# Iterative Schemes for Finite Families of Maximal Monotone Operators Based on Resolvents

Authors: Li Wei; Ruilin Tan
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The purpose of this paper is to present two iterative schemes based
on the relative resolvent and the generalized resolvent, respectively. And, it is
shown that the iterative schemes converge weakly to common solutions for two
finite families of maximal monotone operators in a real smooth and uniformly convex
Banach space and one example is demonstrated to explain that some assumptions
in the main results are meaningful, which extend the corresponding works by some
authors.


---

## Body

## 1. Introduction and Preliminaries

LetE be a real Banach space with norm ∥
·
∥ and let E
* denote the dual space of E. We use “→” and “⇀” to denote strong and weak convergence either in E or E
*, respectively. A Banach space E is said to be strictly convex if
(1)
∥
x
∥
=
∥
y
∥
=
1
,
x
≠
y
⟹
∥
x
+
y
2
∥
<
1
.
Also, E is said to be uniformly convex if, for each ε
∈
(
0,2
], there exists δ
>
0 such that
(2)
∥
x
∥
=
∥
y
∥
=
1
,
∥
x
-
y
∥
≥
ε
⟹
∥
x
+
y
2
∥
≤
1
-
δ
.
A Banach space E is said to be smooth if
(3)
lim
⁡
t
→
0
⁡
∥
x
+
t
y
∥
-
∥
x
∥
t
exists for each x
,
y
∈
{
z
∈
E
:
∥
z
∥
=
1
}
:
=
S
(
E
). In this case, the norm of E is said to be Gâteaux differentiable. The space E is said to have a uniformly Gâteaux differentiable norm if, for each y
∈
S
(
E
), the limit (3) is attained uniformly for x
∈
S
(
E
). The norm of E is said to be Frêchet differentiable if, for each x
∈
S
(
E
), the limit (3) is attained uniformly for y
∈
S
(
E
). The norm of E is said to be uniformly Frêchet differentiable if the limit (3) is attained uniformly for x
,
y
∈
S
(
E
).The normalized duality mappingJ
:
E
→
2
E
* is defined by
(4)
J
x
:
=
{
f
∈
E
*
:
〈
x
,
f
〉
=
∥
x
∥
2
=
∥
f
∥
2
}
,
x
∈
E
.We call thatJ is weakly sequentially continuous if {
x
n
} is a sequence in E which converges weakly to x it follows that {
J
x
n
} converges inweak
* to J
x.We know the following properties ofJ (see [1] for details):(i)
J
x
≠
∅ for each x
∈
E;
(ii)
ifE is smooth, then J is single-valued and strictly monotone;
(iii)
ifE is strictly convex, then J is one to one; that is, x
≠
y
⇒
J
x
⋂
‍
J
y
=
∅;
(iv)
ifE has a uniformly Gâteaux differentiable norm, then J is norm toweak
* uniformly continuous on each bounded subset of E;
(v)
ifE is a smooth and uniformly convex Banach space, then J
-
1
:
E
*
→
E is also a duality mapping and is uniformly continuous on each bounded subset of E
*.An operatorA
⊂
E
×
E
* is said to be monotone if 〈
x
1
-
x
2
,
y
1
-
y
2
〉
≥
0, for ∀
y
i
∈
A
x
i
,
i
=
1,2. A monotone operator A is said to be maximal if its graph G
(
A
)
=
{
(
x
,
y
)
:
y
∈
A
x
} is not properly contained in the graph of any other monotone operator. If A is maximal monotone, then the set A
-
1
0 is closed and convex: moreover, if E is a real smooth and uniformly convex Banach space, then G
(
A
) is demiclosed; that is, ∀
{
x
n
}
⊂
D
(
A
),  x
n
⇀
x, (
n
→
∞
),  ∀
y
n
∈
A
x
n, y
n
→
y
,
(
n
→
∞
)
⇒
x
∈
D
(
A
), and y
∈
A
x. If E is reflexive and strictly convex, then a monotone operator A is maximal if and only if R
(
J
+
λ
A
)
=
E
*, for each λ
>
0 (see [2] for more details).A mappingA
^
:
D
(
A
^
)
⊂
E
→
E is said to be accretive (c.f. [3]) if ∥
x
1
-
x
2
∥
≤
∥
x
1
-
x
2
+
r
(
y
1
-
y
2
)
∥, for ∀
x
i
∈
D
(
A
^
)
,
y
i
∈
A
^
x
i
,
i
=
1,2, and r
>
0. In a Hilbert space H, the m-accretive mapping is exactly the maximal monotone operator.The Lyapunov functionalφ
:
E
×
E
→
R
+ is defined as follows:
(5)
φ
(
x
,
y
)
=
∥
x
∥
2
-
2
〈
x
,
J
y
〉
+
∥
y
∥
2
,
∀
x
,
y
∈
E
.
It is obvious from the definition of Lyapunov functional that
(6)
(
∥
x
∥
-
∥
y
∥
)
2
≤
φ
(
x
,
y
)
≤
(
∥
x
∥
+
∥
y
∥
)
2
,
for each x
,
y
∈
E.We have the following well-known result.Lemma 1 (see [4]).
LetE be a real smooth and uniformly convex Banach space, and let {
x
n
} and {
y
n
} be two sequences in E. If either {
x
n
} or {
y
n
} is bounded and φ
(
x
n
,
y
n
)
→
0
,
n
→
∞, then x
n
-
y
n
→
0,  n
→
∞.Definition 2 (see [5]).
LetE be a real smooth and uniformly convex Banach space and let A
⊂
E
×
E
* be a maximal monotone operator. Then ∀
r
>
0, define Q
r
A
:
E
→
E by Q
r
A
x
=
(
J
+
r
A
)
-
1
J
x, which is called the relative resolvent.We have the following property of the relative resolvent.Lemma 3 (see [5]).
LetE be a real reflexive, strictly convex, and smooth Banach space and let A
⊂
E
×
E
* be a maximal monotone operator such that A
-
1
0
≠
∅. Then ∀
x
∈
E
,
y
∈
A
-
1
0, and r
>
0, we have φ
(
y
,
Q
r
A
x
)
+
φ
(
Q
r
A
x
,
x
)
≤
φ
(
y
,
x
).Definition 4 (see [4]).
LetE be a real reflexive, strictly convex, and smooth Banach space and let C be a nonempty closed and convex subset of E. Then ∀
x
∈
E, there exists a unique element x
0
∈
C satisfying φ
(
x
0
,
x
)
=
inf
⁡
{
φ
(
z
,
x
)
:
z
∈
C
}. In this case, ∀
x
∈
E, define Π
C
:
E
→
C by Π
C
x
=
x
0, and then Π
C is called the generalized projection from E onto C.Lemma 5 (see [4]).
LetE be a real reflexive, strictly convex, and smooth Banach space and let C be a nonempty closed and convex subset of E. Then ∀
x
∈
E
,
∀
y
∈
C,
(7)
φ
(
y
,
Π
C
x
)
+
φ
(
Π
C
x
,
x
)
≤
φ
(
y
,
x
)
.Lemma 6 (see [4]).
LetE be a real smooth Banach space and let C be a nonempty closed and convex subset of E. Let x
∈
E, and x
0
∈
C. Then φ
(
x
0
,
x
)
=
i
n
f
{
φ
(
z
,
x
)
:
z
∈
C
} if and only if 〈
z
-
x
0
,
J
x
0
-
J
x
〉
≥
0
,
∀
z
∈
C.LetE be a smooth Banach space and let C be a nonempty closed and convex subset of E. A mapping T
:
C
→
C is said to be generalized nonexpansive (c.f. [5]) if F
(
T
)
≠
∅ and φ
(
T
x
,
y
)
≤
φ
(
x
,
y
), for ∀
x
∈
C and y
∈
F
(
T
), where F
(
T
) is a set of the fixed points of T; that is, F
(
T
)
:
=
{
x
∈
C
:
T
x
=
x
}.LetC be a nonempty, closed subset of E and let Q be a mapping of E onto C. Then Q is said to be sunny (c.f. [5]) if Q
