Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 439724, 10 pages
doi:10.1155/2011/439724
Research Article
Existence and Uniqueness of
Mild Solutions for Nonlinear Stochastic
Impulsive Differential Equation
L. J. Shen1, 2 and J. T. Sun1
1
2
Department of Mathematics, Tongji University, Shanghai 200092, China
Department of Mathematics, Luoyang Normal University, Luoyang 471022, China
Correspondence should be addressed to J. T. Sun, sunjt@sh163.net
Received 9 May 2011; Accepted 9 October 2011
Academic Editor: Jinhu Lü
Copyright q 2011 L. J. Shen and J. T. Sun. 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.
This paper investigates the existence and uniqueness of mild solutions to the general nonlinear
stochastic impulsive differential equations. By using Schaefer’s fixed theorem and stochastic
analysis technique, we propose sufficient conditions on existence and uniqueness of solution for
stochastic differential equations with impulses. An example is also discussed to illustrate the
effectiveness of the obtained results.
1. Introduction
Impulsive dynamical systems often exhibit continuous evolutions typically described
by ordinary differential equations and instantaneous state jumps. Since many evolution
processes, optimal control models in economics, stimulated neural networks, frequencymodulated systems, and some motions of missiles or aircrafts are characterized by the
impulsive dynamical behavior, the study of impulsive systems is of great importance.
Nowadays, there has been increasing interest in the analysis and synthesis of impulsive
systems due to their significance both in theory and applications; see 1–4 and the references
therein.
On the other hand, stochastic differential equations SDEs have attracted a great
attention since they have been used extensively in many areas of application including
finance and social science. Existence, uniqueness, and qualitative analysis of solutions of
SDEs have been considered by many authors; see, for example, 5, 6 and Oksendal 7.
In particular, 7 has obtained sufficient conditions uniform Lipschitz condition and the
linear growth condition on the existence and uniqueness of solution of SDEs. With the same
2
Abstract and Applied Analysis
conditions, Mao 8 has also proved there exists a unique solution for neutral stochastic
functional differential equations with finite delay, and systems with infinite delay was
investigated by 9 under uniform Lipschitz condition and moment estimate.
Furthermore, besides impulsive effects, stochastic effects likewise exist in real systems.
Therefore, stochastic impulsive differential equations SIDEs describing these dynamical
systems subject to both stochastic and impulse changes have attracted considerable attention
10–14. Liu et al. 10 has proved the existence and uniqueness with Lipschitz and
linear growth condition by continuation method. Results of existence and uniqueness were
obtained by stochastic Lyapunov-type method in 11, 12. Besides that, the existing literature
of SIDEs involved mainly the issue on controllability and stability see 13, 14, and little
investigation, to our best knowledge, has been carried out for existence and uniqueness of
general nonlinear systems with stochastic impulsive structure. Unlike the previous results
in 10–12, this paper will deal with the situation when the drift and diffusion coefficients
do not satisfy Lipschitz condition. Sufficient conditions of existence and uniqueness
of mild solutions for general nonlinear SIDEs are developed based on the Schaefer’s
fixed point theorem instead of employing stochastic Lyapunov-type function. In addition,
some known results can been considered as special cases of the main theorems in this
paper.
The organization of this paper is as follows. In Section 1, we will recall briefly some
basic notations and preliminary facts which will be used throughout the paper. In Section 2,
sufficient conditions are obtained on existence and uniqueness of mild solutions to SIDEs.
A numerical example is given to illustrate the proposed conditions in Section 3. Finally,
concluding remarks are presented in Section 4.
2. Preliminaries
Let · denote the Euclidean norm in Rn . If A is a matrix, its Frobenius norm is represented by
A trAT A. {Ω, F, P} stands for a complete probability space with a filtration {Ft }t≥0
satisfying the usual conditions i.e., right continuous and F0 containing all P-null sets.
Assume that Bt B1 t, B2 t, . . . , Bl tT is an l-dimensional Brownian motion defined on
n
{Ω, F, P}. Denote LF
2 J, R as the family of all square integrable and Ft adapted measurable
processes xt xt, w mapping J 0, b into Rn such that sup0<s≤t Exs2 < ∞.
