# Projection Methods for a System of Nonlinear Mixed Variational Inequalities in Banach Spaces

Authors: Zhong-Bao Wang; Guo-Ji Tang; Hong-Ling Zhang
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

The existence and uniqueness of solution for a system of nonlinear mixed variational inequality in Banach spaces is given firstly. A Mann iterative sequences with errors for this system of nonlinear mixed variational inequalities in Banach spaces is studied, by using the generalizedf-projection operator π
K
f. Our results extend the main results in (Verma (2005); Verma (2001)) from Hilbert spaces to Banach spaces.


---

## Body

## 1. Introduction

Throughout this paper, unless otherwise stated, we letB be a real Banach space with the norm ∥
·
∥, we let B
* be the topological dual space of B, we let 〈
·
,
·
〉 be the duality between B, and we let B
* and K
⊂
B be a nonempty closed convex set.LetA
1
,
A
2
,
S
1
,
S
2
:
K
→
B
* be single-valued mappings. Let J
:
B
→
B
* be the normalized duality mapping, and let f
:
K
⊂
B
→
R
∪
{
+
∞
} be a proper lower semicontinuous mapping. We will consider the following system of nonlinear mixed variational inequalities problems (SNMVIP): find (
x
*
,
y
*
)
∈
K
×
K satisfying
(1)
〈
ρ
1
A
1
(
y
*
)
+
ρ
1
S
1
(
y
*
)
+
J
(
x
*
)
-
J
(
y
*
)
,
x
-
x
*
〉
111111111
+
f
(
x
)
-
f
(
x
*
)
≥
0
,
∀
x
∈
K
,
〈
η
1
A
2
(
x
*
)
+
η
1
S
2
(
x
*
)
+
J
(
y
*
)
-
J
(
x
*
)
,
x
-
y
*
〉
11111111
+
f
(
x
)
-
f
(
y
*
)
≥
0
,
∀
x
∈
K
,
where ρ
1
>
0 and η
1
>
0.IfB
=
H is a real Hilbert space, S
1
=
S
2
=
S, A
1
=
A
2
=
0, and f
=
0, then SNMVIP (1) reduces to the following problem:
(2)
〈
ρ
1
S
(
y
*
)
+
x
*
-
y
*
,
x
-
x
*
〉
≥
0
,
∀
x
∈
K
,
〈
η
1
S
(
x
*
)
+
y
*
-
x
*
,
x
-
y
*
〉
≥
0
,
∀
x
∈
K
.
The problem (2) was introduced and studied by Verma [1, 2]. We note that for ρ
1
=
0 the problem (2) reduces to the nonlinear variational inequalities problem: determine an element x
*
∈
K such that
(3)
〈
S
(
x
*
)
,
x
-
y
*
〉
≥
0
,
∀
x
∈
K
.In brief, the system of nonlinear mixed variational inequalities (1) is more general and includes many systems of variational inequalities and variational inequalities as special cases.It is well known that the projection method and its variant forms have represented an important tool for solving variational inequalities and the system of variational inequalities; see [1–21] and the references therein. Wu and Huang [10] introduced and studied a new class of generalized f-projection operators in Banach spaces, which extends the definition of the generalized projection operators introduced and studied by Alber [3, 4] and Li [7]. Some properties of the generalized f-projection operator are given in [10]. By using the generalized projection operators and the generalized f-projection operators, some authors studied the existence theorems of solution and some iterative algorithms of approximating solutions for variational inequalities; see [3, 4, 6–8, 10–12].In addition, Verma [1, 2] applies the metric projection operator technique and the fixed theorem to suggest some iterative algorithms for the system of variational inequalities with the monotone mappings in Hilbert spaces. By using the sunny nonexpansive retraction, Yao et al. [21] prove that the suggested two-step projection method converges strongly to the solution of a system of variational inequality, which extend the main results in Verma [1] from Hilbert spaces to Banach spaces. Recently, Wang et al. show that the Lipschitz continuity of the generalized f-projection operator and the normalized duality mapping and study a class of system of generalized mixed variational inequalities in Banach spaces.Motivated and inspired by the research work going on this field, in this paper, we firstly show the existence and uniqueness of solution for a system of nonlinear mixed variational inequality in Banach spaces. By using the generalizedf-projection operator π
K
f, we study Mann iterative sequences with errors for this system of nonlinear mixed variational inequalities in Banach spaces. Our results extend and improve the main results in Verma [1, 2] from Hilbert spaces to Banach spaces.