(
Q
(
x
)
+
t
(
x
-
Q
(
x
)
)
)
=
Q
(
x
), for all x
∈
E and t
≥
0. A mapping Q
:
E
→
C is said to be a retraction (c.f. [5]) if Q
(
z
)
=
z for every z
∈
C. If E is smooth and strictly convex, then a sunny generalized nonexpansive retraction of E onto C is uniquely decided (c.f. [5]). Then, if E is smooth and strictly convex, a sunny generalized nonexpansive retraction of E onto C is denoted by R
C.A subsetC of E is said to be a sunny nonexpansive retract of E (c.f. [5]) if there exists a sunny nonexpansive retraction of E onto C and it is called a generalized nonexpansive retract of E if there exists a generalized nonexpansive retraction of E onto C.Definition 7 (see [5]).
LetE be a real reflexive, strictly convex, and smooth Banach space and let B
⊂
E
*
×
E be a maximal monotone operator. Then ∀
r
>
0, define R
r
B
:
E
→
E by R
r
B
x
=
(
I
+
r
B
J
)
-
1
x, which is called the generalized resolvent.Lemma 8 (see [5]).
LetE be a real reflexive and strictly Banach space with a Frêchet differential norm and let B
⊂
E
*
×
E be a maximal monotone operator with B
-
1
0
≠
∅. Then (i)(
B
J
)
-
1
0
=
F
(
R
r
B
);  (ii)   (
B
J
)
-
1
0 is closed; (iii)R
r
B is generalized nonexpansive, for r
>
0.Lemma 9 (see [5]).
LetE be a real reflexive, smooth, and strictly Banach space and let B
⊂
E
*
×
E be a maximal monotone operator with B
-
1
0
≠
∅. Then
(8)
φ
(
x
,
R
r
B
x
)
+
φ
(
R
r
B
x
,
u
)
≤
φ
(
x
,
u
)
,
a
∀
r
>
0
,
u
∈
(
B
J
)
-
1
0
,
x
∈
E
.Lemma 10 (see [6]).
Let{
a
n
} and {
b
n
} be two sequences of nonnegative real numbers and a
n
+
1
≤
a
n
+
b
n for ∀
n
≥
1. If ∑
n
=
0
∞
‍
b
n
<
+
∞, then li
m
n
→
∞
a
n exists.Finding zeros of maximal monotone operators is a hot topic in applied mathematics since it has practical background. One classical method for studying the problem0
∈
A
x, where A is a maximal monotone operator, is the following so-called proximal method (c.f. [7]), presented in a Hilbert space:
(9)
x
0
∈
H
,
x
n
+
1
≈
J
r
n
A
x
n
,
n
≥
0
,
where J
r
n
A
:
=
(
I
+
r
n
A
)
-
1. It was shown that the sequence generated by (9) converges weakly to a point in A
-
1
0 under some conditions.In 2004, Kamimura et al. extended the study on zeros of maximal monotone operators to the following iterative scheme based on the relative resolventQ
r
n
A in Banach spaces (c.f. [8]):
(10)
x
1
∈
E
,
x
n
+
1
=
J
-
1
[
α
n
J
x
n
+
(
1
-
α
n
)
J
Q
r
n
A
x
n
]
.
And, they showed that {
x
n
} generated by (10) converges weakly to a point in A
-
1
0, where A
⊂
E
×
E
* is a maximal monotone operator.In 2007, Ibaraki and Takahashi [9] studied the following iterative scheme based on the generalized resolvent R
r
n
B in Banach spaces:
(11)
x
1
∈
E
,
x
n
+
1
=
α
n
x
n
+
(
1
-
α
n
)
R
r
n
B
x
n
.
And, they showed that {
x
n
} generated by (11) converges weakly to a point in (
B
J
)
-
1
0, where B
⊂
E
*
×
E is a maximal monotone operator.In 2010, Shehu and Ezeora, [10] presented the following iterative scheme for a family of m-accretive mappings {
A
i
}
i
=
1
N in a real uniformly smooth and uniformly convex Banach space E:
(12)
x
1
∈
E
,
y
n
=
(
1
-
α
n
)
x
n
,
x
n
+
1
=
(
1
-
β
n
)
x
n
+
β
n
S
N
y
n
,
n
≥
1
,
where S
N
:
=
a
0
I
+
a
1
J
A
1
+
a
2
J
A
2
+
⋯
+
a
N
J
A
N with J
A
i
=
(
I
+
A
i
)
-
1, for i
=
1,2
,
…
,
N.  0
<
a
k
<
1, for k
=
0,1
,
2
,
…
,
N, and ∑
k
=
0
N
‍
a
k
=
1. Then {
x
n
} converges strongly to the common point in A
i
-
1
0, where i
=
1,2
,
…
,
N.Can we extend the study onm-accretive mappings [10] to maximal monotone operators? Inspired by the work on (10)–(12), in Section 2, we will present the following iterative scheme based on the relative resolvent:
(
A
)
x
1
∈
E
,
u
n
=
J
-
1
[
(
1
-
α
n
)
J
x
n
]
,
v
n
=
J
-
1
[
(
1
-
β
n
)
J
x
n
+
β
n
J
U
n
u
n
]
,
x
n
+
1
=
J
-
1
[
γ
n
J
x
n
+
(
1
-
γ
n
)
J
W
n
v
n
]
,
n
≥
1
,
where A
i
,
B
j
⊂
E
×
E
* are maximal monotone operators, i
=
1,2
,
…
,
N,  j
=
1,2
,
…
,
M. Suppose (
⋂
i
=
1
N
‍
A
i
-
1
0
)
⋂
‍
(
⋂
j
=
1
M
‍
B
j
-
1
0
)
≠
∅.  U
n
:
=
J
-
1
[
a
0
J
+
a
1
J
Q
r
n
,
1
A
1
+
a
2
J
Q
r
n
,
2
A
2
+
⋯
+
a
N
J
Q
r
n
,
N
A
N
], and Q
r
n
,
i
A
i
=
(
J
+
r
n
,
i
A
i
)
-
1
J and r
n
,
i
>
0, for i
=
1,2
,
…
,
N and n
≥
1.  W
n
:
=
J
-
1
[
b
0
J
+
b
1
J
Q
s
n
,
1
B
1
+
b
2
J
Q
s
n
,
2
B
2
Q
s
n
,
1
B
1
+
⋯
+
b
M
J
Q
s
n
,
M
B
M
Q
s
n
,
M
-
1
B
M
-
1
⋯
Q
s
n
,
1
B
1
], and Q
s
n
,
j
B
j
=
(
J
+
s
n
,
j
B
j
)
-
1
J and s
n
,
j
>
0, for j
=
1,2
,
…
,
M and n
≥
1.  a
0
,
a
1
,
…
,
a
N
;
b
0
,
b
1
,
…
,
b
M are real numbers in (
0,1
) with ∑
i
=
0
N
‍
a
i
=
1 and ∑
j
=
0
M
‍
b
j
=
1.In Section3, we will study the following iterative scheme based on generalized resolvent:
(
B
)
x
1
∈
E
,
u
n
=
(
1
-
α
n
)
x
n
,
v
n
=
(
1
-
β
n
)
x
n
+
β
n
S
n
u
n
,
x
n
+
1
=
γ
n
x
n
+
(
1
-
γ
n
)
T
n
v
n
,
n
≥
1
,
where A
i
,
B
j
⊂
E
*
×
E are maximal monotone operators, i
=
1,2
,
…
,
N,  j
=
1,2
,
…
,
M. Suppose (
⋂
i
=
1
N
‍
A
i
-
1
0
)
⋂
‍
(
⋂
j
=
1
M
‍
B
j
-
1
0
)
≠
∅.  S
n
:
=
a
0
I
+
a
1
R
r
n
,
1
A
1
+
a
2
R
r
n
,
2
A
2
+
⋯
+
a
N
R
r
n
,
N
A
N, and T
n
:
=
b
0
I
+
b
1
R
s
n
,
1
B
1
+
b
2
R
s
n
,
2
B
2
R
s
n
,
1
B
1
+
⋯
+
b
M
R
s
n
,
M
B
M
R
s
n
,
M
-
1
B
M
-
1
⋯
R
s
n
,
1
B
1. For i
=
1,2
,
…
,
N,  R
r
n
,
i
A
i
=
(
I
+
r
n
,
i
A
i
J
)
-
1. For j
=
1,2
,
…
,
M,  R
s
n
,
j
B
j
=
(
I
+
s
n
,
j
B
j
J
)
-
1.  a
0
,
a
1
,
…
,
a
N and b
0
,
b
1
,
…
,
b
M are real numbers in (
0,1
) and ∑
i
=
0
N
‍
a
i
=
1,∑
j
=
0
M
‍
b
j
=
1.  r
n
,
i
>
0, for i
=
1,2
,
…
,
N, and s
n
,
j
>
0, for j
=
1,2
,
…
,
M and n
≥
1.In this paper, some weak convergence theorems are obtained, which can be regarded as the extension and complement of the work done in [7–10], and so forth. At the end of Section 3, one example is demonstrated to show that the assumption that (
⋂
i
=
1
N
‍
A
i
-
1
0
)
⋂
‍
(
⋂
j
=
1
M
‍
B
j
-
1
0
)
≠
∅ in the discussions of (
A
) and (
B
) is meaningful.


