# A System of Differential Set-Valued Variational Inequalities in Finite Dimensional Spaces

Authors: Wei Li; Xing Wang; Nan-Jing Huang
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

A system of differential set-valued variational inequalities is introduced and studied in finite dimensional Euclidean spaces. An existence theorem of weak solutions for the system of differential set-valued variational inequalities in the sense of Carathéodory is proved under some suitable conditions. Furthermore, a convergence result on Euler time-dependent procedure for solving the system of differential set-valued variational inequalities is also given.


---

## Body

## 1. Introduction

For a set-valued mappingF
:
R
n
⇉
R
n and a nonempty closed convex set K in R
n, the VI(
K
,
F
), is to find u
∈
K and u
*
∈
F
(
u
) such that 〈
u
*
,
u
′
-
u
〉
≥
0 for all u
′
∈
K. Let SOL
(
K
,
F
) denote the solution set of this problem. We write x
˙
:
=
d
x
/
d
t for the time-derivative of a function x
(
t
). In this paper, we consider the following system of differential set-valued variational inequalities:
(1)
x
˙
(
t
)
=
f
(
t
,
x
(
t
)
)
+
B
1
(
t
,
x
(
t
)
)
u
(
t
)
+
B
2
(
t
,
x
(
t
)
)
v
(
t
)
,
〈
G
1
(
t
,
x
(
t
)
)
+
F
1
(
u
(
t
)
)
,
u
′
-
u
(
t
)
〉
≥
0
,
∀
u
′
∈
K
,
〈
G
2
(
t
,
x
(
t
)
)
+
F
2
(
v
′
(
t
)
)
,
v
′
-
v
(
t
)
〉
≥
0
,
∀
v
′
∈
K
,
x
(
0
)
=
x
0
,
where Ω
≡
[
0
,
T
]
×
R
m, f
:
Ω
→
R
m, B
i
:
Ω
→
R
m
×
n, G
i
:
Ω
→
R
n, and F
i
:
R
n
⇉
R
n (i
=
1,2) are given mappings.In [1], Pang and Stewart introduced a class of differential variational inequalities in finite dimensional Euclidean spaces. For some related results, we refer to [2–17]. Recently, the differential variational inequalities have been used in cellular biology (see [18]). In [18], the authors needed two or more variational inequalities to formulate the switching between the metabolic models. Sometimes it is convenient to apply the differential vector variational inequalities in [19] to show the fermentation dynamics. However, when we study the fermentation model (20) in [18], we find that the system (1) in this paper can help us a lot.In this paper, we establish an existence theorem of weak solutions for the system (1) in the sense of Carathéodory under some suitable conditions. Furthermore, we give a convergence result on Euler time-dependent procedure for solving the system (1).


## 2. Preliminaries

In this section, we will introduce some basic notations and preliminary results.In the rest of this paper, we will use the following assumptions (A) and (B).(A)
f,  B
1, B
2, G
1, and G
2 are Lipschitz continuous functions on Ω with Lipschitz constants L
f, L
B
1, L
B
2, L
G
1, and L
G
2, respectively.
(B)
B
1 is bounded on Ω with σ
B
1
≡
sup
(
t
,
x
)
∈
Ω
∥
B
1
(
t
,
x
)
∥
<
∞; B
2 is bounded on Ω with σ
B
2
≡
sup
(
t
,
x
)
∈
Ω
∥
B
2
(
t
,
x
)
∥
<
∞.Definition 1.
A set-valued mapF
:
R
n
⇉
R
n is said to be(i)
monotone on a convex setK
⊂
R
n if for each pair of points x
,
y
∈
K, and for all x
*
∈
F
(
x
) and y
*
∈
F
(
y
), 〈
x
*
-
y
*
,
x
-
y
〉
≥
0;
(ii)
pseudo monotone on a convex setK
⊂
R
n if for each pair of points x
,
y
∈
K, and for all x
*
∈
F
(
x
) and y
*
∈
F
(
y
), 〈
y
*
,
x
-
y
〉
≥
0 implies that 〈
x
*
,
x
-
y
〉
≥
0.Definition 2.
A functionf
:
Ω
→
R
n  (
resp
.
,
B
:
Ω
→
R
n
×
m
) is said to be Lipschitz continuous if there exists a constant L
f
>
0  (
resp
.
,
L
B
>
0
) such that, for any (
t
1
,
x
)
,
(
t
2
,
y
)
∈
Ω,
(2)
∥
f
(
t
1
,
x
)
-
f
(
t
2
,
y
)
∥
≤
L
f
(
|
t
1
-
t
2
|
+
∥
x
-
y
∥
)
(
resp
.
,
∥
B
(
t
1
,
x
)
-
B
(
t
2
,
y
)
∥
≤
L
B
(
|
t
1
-
t
2
|
+
∥
x
-
y
∥
)
)
.Definition 3.
LetX, Y be topological spaces and let F
:
X
⇉
Y be a set-valued mapping with nonempty values. One says that F is upper semicontinuous at x
0
∈
X if and only if, for any neighborhood 𝒩
(
F
(
x
0
)
) of F
(
x
0
), there exists a neighborhood 𝒩
(
x
0
) of x
0 such that
(3)
F
(
x
)
⊂
𝒩
(
F
(
x
0
)
)
,
∀
x
∈
𝒩
(
x
0
)
.Lemma 4 (see [1]).
Let𝔽
:
Ω
⇉
R
m be an upper semicontinuous set-valued map with nonempty closed convex values. Suppose that there exists a scalar ρ
𝔽
>
0 satisfying
(4)
sup
{
∥
y
∥
:
y
∈
𝔽
(
t
,
x
)
}
≤
ρ
𝔽
(
1
+
∥
x
∥
)
,
∀
(
t
,
x
)
∈
Ω
.
For every x
0
∈
R
n, the D
I
:
x
˙
∈
𝔽
(
t
,
x
)
,
x
(
0
)
=
x
0 has a weak solution in the sense of Carathéodory.Lemma 5 (see [1]).
Leth
:
Ω
×
R
m
→
R
n be a continuous function and let U
:
Ω
⇉
R
m be a closed set-valued map such that, for some constant η
U
>
0,
(5)
sup
u
∈
U
(
t
,
x
)
∥
u
∥
≤
η
U
(
1
+
∥
x
∥
)
,
∀
(
t
,
x
)
∈
Ω
.
Let v
:
[
0
,
T
]
→
R
n be a measurable function and let x
:
[
0
,
T
]
→
R
n be a continuous function satisfying v
(
t
)
∈
h
(
t
,
x
(
t
)
,
U
(
t
,
x
(
t
)
)
) for almost all t
∈
[
0
,
T
]. There exists a measurable function u
:
[
0
,
T
]
→
R
m such that u
(
t
)
∈
U
(
t
,
x
(
t
)
) and v
(
t
)
=
h
(
t
,
x
(
t
)
,
u
(
t
)
) for almost all t
∈
[
0
,
T
].Lemma 6 (see [20]).
Letm
^ denote the Lebesgue measure on R
n and let f
:
R
n
→
R
m be a measurable function. Let L be a measurable set in R
n with m
^
(
L
)
<
∞. Then, for any ɛ
>
0, there exists a compact set K
⊆
L with m
^
(
L
∖
K
)
<
ɛ such that the restriction of f to K is continuous.Definition 7 (see [21]).
An acyclic set is a set whose homology is the same as the homology of the space consisting of just one point. An acyclic map is an upper semicontinuous set-valued map which has compact acyclic values.In [21], we can find that every homeomorphic image of a compact convex set is an acyclic set.Lemma 8 (see [1]).
Every acyclic set-valued mapF
:
X
→
X on a compact convex set X has a fixed point: x
∈
F
(
x
) for some x
∈
X.


