Existence and Stability Results for Semilinear
Systems of Impulsive Stochastic Differential
Equations with Fractional Brownian Motion
T. Blouhi
1
, T. Caraballo
2
and A. Ouahab
1
1
Laboratory of Mathematics, Univ Sidi Bel Abbes
PoBox 89, 22000 Sidi-Bel-Abb`es, Algeria.
E-mail: blouhitayeb@yahoo.com, agh ouahab@yahoo.fr
2
Depto. Ecuaciones Diferenciales y An´alisis Num´erico,
Universidad de Sevilla, Campus de Reina Mercedes
41012-Sevilla, Spain
E-mail: caraball@us.es
Abstract
Some results on the existence and uniqueness of mild solution for a system
of semilinear impulsive differential equations with infinite fractional Brownian
motions are proved. The approach is based on Perov’s fixed point theorem and
a new version of Schaefer’s fixed point theorem in generalized Banach spaces.
The relationship between mild and weak solutions and the exponential stability
of mild solutions are investigated as well. The abstract theory is illustrated with
an example.
Key words and phrases: Mild solutions, fractional Brownian motion, impulsive
differential equations, matrix convergent to zero, generalized Banach space, fixed point.
AMS (MOS) Subject Classifications: 34A37,60H99,47H10.
1 Introduction
Differential equations with impulses were considered for the first time by Milman and
Myshkis [23] and then it was followed by a period of active research which culminated
with the monograph by Halanay and Wexler [15]. Many phenomena and evolution pro-
cesses in physics, chemical technology, population dynamics, and natural sciences may
change their state abruptly or be subject to short-term perturbations. These perturba-
tions may be seen as impulses. Impulsive problems arise also in various applications in
communications, mechanics (jump discontinuities in velocity), electrical engineering,
medicine and biology. A comprehensive introduction to the basic theory is well devel-
oped in the monographs by Benchohra et al. [2], Graef et al. [14], Laskshmikantham
et al. [17], Samoilenko and Perestyuk [35].
Random differential and integral equations play an important role in characteriz-
ing many social, physical, biological and engineering problems; see for instance the
2 Stochastic differential equations
monographs by Da Prato and Zabczyk [11], Gard [12], Gikhman and Skorokhod [13],
Sobzyk [36] and Tsokos and Padgett [37]. For example, a stochastic model for drug
distribution in a biological system was described by Tsokos and Padgett [37] as a closed
system with a simplified heart, one organ or capillary bed, and re-circulation of a blood
with a constant rate of flow, where the heart is considered as a mixing chamber of con-
stant volume. For the basic theory concerning stochastic differential equations see the
monographs by Bharucha-Reid [3], Mao [22], Øksendal, [26], Tsokos and Padgett [37],
Sobczyk [36] and Da Prato and Zabczyk [11].
The study of impulsive stochastic differential equations is a relatively new research
area. The existence and stability of stochastic impulsive differential equations were
recently investigated, for example in [8, 14, 18–20,27,34, 38, 39].
This paper is concerned with a system of stochastic impulsive differential equations
of the following type:
dx(t) = (Ax(t) + f
1
(t, x(t), y(t))dt
+
∞
X
l=1
σ
1
l
(t, x(t)), y(t))dB
H
l
(t), t ∈ [0, T ], t 6= t
k
,
dy(t) = (Ay(t) + f
2
(t, x(t), y(t)))dt
+
∞
X
l=1
σ
2
l
(t, x(t), y(t))dB
H
l
(t), t ∈ J, t 6= t
k
,
∆x(t) = I
k
(x(t
k
)), t = t
k
k = 1, 2, . . . , m
∆y(t) = I
k
(y(t
k
)), t = t
k
k = 1, 2, . . . , m
x(0) = x
0
,
y(t) = y
0
,
(1.1)
where X is a real separable Hilbert space with inner product h·, ·i and induced norm k·k,
A : D(A) ⊂ X −→ X is the infinitesimal generator of a strongly continuous semigroup
of bounded linear operators (S(t))
t≥0
in X and f
1
, f
2
: [0, T ] ×X ×X −→ X are given
functions, B
H
l
is an infinite sequence of mutually independent fractional Brownian
motions, l = 1, 2, . . ., with Hurst parameter H, I
k
,
I
k
∈ C(X, X) (k = 1, 2, . . . , m),
σ
1
l
, σ
2
l
: J × X × X → L
0
Q
(Y, X). Here, L
0
Q
(Y, X) denotes the space of all Q-Hilbert-
Schmidt operators from Y into X, which will be also defined in the next section.