## 2.  Weak Convergence Theorems Based on the Relative Resolvent

Theorem 11.
LetE be a real smooth and uniformly convex Banach space. Let A
i
,
B
j
⊂
E
×
E
* be maximal monotone operators, where i
=
1,2
,
…
,
N
,
j
=
1,2
,
…
,
M. Suppose that both J
:
E
→
E
* and J
-
1
:
E
*
→
E are weakly sequentially continuous and D
:
=
(
⋂
i
=
1
N
‍
A
i
-
1
0
)
⋂
‍
(
⋂
j
=
1
M
‍
B
j
-
1
0
)
≠
∅. Let {
x
n
} be generated by the iterative scheme (
A
), where U
n
:
=
J
-
1
[
a
0
J
+
a
1
J
Q
r
n
,
1
A
1
+
a
2
J
Q
r
n
,
2
A
2
+
⋯
+
a
N
J
Q
r
n
,
N
A
N
], and Q
r
n
,
i
A
i
=
(
J
+
r
n
,
i
A
i
)
-
1
J, for i
=
1,2
,
…
,
N,  0
<
a
k
<
1, for k
=
0,1
,
2
,
…
,
N,  ∑
k
=
0
N
‍
a
k
=
1.  W
n
:
=
J
-
1
[
b
0
J
+
b
1
J
Q
s
n
,
1
B
1
+
b
2
J
Q
s
n
,
2
B
2
Q
s
n
,
1
B
1
+
⋯
+
b
M
J
Q
s
n
,
M
B
M
Q
s
n
,
M
-
1
B
M
-
1
⋯
Q
s
n
,
1
B
1
], where Q
s
,
j
B
j
=
(
J
+
s
n
,
j
B
j
)
-
1
J, for j
=
1,2
,
…
,
M,  0
<
b
k
<
1, for k
=
0,1
,
2
,
…
,
M,  ∑
k
=
0
M
‍
b
k
=
1. Suppose {
α
n
},  {
β
n
}, and {
γ
n
} are three sequences in (
0,1
) and {
r
n
,
i
}, {
s
n
,
j
}
⊂
(
0
,
+
∞
) satisfying the following conditions:(i)
α
n
→
0, as n
→
∞;
(ii)
∑
n
=
1
∞
‍
α
n
β
n
(
1
-
γ
n
)
<
+
∞;
(iii)
limin
f
n
→
+
∞
β
n
>
0 and limsup
⁡
n
→
+
∞
γ
n
<
1;
(iv)
limin
f
n
→
∞
r
n
,
i
>
0 and limin
f
n
→
∞
s
n
,
j
>
0, for i
=
1,2
,
…
,
N and j
=
1,2
,
…
,
M.Then{
x
n
} converges weakly to the unique element v
0
∈
D which satisfies
(13)
v
0
=
lim
⁡
n
→
∞
⁡
Π
D
x
n
.Proof.
We will split the proof into six steps.
Step 1. {
x
n
} is bounded.
For∀
p
∈
D, noticing the definition of the Lyapunov functional and by using Lemma 3 repeatedly, we have
(14)
φ
(
p
,
x
n
+
1
)
≤
γ
n
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
φ
(
p
,
W
n
v
n
)
≤
γ
n
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
×
[
b
0
φ
(
p
,
v
n
)
+
b
1
φ
(
p
,
Q
s
n
,
1
B
1
v
n
)
+
b
2
φ
(
p
,
Q
s
n
,
2
B
2
Q
s
n
,
1
B
1
v
n
)
+
⋯
+
b
M
φ
(
p
,
Q
s
n
,
M
B
M
Q
s
n
,
M
-
1
B
M
-
1
⋯
Q
s
n
,
1
B
1
v
n
)
]
≤
γ
n
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
φ
(
p
,
v
n
)
≤
γ
n
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
×
[
(
1
-
β
n
)
φ
(
p
,
x
n
)
+
β
n
φ
(
p
,
U
n
u
n
)
]
≤
[
1
-
β
n
(
1
-
γ
n
)
]
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
β
n
[
a
0
φ
(
p
,
u
n
)
+
∑
i
=
1
N
‍
a
i
φ
(
p
,
Q
r
n
,
i
A
i
u
n
)
]
≤
[
1
-
β
n
(
1
-
γ
n
)
]
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
β
n
φ
(
p
,
u
n
)
≤
[
1
-
α
n
β
n
(
1
-
γ
n
)
]
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
α
n
β
n
∥
p
∥
2
.
Lemma10 ensures that li
m
n
→
∞
φ
(
p
,
x
n
) exists, which implies that {
x
n
} is bounded in view of (6).
Then from iterative scheme(
A
), {
u
n
} is bounded. Since φ
(
p
,
U
n
u
n
)
≤
φ
(
p
,
u
n
), for ∀
p
∈
D, then {
U
n
u
n
} is bounded, which ensures that {
v
n
} is bounded. For ∀
p
∈
D,  φ
(
p
,
Q
s
n
,
M
B
M
Q
s
n
,
M
-
1
B
M
-
1
Q
s
n
,
1
B
1
v
n
)
≤
φ
(
p
,
v
n
), then we know that {
Q
s
n
,
M
B
M
Q
s
n
,
M
-
1
B
M
-
1
Q
s
n
,
1
B
1
v
n
} is bounded.
Step 2. ω
(
x
n
)
⊂
D, where ω
(
x
n
) is the set of the weak limit points of all of the weakly convergent subsequences of {
x
n
}.
Since{
x
n
} is bounded, then ω
(
x
n
)
≠
∅. And, there exists a subsequence of {
x
n
}; for simplicity, we still denote it by {
x
n
} such that x
n
⇀
x
,
n
→
∞.
For∀
p
∈
D, using Lemma 3 again, we have the following:
(15)
φ
(
p
,
x
n
+
1
)
≤
γ
n
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
φ
(
p
,
W
n
v
n
)
≤
γ
n
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
×
[
(
b
0
+
b
1
+
⋯
+
b
M
-
1
)
φ
(
p
,
v
n
)
+
b
M
φ
(
p
,
Q
s
n
,
M
B
M
Q
s
n
,
M
-
1
B
M
-
1
⋯
Q
s
n
,
1
B
1
v
n
)
]
≤
γ
n
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
(
1
-
b
M
)
φ
(
p
,
v
n
)
+
b
M
(
1
-
γ
n
)
φ
(
p
,
Q
s
n
,
M
-
1
B
M
-
1
Q
s
n
,
M
-
2
B
M
-
2
⋯
Q
s
n
,
1
B
1
v
n
)
-
b
M
(
1
-
γ
n
)
×
φ
(
Q
s
n
,
M
B
M
Q
s
n
,
M
-
1
B
M
-
1
⋯
Q
s
n
,
1
B
1
v
n
,
Q
s
n
,
M
-
1
B
M
-
1
Q
s
n
,
M
-
2
B
M
-
2
⋯
Q
s
n
,
1
B
1
v
n
)
≤
γ
n
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
×
[
(
1
-
β
n
)
φ
(
p
,
x
n
)
+
β
n
φ
(
p
,
U
n
u
n
)
]
-
b
M
(
1
-
γ
n
)
×
φ
(
Q
s
n
,
M
B
M
Q
s
n
,
M
-
1
B
M
-
1
⋯
Q
s
n
,
1
B
1
v
n
,
Q
s
n
,
M
-
1
B
M
-
1
Q
s
n
,
M
-
2
B
M
-
2
⋯
Q
s
n
,
1
B
1
v
n
)
≤
[
1
-
α
n
β
n
(
1
-
γ
n
)
]
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
α
n
β
n
∥
p
∥
2
-
b
M
(
1
-
γ
n
)
×
φ
(
Q
s
n
,
M
B
M
Q
s
n
,
M
-
1
B
M
-
1
⋯
Q
s
n
,
1
B
1
v
n
,
Q
s
n
,
M
-
1
B
M
-
1
Q
s
n
,
M
-
2
B
M
-
2
⋯
Q
s
n
,
1
B
1
v
n
)
.
Then (15) implies that
(16)
b
M
(
1
-
γ
n
)
φ
(
Q
s
n
,
M
B
M
Q
s
n
,
M
-
1
B
M
-
1
⋯
Q
s
n
,
1
B
1
v
n
,
Q
s
n
,
M
-
1
B
M
-
1
Q
s
n
,
M
-
2
B
M
-
2
⋯
Q
s
n
,
1
B
1
v
n
)
≤
[
1
-
α
n
β
n
(
1
-
γ
n
)
]
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
