A System of Differential Set-Valued Variational Inequalities in Finite Dimensional Spaces
Hindawi Publishing Corporation
Journal of Function Spaces
Volume 2014, Article ID 918796, 8 pages
http://dx.doi.org/10.1155/2014/918796
Research Article
A System of Differential Set-Valued Variational Inequalities in
Finite Dimensional Spaces
Wei Li,1 Xing Wang,2 and Nan-Jing Huang1
1
2
Department of Mathematics, Sichuan University, Chengdu 610064, China
School of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330013, China
Correspondence should be addressed to Nan-Jing Huang;
Received 29 May 2013; Revised 5 November 2013; Accepted 19 November 2013; Published 4 february 2014
Academic Editor: John R. Akeroyd
Copyright © 2014 Wei Li et al. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
1. Introduction
For a set-valued mapping 𝐹 : 𝑅𝑛 𝑅𝑛 and a nonempty
closed convex set 𝐾 in 𝑅𝑛 , 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 𝑥̇ := 𝑑𝑥/𝑑𝑡 for the time-derivative of a function 𝑥(𝑡).
In this paper, we consider the following system of differential
set-valued variational inequalities:
𝑥̇ (𝑡) = 𝑓 (𝑡, 𝑥 (𝑡)) + 𝐵1 (𝑡, 𝑥 (𝑡)) 𝑢 (𝑡) + 𝐵2 (𝑡, 𝑥 (𝑡)) V (𝑡) ,
⟨𝐺1 (𝑡, 𝑥 (𝑡)) + 𝐹1 (𝑢 (𝑡)) , 𝑢 − 𝑢 (𝑡)⟩ ≥ 0,
∀𝑢 ∈ 𝐾,
⟨𝐺2 (𝑡, 𝑥 (𝑡)) + 𝐹2 (V (𝑡)) , V − V (𝑡)⟩ ≥ 0,
∀V ∈ 𝐾,
(1)
𝑥 (0) = 𝑥0 ,
where Ω ≡ [0, 𝑇] × 𝑅𝑚 , 𝑓 : Ω → 𝑅𝑚 , 𝐵𝑖 : Ω → 𝑅𝑚×𝑛 ,
𝐺𝑖 : Ω → 𝑅𝑛 , and 𝐹𝑖 : 𝑅𝑛 𝑅𝑛 (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, 𝐵1 , 𝐵2 , 𝐺1 , and 𝐺2 are Lipschitz continuous functions on Ω with Lipschitz constants 𝐿 𝑓 , 𝐿 𝐵1 , 𝐿 𝐵2 , 𝐿 𝐺1 ,
and 𝐿 𝐺2 , respectively.
(B) 𝐵1 is bounded on Ω with 𝜎𝐵1 ≡ sup(𝑡,𝑥)∈Ω ‖𝐵1 (𝑡, 𝑥)‖ <
≡
∞; 𝐵2 is bounded on Ω with 𝜎𝐵2
sup(𝑡,𝑥)∈Ω ‖𝐵2 (𝑡, 𝑥)‖ < ∞.
2
Journal of Function Spaces
Definition 1. A set-valued map 𝐹 : 𝑅𝑛 𝑅𝑛 is said to be
𝑛
(i) monotone on a convex set 𝐾 ⊂ 𝑅 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 set 𝐾 ⊂ 𝑅𝑛 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 function 𝑓 : Ω → 𝑅𝑛 (resp., 𝐵 : Ω → 𝑅𝑛×𝑚 )
is said to be Lipschitz continuous if there exists a constant
𝐿 𝑓 > 0 (resp., 𝐿 𝐵 > 0) such that, for any (𝑡1 , 𝑥), (𝑡2 , 𝑦) ∈ Ω,
𝑓 (𝑡1 , 𝑥) − 𝑓 (𝑡2 , 𝑦) ≤ 𝐿 𝑓 (𝑡1 − 𝑡2 + 𝑥 − 𝑦)
(resp., 𝐵 (𝑡1 , 𝑥) − 𝐵 (𝑡2 , 𝑦) ≤ 𝐿 𝐵 (𝑡1 − 𝑡2 + 𝑥 − 𝑦)) .
