# Strong Convergence Theorems of theC

Authors: Huimin He; Sanyang Liu
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The aim of this paper is to show the strong convergence theorems of theC
Q algorithm for H-monotone operators in Hilbert spaces by hybrid method in the mathematical programming. The main results extend and improve the corresponding results. Moreover, the assumption conditions of our results are weaker than those of the corresponding results.


---

## Body

## 1. Introduction

In this paper, we show aC
Q algorithm for solving the inclusion problem in Hilbert space ℋ. The inclusion problem is finding the zero solutions of T; that is,
(1)
x
∈
ℋ
,
s
.
t
.
0
∈
T
(
x
)
.
This problem is closely related to many problems, such as variational inequalities, fixed points problem, and complementarity problem of mathematical programming, and it plays an important role in convex analysis and some partial differential equations. The inclusion problem (1) on monotone operator and maximal monotone operators is extensively investigated by so many researchers. See Tseng [1], Kamimura et al. [2–19], and so on.In 2003, Fang and Huang [20] firstly introduced H-monotone operators and discussed some properties of this class of operators.Motivated by Fang and Huang [20], very recently, we firstly consider the inclusion problem 0
∈
T
(
x
) of H-monotone operator for finding the solutions of it in a Hilbert space ℋ [21].In [21], we mainly presented strong and weak convergence theorems for Halpern type and Mann type algorithms, respectively, and the relations between maximal monotone operators and H-monotone operators are analyzed in detail. Simultaneously, we apply these results to the minimization problem for T
=
∂
f and provide some numerical examples to support the theoretical findings. These results start a new branch of research for the inclusion problem 0
∈
T
(
x
), and we do further extending study for this subject.Motivated by the main results of Nakajo and Takahashi [22], we propose a so-called C
Q iteration algorithm as follows:
(2)
x
0
=
x
∈
ℋ
,
y
n
=
α
n
(
x
n
+
f
n
)
+
(
1
-
α
n
)
J
H
,
r
n
T
H
(
x
n
+
f
n
)
,
C
n
=
{
z
∈
H
∣
∥
y
n
-
z
∥
≤
∥
x
n
+
f
n
-
z
∥
}
,
Q
n
=
{
z
∈
H
∣
〈
x
n
-
z
,
x
0
-
x
n
〉
≥
0
}
,
x
n
+
1
=
P
C
n
∩
Q
n
x
0
,
where {
α
n
}
⊂
[
0,1
] and {
r
n
}
⊂
(
0
,
+
∞
).The aim of this paper is to establish the strong convergence theorems to approximate the zero point ofH-monotone operator, namely, finding the x
∈
ℋ such that 0
∈
T
(
x
).