α
n
β
n
∥
p
∥
2
-
φ
(
p
,
x
n
+
1
)
.
Sinceli
m
n
→
∞
φ
(
p
,
x
n
) exists and {
Q
s
n
,
M
B
M
Q
s
n
,
M
-
1
B
M
-
1
Q
s
n
,
1
B
1
v
n
} is bounded, then, using Lemma 1, we know that
(17)
Q
s
n
,
M
B
M
Q
s
n
,
M
-
1
B
M
-
1
⋯
Q
s
n
,
1
B
1
v
n
-
Q
s
n
,
M
-
1
B
M
-
1
Q
s
n
,
M
-
2
B
M
-
2
⋯
Q
s
n
,
1
B
1
v
n
⟶
0
,
as n
→
∞. Revise (14) in the following way:
(18)
φ
(
p
,
x
n
+
1
)
≤
γ
n
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
φ
(
p
,
W
n
v
n
)
≤
γ
n
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
×
[
(
b
0
+
b
1
+
⋯
+
b
M
-
2
+
b
M
)
φ
(
p
,
v
n
)
+
b
M
-
1
φ
(
p
,
Q
s
n
,
M
-
1
B
M
-
1
Q
s
n
,
M
-
2
B
M
-
2
⋯
Q
s
n
,
1
B
1
v
n
)
]
≤
γ
n
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
(
1
-
b
M
-
1
)
φ
(
p
,
v
n
)
+
b
M
-
1
(
1
-
γ
n
)
φ
(
p
,
Q
s
n
,
M
-
2
B
M
-
2
Q
s
n
,
M
-
3
B
M
-
3
⋯
Q
s
n
,
1
B
1
v
n
)
-
b
M
-
1
(
1
-
γ
n
)
×
φ
(
Q
s
n
,
M
-
1
B
M
-
1
Q
s
n
,
M
-
2
B
M
-
2
⋯
Q
s
n
,
1
B
1
v
n
,
Q
s
n
,
M
-
2
B
M
-
2
Q
s
n
,
M
-
3
B
M
-
3
⋯
Q
s
n
,
1
B
1
v
n
)
.
Then repeating the above process, we have
(19)
Q
s
n
,
M
-
1
B
M
-
1
Q
s
n
,
M
-
2
B
M
-
2
⋯
Q
s
n
,
1
B
1
v
n
-
Q
s
n
,
M
-
2
B
M
-
2
Q
s
n
,
M
-
3
B
M
-
3
⋯
Q
s
n
,
1
B
1
v
n
⟶
0
,
as n
→
∞. Similarly, we have
(20)
Q
s
n
,
M
-
2
B
M
-
2
Q
s
n
,
M
-
3
B
M
-
3
⋯
Q
s
n
,
1
B
1
v
n
-
Q
s
n
,
M
-
3
B
M
-
3
Q
s
n
,
M
-
4
B
M
-
4
⋯
Q
s
n
,
1
B
1
v
n
⟶
0
,
Q
s
n
,
M
-
3
B
M
-
3
Q
s
n
,
M
-
4
B
M
-
4
⋯
Q
s
n
,
1
B
1
v
n
-
Q
s
n
,
M
-
4
B
M
-
4
Q
s
n
,
M
-
5
B
M
-
5
⋯
Q
s
n
,
1
B
1
v
n
⟶
0
,
⋮
Q
s
n
,
1
B
1
v
n
-
v
n
⟶
0
.
On the other hand, noticing (14) and using Lemma 3, we have
(21)
φ
(
p
,
x
n
+
1
)
≤
[
1
-
β
n
(
1
-
γ
n
)
]
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
β
n
[
a
0
φ
(
p
,
u
n
)
+
∑
i
=
1
N
‍
a
i
φ
(
p
,
Q
r
n
,
i
A
i
u
n
)
]
≤
[
1
-
β
n
(
1
-
γ
n
)
]
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
×
β
n
{
a
0
φ
(
p
,
u
n
)
+
∑
i
=
1
N
‍
a
i
[
φ
(
p
,
u
n
)
-
φ
(
Q
r
n
,
i
A
i
u
n
,
u
n
)
]
}
≤
[
1
-
α
n
β
n
(
1
-
γ
n
)
]
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
α
n
β
n
∥
p
∥
2
-
(
1
-
γ
n
)
β
n
∑
i
=
1
N
‍
a
i
φ
(
Q
r
n
,
i
A
i
u
n
,
u
n
)
.
Similar to the discussion of (20), we have
(22)
Q
r
n
,
i
A
i
u
n
-
u
n
⟶
0
,
as n
→
∞,  i
=
1,2
,
…
,
N.
Since bothJ and J
-
1 are weakly sequentially continuous, J
u
n
=
(
1
-
α
n
)
J
x
n, and x
n
⇀
x, then u
n
⇀
x, as n
→
∞. Now, (22) implies that Q
r
n
,
i
A
i
u
n
⇀
x. If we set z
n
,
i
=
Q
r
n
,
i
A
i
u
n, then from (22) and the fact that J is uniformly norm to norm continuous on each bounded subset of E, we have A
i
z
n
,
i
=
(
J
u
n
-
J
z
n
,
i
)
/
r
n
,
i
→
0, as n
→
∞, for i
=
1,2
,
…
,
N. Since G
(
A
i
) is demiclosed, then x
∈
⋂
i
=
1
N
‍
A
i
-
1
0.
Now, fromQ
r
n
,
i
A
i
u
n
⇀
x, we have U
n
u
n
⇀
x, and then J
U
n
u
n
⇀
J
x, which implies that v
n
⇀
x, as n
→
∞, since J
v
n
=
(
1
-
β
n
)
J
x
n
+
β
n
J
U
n
u
n. Thus (20) implies that Q
s
n
,
1
B
1
v
n
⇀
x. In the same way as the proof of x
∈
A
i
-
1
0, we have x
∈
B
1
-
1
0.
From the fact thatQ
s
n
,
1
B
1
v
n
⇀
x and (20), we have Q
s
n
,
2
B
2
Q
s
n
,
1
B
1
v
n
⇀
x, and Q
s
n
,
2
B
2
Q
s
n
,
1
B
1
v
n
-
Q
s
n
,
1
B
1
v
n
→
0, as n
→
∞. Then, if we set w
n
,
2
=
Q
s
n
,
2
B
2
Q
s
n
,
1
B
1
v
n
=
Q
s
n
,
2
B
2
w
n
,
1, we have B
2
w
n
,
2
=
(
J
w
n
,
1
-
J
w
n
,
2
)
/
s
n
,
2
→
0, which ensures that x
∈
B
2
-
1
0. By induction, using (20) repeatedly, we know that x
∈
⋂
j
=
1
M
‍
B
j
-
1
0.
Therefore,x
∈
D, and then ω
(
x
n
)
⊂
D.
Step 3. There exists a unique element v
0
∈
D such that
(23)
lim
⁡
n
→
∞
⁡
φ
(
v
0
,
x
n
)
=
min
⁡
y
∈
D
⁡
lim
⁡
n
→
∞
⁡
φ
(
y
,
x
n
)
.
In fact, leth
(
y
)
=
li
m
n
→
∞
φ
(
y
,
x
n
),  ∀
y
∈
D. Then h
:
D
→
R
+ is proper, convex, and lower-semicontinuous and h
(
y
)
→
+
∞, as ∥
y
∥
→
+
∞. Thus there exists v
0
∈
D such that h
(
v
0
)
=
mi
n
y
∈
D
h
(
y
). Since h is strictly convex, then v
0 is unique.
Step 4. li
m
n
→
∞
φ
(
Π
D
x
n
,
x
n
) exists.
From the definition ofΠ
D, we have φ
(
Π
D
x
n
+
1
,
x
n
+
1
)
≤
φ
(
Π
D
x
n
,
x
n
+
1
).
Using (14), we have
(24)
φ
(
Π
D
x
n
,
x
n
+
1
)
≤
[
1
-
α
n
β
n
(
1
-
γ
n
)
]
φ
(
Π
D
x
n
,
x
n
)
+
(
1
-
γ
n
)
α
n
β
n
∥
p
∥
2
≤
φ
(
Π
D
x
n
,
x
n
)
+
(
1
-
γ
n
)
α
n
β
n
∥
p
∥
2
.
Thus(25)
φ
(
Π
D
x
n
+
1
,
x
n
+
1
)
≤
φ
(
Π
D
x
n
,
x
n
)
+
(
1
-
γ
n
)
α
n
β
n
∥
p
∥
2
.
Then Lemma10 ensures that li
m
n
→
∞
φ
(
Π
D
x
n
,
x
n
) exists.
Step 5. li
m
n
→
∞
Π
D
x
n
=
v
0, where v
0 is the same as that in Step 3.
From Lemma5, we have φ
(
v
0
,
Π
D
x
n
)
≤
φ
(
v
0
,
x
n
)
-
φ
(
Π
D
x
n
,
x
n
). Thus
(26)
limsup
⁡
n
→
∞
⁡
φ
(
v
0
,
Π
D
x
n
)
≤
lim
⁡
n
→
∞
⁡
φ
(
v
0
,
x
n
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
-
lim
⁡
n
→
∞
⁡
φ
(
Π
D
x
n
,
x
n
)
h
h
h
h
h
h
h
h
h
h
h
h
h
i
h
=