## 3. Main Results

In this section, we obtain existence theorem for weak solutions of the differential set-valued variational inequality in the sense of Carathéodory. Furthermore, we establish a convergence result for solving differential set-valued variational inequality.Theorem 9.
Assume that(
f
,
B
1
,
B
2
,
G
1
,
G
2
) satisfy conditions (A) and (B) and F
i
:
R
n
⇉
R
n  (
i
=
1,2
) are upper semicontinuous with nonempty and compact values such that q
i
+
F
i  (
i
=
1,2
) are pseudo monotone on R
n for each q
i
∈
G
i
(
Ω
)
(
i
=
1,2
). If K is a bounded, closed, and convex subset of R
n, then initial-value system (1) has a weak solution.Proof.
From the proofs of Lemmas 3.2, 3.3, and 3.4 and Theorem 3.1 [19], it is easy to see that the assumption “F is pseudo monotone on R
n” in there should be replaced by the assumption “q
+
F is pseudo monotone on R
n for each q
∈
G
(
Ω
).” Since K is a bounded, closed, and convex subset of R
n, it follows from Lemma 3.3 in [19] that SOL
(
K
,
q
i
+
F
i
)
(
i
=
1,2
) are nonempty and bounded. Let u
=
(
u
1
,
u
2
), where u
i
∈
SOL
(
K
,
q
i
+
F
i
), (
i
=
1,2
). Then it follows that u is bounded on  R
2
n. Moreover, Lemma 3.4 in [19] shows that SOL
(
K
,
q
i
+
F
i
)  (
i
=
1,2
) are closed and convex for all q
i
∈
G
i
(
Ω
). Therefore, SOL
(
K
,
q
1
+
F
2
)
×
SOL
(
K
,
q
2
+
F
2
) is closed and convex. Let
(6)
𝔽
(
t
,
x
)
≡
{
f
(
t
,
x
)
+
B
1
(
t
,
x
)
u
1
+
B
2
(
t
,
x
)
u
2
:
u
i
∈
SOL
(
K
,
G
i
(
t
,
x
)
+
F
i
)
}
.
We can prove in a similar way as Lemma 6.3 in [1] that 𝔽 has linear growth and it is upper semicontinuous on Ω. Now it follows from Lemmas 4 and 5 that system (1) has a weak solution. This completes the proof.Remark 10.
IfF
i
:
R
n
⇉
R
n
(
i
=
1,2
) are monotone, then it is easy to see that q
i
+
F
i
(
i
=
1,2
) are pseudo monotone on R
n for each q
i
∈
G
i
(
Ω
)  (
i
=
1,2
).Lemma 11.
LetG
:
Ω
×
R
m
→
R
n be a continuous function, F
:
L
2
[
0
,
T
]
⇉
L
2
[
0
,
T
] a set-valued function, and u
(
t
)
∈
K with u
∈
L
2
[
0
,
T
]. Suppose there exists u
*
∈
F
(
u
) such that, for any continuous function u
~
:
[
0
,
T
]
→
K, one has
(7)
∫
0
T
‍
〈
G
(
t
,
x
(
t
)
)
+
u
*
(
t
)
,
u
~
(
t
)
-
u
(
t
)
〉
d
t
≥
0
.
Then, for almost all t
∈
[
0
,
T
],  u
(
t
)
∈
S
O
L
(
K
,
G
(
t
,
x
(
t
)
)
+
F
(
·
)
).Proof.
We assume that the contrary holds. Then there exists a setE
⊂
[
0
,
T
] with m
^
(
E
)
>
0 (where m
^
(
E
) denotes the Lebesgue measure of E such that, for all t
∈
E, u
(
t
)
∉
SOL
(
K
,
G
(
t
,
x
(
t
)
)
+
F
(
·
)
). By Lemma 6, we know that there exists a closed subset E
1 of E with m
^
(
E
1
)
>
0 such that u
(
t
) and u
*
(
t
) are continuous on E
1, where u
*
(
t
)
∈
F
(
u
(
t
)
). Then there exists a closed subset E
2 of E
1 with m
^
(
E
2
)
>
0 and v
0
∈
K such that
(8)
〈
G
(
t
,
x
(
t
)
)
+
u
*
(
t
)
,
v
0
-
u
(
t
)
〉
<
0
,
and so
(9)
∫
E
2
‍
〈
G
(
t
,
x
(
t
)
)
+
u
*
(
t
)
,
v
0
-
u
(
t
)
〉
d
t
<
0
.
Let
(10)
u
0
(
t
)
=
{
v
0
,
t
∈
E
2
,
u
(
t
)
,
t
∈
[
0
,
T
]
∖
E
2
.
We know that u
0
(
t
)
∈
K is an integrable function on [
0
,
T
]. Since the space of continuous functions C
(
[
0
,
T
]
;
R
m
) is dense in L
1
(
[
0
,
T
]
;
R
m
), we can approximate u
0
(
t
)
∈
L
1
(
[
0
,
T
]