(2)
Definition 3. Let 𝑋, 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 𝑥0 ∈ 𝑋 if and only
if, for any neighborhood N(𝐹(𝑥0 )) of 𝐹(𝑥0 ), there exists a
neighborhood N(𝑥0 ) of 𝑥0 such that
𝐹 (𝑥) ⊂ N (𝐹 (𝑥0 )) ,
∀𝑥 ∈ N (𝑥0 ) .
(3)
Lemma 4 (see [1]). Let F : Ω 𝑅𝑚 be an upper semicontinuous set-valued map with nonempty closed convex
values. Suppose that there exists a scalar 𝜌F > 0 satisfying
sup {𝑦 : 𝑦 ∈ F (𝑡, 𝑥)} ≤ 𝜌F (1 + ‖𝑥‖) , ∀ (𝑡, 𝑥) ∈ Ω.
(4)
For every 𝑥0 ∈ 𝑅𝑛 , the 𝐷𝐼 : 𝑥̇ ∈ F(𝑡, 𝑥), 𝑥(0) = 𝑥0 has a weak
solution in the sense of Carathéodory.
Lemma 5 (see [1]). Let ℎ : Ω × 𝑅𝑚 → 𝑅𝑛 be a continuous
function and let 𝑈 : Ω 𝑅𝑚 be a closed set-valued map such
that, for some constant 𝜂𝑈 > 0,
sup ‖𝑢‖ ≤ 𝜂𝑈 (1 + ‖𝑥‖) ,
𝑢∈𝑈(𝑡,𝑥)
∀ (𝑡, 𝑥) ∈ Ω.
(5)
Let V : [0, 𝑇] → 𝑅𝑛 be a measurable function and let 𝑥 : [0, 𝑇]
→ 𝑅𝑛 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] → 𝑅𝑚 such that 𝑢(𝑡) ∈ 𝑈(𝑡, 𝑥(𝑡)) and
V(𝑡) = ℎ(𝑡, 𝑥(𝑡), 𝑢(𝑡)) for almost all 𝑡 ∈ [0, 𝑇].
̂ denote the Lebesgue measure on
Lemma 6 (see [20]). Let 𝑚
𝑅𝑛 and let 𝑓 : 𝑅𝑛 → 𝑅𝑚 be a measurable function. Let 𝐿 be
̂
< ∞. Then, for any 𝜀 > 0,
a measurable set in 𝑅𝑛 with 𝑚(𝐿)
̂ \ 𝐾) < 𝜀 such that
there exists a compact set 𝐾 ⊆ 𝐿 with 𝑚(𝐿
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 map 𝐹 : 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, 𝐵1 , 𝐵2 , 𝐺1 , 𝐺2 ) satisfy conditions
(A) and (B) and 𝐹𝑖 : 𝑅𝑛 𝑅𝑛 (𝑖 = 1, 2) are upper
semicontinuous with nonempty and compact values such that
𝑞𝑖 + 𝐹𝑖 (𝑖 = 1, 2) are pseudo monotone on 𝑅𝑛 for each 𝑞𝑖 ∈
𝐺𝑖 (Ω) (𝑖 = 1, 2). If 𝐾 is a bounded, closed, and convex subset
of 𝑅𝑛 , 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 in [19], it is easy to see that the assumption “𝐹
is pseudo monotone on 𝑅𝑛 ” in there should be replaced by
the assumption “𝑞 + 𝐹 is pseudo monotone on 𝑅𝑛 for each
𝑞 ∈ 𝐺(Ω).” Since 𝐾 is a bounded, closed, and convex subset
of 𝑅𝑛 , it fol (...truncated)