h
(
v
0
)

-

lim
⁡
n
→
∞
⁡
φ
(
Π
D
x
n
,
x
n
)


≤

0
Therefore, Lemma1 implies that Π
D
x
n
→
v
0, as n
→
∞.
Step 6. x
n
⇀
v
0 where v
0 is the same as that in Step 3.
From Lemma6, we know that,
(27)
∀
y
∈
D
,
〈
Π
D
x
n
-
y
,
J
Π
D
x
n
-
J
x
n
〉
≤
0
.
Since J is weakly sequentially continuous, then from Step 5, we have J
Π
D
x
n
⇀
J
v
0, as n
→
∞.
Since{
x
n
} is bounded, then there exists a subsequence {
x
n
j
} of {
x
n
} such that x
n
j
⇀
x
0, as j
→
∞. From Step 2, x
0
∈
D. And, J
x
n
j
⇀
J
x
0, as j
→
∞. Substituting {
x
n
} by {
x
n
j
} in (27) and taking limits on both sides, we have
(28)
∀
y
∈
D
,
〈
v
0
-
y
,
J
v
0
-
J
x
0
〉
≤
0
.
Lettingy
=
x
0 in (28), then 〈
v
0
-
x
0
,
J
v
0
-
J
x
0
〉
≤
0, which implies that x
0
=
v
0, since J is strictly monotone.
Suppose there exists another subsequence{
x
n
l
} of {
x
n
} such that x
n
l
⇀
x
1, as l
→
∞. Then x
1
∈
D and J
x
n
l
⇀
J
x
1, as l
→
+
∞. Repeating the above process, we know that x
1
=
v
0. Therefore, all of the weakly convergent subsequences of {
x
n
} converge weakly to the same element v
0, and then x
n
⇀
v
0 which satisfies (13), as n
→
∞.
This completes the proof.If, in Theorem11, the Banach space E reduces to the Hilbert space H, then we have the following theorem.Theorem 12.
LetH be a Hilbert space and let D be the same as that in Theorem 11. Let A
i
,
B
j
⊂
H
×
H be m-accretive mappings, where i
=
1,2
,
…
,
N
;
j
=
1,2
,
…
,
M. Let {
a
n
}
n
=
0
∞ and {
b
n
}
n
=
0
∞ be the same as those in Theorem 11. Let {
α
n
},  {
β
n
}
,
{
γ
n
}
⊂
(
0,1
) and {
r
n
,
i
}
,
{
s
n
,
j
}
⊂
(
0
,
+
∞
) satisfy some conditions presented in Theorem 11.Let{
x
n
} be generated by the following scheme:
(
C
)
x
1
∈
H
,
u
n
=
(
1
-
α
n
)
x
n
,
v
n
=
(
1
-
β
n
)
x
n
+
β
n
U
n
u
n
,
x
n
+
1
=
γ
n
x
n
+
(
1
-
γ
n
)
W
n
v
n
,
n
≥
1
,
where U
n
=
a
0
I
+
a
1
(
I
+
r
n
,
1
A
1
)
-
1
+
⋯
+
a
N
(
I
+
r
n
,
N
A
N
)
-
1 and W
n
=
b
0
I
+
b
1
(
I
+
s
n
,
1
B
1
)
-
1
+
b
2
(
I
+
s
n
,
2
B
2
)
-
1
(
I
+
s
n
,
1
B
1
)
-
1
+
⋯
+
b
M
(
I
+
s
n
,
M
B
M
)
-
1
(
I
+
s
n
,
M
-
1
B
M
-
1
)
-
1
⋯
(
I
+
s
n
,
1
B
1
)
-
1. Then {
x
n
} converges weakly to the unique element p
0
∈
D, where p
0
=
li
m
n
→
∞
P
D
x
n and P
D is the metric projection from H onto D.Remark 13.
Compared to the work in [10], we may find that Theorem 11 is not a simple extension from the case of m-accretive mappings to maximal monotone operators. In (
A
), different A
i and B
j have different coefficients while in (12), different A
i have the same coefficients.