;
R
m
) by continuous functions u
¯
(
t
)
∈
K and obtain that there exists a continuous function u
¯
(
t
) such that
(11)
∫
0
T
‍
〈
G
(
t
,
x
(
t
)
)
+
u
*
(
t
)
,
u
¯
(
t
)
-
u
(
t
)
〉
d
t
<
0
,
which contradicts (7). This completes the proof.Remark 12.
Ifu
(
t
) is an integrable function satisfying, for almost all t
∈
[
0
,
T
],
(12)
u
(
t
)
∈
SOL
(
K
,
G
(
t
,
x
(
t
)
,
·
)
+
F
(
·
)
)
,
then the integral inequality (7) must hold for any continuous u
^
:
[
0
,
T
]
→
K.Now we begin to design a computational method for solving DVI (1). With x
h
,
0
:
=
x
0, we compute
(13)
{
x
h
,
1
,
x
h
,
2
,
…
,
x
h
,
N
h
+
1
}
⊂
R
n
,
{
u
h
,
1
,
u
h
,
2
,
…
,
u
h
,
N
h
+
1
}
⊂
K
,
{
v
h
,
1
,
v
h
,
2
,
…
,
v
h
,
N
h
+
1
}
⊂
K
,
by the recursion, for i
=
0,1
,
…
,
N
h, where N
h
=
(
T
/
h
)
-
1,
(14)
x
h
,
i
+
1
=
x
h
,
i
+
h
[
f
(
t
h
,
i
+
1
,
θ
x
h
,
i
+
(
1
-
θ
)
x
h
,
i
+
1
)
=
x
h
,
i
+
h
+
B
1
(
t
h
,
i
,
x
h
,
i
)
u
h
,
i
+
1
+
B
2
(
t
h
,
i
,
x
h
,
i
)
v
h
,
i
+
1
]
,
u
h
,
i
+
1
∈
SOL
(
K
,
G
1
(
t
h
,
i
+
1
,
x
h
,
i
+
1
)
+
F
1
)
,
v
h
,
i
+
1
∈
SOL
(
K
,
G
2
(
t
h
,
i
+
1
,
x
h
,
i
+
1
)
+
F
2
)
;
that is,
(15)
x
h
,
i
+
1
=
x
h
,
i
+
h
[
f
(
t
h
,
i
+
1
,
θ
x
h
,
i
+
(
1
-
θ
)
x
h
,
i
+
1
)
=
x
h
,
i
+
h
+
B
1
(
t
h
,
i
,
x
h
,
i
)
u
h
,
i
+
1
+
B
2
(
t
h
,
i
,
x
h
,
i
)
v
h
,
i
+
1
]
,
〈
G
1
(
t
h
,
i
+
1
,
x
h
,
i
+
1
)
+
F
1
(
u
h
,
i
+
1
)
,
u
′
-
u
h
,
i
+
1
〉
≥
0
,
∀
u
′
∈
K
,
〈
G
2
(
t
h
,
i
+
1
,
x
h
,
i
+
1
)
+
F
2
(
v
h
,
i
+
1
)
,
v
′
-
v
h
,
i
+
1
〉
≥
0
,
∀
v
′
∈
K
.Lemma 13.
Let(
f
,
B
1
,
B
2
,
G
1
,
G
2
) satisfy conditions (A) and (B). Then there exists an h
0
>
0 such that, for any h
∈
(
0
,
h
0
], (
x
r
e
f
,
u
,
v
)
∈
R
n
+
m
+
m with θ
∈
[
0,1
] and t, t
r
e
f in [
0
,
T
], there exists a unique vector x
u
v satisfying
(16)
x
u
v
-
x
r
e
f
=
h
[
f
(
t
,
θ
x
r
e
f
+
(
1
-
θ
)
x
u
v
)
+
B
1
(
t
r
e
f
,
x
r
e
f
)
u
+
B
2
(
t
r
e
f
,
x
r
e
f
)
v
]
.
Moreover, for any u
,
v
,
u
′
,
v
′
∈
R
m, one has
(17)
∥
x
u
v
-
x
u
′
v
′
∥
≤
h
σ
B
1
∥
u
-
u
′
∥
+
h
σ
B
2
∥
v
-
v
′
∥
1
-
h
(
1
-
θ
)
L
f
,
∥
x
u
v
-
x
r
e
f
∥
≤
ρ
f
(
1
+
∥
x
r
e
f
∥
)
+
σ
B
1
∥
u
∥
+
σ
B
2
∥
v
∥
1
-
h
(
1
-
θ
)
ρ
f
.Proof.
It suffices to chooseh
0 satisfying
(18)
0
<
h
0
<
min
{
1
(
1
-
θ
)
L
f
,
1
(
1
-
θ
)
ρ
f
}
.
The right-hand side is taken to be ∞ if θ
=
1. Under this choice, consider any tuple (
h
,
x
ref
,
u
,
v
,
t
,
t
ref
) as specified. Let
(19)
ℱ
(
x
)
=
h
f
(
t
,
θ
x
ref
+
(
1
-
θ
)
x
)
+
h
B
1
(
t
ref
,
x
ref
)
u
+
h
B
2
(
t
ref
,
x
ref
)
v
+
x
ref
.
Then
(20)
∥
ℱ
(
x
1
)
-
ℱ
(
x
2
)
∥
=
∥
h
f
(
t
,
θ
x
ref
+
(
1
-
θ
)
x
1
)
-
h
f
(
t
,
θ
x
ref
+
(
1
-
θ
)
x
2
)
∥
≤
h
L
f
(
1
-
θ
)
∥
x
1
-
x
2
∥
,
with 0
<
h
L
f
(
1
-
θ
)
<
1. This shows that the map ℱ is contractive and so there exists a unique vector x
u
v such that
(21)
x
u
v
-
x
ref
=
h
[
f
(
t
,
θ
x
ref
+
(
1
-
θ
)
x
u
v
)
+
B
1
(
t
ref
,
x
ref
)
u
+
B
2
(
t
ref
,
x
ref
)
v
]
.
It implies that, for any (
u
1
,
v
1
)
,
(
u
2
,
v
2
)
∈
R
m
×
m, there exist x
u
1
v
1 and x
u
2
v
2 such that
(22)
x
u
1
v
1
-
x
ref
=

h
[
f
(
t
,
θ
x
ref
+
(
1
-
θ
)
x
u
1
v
1
)
+

B
1
(
t
ref
,
x
ref
)
u
1
+
B
2
(
t
ref
,
x
ref
)
v
1
]
,
x
u
2
,
v
2
-
x
ref
=

h
[
f
(
t
,
θ
x
ref
+
(
1
-
θ
)
x
u
2
,
v
2
)