## 2. Preliminaries

Definition 1.
A multivalued operatorT
:
ℋ
→
2
ℋ is said to be(i)
monotone if(3)
〈
x
-
y
,
u
-
v
〉
≥
0
,
∀
u
,
v
∈
ℋ
,
x
∈
T
u
,
y
∈
T
v
;
(ii)
maximal monotone ifT is monotone and (
I
+
λ
T
)
(
ℋ
)
=
ℋ for all λ
>
0, where I denotes the identity mapping on ℋ.We note that theT is maximal monotone if and only if T is monotone and the graph
(4)
G
(
T
)
=
{
(
z
,
w
)
∈
ℋ
×
ℋ
∣
w
∈
T
z
}
is not properly contained in the graph of any other monotone operator T
′
:
ℋ
→
2
ℋ.Definition 2 (see [20]).
LetH
:
ℋ
→
ℋ be a single mapping and T
:
ℋ
→
2
ℋ a multivalued mapping. T is said to be(i)
H-monotone if T is monotone and (
H
+
λ
T
)
(
ℋ
)
=
ℋ holds for every λ
>
0;
(ii)
stronglyH-monotone if T is strongly monotone and (
H
+
λ
T
)
(
ℋ
)
=
ℋ holds for every λ
>
0.Definition 3.
LetT
:
ℋ
→
ℋ be a single-valued operator. T is said to be(i)
strictly monotone ifT is monotone and
(5)
〈
T
x
-
T
y
,
x
-
y
〉
=
0
,
iff
x
=
y
;
(ii)
strongly monotone if there exists some constantr
>
0 such that
(6)
〈
T
x
-
T
y
,
x
-
y
〉
≥
r
∥
x
-
y
∥
2
,
∀
x
,
y
∈
ℋ
;
(iii)
Lipschitz continuous if there exists some constants
>
0 such that
(7)
∥
T
x
-
T
y
∥
≤
s
∥
x
-
y
∥
,
∀
x
,
y
∈
ℋ
.Remark 4.
We note that ifT is strongly monotone, then T is strictly monotone, but vice is not. If T is strongly monotone with constant r and Lipschitz continuous with constant s, then we have r
≤
s. As r
=
s
>
0, then T satisfies
(8)
∥
T
x
-
T
y
∥
=
r
∥
x
-
y
∥
.LetT
:
ℋ
→
ℋ be a single-valued operator and let H
:
ℋ
→
ℋ be a strongly monotone and Lipschitz continuous operator with constant γ. Let T
:
ℋ
→
2
ℋ be an H-monotone operator and the resolvent operator J
H
,
ρ
T
:
ℋ
→
ℋ is defined by
(9)
J
H
,
ρ
T
(
u
)
=
(
H
+
ρ
T
)
-
1
(
u
)
,
∀
u
∈
ℋ
,
for each ρ
>
0. We can define the following operators which are calledYosida approximations:
(10)
A
ρ
=
1
ρ
(
I
-
H
·
J
H
,
ρ
T
)
,
∀
ρ
>
0
.
We give some elementary properties of J
H
,
ρ
T and A
ρ.Lemma 5 (Proposition  4.1 in [21]).
LetH
:
ℋ
→
ℋ be a strongly monotone and Lipschitz continuous operator with constant γ and let T
:
ℋ
→
2
ℋ be an H-monotone operator. Then the following properties hold:(i)
∥
J
H
,
ρ
T
(
x
)
-
J
H
,
ρ
T
(
y
)
∥
≤
(
1
/
γ
)
∥
x
-
y
∥
, for  all
x
,
y
∈
R
(
H
+
ρ
T
)
;
(ii)
∥
H
·
J
H
,
ρ
T
(
x
)
-
H
·
J
H
,
ρ
T
(
y
)
∥
≤
∥
x
-
y
∥
, for  all
x
,
y
∈
ℋ, or ∥
J
H
,
ρ
T
·
H
(
x
)
-
J
H
,
ρ
T
·
H
(
y
)
∥  ≤  ∥
x
-
y
∥
, for  all
x
,
y
∈
ℋ
;
(iii)
A
ρ is monotone and
(11)
∥
A
ρ
x
-
A
ρ
y
∥
≤
2
ρ
∥
x
-
y
∥
,
∀
x
,
y
∈
R
(
H
+
ρ
T
)
;
(iv)
A
ρ
x
∈
T
J
H
,
ρ
T
(
x
)
, for  all
x
∈
R
(
H
+
ρ
T
).Lemma 6 (Proposition  4.2 in [21]).
Consideru
∈
T
-
1
0 if and only if u satisfies the relation
(12)
u
=
J
H
,
ρ
T
(
H
(
u
)
)
,
where ρ
>
0 is a constant and J
H
,
ρ
T is the resolvent operator defined by (9).Lemma 7 (Proposition  2.1 in [20]).
LetH
:
ℋ
→
ℋ be a strictly monotone single-valued operator and T
:
ℋ
→
2
ℋ an H-monotone operator. Then T is maximal monotone.Lemma 8 (see [23]).
LetE be a real Banach space. Then for all x
,
y
∈
E
(13)
∥
x
+
y
∥
2
≤
∥
x
∥
2
+
2
〈
y
,
j
(
x
+
y
)
〉
,
∀
j
(
x
+
y
)

∈

J
(
x
+
y
)

.