2
n
It is clear that LF
2 J, R is the Banach space with the norm topology given by xt∗ sup0≤s≤t Exs2 .
Consider stochastic differential equations with impulsive effects as follows:
dxt ft, xtdt gt, xtdBt,
Δx τk
Ik xτk ,
t ∈ J, t /
τk ,
k 1, 2, . . . , m,
2.1
x0 z,
where f : J × Rn → Rn , g : J × Rn → Rn×l are Borel measurable functions, Δxτk
xτk
− xτk xτk
− xτk− , Ik ∈ CRn , Rn . The initial value z is a F0 -measurable random
variable independent of Bt such that Ez2 < ∞.
Abstract and Applied Analysis
3
Definition 2.1. The functions f and g are said to be L2 -Carathéodory if
1 ft, x and gt, x are measurable in t and continuous in the second variable a.e.
t ∈ J,
2 for each positive constant q, there exists a square integrable function hq : J → R
such that
ft, u ∨ gt, u ≤ hq t,
u ≤ q,
a.e. t ∈ J.
2.2
Motivated by 1, 7, let us start by defining what we mean by a mild solution of 2.1.
n
Definition 2.2. A Rn -valued stochastic process xt xt, w ∈ LF
2 J, R defined on J is said to
be a mild solution of 2.1 with initial value x0 z if the following conditions are satisfied
1 xt, w is measurable as a function from J × Ω to Rn and xt is Ft -adapted;
2 Ex2 < ∞ for each t ∈ J;
n
3 for each z ∈ LF
2 J, R , the process x satisfies the following integral equation:
xt z t
fs, xsds t
0
gs, xsdBs 0
Ik xτk .
2.3
0<τk <t
Our main result is based on the following fixed point theorem of Schaefer.
Lemma 2.3 Schaefer fixed-point theorem. Let C, · be a convex subset of a normal linear
space, and let the operator A : C → C be completely continuous. Define
2.4
FA {x ∈ C : x λAx, λ ∈ 0, 1}.
Then, either
1 the set FA is unbounded or
2 the operator A has a fixed point in C.
3. Main Results
We are now in a position to state and prove our existence result of 2.1. Throughout this
paper, we make the following assumptions.
H1 The functions f : J × Rn → Rn and g : J × Rn → Rn×l are L2 -Carathéodory.
H2 There exist positive constants ck , k 1, 2, . . . , m, such that
Ik x ≤ ck ,
n
x ∈ LF
2 J, R .
H3 There exists a square integrable function p : J
nondecreasing function ψ : 0, ∞ → 0, ∞ such that
ft, x ∨ gt, x ≤ ptψx,
3.1
→ R
and a continuous
n
t ∈ J, x ∈ LF
2 J, R ,
3.2
4
Abstract and Applied Analysis
with
b
4
b 1p2 sds <
0
∞
c
du
,
ψ 2 u
3.3
where c 4Ez2 .
Theorem 3.1. Assume that H1 -H3 hold, then there is at least one solution of 2.1 on J.
n
Proof. For any xt ∈ LF
2 J, R , define the operator P by
Px z t
fs, xsds t
0
0
gs, xsdBs Ik xτk .
3.4
0<τk <t
By Lemma 2.3, we will first show that the operator P is completely continuous.
F
n
n
Step 1. P is continuous from LF
2 J, R to L2 J, R . Let xn be a sequence such that xn → x as
n
n → ∞ in LF
2 J, R , then
EP xn − P x2
2
t
t
E fs, xn − fs, x ds gs, xn − gs, x dBs Ik xn τk − Ik xτk 0
0
0<τ <t
k
2
2
t
≤ 3E fs, xn − fs, x ds 3E
Ik xn τk − Ik xτk 0
0<τ <t
k
2
t
3E gs, xn − gs, x dBs .
0
3.5
By the Bunyakovskii inequality, one has
2
t
t
2
E fs, xn − fs, x ds ≤ bE fs, xn − fs, x ds,
0
0
3.6
and by the Itô isometry it follows that
2
t
t
2
E gs, xn − gs, x dBs ≤ E gs, xn − gs, x ds.
0
0
3.7
Abstract and Applied Analysis
5
Hence, it can be obtained directly from 3.5–3.7 that
2
EP xn − P x ≤ 3bE
t
fs, xn − fs, x2 ds 3E
t
0
0
3m
gs, xn − gs, x2 ds
3.8
EIk xn τk − Ik xτk 2 .