## 2. Preliminaries

Recall thatB is said to be strictly convex if ∥
(
x
+
y
)
/
2
∥
<
1 for all x, y
∈
B with ∥
x
∥
=
∥
y
∥
=
1 and x
≠
y. It is also said to be uniformly convex if lim
⁡
n
→
∞
∥
x
n
-
y
n
∥
=
0 for any two sequences {
x
n
}, {
y
n
} in B with ∥
x
n
∥
=
∥
y
n
∥
=
1 and lim
⁡
n
→
∞
∥
(
x
n
+
y
n
)
/
2
∥
=
1. The function
(4)
δ
B
(
ϵ
)
=
inf
⁡
⁡
{
1
-
∥
x
+
y
∥
2
:
∥
x
∥
=
1
,
∥
y
∥
=
1
,
∥
x
-
y
∥
≥
ϵ
}
is called the modulus of convexity of B. It is well known that B is 2-uniformly convex if and only if there exists a constant k
B
>
0 such that δ
(
ϵ
)
≥
k
B
ϵ
2 for all ϵ
∈
(
0,2
]. Recall that B is said to be smooth if lim
⁡
t
→
0
(
∥
x
+
t
y
∥
-
∥
x
∥
)
/
t exists for all x
,
y
∈
B with ∥
x
∥
=
∥
y
∥
=
1. It is also said to be uniformly smooth if the limit is attained uniformly for ∥
x
∥
=
∥
y
∥
=
1. The function
(5)
ρ
B
(
t
)
=
sup
⁡
⁡
{
∥
x
+
y
∥
+
∥
x
-
y
∥
2
-
1
:
∥
x
∥
=
1
,
∥
y
∥
≤
t
}
is called the modulus of smoothness of B. It is known that B is said to be 2-uniformly smooth if and only if there exists a constant c
B
>
0 such that ρ
B
(
t
)
≤
c
B
t
2. Moreover, B
* is uniformly convex if and only if B is uniformly smooth. In this case, B is reflexive by the Milman theorem.Next we recall the concept of the normalized duality mapping. The normalized duality mappingJ
:
B
→
2
B
* is defined by
(6)
J
(
x
)
=
{
∥
j
(
x
)
∥
2
j
(
x
)
∈
B
*
:
〈
j
(
x
)
,
x
〉
=
∥
j
(
x
)
∥
∥
x
∥
=
∥
x
∥
2
=
∥
j
(
x
)
∥
2
}
,
∀
x
∈
B
.
Many properties of the normalized duality mapping J can be found in [18].We list some properties ofJ as follows:(
a
1
)
ifB is a smooth Banach space, then J is single-valued and continuous from the strong topology of B to the weak * topology of B
*;
(
a
2
)
J is the identity operator in Hilbert spaces;
(
a
3
)
ifB is a reflexive, smooth, and strictly convex Banach space, and J
*
:
B
*
→
B is the normalized duality mapping on B
*, then J
-
1
=
J
*, J
J
*
=
I
B
* and J
*
J
=
I
B.For any fixedρ
>
0, let G
:
B
*
×
K
→
(
-
∞
,
+
∞
] be a function defined as follows:
(7)
G
(
φ
,
x
)
=
∥
x
∥
2
-
2
〈
φ
,
x
〉
+
∥
φ
∥
2
+
2
ρ
f
(
x
)
,
11111111111
∀
φ
∈
B
*
,
∀
x
∈
K
,
where φ
∈
B
*, x
∈
K and f
:
K
⊂
B
→
R
∪
{
+
∞
} is proper, convex, and lower semicontinuous. It is easy to see that G
(
φ
,
x
)
≥
(
∥
x
∥
-
∥
φ
∥
)
2
+
2
ρ
f
(
x
) for all φ
∈
B
* and x
∈
K.Definition 1.
We say thatB has the property (
h
) if x
n
⇀
x weakly and ∥
x
n
∥
→
∥
x
∥ implies x
n
→
x.Remark 2.
It is well known that any locally uniformly convex Banach space has the property(
h
) and B
* has a Fréchet differentiable norm if and only if B is reflexive and strictly convex and has the property (
h
) (see, e.g., [19]).Definition 3 (see [10]).
We say thatπ
K
f
:
B
*
→
2
K is a generalized f-projection operator if
(8)
π
K
f
φ
=
{
u
∈
K
:
G
(
φ
,
u
)
=
inf
⁡
y
∈
K
⁡
G
(
φ
,
y
)
}
,
∀
φ
∈
B
*
.Remark 4.
(i) Iff
(
x
)
=
0 for all x
∈
K, then the generalized f-projection operator π
K
f reduces to the generalized projection operator π
k defined by Alber [4] and Li [7]; that is,
(9)
π
K
φ
=
{
u
∈
K
:
G
1
(
φ
,
u
)
=
inf
⁡
y
∈
K
⁡
G
1
(
φ
,
y
)
}
,
∀
φ
∈
B
*
,
where G
1
(
φ
,
x
)
=
∥
x
∥
2
-
2
〈
φ
,
x
〉
+
∥
φ
∥
2 for all φ
∈
B
* and x
∈
K.