+
B
1
(
t
ref
,
x
ref
)
u
2
+
B
2
(
t
ref
,
x
ref
)
v
2
]
.
By (22), we have
(23)
∥
x
u
1
v
1
-
x
u
2
,
v
2
∥
≤
h
L
f
(
1
-
θ
)
∥
x
u
1
v
1
-
x
u
2
,
v
2
∥
+
h
σ
B
1
∥
u
1
-
u
2
∥
+
h
σ
B
2
∥
v
1
-
v
2
∥
and so
(24)
∥
x
u
1
v
1
-
x
u
2
,
v
2
∥
≤
h
σ
B
1
∥
u
1
-
u
2
∥
+
h
σ
B
2
∥
v
1
-
v
2
∥
1
-
h
L
f
(
1
-
θ
)
.
Now the Lipschitz continuity of f implies that there exists ρ
f satisfying
(25)
∥
f
(
t
,
x
)
∥
≤
ρ
f
(
1
+
∥
x
∥
)
.
It follows that
(26)
∥
x
u
v
-
x
ref
∥
=
h
∥
f
(
t
,
θ
x
ref
+
(
1
-
θ
)
x
u
v
)
+
B
1
(
t
ref
,
x
ref
)
u
+
B
2
(
t
ref
,
x
ref
)
v
∥
≤
h
ρ
f
(
1
+
∥
θ
x
ref
+
(
1
-
θ
)
x
u
v
∥
)
+
h
σ
B
1
∥
u
∥
+
h
σ
B
2
∥
v
∥
≤
h
ρ
f
(
1
+
(
1
-
θ
)
∥
x
u
v
-
x
ref
∥
+
∥
x
ref
∥
)
+
h
σ
B
1
∥
u
∥
+
h
σ
B
2
∥
v
∥
and so
(27)
∥
x
u
v
-
x
ref
∥
≤
h
ρ
f
(
1
+
∥
x
ref
∥
)
+
h
σ
B
1
∥
u
∥
+
h
σ
B
2
∥
v
∥
1
-
h
ρ
f
(
1
-
θ
)
.
This completes the proof.Lemma 14.
Let(
f
,
B
1
,
B
2
,
G
1
,
G
2
) satisfy conditions (A) and (B). Suppose that S
O
L
(
K
,
q
1
+
F
1
) and S
O
L
(
K
,
q
2
+
F
2
) satisfy the linear growth properties
(28)
sup
{
∥
u
∥
:
u
∈
S
O
L
(
K
,
q
1
+
F
1
)
}
≤
ρ
1
(
1
+
∥
q
1
∥
)
,
up
{
∥
u
∥
:
u
∈
S
O
L
(
K
,
q
1
+
F
1
)
}
1
≤
∀
q
1
∈
G
1
(
Ω
)
,
sup
{
∥
u
∥
:
u
∈
S
O
L
(
K
,
q
2
+
F
2
)
}
≤
ρ
2
(
1
+
∥
q
2
∥
)
,
sup
{
∥
u
∥
:
u
∈
S
O
L
(
K
,
q
2
+
F
2
)
}
≤
∀
q
2
∈
G
2
(
Ω
)
.
Then there exist positive scalars C
0
x, C
1
x, C
0
u, C
1
u, C
0
v, C
1
v, and h
1 such that, for any h
∈
(
0
,
h
1
] and i
=
0,1
,
…
,
N
h,
(29)
∥
x
h
,
i
+
1
∥
≤
C
0
x
+
C
1
x
∥
x
0
∥
,
∥
u
h
,
i
+
1
∥
≤
C
0
u
+
C
1
u
∥
x
0
∥
,
∥
v
h
,
i
+
1
∥
≤
C
0
v
+
C
1
v
∥
x
0
∥
.Proof.
Throughout the proof below, the scalarh
>
0 is taken to be sufficiently small. Let
(30)
ρ
x
=
ρ
f
+
σ
B
1
+
σ
B
2
1
-
h
(
1
-
θ
)
ρ
f
.
It follows from Lemma 13 that
(31)
∥
x
h
,
i
+
1
-
x
h
,
i
∥
≤
h
ρ
x
(
1
+
∥
x
h
,
i
∥
+
∥
u
h
,
i
+
1
∥
+
∥
v
h
,
i
+
1
∥
)
.
By the linear growth of solutions to VI, we have
(32)
∥
u
h
,
i
+
1
∥
≤
ρ
1
(
1
+
∥
G
1
(
t
h
,
i
+
1
,
x
h
,
i
+
1
)
∥
)
≤
ρ
1
(
1
+
ρ
G
1
(
1
+
∥
x
h
,
i
+
1
∥
)
)
≤
ρ
1
[
1
+
ρ
G
1
{
1
+
∥
x
h
,
i
∥
+
h
ρ
x
≤
ρ
1
[
1
+
ρ
G
1
×
(
1
+
∥
x
h
,
i
∥
+
∥
u
h
,
i
+
1
∥
+
∥
v
h
,
i
+
1
∥
)
}
]
≤
(
ρ
1
+
ρ
1
ρ
G
1
+
h
ρ
1
ρ
G
1
ρ
x
)
+
(
ρ
1
ρ
G
1
+
h
ρ
1
ρ
G
1
ρ
x
)
∥
x
h
,
i
∥
+
h
ρ
1
ρ
G
1
ρ
x
∥
u
h
,
i
+
1
∥
+
h
ρ
1
ρ
G
1
ρ
x
∥
v
h
,
i
+
1
∥
,
∥
v
h
,
i
+
1
∥
≤
ρ
2
(
1
+
∥
G
2
(
t
h
,
i
+
1
,
x
h
,
i
+
1
)
∥
)
≤
ρ
2
(
1
+
ρ
G
2
(
1
+
∥
x
h
,
i
+
1
∥
)
)
≤
ρ
2
[
1
+
ρ
G
2
{
1
+
∥
x
h
,
i
∥
+
h
ρ
x
≤
ρ
1
[
1
+
ρ
G
1
×
(
1
+
∥
x
h
,
i
∥
+
∥
u
h
,
i
+
1
∥
+
∥
v
h
,
i
+
1
∥
)
}
]
≤
(
ρ
2
+
ρ
2
ρ
G
2
+
h
ρ
2
ρ
G
2
ρ
x
)
+
(
ρ
2
ρ
G
2
+
h
ρ
2
ρ
G
2
ρ
x
)
∥
x
h
,
i
∥
+
h
ρ
2
ρ
G
2
ρ
x
∥
u
h
,
i
+
1
∥
+
h
ρ
2
ρ
G
2
ρ
x
∥
v
h
,
i
+
1
∥
.
Let
(33)
M
1
=
ρ
1
+
ρ
1
ρ
G
1
+
h
ρ
1
ρ
G
1
ρ
x
,
N
1
=
ρ
1
ρ
G
1
ρ
x
,
M
2
=
ρ
2
+
ρ
2
ρ
G
2
+
h
ρ
2
ρ
G
2
ρ
x
,
N
2
=
ρ
2
ρ
G
2
ρ
x
.
Then, we have
(34)
(
1
-
h
N
1
)
∥
u
h
,
i
+
1
∥
≤
M
(
1
+
∥
x
h
,
i
∥
)
+
h
N
1
∥
v
h
,
i
+
1
∥
,
(
1
-
h
N
2
)
∥
v
h
,
i
+
1
∥
≤
M
(
1
+
∥
x
h
,
i
∥
)
+
h
N
2
∥
u
h
,
i
+
1
∥
.
Letting 0
<
h
<
min
{
1
/
N
1
,
1
/
N
2
}, one has
(35)
∥
u
h
,
i
+
1
∥
≤
1
1
-
h
N
1
×
[
(
1
1
-
h
N
2
(
M
(
1
+
∥
x
h
,
i
∥
)
+
h
N
2
∥
u
h
,
i
+
1
∥
)
)
M
(
1
+
∥
x
h
,
i
∥
)
+
h
N
1
×