0<τk <t
Since f is L2 -Carathéodory and Ik are continuous functions, by the Lebesgue dominated
convergence theorem, we see
P xn − P x2∗ ≤ 3bE
t
fs, xn − fs, x2 ds 3E
t
0
0
3m
gs, xn − gs, x2 ds
3.9
2
EIk xn τk − Ik xτk −→ 0
as n −→ ∞.
0<τk <t
Step 2. P maps bounded sets into bounded sets. That is enough to prove that there exists a
n
positive constant θ such that for each xt ∈ Bq {x ∈ LF
2 J, R , x ≤ q} one has P x∗ ≤ θ.
By the definition in 3.4, for each t ∈ J, xt ∈ Bq ,
EP x2 ≤ 4Ez2 4bE
t
fs, xs2 ds 4E
t
0
≤ 4Ez2 4b
t
0
2
0
h2q sds 4
≤ 4Ez 4b 4
t
0
t
0
gs, xs2 ds 4m
ck2
h2q sds 4m
0<τk <t
ck2
3.10
0<τk <t
h2q sds 4m
ck2 : θ.
0<τk <t
n
Step 3. P maps bounded sets into equicontinuous sets of LF
2 J, R .
Let t1 , t2 ∈ J, t1 < t2 , x ∈ Bq . Then, it yields
EP xt2 − P xt1 2
≤ 3m
t1 ≤t<t2
≤ 3m
t1 ≤t<t2
max{ck } 3t2 − t1 E
k
t2
fs, xs2 ds 3E
t1
max{ck } 3t2 − t1 1
t2
t1
t2
t1
gs, xs2 ds
3.11
h2q sds.
As t2 → t1 , the right-hand side of the inequality above tends to zero. The case 0 < t2 < t1 ≤ b
can be considered in a similar manner.
From Ascoli-Arzela theorem and Steps 1 to 3, the operator P is completely continuous.
To complete the proof of the theorem, it suffices to prove the following step.
6
Abstract and Applied Analysis
Step 4. There exists a priori bound of the set
n
FP x ∈ LF
2 J, R , x λP x for some λ ∈ 0, 1 .
3.12
Let xt ∈ FP , then xt λP xt for some 0 < λ < 1. Thus, for each t ∈ J,
xt λ z t
fs, xsds t
0
gs, xsdBs 0
Ik xτk ,
3.13
0<τk <t
which implies that
Ext2 ≤ 4Ez2 4bE
t
fs, xs2 ds 4E
0
t
0
gs, xs2 ds 4m
ck2 .
3.14
0<τk <t
Consider the function μt defined by
μt sup Exs2 , 0 ≤ s ≤ t ,
0 ≤ t ≤ b.
3.15
With the previous inequality, we have for t ∈ 0, b
μt ≤ 4Ez2 4b 4
t
0
p2 sψ 2 μs ds 4m
ck2 .
3.16
0<τk <t
Take the right-hand side of the inequality above as vt, then we get
v0 c 4Ez2 ,
3.17
μt ≤ vt ∀ t ∈ J,
and, moreover,
v t 4b 4p2 tψ 2 μt ≤ 4b 4p2 tψ 2 vt,
3.18
or, equivalently, by H3 , for each t ∈ J,
vt
v0
ds
≤
ψ 2 s
t
0
4b 4p2 sds ≤
b
0
4b 4p2 sds <
∞
c
ds
.
ψ 2 s
3.19
Hence, there exists a constant k such that vt ≤ k, t ∈ J, that is, μt xt2∗ ≤ vt ≤ k, t ∈ J,
which implies that FP is bounded. From the Schaefer’s fixed theorem, we deduce that P has
a fixed point xt which is a solution to 2.1. The proof is completed.
In this part, in order to attain the uniqueness result of 2.1, we propose the following
assumptions.
Abstract and Applied Analysis
7
A1 There exists a square integrable function lt : J → R
such that, for all x, y ∈
n
LF
2 R and t ∈ J,
ft, x − f t, y ∨ gt, x − g t, y ≤ ltx − y.
3.20
A2 There exist positive constants dk , k 1, 2, . . . , m satisfying
Ik x − Ik y ≤ dk x − y,
n
x, y ∈ LF
2 J, R .
3.21
Theorem 3.2. Assume that (A1 )-(A2 ) hold, then there exists a unique solution to 2.1 if
m
k1 dk
< 1.