Moreover, the fixed times t
k
satisfy 0 < t
1
< t
2
< . . . < t
m
< T , and y(t
−
k
) and y(t
+
k
)
denote the left and right limits of y(t) at t = t
k
.
(
σ(t, x, y) = (σ
1
(t, x, y), σ
2
(t, x, y), . . .),
kσ(t, x, y)k
2
=
P
∞
j=1
kσ
j
(t, x, y)k
2
L
0
Q
< ∞
(1.2)
where σ(·, ·, ·) ∈ `
2
, and `
2
is given by
`
2
= {φ = (φ
j
)
j≥1
: [0, T ]×X×X → L
0
Q
(Y, X) : kφ(t, x, y)k
2
=
∞
X
j=1
kφ
j
(t, x, y)k
2
L
0
Q
< ∞}.
T. Blouhi, T. Caraballo and A. Ouahab 3
We denote a ∧b = min(a, b) and a ∨ b = max(a, b). It is obvious that system (1.1) can
be seen as a fixed point problem for the model
dz(t) = A
∗
z(t) + f(t, z(t))dt +
∞
X
l=1
σ
l
(t, z))dB
H
l
(t), t ∈ [0, T ], t 6= t
k
,
∆z(t) = I
∗
k
(z(t
k
)), t = t
k
k = 1, 2, . . . , m
z(0) = z
0
,
(1.3)
where
z(t) =
x(t)
y(t)
, A
∗
=
A 0
0 A
, f(t, z) =
f
1
(t, x(t), y(t))
f
2
(t, x(t), y(t))
, σ
l
(t, z) =
σ
1
l
(t, x, y)
σ
2
l
(t, x, y)
and z
0
=
x
0
y
0
.
In the deterministic framework, the above system was used to study initial value
problems and boundary value problems for nonlinear competitive or cooperative differ-
ential systems from mathematical biology [24] and mathematical economics [16] where
the model is usually considered in the operator form (1.1).
Some results on the existence of solutions for differential equations with infinite
Brownian motions were obtained in [7, 38]. Existence and uniqueness of mild solu-
tions to neutral stochastic delay functional integro-differential equations perturbed by
a fractional Brownian motion can be found in Caraballo and Diop [10].
Very recently, for A ≡ 0, X = R
n
and B
H
l
a standard Brownian motion, problem
(1.1) was studied by Blouhi et. al. [4]. In [5,6,31,32] the authors present existence and
uniqueness results for systems of semilinear differential equations without impulses.
Recently, Precup [31] proved the role of matrix convergence and vector metrics in the
analysis of semilinear operator systems.
The aim of this paper is to study existence, uniqueness and exponential stability
of mild solutions of semilinear systems of stochastic differential equations with infinite
fractional Brownian motions and impulses. The paper is organized as follows. In
sections 2 and 3 we introduce all the background material used in this paper such
as stochastic calculus and some properties of generalized Banach spaces. In Section
4 we state and prove our main results by using Perov’s and Schaefer’s fixed point
theorems in generalized Banach spaces. Finally, sections 5 and 6 are devoted to prove
the relationship between mild and weak solutions, and exponential stability of solutions
for Problem (1.1). Some application examples are finally considered in the last section.
2 Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are
used throughout this paper.
4 Stochastic differential equations
Let (Ω, F, P) be a complete probability space with a filtration (F = F
t
)
t≥0
satisfying
the usual conditions (i.e. right continuous and F
0
containing all P-null sets).
For a stochastic process x(·, ·) : [0, T ]×Ω → X we will write x(t) (or simply x when
no confusion is possible) instead of x(t, ω).
Definition 2.1. Given H ∈ (0, 1), a continuous centered Gaussian process B
H
is
said to be a two-sided one-dimensional fractional Brownian motion (fBm) with Hurst
parameter H, if its covariance function R
H
(t, s) = E[B
H
(t))B
H
(s)] satisfies
R
H
(t, s) =
1
2
(|t|
2H
+ |s|
2H
− |t − s|
2H
) t, s ∈ [0, T ].