(
1
1
-
h
N
2
(
M
(
1
+
∥
x
h
,
i
∥
)

+

h
N
2
∥
u
h
,
i
+
1
∥
)
)
]
.
When h is sufficiently small, there exists ρ
M
1
>
0 such that
(36)
∥
u
h
,
i
+
1
∥
≤
ρ
M
1
(
1
+
∥
x
h
,
i
∥
)
.
In a similar way, we can prove that there exists ρ
M
2
>
0 such that
(37)
∥
v
h
,
i
+
1
∥
≤
ρ
M
2
(
1
+
∥
x
h
,
i
∥
)
.
It follows from (31) that
(38)
∥
x
h
,
i
+
1
-
x
h
,
i
∥
≤
h
ρ
x
(
1
+
∥
x
h
,
i
∥
+
ρ
M
1
(
1
+
∥
x
h
,
i
∥
)
+
ρ
M
2
(
1
+
∥
x
h
,
i
∥
)
)
=
(
h
ρ
x
+
h
ρ
x
ρ
M
1
+
h
ρ
x
ρ
M
2
)
(
1
+
∥
x
h
,
i
∥
)
.
Let
(39)
ψ
x
=
ρ
x
+
ρ
x
ρ
M
1
+
ρ
x
ρ
M
2
.
Then
(40)
∥
x
h
,
i
+
1
-
x
h
,
i
∥
≤
h
ψ
x
(
1
+
∥
x
h
,
i
∥
)
.
It follows from Lemma 7.2 in [1] that there exist positive scalars C
0
x, C
1
x, C
0
u, C
1
u, C
0
v, C
1
v, and h
1 such that, for any h
∈
(
0
,
h
1
] and i
=
0,1
,
…
,
N
h,
(41)
∥
x
h
,
i
+
1
∥
≤
C
0
x
+
C
1
x
∥
x
0
∥
,
∥
u
h
,
i
+
1
∥
≤
C
0
u
+
C
1
u
∥
x
0
∥
,
∥
v
h
,
i
+
1
∥
≤
C
0
v
+
C
1
v
∥
x
0
∥
.
This completes the proof.Lemma 15.
LetK
⊂
R
n be a nonempty, closed, and convex set and let (
f
,
B
1
,
B
2
,
G
1
,
G
2
) satisfy conditions (A) and (B). Suppose that the set-valued maps F
1
,
F
2 are upper semicontinuous with nonempty compact values such that q
i
+
F
i  (
i
=
1,2
) are pseudo monotone on R
n for each q
i
∈
G
i
(
Ω
)  (
i
=
1,2
). For some constant ρ
>
0, S
O
L
(
K
,
q
1
+
F
1
) and S
O
L
(
K
,
q
2
+
F
2
) satisfy the linear growth properties
(42)
sup
{
∥
u
∥
:
u
∈
S
O
L
(
K
,
q
1
+
F
1
)
}
≤
ρ
(
1
+
∥
q
1
∥
)
,
sup
{
∥
u
∥
:
u
∈
S
O
L
(
K
,
q
1
+
F
1
)
}
≤
∀
q
1
∈
G
1
(
Ω
)
,
(43)
sup
{
∥
u
∥
:
u
∈
S
O
L
(
K
,
q
2
+
F
2
)
}
≤
ρ
(
1
+
∥
q
2
∥
)
,
sup
{
∥
u
∥
:
u
∈
S
O
L
(
K
,
q
2
+
F
2
)
}
≤
∀
q
2
∈
G
2
(
Ω
)
.
Then there exists a scalar h
R
>
0 such that, for any h
∈
(
0
,
h
R
] with θ
∈
[
0,1
] and x
0
∈
R, there exists (
x
h
,
i
+
1
,
u
h
,
i
+
1
,
v
h
,
i
+
1
) satisfying (15) for every i
=
0,1
,
…
,
N
h.Proof.
Assume thatψ
x is defined by (39). For any scalar h
>
0 sufficiently small, we define the scalars ρ
1
,
ρ
2
,
…
,
ρ
N
h
+
1 by
(44)
ρ
i
+
1
≡
(
1
+
h
ψ
x
)
ρ
i
+
h
ψ
x
,
i
=
0,1
,
…
,
N
h
,
where ρ
0 is arbitrary. By the proof of Lemma 7.2 in [1], we can show that
(45)
ρ
i
≤
e
T
ψ
x
ρ
0
+
e
T
ψ
x
-
1
,
∀
i
=
0,1
,
…
,
N
i
+
1
.
Let α denote the quantity on the right-hand side, which depends on ρ
0 but is independent of h. Let 0
<
h
R
<
min
{
h
0
,
h
1
} satisfy
(46)
h
R
ρ
f
(
1
+
α
)
+
(
σ
B
1
+
σ
B
2
)
ρ
ρ
G
1
(
1
+
2
α
)
1
-
h
R
(
1
-
θ
)
ρ
f
<
α
,
where h
0 and h
1 are as described in Lemmas 13 and 14, respectively.
Next we show that, for any fixedh
∈
(
0
,
h
R
], there exists a triple (
x
h
,
i
+
1
,
u
h
,
i
+
1
,
v
h
,
i
+
1
) satisfying (15) with ∥
x
h
,
i
+
1
∥
≤
ρ
i
+
1 for all i
=
0,1
,
…
,
N
h. Let B
α denote the Euclidean ball in R
n with center at the origin and radius 2
α. For any x
∈
B
α, let S
j
(
t
,
x
) denote the nonempty set SOL
(
K
,
G
j
(
t
,
x
)
+
F
j
). Since G
j is Lipschitz continuous on Ω, we know that G
j have linear growth on Ω in x; that is, for some positive constants ρ
G
j and for all (
t
,
x
)
∈