(ii) Iff
(
x
)
=
0 for all x
∈
K and B
=
H is a Hilbert space, then the generalized f-projection operator π
K
f is equivalent to the following metric projection operator:
(10)
P
K
(
φ
)
=
{
u
∈
K
:
∥
u
-
φ
∥
=
inf
⁡
y
∈
K
⁡
∥
y
-
φ
∥
}
,
∀
φ
∈
H
.Definition 5.
LetB be a reflexive, smooth, and strictly convex Banach space with the dual space B
*, and 2
B
* denote the family of all nonempty subsets of B
*. T
:
B
→
2
B
* is said to be J-strongly monotone if there exists a constant S
>
0 satisfying
(11)
〈
u
-
v
,
J
*
(
J
(
x
)
-
J
(
y
)
)
〉
≥
S
∥
x
-
y
∥
2
,
1111111111
∀
u
∈
T
(
x
)
,
∀
v
∈
T
(
y
)
.Remark 6.
IfB
=
H is a Hilbert space, then J-strongly monotonicity reduce to strongly monotonicity in [15].Theorem 77 (see [6, 10]).
IfB is a reflexive Banach space with dual space B
* and K is a nonempty closed convex subset of B, then the following conclusions hold:(i)
for any givenφ
∈
B
*, π
K
f
φ is a nonempty, closed, and convex subset of K;
(ii)
if B is smooth, then for any givenφ
∈
B
*, x
∈
π
K
f
φ if and only if
(12)
〈
φ
-
J
(
x
)
,
x
-
y
〉
+
ρ
f
(
y
)
-
ρ
f
(
x
)
≥
0
,
∀
y
∈
K
;
(iii)
ifB is strictly convex, then the operator π
K
f
:
B
*
→
K is single-valued.Lemma 8 (see [13]).
B is 2-uniformly smooth if and only if there exists a constant C
B
>
0 such that, for all x
,
y
∈
B,
(13)
∥
x
+
y
∥
2
≤
∥
x
∥
2
+
2
〈
y
,
J
x
〉
+
C
B
∥
y
∥
2
.Lemma 99 (see [20]).
Let{
a
n
}, {
b
n
}, and {
c
n
} be three sequences of nonnegative numbers satisfying the following conditions: there exists n
0 such that
(14)
a
n
+
1
≤
(
1
-
t
n
)
a
n
+
b
n
t
n
+
c
n
,
∀
n
≥
n
0
,
where
(15)
t
n
∈
[
0,1
]
,
∑
n
=
0
∞
‍
t
n
=
+
∞
;
lim
⁡
n
→
∞
b
n
=
0
,
∑
n
=
0
∞
‍
c
n
<
+
∞
.
Then a
n
→
0 as n
→
+
∞.Proposition 1010 (see [14]).
LetB be a reflexive, smooth, and strictly convex Banach space with the dual space B
*. Then for any x
,
y
∈
B,
(16)
〈
J
x
-
J
y
,
x
-
y
〉
≥
2
C
1
2
δ
B
(
∥
x
-
y
∥
2
C
1
)
,
where C
1
=
(
∥
x
∥
2
+
∥
y
∥
2
)
/
2.Proposition 1111 (see [14]).
LetB be a real uniformly convex and uniformly smooth Banach space. Then
(17)
∥
J
x
-
J
y
∥
≤
8
M
2
h
B
(
4
∥
x
-
y
∥
M
2
)
,
∀
x
,
y
∈
B
,
where M
2
=
(
∥
x
∥
2
+
∥
y
∥
2
)
/
2 and h
B
(
τ
)
=
ρ
B
(
τ
)
/
τ.Proposition 12.
LetB be a reflexive, smooth, and strictly convex Banach space with the dual space B
*, let B have the property (h), and let K be a nonempty closed convex subset of B. For any x
1
*
,
x
2
*
∈
B
*, let x
1
=
π
K
f
x
1
* and x
2
=
π
K
f
x
2
*. Then
(18)
〈
x
1
*
-
x
2
*
,
x
1
-
x
2
〉
≥
2
C
1
2
δ
B
(
∥
x
1
-
x
2
∥
2
C
1
)
,
where C
1
=
(
∥
x
1
∥
2
+
∥
x
2
∥
2
)
/
2.Proof.
According to Proposition10,
(19)
〈
J
x
1
-
J
x
2
,
x
1
-
x
2
〉
≥
2
C
1
2
δ
B
(
∥
x
1
-
x
2
∥
2
C
1
)
,
where C
1
=
(
∥
x
1
∥
2
+
∥
x
2
∥
2
)
/
2. Since x
1
=
π
K
f
x
1
* and x
2
=
π
K
f
x
2
*, Theorem 7 yields
(20)
〈
J
x
1
-
x
1
*
,
x
2
-
x
1
〉
+
ρ
f
(
x
2
)
-
ρ
f
(
x
1
)
≥
0
,
〈
J
x
2
-
x
2
*
,
x
1
-
x
2
〉
+
ρ
f
(
x
1
)
-
ρ
f
(
x
2
)
≥
0
.