Proof. Let P be an operator defined as in 3.4. To prove that P is contractive, introduce the
n
Bielecki-type norm on LF
2 J, R defined by
xB sup e−τLt x∗ ,
3.22
t∈J
t
where Lt 0 l2 sds and τ is sufficiently large constant.
n
Let x, y ∈ LF
2 J, R , then we have that, for each t ∈ J,
P x − P y
∗
t
Ik xτk − Ik yτk fs, x − f s, y ∗ ds gs, x − g s, y dBs ≤
∗
0
0
0<τ <t
t
∗
≤
t
0
t
2
bl sx − y∗ ds
t
0
1/2
gt, x − g t, y 2 ds
∗
1/2
bl se
2
m
k1
2
x − y∗ ds
2τLs −2τLs e
0
1/2
2
k
t
0
m
k1
dk x − y ∗
1/2
l se
2
2
x − y∗ ds
2τLs −2τLs e
dk eτLt e−τLt x − y∗
≤ x − yB
1
2τ
t
1/2
1/2
t 1
2τLs
2τLs
e
b e
ds
x − yB
ds
2τ 0
0
m
x − yB eτLt dk
k1
8
Abstract and Applied Analysis
m
1 ≤ x − yB √
b 1 eτLt dk eτLt x − yB
2τ
k1
⎛
⎞
m
1 ≤ eτLt ⎝
b 1 dk ⎠x − yB .
2τ
k1
3.23
Thus,
⎞
⎛
m
1
e−τLt P x − P y∗ ≤ ⎝
b 1 dk ⎠x − yB ,
2τ
k1
3.24
⎛
⎞
m
1
P x − P y ≤ ⎝
b 1 dk ⎠x − yB .
B
2τ
k1
3.25
therefore,
Hence, P is a contraction and it has a unique fixed point which is a solution to 2.1. The proof
is completed.
Remark 3.3. Note that the existence and uniqueness result in 7 can be inferred from
Theorems 3.1 and 3.2 in case that Ik 0 in 2.1. Furthermore, if the systems in 9 and
2.1 are both reduced to the impulsive differential equations, then we can deduce the results
on existence and uniqueness in 9 directly from Theorems 3.1 and 3.2. And, if the delay 0
in 2, then the conditions there can be inferred from Theorems 3.1 and 3.2 in case that g 0
in 2.1.
4. Example
Consider the following stochastic impulsive differential equation:
dx 1 tx dt t3 1 x dBt,
1
x
2
1−
−x
2
t ∈ 0, 1 \
1
,
2
− 1
1
arctan x
,
4
2
4.1
x0 0,
where I1 x 1/4 arctan x with I1 x ≤ π/8 and
I1 x − I1 y ≤ 1 x − y
4
n
∀ x, y ∈ LF
2 J, R .
4.2
Abstract and Applied Analysis
9
Assume that pt 1 t t3 and ψx x, then it can be easily verified that the conditions
H1 –H3 are satisfied, and by, Theorem 3.1, 4.1 has at least one solution.
n
3
For all x, y ∈ LF
2 J, R and t ∈ 0, 1, set lt 1 t t , then
ft, x − f t, y ∨ gt, x − g t, y ≤ ltx − y.
4.3
As stated in 4.2, dk 1/4 < 1; hence, 4.1 has a unique solution by Theorem 3.2.
As demonstrated in the example, if f or g takes the form such as ptx2 , then inequality
3.3 in hypothesis H3 cannot hold. Based on this observation, in the subsequent research,
some useful tools and conditions such as locally Lipschitz condition will be considered to
relax restrictions on f and g.
5. Conclusion
By using Schaefer’s fixed theorem, this paper has obtained sufficient conditions of existence
and uniqueness of mild solutions to general nonlinear stochastic impulsive differential
equations. In addition, some known results about systems without delay or impulse can be
considered as special cases of the main theorems.
Acknowledgment
This work is supported by the NSF of China under Grants 60874027,61174039, the Fundamental Research Funds for the Central Universities of China, and by Youth Science Foundation of Luoyang Normal University 2010-qnjj-002.
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Abstract and Applied Analysis
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