Ω,
(47)
∥
G
j
(
t
,
x
)
∥
≤
ρ
G
j
(
1
+
∥
x
∥
)
.
By the linear growth assumption, for any x
∈
B
α, we have
(48)
sup
{
∥
u
∥
:
u
∈
S
j
(
t
,
x
)
}
≤
ρ
(
1
+
∥
G
j
(
t
,
x
)
∥
)
≤
ρ
(
1
+
ρ
G
j
(
1
+
∥
x
∥
)
)
≤
ρ
(
1
+
ρ
G
j
)
(
1
+
2
α
)
,
j
=
1,2
.
Define mappings S
i from B
α to subset of B
α as follows: for any x
∈
B
α,
(49)
S
i
(
x
)
≡
(
I
-
h
f
(
t
h
,
i
+
1
,
θ
x
h
,
i
+
(
1
-
θ
)
x
)
)
-
1
×
[
x
h
,
i
+
h
B
1
(
t
h
,
i
,
x
h
,
i
)
S
1
(
t
h
,
i
+
1
,
x
)
+
h
B
2
(
t
h
,
i
,
x
h
,
i
)
S
2
(
t
h
,
i
+
1
,
x
)
]
.
Since F
1 and F
2 are upper semicontinuous with nonempty compact values such that q
i
+
F
i  (
i
=
1,2
) are pseudo monotone on R
n for each q
i
∈
G
i
(
Ω
)  (
i
=
1,2
), it follows from Lemmas 3.3 and 3.4 in [19] that SOL
(
K
,
G
1
(
t
,
x
)
+
F
1
) and SOL
(
K
,
G
2
(
t
,
x
)
+
F
2
) are nonempty, closed, and convex sets. By (48), we obtain that SOL
(
K
,
G
1
(
t
,
x
)
+
F
1
) and SOL
(
K
,
G
2
(
t
,
x
)
+
F
2
) are compact and convex. Consider the map
(50)
(
x
,
y
)
⟼
x
h
,
i
+
h
B
1
(
t
h
,
i
,
x
h
,
i
)
x
+
h
B
2
(
t
h
,
i
,
x
h
,
i
)
y
.
It is easy to see that this map is continuous. Therefore, by the Tychonoff theorem, we know that S
1
(
t
,
x
)
×
S
2
(
t
,
x
) is compact and so
(51)
x
h
,
i
+
h
B
1
(
t
h
,
i
,
x
h
,
i
)
S
1
(
t
h
,
i
+
1
,
x
)
+
h
B
2
(
t
h
,
i
,
x
h
,
i
)
S
2
(
t
h
,
i
+
1
,
x
)
is compact. Since the mapping (
I
-
h
f
(
t
h
,
i
+
1
,
θ
x
h
,
i
+
(
1
-
θ
)
·
)
)
-
1 is a homeomorphism for all h
>
0 sufficiently small, it follows that S
i
(
x
) is a compact acyclic set. We need to show that S
i
(
x
) is a subset of B
α. Let x
~ be an arbitrary element in S
i
(
x
) and let u
∈
S
1
(
t
h
,
i
+
1
,
x
)
,
v
∈
S
2
(
t
h
,
i
+
1
,
x
) be such that
(52)
x
~
=
x
h
,
i
+
h
[
f
(
t
h
,
i
+
1
,
θ
x
i
+
(
1
-
θ
)
x
~
)
+
B
1
(
t
h
,
i
,
x
h
,
i
)
u
+
B
2
(
t
h
,
i
,
x
h
,
i
)
v
]
.
From Lemma 13, we have
(53)
∥
x
~
-
x
h
,
i
∥
≤
h
ρ
f
(
1
+
∥
x
h
,
i
∥
)
+
σ
B
1
∥
u
∥
+
σ
B
2
∥
v
∥
1
-
h
(
1
-
θ
)
ρ
f
.
By induction hypothesis and ∥
x
h
,
i
∥
≤
ρ
i
≤
α, one has
(54)
∥
x
~
∥
≤
ρ
i
+
h
ρ
f
(
1
+
ρ
i
)
+
(
σ
B
1
+
σ
B
2
)
ρ
ρ
G
1
(
1
+
2
α
)
1
-
h
(
1
-
θ
)
ρ
f
<
2
α
.
Now we need to prove that the solution mapping S
1
(
t
h
,
i
+
1
,
x
) is upper semicontinuous. To prove the upper semicontinuity of S
1
(
t
h
,
i
+
1
,
x
), it suffices to show that S
1
(
t
h
,
i
+
1
,
x
) is closed. Suppose that {
x
n
}
⊂
R
n is a sequence converging to x
0
∈
R
n and u
n
∈
S
1
(
t
h
,
i
+
1
,
x
n
). Then the linear growth condition implies that {
u
n
} is bounded and so it has a convergent subsequence with a limit u
0. Since u
n
∈
S
1
(
t
h
,
i
+
1
,
x
n
), there exists u
n
′
∈
F
1
(
u
n
) such that
(55)
〈
G
1
(
t
h
,
i
+
1
,
x
n
)
+
u
′
n
,
u
′
-
u
n
〉
,
∀
u
′
∈
K
.
Since F
1 is upper semicontinuous on R
n with compact values, it follows that there exists a subsequence of {
u
n
′
}, denoted again by {
u
n
′
}, such that u
n
′
→
u
0
′
∈
F
1
(
u
0
). Letting n
→
∞, we have
(56)
〈
G
1
(
t
h
,
i
+
1
,
x
0
)
+
u
0
′
,
u
′
-
u
0
〉
≥
0
,
∀
u
′
∈
K
and so u
0
∈
S
1