It follows from (20) that
(21)
〈
J
x
1
-
x
1
*
,
x
2
-
x
1
〉
+
〈
J
x
2
-
x
2
*
,
x
1
-
x
2
〉
≥
0
and so
(22)
〈
x
1
*
-
x
2
*
,
x
1
-
x
2
〉
≥
〈
J
x
1
-
J
x
2
,
x
1
-
x
2
〉
.
By (19), we have
(23)
〈
x
1
*
-
x
2
*
,
x
1
-
x
2
〉
≥
2
C
1
2
δ
B
(
∥
x
1
-
x
2
∥
2
C
1
)
,
where C
1
=
(
∥
x
1
∥
2
+
∥
x
2
∥
2
)
/
2. This completes the proof.Theorem 13.
LetB be a reflexive, smooth, and 2-uniformly convex Banach space with the dual space B
* and δ
(
ϵ
)
≥
k
B
ϵ
2
,
∀
ϵ
∈
(
0,2
]for some k
B
>
0. Then there exist a constant K
B
=
(
2
/
k
B
) such that
(24)
∥
π
K
f
x
1
*
-
π
K
f
x
2
*
∥
≤
K
B
∥
x
1
*
-
x
2
*
∥
,
∀
x
1
*
,
x
2
*
∈
B
*
.Proof.
From Proposition12, it follows that
(25)
〈
x
1
*
-
x
2
*
,
π
K
f
x
1
*
-
π
K
f
x
2
*
〉
≥
2
C
1
2
δ
(
∥
π
K
f
x
1
*
-
π
K
f
x
2
*
∥
2
C
1
)
,
where C
1
=
(
∥
π
K
f
x
1
*
∥
2
+
∥
π
K
f
x
2
*
∥
2
)
/
2. Since δ
B
(
ϵ
)
≥
k
B
ϵ
2, (25) implies that
(26)
〈
x
1
*
-
x
2
*
,
π
K
f
x
1
*
-
π
K
f
x
2
*
〉
≥
2
C
1
2
δ
B
(
∥
π
K
f
x
1
*
-
π
K
f
x
2
*
∥
2
C
1
)
≥
k
B
2
∥
π
K
f
x
1
*
-
π
K
f
x
2
*
∥
2
,
and so
(27)
∥
π
K
f
x
1
*
-
π
K
f
x
2
*
∥
≤
2
k
B
∥
x
1
*
-
x
2
*
∥
=
K
B
∥
x
1
*
-
x
2
*
∥
.
This completes the proof.Theorem 14.
LetB be a real 2-uniformly smooth and uniformly convex Banach space with the dual space B
* and ρ
B
(
t
)
≤
c
B
t
2 for some c
B
>
0. Then there exist a constant C
=
32
c
B
>
0 such that
(28)
∥
J
x
-
J
y
∥
≤
C
∥
x
-
y
∥
,
∀
x
,
y
∈
B
.Proof.
From Proposition11, it follows that
(29)
∥
J
x
-
J
y
∥
≤
8
M
2
h
B
(
4
∥
x
-
y
∥
M
2
)
,
x
,
y
∈
B
,
where M
2
=
(
∥
x
∥
2
+
∥
y
∥
2
)
/
2 and h
B
(
τ
)
=
ρ
B
(
τ
)
/
τ. Since ρ
B
(
t
)
≤
c
B
t
2,
(30)
∥
J
x
-
J
y
∥
≤
8
M
2
h
B
(
4
∥
x
-
y
∥
M
2
)
≤
32
c
B
∥
x
-
y
∥
=
C
∥
x
-
y
∥
.
This completes the proof.


## 3. Main Results

From Theorem7, we know that the following theorem holds.Theorem 15.
LetB be a 2-uniformly smooth and 2-uniformly convex Banach space. (
x
*
,
y
*
) is a solution of the S
N
M
V
I
P (1) if and only if (
x
*
,
y
*
)
∈
K
×
K satisfying
(31)
x
*
=
π
K
f
[
J
(
y
*
)
-
(
ρ
1
A
1
(
y
*
)
+
ρ
1
S
1
(
y
*
)
)
]
,
y
*
=
π
K
f
[
J
(
x
*
)
-
(
η
1
A
2
(
x
*
)
+
η
1
S
2
(
x
*
)
)
]
.In the sequel, we shall show the existence and uniqueness of solution for the problems (1), (2), and (3), respectively. Next, we construct some new iterative algorithms for the problems (1), (2), and (3). We also give the convergence analysis of the iterative sequences generated by the algorithms.Theorem 16.