## 3. Weak Convergence Theorems Based on the Generalized Resolvent

Theorem 14.
LetE be a real smooth and uniformly convex Banach space. Let A
i
,
B
j
⊂
E
*
×
E be maximal monotone operators, where i
=
1,2
,
…
,
N
,
j
=
1,2
,
…
,
M. Suppose that both J
:
E
→
E
* and J
-
1
:
E
*
→
E are weakly sequentially continuous and D
~
~
:
=
(
⋂
i
=
1
N
‍
A
i
-
1
0
)
⋂
‍
(
⋂
j
=
1
M
‍
B
j
-
1
0
)
≠
∅. Let {
x
n
} be generated by the iterative scheme (
B
), where S
n
:
=
a
0
I
+
a
1
R
r
n
,
1
A
1
+
a
2
R
r
n
,
2
A
2
+
⋯
+
a
N
R
r
n
,
N
A
N, and R
r
n
,
i
A
i
=
(
I
+
r
n
,
i
A
i
J
)
-
1, for i
=
1,2
,
…
,
N,  0
<
a
k
<
1, for k
=
0,1
,
2
,
…
,
N,  ∑
k
=
0
N
‍
a
k
=
1.  W
n
:
=
b
0
I
+
b
1
R
s
n
,
1
B
1
+
b
2
R
s
n
,
2
B
2
R
s
n
,
1
B
1
+
⋯
+
b
M
R
s
n
,
M
B
M
R
s
n
,
M
-
1
B
M
-
1
⋯
R
s
n
,
1
B
1, where R
s
,
j
B
j
=
(
I
+
s
n
,
j
B
j
J
)
-
1,  for j
=
1,2
,
…
,
M,  0
<
b
k
<
1, for k
=
0,1
,
2
,
…
,
M,  ∑
k
=
0
M
‍
b
k
=
1. Suppose {
α
n
},{
β
n
}, and {
γ
n
} are three sequences in (
0,1
) and {
r
n
,
i
}, {
s
n
,
j
}
⊂
(
0
,
+
∞
) satisfy the following conditions:(i)
α
n
→
0, as n
→
∞;
(ii)
∑
n
=
1
∞
‍
α
n
β
n
(
1
-
γ
n
)
<
+
∞;
(iii)
limin
f
n
→
+
∞
β
n
>
0 limsu
p
n
→
+
∞ γ
n
<
1;
(iv)
limin
f
n
→
∞
r
n
,
i
>
0 and
limin
f
n
→
∞
s
n
,
j
>
0, for i
=
1,2
,
…
,
N and j
=
1,2
,
…
,
M.
Then{
x
n
} converges weakly to the unique element v
0
∈
D
~, where
(29)
D
~
:
=
[
⋂
i
=
1
N
‍
(
A
i
J
)
-
1
0
]
⋂
‍
[
⋂
j
=
1
M
‍
(
B
j
J
)
-
1
0
]
.Proof.
We will split the proof into four steps.
Step 1. D
~
≠
∅.
SinceD
~
~
≠
∅, then we may choose p
∈
D
~
~, which implies that p
∈
A
i
-
1
0 and p
∈
B
j
-
1
0, for i
=
1,2
,
…
,
N; j
=
1,2
,
…
,
M. Thus 0
∈
A
i
p
=
A
i
J
J
-
1
p and 0
∈
B
j
p
=
B
j
J
J
-
1
p, for i
=
1,2
,
…
,
N; j
=
1,2
,
…
,
M. And then J
-
1
p
∈
(
A
i
J
)
-
1
0 and J
-
1
p
∈
(
B
j
J
)
-
1
0, for i
=
1,2
,
…
,
N; j
=
1,2
,
…
,
M. Therefore, J
-
1
p
∈
D
~ which implies that D
~
≠
∅.
Step 2. {
x
n
} is bounded.
For∀
p
∈
D
~, noticing the definition of the Lyapunov functional and by using Lemma 9 repeatedly, we have
(30)
φ
(
x
n
+
1
,
p
)
≤
γ
n
φ
(
x
n
,
p
)
+
(
1
-
γ
n
)
φ
(
T
n
v
n
,
p
)
≤
γ
n
φ
(
x
n
,
p
)
+
(
1
-
γ
n
)
×
[
b
0
φ
(
v
n
,
p
)
+
b
1
φ
(
R
s
n
,
1
B
1
v
n
,
p
)
+
b
2
φ
(
R
s
n
,
2
B
2
R
s
n
,
1
B
1
v
n
,
p
)
+
⋯
+
b
M
φ
(
R
s
n
,
M
B
M
R
s
n
,
M
-
1
B
M
-
1
⋯
R
s
n
,
1
B
1
v
n
,
p
)
]
≤
γ
n
φ
(
x
n
,
p
)
+
(
1
-
γ
n
)
φ
(
v
n
,
p
)
≤
γ
n
φ
(
x
n
,
p
)
+
(
1
-
γ
n
)
×
[
(
1
-
β
n
)
φ
(
x
n
,
p
)
+
β
n
φ
(
S
n
u
n
,
p
)
]
≤
[
1
-
β
n
(
1
-
γ
n
)
]
φ
(
x
n
,
p
)
+
(
1
-
γ
n
)
β
n
[
a
0
φ
(
u
n
,
p
)
+
∑
i
=
1
N
‍
a
i
φ
(
R
r
n
,
i
A
i
u
n
,
p
)
]
≤
[
1
-
β
n
(
1
-
γ
n
)
]
φ
(
x
n
,
p
)
+
(
1
-
γ
n
)
β
n
φ
(
u
n
,
p
)
≤
[
1
-
α
n
β
n
(
1
-
γ
n
)
]
φ
(
x
n
,
p
)
+
(
1
-
γ
n
)
α
n
β
n
∥
p
∥
2
.
Lemma 10 ensures that li
m
n
→
∞
φ
(
x
n
,
p
) exists, which ensures that {
x
n
} is bounded.
Step 3. ω
(
x
n
)
⊂
D
~, where ω
(
x
n
) is the set of weak limit points of all of the weakly convergent subsequences of {
x
n
}.
Since{
x
n
} is bounded, then ω
(
x
n
)
≠
∅. So there exists a subsequence of {
x
n
}; for simplicity, we still denote it by {
x
n
} such that x
n
⇀
x
,
n
→
∞.
Using Lemma9 again, we have for ∀
p
∈
D
~
(31)
φ
(
x
n
+
1
,
p
)
≤
γ
n
φ
(
x
n
,
p
)
+
(
1
-
γ
n
)
φ
(
T
n
v
n
,
p
)
≤
γ
n
φ
(
x
n
,