(
t
h
,
i
+
1
,
x
0
). It follows that S
1
(
t
h
,
i
+
1
,
x
) is closed and so upper semicontinuous. In a similar way, we can prove that S
2
(
t
h
,
i
+
1
,
x
) is upper semicontinuous. Thus, we know that S
i
:
B
α
→
B
α is a closed set-valued mapping with compact acyclic values. By Lemma 8, S
i has a fixed point and so there exists a triple (
x
h
,
i
+
1
,
u
h
,
i
+
1
,
v
h
,
i
+
1
) satisfying (15). Now we show that ∥
x
h
,
i
+
1
∥
≤
ρ
i
+
1. In fact, by (40) and Lemma 7.2 in [1], one has
(57)
∥
x
h
,
i
+
1
∥
≤
e
T
ψ
x
∥
x
0
∥
+
e
T
ψ
x
-
1
.
The definition of ρ
i
+
1 implies that ∥
x
h
,
i
+
1
∥
≤
ρ
i
+
1. This completes the proof.Letx
^
h
(
·
) be the continuous piecewise linear interpolant of the family {
x
h
,
i
+
1
},  u
^
h
(
·
) the constant piecewise interpolant of the family {
u
h
,
i
+
1
}, and v
^
h
(
·
) the constant piecewise interpolant of the family {
v
h
,
i
+
1
}; that is,
(58)
x
^
h
(
t
)
=
x
h
,
i
+
t
-
t
i
h
(
x
h
,
i
+
1
-
x
h
,
i
)
,
∀
t
∈
[
t
h
,
i
,
t
h
,
i
+
1
]
,
u
^
h
(
t
)
=
u
h
,
i
+
1
,
∀
t
∈
(
t
i
,
t
i
+
1
]
,
v
^
h
(
t
)
=
v
h
,
i
+
1
,
∀
t
∈
(
t
i
,
t
i
+
1
]
,
for i
=
0,1
,
…
,
N
h.Theorem 16.
Let(
f
,
B
1
,
B
2
,
G
1
,
G
2
) satisfy conditions (A) and (B) and let K
⊂
R
n be a nonempty, closed, and convex set. Suppose that S
O
L
(
K
,
q
1
+
F
1
) and S
O
L
(
K
,
q
2
+
F
2
) satisfy the linear growth properties. Then there exists a sequence {
h
n
}
↓
0 such that x
^
h
n
→
x
^ uniformly on [
0
,
T
] and u
^
h
n
→
u
^ weakly in L
2
[
0
,
T
] with v
^
h
n
→
v
^ weakly in L
2
[
0
,
T
]. Furthermore, assume that F
1
(
u
)
=
ψ
1
(
E
1
u
),  F
2
(
v
)
=
ψ
2
(
E
2
v
), E
j
∈
R
m
×
m, j
=
1,2 and
(59)
ψ
j
:
L
2
(
[
0
,
T
]
,
R
m
)
⇉
L
2
(
[
0
,
T
]
,
R
m
)
,
j
=
1,2
are upper semicontinuous set-valued mappings with nonempty compact values and there exist constants C
1
,
C
2 such that, for any h sufficiently small,
(60)
∥
E
1
u
h
,
i
+
1
-
E
1
u
h
,
i
∥
≤
h
C
1
,
∥
E
2
v
h
,
i
+
1
-
E
2
v
h
,
i
∥
≤
h
C
2
.
Then the limit (
x
^
,
u
^
,
v
^
) is a weak solution of the system (1).Proof.
By (31) and Lemma 14, we deduce that, for h
>
0 sufficiently small, there exists an L
x
0
>
0, which is independent of h, such that
(61)
∥
x
h
,
i
+
1
-
x
h
,
i
∥
≤
L
x
0
h
,
i
=
0,1
,
…
,
N
h
.
It follows from (58) that x
^
h is also Lipschitz continuous on [
0
,
T
] and the Lipschitz constant is independent of h. Thus, there exists an h
0
>
0 such that the family of functions {
x
^
h
}  (
h
∈
(
0
,
h
0
]
) is an equicontinuous family of functions. Let
(62)
∥
x
^
h
∥
L
∞
=
sup
t
∈
[
0
,
T
]
∥
x
^
h
(
t
)
∥
.
From (58) and Lemma 14, we deduce that {
x
^
h
} is uniformly bounded. By using the Arzelá-Ascoli theorem, there exists a sequence {
h
n
} with h
n
↓
0 such that {
x
^
h
n
} converges in the supremum norm to a Lipschitz function x
^ on [
0
,
T
]. Since SOL
(
K
,
q
1
+
F
1
) and SOL
(
K
,
q
2
+
F
2
) satisfy the linear growth properties, it follows from Lemma 14 that {
u
h
,
i
+
1
} is uniformly bounded in the L
∞ norm on [
0
,
T
]. From (58), we know that {