LetB be a 2-uniformly smooth and 2-uniformly convex Banach space with ρ
B
(
t
)
≤
c
B
t
2 for some c
B
>
0 and δ
(
ϵ
)
≥
k
B
ϵ
2
,
∀
ϵ
∈
(
0,2
] for some k
B
>
0. Let A
1
,
A
2
,
S
1
,
S
2
:
K
→
B
* be single-valued mappings. Let J
:
B
→
B
* and J
*
:
B
*
→
B be the normalized duality mapping, and let f
:
K
⊂
B
→
R
∪
{
+
∞
} be a proper lower semicontinuous mapping. Suppose that the following conditions are satisfied: (i)
for eachi
∈
{
1,2
}, S
i is J-strongly monotone with constants k
i and s
i-Lipschitz continuous;
(ii)
for eachi
∈
{
1,2
}, A
i is l
i-Lipschitz continuous; and that there exist constants ρ
1
,
η
1
>
0 satisfying
(32)
K
B
=
2
k
B
,
C
=
32
c
B
,
Θ
1
=
K
B
(
C
2
-
2
η
1
k
2
+
C
B
η
1
2
s
2
2
+
η
1
l
2
)
<
1
,
Θ
2
=
K
B
(
C
2
-
2
ρ
1
k
1
+
ρ
1
2
s
1
2
C
B
+
ρ
1
l
1
)
<
1
.
Then SNMVIP (1) has a unique solution (
x
*
,
y
*
).Proof.
First, we prove the existence of the solution. Define a mappingF
:
K
→
K as follows:
(33)
F
(
x
)
=
π
K
f
[
J
(
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
)
-
ρ
1
(
A
1
+
S
1
)
11111111
×
(
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
)
]
.
For any x
,
y
∈
K, by Theorem 13, we know that there exists K
B
=
(
2
/
k
B
)
>
0 such that
(34)
∥
F
(
x
)
-
F
(
y
)
∥
=
∥
π
K
f
[
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
)
]
J
(
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
)
1111111
-
ρ
1
(
A
1
+
S
1
)
(
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
)
]
111
-
π
K
f
[
J
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
11111111
-
ρ
1
(
A
1
+
S
1
)
111111111
×
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
]
∥
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
)
]
∥
.
≤
K
B
∥
{
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
)
}
[
J
(
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
)
1111111
-
J
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
1111111
-
ρ
1
(
S
1
(
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
)
11111111111
-
S
1
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
)
]
∥
+
ρ
1
∥
A
1
(
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
)
1111111
-
A
1
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
∥
.
Theorem 14 implies that there exists C
=
32
c
B
>
0 such that
(35)
∥
J
(
x
)
-
J
(
y
)
∥
≤
C
∥
x
-
y
∥
,
∀
x
,
y
∈
K
.
From condition (i) and Lemma 8, it follows that
(36)
∥
{
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
)
}
J
(
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
)
-
J
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
-
ρ
1
(
S
1
(
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
)
11111111
-
S
1
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
)
∥
2
=
∥
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
J
(
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
)
-
J
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
∥
2
+
ρ
1
2
C
B
∥
S
1
(
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
)
1111111111
-
S
1
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
∥
2
-
2
ρ
1
〈
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
)
S
1
(
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
)
11111111
-
S
1
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
,
11111111
J
*
(
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
)
〉
J
(
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
)
11111111
-
J
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
)
〉
≤
(
C
2
-
2
ρ
1
k
1
+
ρ
1
2
s
1
2
C
B
)
×
∥
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
111111
-
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
∥
2
.
By condition (ii) and Theorem 13, we know that
(37)
∥
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
-
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
∥
.
≤
K
B
[
∥
J
(
x
)
-
J
(
y
)
-
η
1
(
S
2
(
x
)
-
S
2
(
y
)
)
∥
1111111
+
η
1
∥
A
2
(
x
)
-
A
2
(
y
)
∥
]
≤
K
B
[
∥
J
(
x
)
-
J
(
y
)
-
η
1
(
S
2
(
x
)
-
S
2
(
y
)
)
∥
1111111
+
η
1
l
2
∥
x
-
y
∥
]
.
By (35), condition (i), and Lemma 8, we have
(38)
∥
J
(
x
)
-
J
(
y
)
-
η
1
(
S
2
(
x
)
-
S
2
(
y
)
)
∥
2
≤
∥
J
(
x
)
-
J
(
y
)
∥
2
-
2
η
1
×
〈
S
2
(
x
)
-
S
2
(
y
)
,
J
*
(
J
(
x
)
-
J
(
y
)
)
〉
+
C
B
η
1
2
∥
S
2
(
x
)
-
S
2
(
y
)
∥
2
≤
C
2
∥
x
-
y
∥
2
-
2
η
1
k
2
∥
x
-
y
∥
1
2
+
C
B
η
1
2
s
2
2
∥
x
-
y
∥
2
=
(
C
2
-
2
η
1
k
2
+
C
B
η
1
2
s
2
2
)
∥
x
-
y
∥
2
.