p
)
+
(
1
-
γ
n
)
×
[
(
b
0
+
b
1
+
⋯
+
b
M
-
1
)
φ
(
v
n
,
p
)
+
b
M
φ
(
R
s
n
,
M
B
M
R
s
n
,
M
-
1
B
M
-
1
⋯
R
s
n
,
1
B
1
v
n
,
p
)
]
≤
γ
n
φ
(
x
n
,
p
)
+
(
1
-
γ
n
)
(
1
-
b
M
)
φ
(
v
n
,
p
)
+
b
M
(
1
-
γ
n
)
φ
(
R
s
n
,
M
-
1
B
M
-
1
R
s
n
,
M
-
2
B
M
-
2
⋯
R
s
n
,
1
B
1
v
n
,
p
)
-
b
M
(
1
-
γ
n
)
×
φ
(
R
s
n
,
M
-
1
B
M
-
1
R
s
n
,
M
-
2
B
M
-
2
⋯
R
s
n
,
1
B
1
v
n
,
R
s
n
,
M
B
M
R
s
n
,
M
-
1
B
M
-
1
⋯
R
s
n
,
1
B
1
v
n
)
≤
γ
n
φ
(
x
n
,
p
)
+
(
1
-
γ
n
)
×
[
(
1
-
β
n
)
φ
(
x
n
,
p
)
+
β
n
φ
(
S
n
u
n
,
p
)
]
-
b
M
(
1
-
γ
n
)
×
φ
(
R
s
n
,
M
-
1
B
M
-
1
R
s
n
,
M
-
2
B
M
-
2
⋯
R
s
n
,
1
B
1
v
n
,
R
s
n
,
M
B
M
R
s
n
,
M
-
1
B
M
-
1
⋯
R
s
n
,
1
B
1
v
n
)
≤
[
1
-
α
n
β
n
(
1
-
γ
n
)
]
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
α
n
β
n
∥
p
∥
2
-
b
M
(
1
-
γ
n
)
×
φ
(
R
s
n
,
M
-
1
B
M
-
1
R
s
n
,
M
-
2
B
M
-
2
⋯
R
s
n
,
1
B
1
v
n
,
R
s
n
,
M
B
M
R
s
n
,
M
-
1
B
M
-
1
⋯
R
s
n
,
1
B
1
v
n
)
.
Then (31) implies that
(32)
b
M
(
1
-
γ
n
)
φ
(
R
s
n
,
M
-
1
B
M
-
1
R
s
n
,
M
-
2
B
M
-
2
⋯
R
s
n
,
1
B
1
v
n
,
R
s
n
,
M
B
M
R
s
n
,
M
-
1
B
M
-
1
⋯
R
s
n
,
1
B
1
v
n
)
≤
[
1
-
α
n
β
n
(
1
-
γ
n
)
]
φ
(
x
n
,
p
)
+
(
1
-
γ
n
)
α
n
β
n
∥
p
∥
2
-
φ
(
x
n
+
1
,
p
)
.
Similar to the discussion of (17) in Step 2 in Theorem 11, we have
(33)
R
s
n
,
M
B
M
R
s
n
,
M
-
1
B
M
-
1
⋯
R
s
n
,
1
B
1
v
n
-
R
s
n
,
M
-
1
B
M
-
1
R
s
n
,
M
-
2
B
M
-
2
⋯
R
s
n
,
1
B
1
v
n
⟶
0
,
as n
→
∞.
Then, similar to the discussions of (19) and (20), we have
(34)
R
s
n
,
M
-
1
B
M
-
1
R
s
n
,
M
-
2
B
M
-
2
⋯
R
s
n
,
1
B
1
v
n
-
R
s
n
,
M
-
2
B
M
-
2
R
s
n
,
M
-
3
B
M
-
3
⋯
R
s
n
,
1
B
1
v
n
⟶
0
,
R
s
n
,
M
-
2
B
M
-
2
R
s
n
,
M
-
3
B
M
-
3
⋯
R
s
n
,
1
B
1
v
n
-
R
s
n
,
M
-
3
B
M
-
3
R
s
n
,
M
-
4
B
M
-
4
⋯
R
s
n
,
1
B
1
v
n
⟶
0
,
⋮
R
s
n
,
1
B
1
v
n
-
v
n
⟶
0
.
On the other hand, noticing (30) and using Lemma 9, we have
(35)
φ
(
x
n
+
1
,
p
)
≤
[
1
-
β
n
(
1
-
γ
n
)
]
φ
(
x
n
,
p
)
+
(
1
-
γ
n
)
β
n
[
a
0
φ
(
u
n
,
p
)
+
∑
i
=
1
N
‍
a
i
φ
(
R
r
n
,
i
A
i
u
n
,
p
)
]
≤
[
1
-
β
n
(
1
-
γ
n
)
]
φ
(
x
n
,
p
)
+
(
1
-
γ
n
)
×
β
n
{
a
0
φ
(
u
n
,
p
)
+
∑
i
=
1
N
‍
a
i
[
φ
(
u
n
,
p
)
-
φ
(
u
n
,
R
r
n
,
i
A
i
u
n
)
]
}
≤
[
1
-
α
n
β
n
(
1
-
γ
n
)
]
φ
(
p
,
x
n
)
+
(
1
-
γ
n
)
α
n
β
n
∥
p
∥
2
-
(
1
-
γ
n
)
β
n
∑
i
=
1
N
‍
a
i
φ
(
u
n
,
R
r
n
,
i
A
i
u
n
)
,
which implies that
(36)
R
r
n
,
i
A
i
u
n
-
u
n
⟶
0
,
as n
→
∞,  i
=
1,2
,
…
,
N.
Sinceu
n
=
(
1
-
α
n
)
x
n and u
n
⇀
x, then u
n
⇀
x, as n
→
∞. Now, (36) implies that R
r
n
,
i
A
i
u
n
⇀
x. Let z
i
*
∈
A
i
z
i, then
(37)
〈
z
i
*
-
u
n
-
R
r
n
,
i
A
i
u
n
r
n
,
i
,
z
i
-
J
R
r
n
,
i
A
i
u
n
〉
≥
0
.
Since J is weakly sequentially continuous, then, letting n
→
∞, (37) ensures that
(38)
〈
z
i
*
,
z
i
-
J
x
〉
≥
0
.
Since A
i is maximal monotone, then J
x
∈
A
i
-
1
0, which implies that x
∈
⋂
i
=
1
N
‍
(
A
i
J
)
-
1
0.
FromR
r
n
,
i
A
i
u
n
⇀
x, we have S
n
u
n
⇀
x, as n
→
∞. Since v
n
=
(
1
-
β
n
)
x
n
+
β
n
S
n
u
n, then v
n
⇀
x, as n
→
∞. Using (34), we have R
s
n
,
1
B
1
v
n
⇀
x
,
R
s
n
,
2
B
2
R
s
n
,
1
B
1
v
n
⇀
x
,
…
,
R
s
n
,
M
B
M
R
s
n
,
M
-
1
B
M
-
1
⋯
R
s
n
,
1
B
1
v
n
⇀
x, as n
→
∞. By induction, similar to the proof of x
∈
(
A
i
J
)
-
1
0, we know that x
∈
⋂
j
=
1
M
‍
(
B
j
J
)
-
1
0.
Therefore,x
∈
D
~, and then ω
(
x
n
)
⊂
D
~.
Step 4. x
n
⇀
v
0, as n
→
∞, where v