u
^
h
} is uniformly bounded in the L
∞ norm on [
0
,
T
], which means that there exists a scalar γ
>
0 such that
(63)
∥
u
^
h
∥
L
∞
≤
γ
.
Since L
2
[
0
,
T
] is a reflective Banach space, every bounded sequence has a weakly convergent subsequence and so there is a sequence {
h
n
}
↓
0 such that u
^
h
n
→
u
^ weakly in L
2
[
0
,
T
]. In a similar way, we obtain that v
^
h
n
→
v
^ weakly in L
2
[
0
,
T
].
Next, we show that(
x
^
,
u
^
,
v
^
) is a weak solution of the system (1). By Lemma 11, it is sufficient to prove the following three assertions:(i)
for any0
≤
s
≤
t
≤
T
(64)
x
^
(
t
)
-
x
^
(
s
)
=
∫
s
t
‍
[
f
(
τ
,
x
^
(
τ
)
)
+
B
1
(
τ
,
x
^
(
τ
)
)
u
^
(
τ
)
+
B
2
(
τ
,
x
^
(
τ
)
)
v
^
(
τ
)
]
d
τ
;
(ii)
there existu
0
*
∈
F
1
(
u
^
) and v
0
*
∈
F
2
(
v
^
) such that, for all continuous functions: u
:
[
0
,
T
]
→
K,
(65)
∫
0
T
‍
〈
G
1
(
t
,
x
^
(
t
)
)
+
u
0
*
(
t
)
,
u
(
t
)
-
u
^
(
t
)
〉
d
t
≥
0
,
∫
0
T
‍
〈
G
2
(
t
,
x
^
(
t
)
)
+
u
0
*
(
t
)
,
u
(
t
)
-
v
^
(
t
)
〉
d
t
≥
0
;
(iii)
the initial conditionx
^
(
0
)
=
x
0.Since
(66)
x
h
,
i
+
1
-
x
h
,
i
=
h
[
f
(
t
h
,
i
+
1
,
θ
x
h
,
i
+
(
1
-
θ
)
x
h
,
i
+
1
)
+
B
1
(
t
h
,
i
,
x
h
,
i
)
u
h
,
i
+
1
+
B
2
(
t
h
,
i
,
x
h
,
i
)
v
h
,
i
+
1
]
=
∫
t
h
,
i
t
h
,
i
+
1
‍
[
f
(
τ
,
x
^
h
(
τ
)
)
+
B
1
(
τ
,
x
^
h
(
τ
)
)
u
h
,
i
+
1
+
B
2
(
τ
,
x
^
h
(
τ
)
)
v
h
,
i
+
1
]
d
τ
+
h
2
ξ
,
where
(67)
∥
ξ
∥
≤
L
f
+
L
f
L
x
+
L
B
1
ψ
u
+
L
B
2
ψ
u
,
L
x and ψ
u are the same as described in Theorem 7.1 in [1]; it follows that, for any 0
≤
s
≤
t
≤
T,
(68)
x
h
(
t
)
-
x
h
(
s
)
=
∫
s
t
‍
[
f
(
τ
,
x
^
h
(
τ
)
)
+
B
1
(
τ
,
x
^
h
(
τ
)
)
u
^
h
(
τ
)
+
B
2
(
τ
,
x
^
h
(
τ
)
)
v
^
h
(
τ
)
]
d
τ
+
O
(
h
)
.
By a similar proof to that in Theorem 7.1 of [1], we can obtain that
(69)
lim
h
→
0
∫
s
t
‍
f
(
τ
,
x
^
h
(
τ
)
)
d
τ
=
∫
s
t
‍
f
(
τ
,
x
^
(
τ
)
)
d
τ
;
lim
h
→
0
∫
s
t
‍
B
1
(
τ
,
x
^
h
(
τ
)
)
u
^
h
(
τ
)
d
τ
=
∫
s
t
‍
B
1
(
τ
,
x
^
(
τ
)
)
u
^
(
τ
)
d
τ
;
lim
h
→
0
∫
s
t
‍
B
2
(
τ
,
x
^
h
(
τ
)
)
v
^
h
(
τ
)
d
τ
=
∫
s
t
‍
B
2
(
τ
,
x
^
(
τ
)
)
v
^
(
τ
)
d
τ
.
Noting the proof of Theorem 7.1 in [1], we have x
^
h
n
→
x
^ and E
1
u
^
h
n
→
E
1
u
^ as n
→
∞. Let u
n
*
∈
ψ
1
(
E
1
u
^
h
n
). Since ψ
1 is upper semicontinuous with nonempty compact values, there exists a subsequence of {
u
n
*
}, denoted again by {
u
n
*
}, such that u
n
*
→
u
0
* with u
0
*
∈
ψ
1
(
E
1
u
^
). This implies that, for any continuous functions: u
~
:
[
0
,
T
]
→
K,
(70)
lim
n
→
∞
∫
0
T
‍
〈
G
1
(
t
,
x
^
h
n
(
t
)
)
+
u
n
*
(
t
)
,
u
~
(
t
)
-
u
^
h
n
(
t
)
〉
d
t
=
∫
0
T
‍
〈
G
1
(
t
,
x
^
)
+
u
0
*
(
t
)
,
u
~
-
u
^
〉
d
t
.
Then, in a similar way of Theorem 7.1 in [1], we can prove that
(71)
∫
0
T
‍
〈
G
1
(
t
,
x
^
)
+
u
0
*
(
t
)
,
u
~
-
u
^
〉
d
t
≥
0
.
The proof in the case j
=
2 is similar and so we omit it here. This completes the proof.Remark 17.
Theorem16 generalizes Theorem 7.1 in [1] from the differential variational inequality to the system of differential set-valued variational inequalities.



---

*Source: 918796-2014-02-04.xml*