Thus
(39)
∥
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
-
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
∥
.
≤
K
B
[
∥
J
(
x
)
-
J
(
y
)
-
η
1
(
S
2
(
x
)
-
S
2
(
y
)
)
∥
111111
+
η
1
l
2
∥
x
-
y
∥
]
≤
K
B
(
C
2
-
2
η
1
k
2
+
C
B
η
1
2
s
2
2
+
η
1
l
2
)
∥
x
-
y
∥
.
Condition (ii) and (39) imply that
(40)
∥
A
1
(
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
)
-
A
1
(
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
)
∥
≤
l
1
∥
π
K
f
(
J
(
x
)
-
η
1
(
A
2
(
x
)
+
S
2
(
x
)
)
)
-
π
K
f
(
J
(
y
)
-
η
1
(
A
2
(
y
)
+
S
2
(
y
)
)
)
∥
≤
l
1
K
B
(
C
2
-
2
η
1
k
2
+
C
B
η
1
2
s
2
2
+
η
1
l
2
)
∥
x
-
y
∥
.
From (34)–(40), we have
(41)
∥
F
(
x
)
-
F
(
y
)
∥
≤
K
B
2
(
C
2
-
2
η
1
k
2
+
C
B
η
1
2
s
2
2
+
η
1
l
2
)
×
[
C
2
-
2
ρ
1
k
1
+
ρ
1
2
s
1
2
C
B
+
ρ
1
l
1
]
∥
x
-
y
∥
=
Θ
1
Θ
2
∥
x
-
y
∥
.
It follows from (32) that 0
<
Θ
1
<
1 and 0
<
Θ
2
<
1. Thus, (41) implies that F is a contractive mapping and so there exists a point x
*
∈
K such that x
*
=
F
(
x
*
). Let
(42)
y
*
=
π
K
f
[
J
(
x
*
)
-
(
η
1
A
2
(
x
*
)
+
η
1
S
2
(
x
*
)
)
]
.
From the definition of F, we have
(43)
x
*
=
π
K
f
[
J
(
y
*
)
-
(
ρ
1
A
1
(
y
*
)
+
ρ
1
S
1
(
y
*
)
)
]
,
y
*
=
π
K
f
[
J
(
x
*
)
-
(
η
1
A
2
(
x
*
)
+
η
1
S
2
(
x
*
)
)
]
.
By Theorem 15, we know that (
x
*
,
y
*
) is a solution of SNMVIP (1).
Next, we show the uniqueness of the solution. Let(
u
*
v
*
) be another solution of SNMVIP (1). It follows from Theorem 15 that
(44)
u
*
=
π
K
f
[
J
(
v
*
)
-
(
ρ
1
A
1
(
v
*
)
+
ρ
1
S
1
(
v
*
)
)
]
,
v
*
=
π
K
f
[
J
(
u
*
)
-
(
η
1
A
2
(
u
*
)
+
η
1
S
2
(
u
*
)
)
]
.
As the proof of (41), we have
(45)
∥
x
*
-
u
*
∥
≤
Θ
1
Θ
2
∥
x
*
-
u
*
∥
.
Since 0
<
Θ
1
<
1 and 0
<
Θ
2
<
1, it follows that u
*
=
x
* and so v
*
=
y
*. This completes the proof.It is clear that ifB is a Hilbert space, then we can take K
B
=
C
=
C
B
=
1 in Theorem 16. According to Theorem 16, it is easy to get the following corollaries.Corollary 17.
LetH be a real Hilbert space, let K
⊂
H be a nonempty closed convex set, and let S
:
K
→
H be strongly monotone with constants k and s-Lipschitz continuous. There exist ρ
1
,
η
1
>
0 satisfying
(46)
1
-
2
η
1
k
+
η
1
2
s
2
<
1
,
1
-
2
ρ
1
k
+
ρ
1
2
s
2
<
1
.
Then the problem (2) has a unique solution (
x
*
,
y
*
).Corollary 18.
LetH be a real Hilbert space, let K
⊂
H be a nonempty closed convex set, and let S
:
K
→
H be strongly monotone with constants k and s-Lipschitz continuous. There exists η
1
>
0 satisfying
(47)
1
-
2
η
1
k
+
η
1
2
s
2
<
1
.
Then the problem (3) has a unique solution (
x
*
,
y
*
).Algorithm 19.