## 3. Strong Convergence Theorems forC
Q Algorithm

We consider the following algorithm, and the sequence{
x
n
} is generated by
(14)
x
0
=
x
∈
ℋ
,
y
n
=
α
n
(
x
n
+
f
n
)
+
(
1
-
α
n
)
J
H
,
r
n
T
H
(
x
n
+
f
n
)
,
C
n
=
{
z
∈
H
:
∥
y
n
-
z
∥
≤
∥
x
n
+
f
n
-
z
∥
}
,
Q
n
=
{
z
∈
H
:
〈
x
n
-
z
,
x
0
-
x
n
〉
≥
0
}
,
x
n
+
1
=
P
C
n
∩
Q
n
x
0
,
where {
α
n
}
⊂
[
0,1
] and {
r
n
}
⊂
(
0
,
+
∞
). Motivated by Nakajo and Takahashi [22] and Fang et al. [12, 13, 20], we get the following results.Theorem 9.
LetH
:
ℋ
→
ℋ be a strongly monotone and Lipschitz continuous operator with constant γ. Let T
:
ℋ
→
2
ℋ be an H-monotone operator; let x
∈
ℋ and {
x
n
} be a sequence defined by (14), where {
α
n
}
⊂
[
0,1
] and {
r
n
}
⊂
(
0
,
+
∞
) satisfy lim
⁡
n
→
∞
⁡
α
n
=
0, ∑
n
=
0
∞
‍
α
n
=
∞, liminf
⁡
n
→
∞
⁡
r
n
>
0, and lim
⁡
n
→
∞
⁡
∥
f
n
∥
=
0. If T
-
1
0
≠
∅, then {
x
n
} converges strongly to P
x, where P is the metric projection of ℋ onto T
-
1
0.Proof.
The proofs can be divided into three steps.
Step  1 (
{
x
n
}
is well defined and
T
-
1
0
⊂
C
n
∩
Q
n
). Based on the definitions of C
n and Q
n, we can get that Q
n is a closed and convex set of ℋ for every n
∈
ℕ.
And since the inequality(15)
∥
y
n
-
z
∥
≤
∥
x
n
+
f
n
-
z
∥
is equivalent to
(16)
∥
y
n
-
x
n
-
f
n
∥
2
+
2
〈
y
n
-
x
n
-
f
n
,
x
n
+
f
n
-
z
〉
≤
0
,
hence, C
n is closed and convex, so is C
n
∩
Q
n for every n
∈
ℕ.
From Lemma6, there exists u
∈
T
-
1
0 such that u
=
J
H
,
ρ
T
(
H
(
u
)
) for all ρ
>
0, and based on Lemma 5, we know J
H
,
ρ
T
·
H is a nonexpansive mapping from ℋ into itself.
For allu
∈
T
-
1
0, it follows that
(17)
∥
y
n
-
u
∥
=
∥
α
n
(
x
n
+
f
n
)
+
(
1
-
α
n
)
J
H
,
r
n
T
H
(
x
n
+
f
n
)
-
u
∥
≤
α
n
∥
x
n
+
f
n
-
u
∥
+
(
1
-
α
n
)
∥
J
H
,
r
n
T
H
(
x
n
+
f
n
)
-
u
∥
≤
∥
x
n
+
f
n
-
u
∥
.
Then u
∈
C
n for each n
∈
ℕ. Therefore, T
-
1
0
⊂
C
n for every n
∈
ℕ.
Next, we show that{
x
n
} is well defined and T
-
1
0
⊂
C
n
∩
Q
n by mathematical induction.
Forn
=
0, we have x
0
=
x
∈
ℋ and Q
0
=
ℋ. Hence, T
-
1
0
⊂
C
0
∩
Q
0, because T
-
1
0
⊂
C
n.
Forn
=
k, suppose that {
x
k
} is given and T
-
1
0
⊂
C
k
∩
Q
k for all k
∈
ℕ. Then, there exists a unique x
k
+
1
∈
C
k
∩
Q
k such that x
k
+
1
=
P
C
k
∩
Q
k
x
0, because C
k
∩
Q
k is closed and convex.
Based on the property of the projection operators, we can get thatx
k
+
1
=
P
C
k
∩
Q
k
x
0 is equivalent to
(18)
〈
x
k
+
1
-
z
,
x
0
-
x
k
+
1
〉
≥
0
,
for all z
∈
C
k
∩
Q
k. By the assumption T
-
1
0
⊂
C
k
∩
Q
k, we have T
-
1
0
⊂
Q
k
+
1.
Hence,(19)
T
-
1
0
⊂
C
k
+
1
∩
Q
k
+
1
.
Step  2 (
{
x
n
}
is bounded and
∥
x
n
+
1
-
x
n
∥
→
0
). By Lemma 5, we get that J
H
,
r
n
T
H is nonexpansive. And from Lemma 6, we have that u
∈
T
-
1
0 is equivalent to u
=
J
H
,
ρ
T
(
H
(
u
)
). So, u
∈
T
-
1
0 is a closed and convex subset of ℋ.
