξ
F into (69), we obtain that the sequences {
x
n
}, {
y
n
}, and {
z
n
} converge to (
x
*
,
y
*
,
z
*
)
∈
Ω
1
×
Ω
2
×
Ω
3 such that the following three inequalities are satisfied:
(70)
ρ
F
y
*
+
x
*
-
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
,
η
F
z
*
+
y
*
-
z
*
,
y
-
y
*
≥
0
,
∀
y
∈
Ω
2
,
ξ
F
x
*
+
z
*
-
x
*
,
z
-
z
*
≥
0
,
∀
z
∈
Ω
3
.
This completes the proofSettingA
1
=
A
2
=
A
3 in Theorem 11, we obtain the following corollary.Corollary 13.
LetA
1, M
1, Ω
1, K
1, and K
1
,
β satisfy conditions (C1) and (C2), and let f
i
:
H
→
H be contractions with a contractive constant h
i
∈
(
0,1
), for all i
=
1,2
,
3. Let {
x
n
}, {
y
n
}, and {
z
n
} be three sequences defined by
(71)
x
0
,
y
0
,
z
0
∈
H
,
x
n
+
1
=
(
1
-
α
n
)
K
1
,
β
x
n
+
α
n
f
1
(
K
1
,
β
y
n
)
,
y
n
+
1
=
1
-
α
n
K
1
,
β
y
n
+
α
n
f
2
K
1
,
β
z
n
,
z
n
+
1
=
(
1
-
α
n
)
K
1
,
β
z
n
+
α
n
f
3
(
K
1
,
β
x
n
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
=
0,1
,
2
,
…
,
where {
α
n
} is a sequence in (
0,1
) satisfying α
n
→
0 and ∑
n
=
0
∞
‍
α
n
=
∞. Then the sequences {
x
n
}, {
y
n
}, and {
z
n
} generated to be (42) converge to x
*, y
*, and z
*, respectively, where (
x
*
,
y
*
,
z
*
) is the unique element in Ω
1
×
Ω
1
×
Ω
1 such that the following three inequalities are satisfied:
(72)
x
*
-
f
1
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
,
y
*
-
f
2
z
*
,
x
-
y
*
≥
0
,
∀
x
∈
Ω
1
,
z
*
-
f
3
x
*
,
x
-
z
*
≥
0
,
∀
x
∈
Ω
1
.Corollary 14.
LetA
1, M
1, Ω, K
1, and K
1
,
β satisfy conditions (C1) and (C2), and let F
:
H
→
H be μ-Lipschitzian and r-strongly monotone mapping. Let {
x
n
}, {
y
n
}, and {
z
n
} be three sequences defined by
(73)
x
0
,
y
0
,
z
0
∈
H
,
x
n
+
1
=
(
1
-
α
n
)
K
1
,
β
x
n
+
α
n
f
1
(
K
1
,
β
y
n
)
,
y
n
+
1
=
(
1
-
α
n
)
K
1
,
β
y
n
+
α
n
f
2
(
K
1
,
β
z
n
)
,
z
n
+
1
=
1
-
α
n
K
1
,
β
z
n
+
α
n
f
3
K
1
,
β
x
n
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
=
0,1
,
2
,
…
,
where f
1
∶
=
I
-
ρ
F, f
2
∶
=
I
-
η
F, f
3
∶
=
I
-
ξ
F with ρ
,
η
,
ξ
∈
(
0
,
2
r
/
μ
2
), and {
α
n
} is a sequence in (
0,1
) satisfying α
n
→
0 and ∑
n
=
0
∞
‍
α
n
=
∞. Then the sequences {
x
n
}, {
y
n
}, and {
z
n
} converge to x
*, y
*, and z
*, respectively, where (
x
*
,
y
*
,
z
*
) is the unique element in Ω
1
×
Ω
1
×
Ω
1 such that the following three inequalities are satisfied:
(74)
ρ
F
y
*
+
x
*
-
y
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
,
η
F
z
*
+
y
*
-
z
*
,
x
-
y
*
≥
0
,
∀
x
∈
Ω
1
,
ξ
F
x
*
+
z
*
-
x
*
,
x
-
z
*
≥
0
,
∀
x
∈
Ω
1
.SettingA
1
=
A
2
=
A
3, f
1
=
f
2
=
f
3, and x
0
=
y
0
=
z
0 in Theorem 11, we obtain the following corollary.Corollary 15.
LetA
1, M
1, Ω
1, K
1, and K
1
,
β satisfy conditions (C1) and (C2), and let f
:
H
→
H be contractions with a contractive constant h
∈
(
0,1
). Let {
x
n
} be the sequences defined by
(75)
x
0
∈
H
,
x
n
+
1
=
(
1
-
α
n
)
K
1
,
β
x
n
+
α
n
f
(
K
1
,
β
x
n
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
=
0,1
,
2
,
…
,
where {
α
n
} is a sequence in (
0,1
) satisfying α
n
→
0 and ∑
n
=
0
∞
‍
α
n
=
∞. Then the sequences {
x
n
} converge to x
*
∈
Ω
1 such that the following three inequalities are satisfied:
(76)
x
*
-
f
1
x
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
.Corollary 16.
LetA
1, M
1, Ω
1, K
1, and K
1
,
β satisfy conditions (C1) and (C2), and let F
:
H
→
H be μ-Lipschitzian and r-strongly monotone mapping. Let {
x
n
} be the sequences defined by
(77)
x
0
∈
H
,
x
n
+
1
=
(
1
-
α
n
)
K
1
,
β
x
n
+
α
n
(
I
-
ρ
F
)
(
K
1
,
β
x
n
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
n
=
0,1
,
2
,
…
,
where ρ
∈
(
0
,
2
r
/
μ
2
) and {
α
n
} is a sequence in (
0,1
) satisfying α
n
→
0 and ∑
n
=
0
∞
‍
α
n
=
∞. Then the sequences {
x
n
} converge to x
*
∈
Ω
1 such that the following three inequalities are satisfied:
(78)
F
x
*
,
x
-
x
*
≥
0
,
∀
x
∈
Ω
1
.



---

*Source: 589679-2014-12-23.xml*