For any givenx
0
∈
K, define the Mann iterative sequences {
x
n
} and {
y
n
} with errors as follows:
(48)
x
n
+
1
=
(
1
-
α
n
)
x
n
+
α
n
π
K
f
×
[
J
(
y
n
)
-
ρ
1
(
A
1
(
y
n
)
+
S
1
(
y
n
)
)
]
+
α
n
u
n
+
w
n
,
y
n
=
π
K
f
[
J
(
x
n
)
-
η
1
(
A
2
(
x
n
)
+
S
2
(
x
n
)
)
]
+
v
n
,
where {
α
n
} is a sequence in [
0,1
] and {
u
n
}, {
v
n
}, and {
w
n
} are three sequences in B satisfying the following conditions:
(49)
∑
n
=
0
∞
‍
α
n
=
+
∞
,
lim
⁡
n
→
∞
∑
n
=
0
∞
‍
∥
w
n
∥
<
+
∞
,
lim
⁡
n
→
∞
∥
u
n
∥
=
lim
⁡
n
→
∞
∥
v
n
∥
=
0
.IfB
=
H is a real Hilbert space, S
1
=
S
2
=
S, A
1
=
A
2
=
0, and f
=
0, then Algorithm 19 reduces to the following iterative algorithm.Algorithm 20 (see [2, Algorithm 2.1]).
For any givenx
0
∈
K, define the iterative sequences {
x
n
} and {
y
n
} as follows:
(50)
x
n
+
1
=
(
1
-
α
n
)
x
n
+
α
n
P
K
[
y
n
-
ρ
1
S
(
y
n
)
]
,
y
n
=
P
K
[
x
n
-
η
1
S
(
x
n
)
]
,
where {
α
n
} is a sequence in [
0,1
] satisfying the following conditions:
(51)
∑
n
=
0
∞
‍
α
n
=
+
∞
.IfB
=
H is a real Hilbert space, S
1
=
S
2
=
S, A
1
=
A
2
=
0, η
1
=
0, and f
=
0, then Algorithm 19 reduces to the following iterative algorithm.Algorithm 2121 (see [1, Algorithm 2.2]).
For any givenx
0
∈
K, define the iterative sequences {
x
n
} as follows:
(52)
x
n
+
1
=
(
1
-
α
n
)
x
n
+
α
n
P
K
[
x
n
-
η
1
S
(
x
n
)
]
,
where {
α
n
} is a sequence in [
0,1
] satisfying the following conditions:
(53)
∑
n
=
0
∞
‍
α
n
=
+
∞
.Theorem 22.
LetA
1, A
2, S
1, S
2, J, and J
* be the same as in Theorem 16, and let {
x
n
,
y
n
} be the iterative sequence generated by Algorithm 19. If the condition (32) holds, then (
x
n
,
y
n
) converges strongly to the unique solution (
x
*
;
y
*
) of SNMVIP (1).Proof.
By Theorem16, we know that SNMVIP (1) has a unique solution (
x
*
;
y
*
). It follows from Theorem 15 that
(54)
x
*
=
π
K
f
[
J
(
y
*
)
-
(
ρ
1
A
1
(
y
*
)
+
ρ
1
S
1
(
y
*
)
)
]
,
y
*
=
π
K
f
[
J
(
x
*
)
-
(
η
1
A
2
(
x
*
)
+
η
1
S
2
(
x
*
)
)
]
.
By Lipschitz property of A
1, Theorem 13, (48), and (54), we have
(55)
∥
x
n
+
1
-
x
*
∥
=
∥
(
1
-
α
n
)
x
n
+
α
n
π
K
f
×
[
J
(
y
n
)
-
ρ
1
(
A
1
(
y
n
)
+
S
1
(
y
n
)
)
]
11111
+
α
n
u
n
+
w
n
-
x
*
∥
(
1
-
α
n
)
x
n
+
α
n
π
K
f
∥
≤
(
1
-
α
n
)
∥
x
n
-
x
*
∥
+
α
n
∥
π
K
f
[
J
(
y
n
)
-
ρ
1
(
A
1
(
y
n
)
+
S
1
(
y
n
)
)
]
-
π
K
f
[
J
(
y
*
)
-
(
ρ
1
A
1
(
y
*
)
+
ρ
1
S
1
(
y
*
)
)
]
∥
+
α
n
∥
u
n
∥
+
∥
w
n
∥
≤
(
1
-
α
n
)
∥
x
n
-
x
*
∥
+
α
n
K
B
∥
J
(
y
n
)
-
J
(
y
*
)
1111111
-
ρ
1
(
S
1
(
y
n
)
-
S
1
(
y
*
)
)
1111111
-
ρ
1
(
A
1
(
y
n
)
-
A
1
(
y
*
)
)
∥
+
α
n
∥
u
n
∥
+
∥
w
n
∥
≤
(
1
-
α
n
)
∥
x
n
-
x
*
∥
+
α
n
K
B
×
∥
J
(
y
n
)
-
J
(
y
*
)
-
ρ
1
(
S
1
(
y
n
)
-
S
1
(
y
*
)
)
∥
+
K
B
ρ
1
l
1
∥
y
n
-
y
*
∥
+
α
n
∥
u
n
∥
+
∥
w
n
∥
.