The rest of the proofs of this step can follow Lemmas 3.2 and 3.3 of [22].
Step  3 (
x
n
→
z
0
=
P
T
-
1
0
x
0
). From lim
⁡
n
→
∞
⁡
∥
f
n
∥
=
0, we get that {
f
n
} is bounded. So, {
y
n
} is bounded.
Now, we suppose a subsequence{
x
n
i
} of {
x
n
} converges weakly to w
0. Since x
n
+
1
∈
C
n, we have
(20)
∥
y
n
-
x
n
∥
≤
∥
y
n
-
x
n
+
1
∥
+
∥
x
n
+
1
-
x
n
∥
≤
∥
x
n
+
f
n
-
x
n
+
1
∥
+
∥
x
n
+
1
-
x
n
∥
≤
2
∥
x
n
+
1
-
x
n
∥
+
∥
f
n
∥
.
From Step  2 ∥
x
n
+
1
-
x
n
∥
→
0 and the assumption lim
⁡
n
→
∞
⁡
∥
f
n
∥
=
0, we obtain
(21)
∥
y
n
-
x
n
∥
⟶
0
.
Hence,
(22)
y
n
i
⇀
w
0
.
And similarly, it follows from x
n
+
1
∈
C
n and Step  2 that
(23)
∥
J
H
,
r
n
T
H
(
x
n
+
f
n
)
-
x
n
-
f
n
∥
=
1
1
-
α
n
∥
y
n
-
x
n
-
f
n
∥
≤
1
1
-
α
n
(
∥
y
n
-
x
n
+
1
∥
+
∥
x
n
+
1
-
x
n
-
f
n
∥
)
≤
2
1
-
α
n
∥
x
n
+
1
-
x
n
-
f
n
∥
≤
2
1
-
α
n
(
∥
x
n
+
1
-
x
n
∥
+
∥
f
n
∥
)
.
Therefore,
(24)
∥
J
H
,
r
n
T
H
(
x
n
+
f
n
)
-
x
n
-
f
n
∥
⟶
0
,
(25)
∥
J
H
,
r
n
T
H
(
x
n
+
f
n
)
-
x
n
∥
≤
∥
J
H
,
r
n
T
H
(
x
n
+
f
n
)
-
x
n
-
f
n
∥
+
∥
f
n
∥
.
This implies that
(26)
∥
J
H
,
r
n
T
H
(
x
n
+
f
n
)
-
x
n
∥
⟶
0
.
Thus,
(27)
J
H
,
r
n
i
T
H
(
x
n
i
+
f
n
i
)
⇀
w
0
∈
ℋ
.
Next, we show that w
0
∈
T
-
1
0. Since T is monotone and A
r
n
(
H
(
x
n
+
f
n
)
)
∈
T
J
H
,
r
n
T
H
(
x
n
+
f
n
) due to A
ρ
(
x
)
∈
T
J
H
,
ρ
T
(
x
) for all x
∈
ℋ, we obtain
(28)
〈
z
-
J
H
,
r
n
i
T
H
(
x
n
i
+
f
n
i
)
,
z
′
-
A
r
n
i
(
H
(
x
n
i
+
f
n
i
)
)
〉
≥
0
,
for all z
′
∈
T
z. Due to the assumption liminf
⁡
n
→
∞
⁡
r
n
>
0 and inequality (24), we have
(29)
∥
A
r
n
(
H
(
x
n
+
f
n
)
)
∥
=
∥
H
(
x
n
+
f
n
)
-
H
J
H
,
r
n
T
H
(
x
n
+
f
n
)
r
n
∥
≤
γ
r
n
∥
J
H
,
r
n
T
H
(
x
n
+
f
n
)
-
x
n
-
f
n
∥
.
So, A
r
n
(
H
(
x
n
+
f
n
)
)
→
0 and
(30)
〈
z
-
w
0
,
z
′
〉
≥
0
,
∀
z
′
∈
T
z
.
By Lemma 5, we know T is a maximal monotone operator; from the maximality property of T; we have
(31)
w
0
∈
T
-
1
0
.
If z
0
=
P
T
-
1
0
x
0, by the lower semicontinuity of the norm, we get
(32)
∥
x
0
-
z
0
∥
≤
∥
x
0
-
w
0
∥
≤
liminf
⁡
i
→
∞
∥
x
0
-
x
n
i
∥
≤
limsup
⁡
i
→
∞
∥
x
0
-
x
n
i
∥
≤
∥
x
0
-
z
0
∥
.
Therefore, we obtain
(33)
lim
⁡
i
→
∞
∥
x
n
i
-
x
0
∥
=
∥
x
0
-
w
0
∥
=
∥
x
0
-
z
0
∥
.
Adding x
n
i
⇀
w
0, we can get that x
n
i
→
w
0
=
z
0.
Thus,x
n
→
z
0
=
P
T
-
1
0
x
0.Remark 10.
In the main results, Theorem9 of this paper, the assumption condition is liminf
⁡
n
→
∞
⁡
r
n
>
0, and we can obtain the conclusion A
r
n
(
H
(
x
n
+
f
n
)
)
→
0. However, the conclusion A
r
n
(
H
(
x
n
+
f
n
)
)
→
0 is obtained by assuming the condition liminf
⁡
n
→
∞
⁡
r
n
=
∞ that appeared in [2, 20].
It is worth noting that conditionliminf
⁡
n
→
∞
⁡
r
n
>
0 is weaker than condition
liminf
⁡
n
→
∞
⁡
r
n
=
∞ by observing
(34)
liminf
⁡
n
→
∞
r
n
=
∞
⟹
liminf
⁡
n
→
∞
r
n
>
0
.



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*Source: 393512-2014-04-30.xml*