It is known that B
H
(t) with H >
1
2
admits the following Volterra representation
B
H
(t) =
Z
t
0
K
H
(t, s)dB(s) (2.1)
where B is a standard Brownian motion given by
B(t) = B
H
((K
∗
H
)
−1
ξ
[0,t]
),
and the Volterra kernel the kernel K(t, s) is given by
K
H
(t, s) = c
H
s
1/2−H
Z
t
s
(u − s)
H−
3
2
u
s
H−
1
2
du, t ≥ s,
where c
H
=
q
H(2H−1)
β(2H−2,H−
1
2
)
and β(·, ·) denotes the Beta function, K(t, s) = 0 if t ≤ s,
and it holds
∂K
H
∂t
(t, s) = c
H
t
s
H−
1
2
(t − s)
H−
3
2
,
and the kernel K
∗
H
is defined as follows. Denote by E the set of step functions on
[0, T ]. Let H be the Hilbert space defined as the closure of E with respect to the scalar
product
hχ
[0,t]
, χ
[0,s]
i
H
= R
H
(t, s),
and consider the linear operator K
∗
H
from E to L
2
([0, T ]) defined by,
(K
∗
H
φ)(t) =
Z
T
s
φ(t)
∂K
H
∂t
(t, s)dt.
Notice that,
(K
∗
H
χ
[0,t]
)(s) = K
H
(t, s)χ
[0,t]
(s).
The operator K
∗
H
is an isometry between E and L
2
([0, T ]) which can be extended to
the Hilbert space H. In fact, for any s, t ∈ [0, T ] we have
hK
∗
H
χ
[0,t]
, K
∗
H
χ
[0,t]
i
L
2
([0,T ])
= hχ
[0,t]
, χ
[0,s]
i
H
= R
H
(t, s).
T. Blouhi, T. Caraballo and A. Ouahab 5
In addition, for any φ ∈ H,
Z
T
0
φ(s)dB
H
(s) =
Z
T
0
(K
∗
H
φ)(s)dB(s),
if and only if K
∗
H
φ ∈ L
2
([0, T ]). Moreover, the following useful result holds
Lemma 2.1. [25] There exists a positive constant c
1
(H) such that for any φ ∈
L
1/H
([0, T ]) it holds
H(2H − 1)
Z
T
0
Z
T
0
|φ(y)||φ(z)||y − z|
2H−2
dydz ≤ c
1
(H)kφk
2
L
1/H
([0,T ])
. (2.2)
Next we are interested in considering an fBm with values in a Hilbert space and
giving the definition of the corresponding stochastic integral.
Definition 2.2. An F
t
-adapted process φ on [0, T ]×Ω → X is an elementary or simple
process if for a partition ψ = {
¯
t
0
= 0 <
¯
t
1
< . . . <
¯
t
n
= T } and (F
¯
t
i
)-measurable X-
valued random variables (φ
¯
t
i
)
1≤i≤n
, φ
t
satisfies
φ
t
(ω) =
n
X
i=1
φ
i
(ω)χ
(
¯
t
i−1
,
¯
t
i
]
(t), for 0 ≤ t ≤ T, ω ∈ Ω.
The Itˆo integral of the simple process φ is defined as
I
H
(φ) =
Z
T
0
φ
l
(s)dB
H
l
(s) =
n
X
i=1
φ
l
(
¯
t
i
)(B
H
l
(
¯
t
i
) − B
H
l
(
¯
t
i−1
)), (2.3)
whenever φ
¯
t
i
∈ L
2
(Ω, F
¯
t
i
P, X) for all i ≤ n.
Let (X, h·, ·i, |·|
X
), (Y, h·, ·i, |·|
Y
) be separable Hilbert spaces. Let L(Y, X) denote
the space of all linear bounded operators from Y into X. Let e
n
, n = 1, 2, . . . be a com-
plete orthonormal basis in Y and Q ∈ L(Y, X) be an operator defined by Qe
n
= λ
n
e
n
with finite trace trQ =
P
∞
n=1
λ
n
< ∞ where λ
n
, n = 1, 2, . . ., are non-negative real
numbers. Let (β
H
n
)
n∈N
be a sequence of two-sided one-dimensional standard fractional
Brownian motions mutually independent on (Ω, F, P). If we define the infinite dimen-
sional fBm on Y with covariance Q as
B
H
(t) =
∞
X
n=1
p
λ
n
β
H
n
(t)e
n
, (2.4)
then it is well defined as an Y -valued Q-cylindrical fractional Brownian motion (see [9])
and we have
Ehβ
H
l
(t), xihβ
H
k
(s), yi = R
H
lk
(t, s)hQ(x), yi, x, y ∈ Y and s, t ∈ [0, T ]