Since S
1 is J-strongly monotone with constants k
1 and s
1-Lipschitz continuous, by Theorem 14 and Lemma 8, we obtain
(56)
∥
J
(
y
n
)
-
J
(
y
*
)
-
ρ
1
(
S
1
(
y
n
)
-
S
1
(
y
*
)
)
∥
≤
C
2
-
2
ρ
1
k
1
+
C
B
ρ
1
2
s
1
2
∥
y
n
-
y
*
∥
.
Combining (55) and (56), we obtain
(57)
∥
x
n
+
1
-
x
*
∥
≤
(
1
-
α
n
)
∥
x
n
-
x
*
∥
+
α
n
Θ
2
∥
y
n
-
y
*
∥
+
α
n
∥
u
n
∥
+
∥
w
n
∥
.
From Lipschitz property ofA
2, (48), and (54), it follows that
(58)
∥
y
n
-
y
*
∥
=
∥
π
K
f
[
J
(
x
n
)
-
η
1
(
A
2
(
x
n
)
+
S
2
(
x
n
)
)
]
+
v
n
-
π
K
f
[
J
(
x
*
)
-
(
η
1
A
2
(
x
*
)
+
η
1
S
2
(
x
*
)
)
]
∥
≤
K
B
∥
J
(
x
n
)
-
J
(
x
*
)
-
η
1
(
S
2
(
x
n
)
-
S
2
(
x
*
)
)
-
η
1
(
A
2
(
x
n
)
-
A
2
(
x
*
)
)
∥
+
∥
v
n
∥
≤
K
B
∥
J
(
x
n
)
-
J
(
x
*
)
-
η
1
(
S
2
(
x
n
)
-
S
2
(
x
*
)
)
∥
+
∥
v
n
∥
+
K
B
η
1
l
2
∥
x
n
-
x
*
∥
.
Since S
2 is J-strongly monotone with constants k
2 and s
2-Lipschitz continuous, Theorem 14 and Lemma 8 imply that
(59)
∥
J
(
x
n
)
-
J
(
x
*
)
-
η
1
(
S
2
(
x
n
)
-
S
2
(
x
*
)
)
∥
≤
C
2
-
2
η
1
k
2
+
C
B
η
1
2
s
2
2
∥
x
n
-
x
*
∥
.
It follows from (57)–(59) that
(60)
∥
x
n
+
1
-
x
*
∥
≤
(
1
-
α
n
)
∥
x
n
-
x
*
∥
+
α
n
Θ
2
Θ
1
∥
x
n
-
x
*
∥
+
α
n
Θ
2
∥
v
n
∥
+
α
n
∥
u
n
∥
+
∥
w
n
∥
=
(
1
-
α
n
(
1
-
Θ
2
Θ
1
)
)
∥
x
n
-
x
*
∥
+
∥
w
n
∥
+
α
n
(
1
-
Θ
2
Θ
1
)
1
(
1
-
Θ
2
Θ
1
)
(
Θ
2
∥
v
n
∥
+
∥
u
n
∥
)
.
Taking t
n
=
α
n
(
1
-
Θ
2
Θ
1
), a
n
=
∥
x
n
+
1
-
x
*
∥, b
n
=
1
/
(
1
-
Θ
2
Θ
1
)
(
Θ
2
∥
v
n
∥
+
∥
u
n
∥
), and c
n
=
∥
w
n
∥, Lemma 9 implies that a
n
→
0 and so x
n
→
x
* as n
→
∞. Since x
n
→
x
*, by (58) and (59), we know that y
n
→
y
* as n
→
∞. This completes the proof.Corollary 23 (see [2, Theorem 2.1]).
LetH, K, and S be the same as in Corollary 17, and let {
x
n
} and {
y
n
} be the iterative sequence generated by Algorithm 20. If there exist constants ρ
1
,
η
1
>
0 satisfying
(61)
1
-
2
η
1
k
+
η
1
2
s
2
<
1
,
1
-
2
ρ
1
k
+
ρ
1
2
s
2
<
1
,
then (
x
n
,
y
n
) converges strongly to the unique solution (
x
*
,
y
*
) of the problem (2).Corollary 24 (see [1, Theorem 3.3]).
LetH, K, and S be the same as in Corollary 18, and let {
x
n
} be the iterative sequence generated by Algorithm 21. If there exists η
1
>
0 satisfying
(62)
1
-
2
η
1
k
+
η
1
2
s
2
<
1
,
then x
n converges strongly to the unique solution x
* of the problem (3).



---

*Source: 606109-2014-05-15.xml*