0 is the unique element in D
~.
From Steps 2 and 3, we know that there exists a subsequence{
x
n
i
} of {
x
n
} such that x
n
i
⇀
v
0
∈
D
~, as i
→
∞. If there exists another subsequence {
x
n
j
} of {
x
n
} such that x
n
j
⇀
v
1
∈
D
~, as j
→
∞, then from Step 1, we know that
(39)
lim
⁡
n
→
∞
⁡
[
φ
(
x
n
,
v
0
)
-
φ
(
x
n
,
v
1
)
]
=
lim
⁡
i
→
∞
⁡
[
φ
(
x
n
i
,
v
0
)
-
φ
(
x
n
i
,
v
1
)
]
=
lim
⁡
i
→
∞
⁡
[
∥
v
0
∥
2
-
∥
v
1
∥
2
+
2
〈
x
n
i
,
J
v
1
-
J
v
0
〉
]
=
∥
v
0
∥
2
-
∥
v
1
∥
2
+
2
〈
v
0
,
J
v
1
-
J
v
0
〉
.
Similarly,
(40)
lim
⁡
n
→
∞
⁡
[
φ
(
x
n
,
v
0
)
-
φ
(
x
n
,
v
1
)
]
=
lim
⁡
j
→
∞
⁡
[
φ
(
x
n
j
,
v
0
)
-
φ
(
x
n
j
,
v
1
)
]
=
lim
⁡
i
→
∞
⁡
[
∥
v
0
∥
2
-
∥
v
1
∥
2
+
2
〈
x
n
j
,
J
v
1
-
J
v
0
〉
]
=
∥
v
0
∥
2
-
∥
v
1
∥
2
+
2
〈
v
1
,
J
v
1
-
J
v
0
〉
.
From (39) and (40), we have 〈
v
1
-
v
0
,
J
v
1
-
J
v
0
〉
=
0, which implies that v
0
=
v
1.
This completes the proof.Remark 15.
If, in Theorem14, the Banach space E reduces to the Hilbert space H, then the result of Theorem 12 is still true. That is, Theorems 14 and 11 are the same in the frame of Hilbert spaces.Remark 16.
Next, we will present an example to show that the assumptions thatD
≠
∅ in Theorem 11 and D
~
≠
∅ in Theorem 14 are meaningful.
Consider the followingp
i
-
Laplacian Dirichlet boundary value problem:
(
D
)
-
div
⁡
(
|
∇
u
|
p
i
-
2
∇
u
)
=
f
(
x
)
,
a
.
e
.
x
∈
Ω
;
ν
u
=
w
,
a
.
e
.
x
∈
Γ
,
where Ω is a bounded conical domain of the Euclidean space R
N
(
N
≥
1
) with its boundary Γ
∈
C
1,  2
N
/
(
N
+
1
)
<
p
i
<
+
∞,  f
∈
W
1
,
p
i
(
Ω
), and w
∈
W
1
-
1
/
p
i
,
p
i
(
Γ
) are given functions, where W
1
-
1
/
p
i
,
p
i
(
Γ
) is the trace space of W
1
,
p
i
(
Ω
), ν
:
W
1
,
p
i
(
Ω
)
→
W
1
-
1
/
p
i
,
p
i
(
Γ
) is the trace operator, i
=
1,2
,
…
,
M.Similar to the discussion in [11], we have the following results.Proposition 17 (see [11]).
Define the mappingC
p
i
:
W
1
,
p
i
(
Ω
)
→
(
W
1
,
p
i
(
Ω
)
)
* by
(41)
(
v
,
C
p
i
u
)
=
∫
Ω
‍
〈
|
∇
u
|
p
i
-
2
∇
u
,
∇
v
〉
d
x
-
∫
Ω
‍
f
v
d
x
,
for ∀
u
,
v
∈
W
1
,
p
i
(
Ω
). Define the mapping B
p
i
:
W
1
,
p
i
(
Ω
)
→
(
W
1
,
p
i
(
Ω
)
)
* by
(42)
(
v
,
B
p
i
u
)
=
∫
Ω
‍
〈
|
∇
u
|
p
i
-
2
∇
u
,
∇
v
〉
d
x
,
for ∀
u
,
v
∈
W
1
,
p
i
(
Ω
).
Then bothB
p
i and C
p
i are maximal monotone, for each i
=
1,2
,
…
,
M.Proposition 18 (see [11]).
Forf
∈
W
1
,
p
i
(
Ω
) and w
∈
W
1
-
1
/
p
i
,
p
i
(
Γ
), nonlinear boundary problem (
D
) has a unique solution u
(
x
)
∈
W
1
,
p
i
(
Ω
), for each i
=
1,2
,
…
,
M. Moreover, u
(
x
)
∈
C
p
i
-
1
(
0
) if and only if u
(
x
) is the solution of (
D
), which ensures that C
p
i
-
1
(
0
)
≠
∅, for each i
=
1,2
,
…
,
M.We can easily get the following result.Proposition 19.
B
p
i
-
1
(
0
)
=
{
u
(
x
)
∈
W
1
,
p
i
(
Ω
)
:
u
(
x
)
≡
C
o
n
s
t
a
n
t
}, for each i
=
1,2
,
…
,
M, which ensures that ⋂
i
=
1
M
‍
B
p
i
-
1
(
0
)
≠
∅. And, ⋂
i
=
1
M
‍
C
p
i
-
1
(
0
)
≠
∅, which means the following nonlinear (
p
1
,
p
2
,
…
,
p
M
)
-Laplacian elliptic systems (
E
) with Dirichlet boundary share the same solution:
(
E
)
-
div
⁡
(
|
∇
u
|
p
1
-
2
∇
u
)
=
f
(
x
)
,
a
.
e
.
x
∈
Ω
;
-
div
⁡
(
|
∇
u
|
p
2
-
2
∇
u
)
=
f
(
x
)
,
a
.
e
.
x
∈
Ω
;
⋮
-
div
⁡
(
|
∇
u
|
p
M
-
2
∇
u
)
=
f
(
x
)
,
a
.
e
.
x
∈
Ω
;
ν
u
=
w
,
a
.
e
.
x
∈
Γ
.



---

*Source: 451279-2014-04-14.xml*