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Abstract
In this work, we study the asymptotic behavior of the mild solutions of a class of stochastic partial functional integrodifferential equation on Hilbert spaces. Using the stochastic convolution developed, we establish the exponential stability in mean square with p ≥ 2. Also, pathwise exponential stability is proved for p> 2. We extend the result of an example is provided for illustration.
1. Introduction
Stochastic delay differential equations (SDDEs) play an important role in many branches of science and industry. Such models have been used with great success in a variety of application areas, including biology, epidemiology, mechanics, economics, and finance. In the past few decades, qualitative theory of SDDEs has been studied intensively by many scholars. Here, we refer to Da Prato and Zabczyk (1992a) and references therein. In recent years, existence, uniqueness, stability, and other quantitative and qualitative properties of solutions to stochastic partial differential equations have been extensively investigated by several authors.
The existence, uniqueness and asymptotic behavior of mild solutions of some stochastic partial differential equations on Hilbert spaces were considered by applying the comparison theorem in Govindan (2002, 2003). Taniguchi (1995) and Taniguchi, Liu, and Truman (2002) have studied the existence, uniqueness and asymptotic behavior of mild solutions of stochastic partial functional differential equation on Hilbert space by using the semigroup approach. Liu and Truman (2000) and Taniguchi (1998) have proved the almost sure exponential stability of mild solution for stochastic partial functional differential equation by using the analytic technique. Liu and Shi (2006), and Liu (2006) have considered the exponential stability for stochastic partial functional differential equations by means of the Razuminkhin-type theorem.
The stochastic integrodifferential equations are more general and still in a state of flux, with new basic results continuously emerging. Integrodifferential equations are important for investigating some problems raised from natural phenomena. They have applications in many areas such as physics, chemistry, economics, social sciences, finance, population dynamics, electrical engineering, medicine biology, ecology and other areas of science and engineering. Qualitative properties such as existence, uniqueness, optimality conditions, controllability and stability for various linear and nonlinear stochastic partial integrodifferential equations have been extensively studied by many researchers, see for instance (Balachandran & Sakthivel, 2001; Diagana, Hernàndez, & Dos Santos, 2009; Dieye, Diop, & Ezzinbi, 2016a, 2016b, 2017; Diop, Ezzinbi, & Lo, 2012; Dos Santos, Guzzo, & Rabelo, 2010; Ezzinbi & Ghnimi, 2010; Sathya & Balachandran, 2012) and the references therein.
As the motivation of above-discussed works, we consider the following stochastic partial functional integrodifferential equation: (1)
(1)
where generates a
-semigroup on a separable Hilbert
,
a closed linear operator on
with time-independent domain
.
,
are Lipschitzian functions in
and continuous in
.
is a Wiener process on the separable Hilbert space
with covariance operator
.
is a
-measurable
-valued square-integrable random variable.
In this work, our main aim is to study the exponential stability in -th mean and also almost sure stability property of mild solutions for the system (1) by using the theory of resolvent operator as developed by Grimmer (1982) and the properties of stochastic convolution developed in Dieye, Diop, and Ezzinbi (2016c). The analysis of (1) when
was initiated in Taniguchi (1995), where the authors proved the existence and stability of solutions by using a strict contraction principle. The main contribution of this paper is on finding conditions to assure the existence, uniqueness, and stability of impulsive neutral stochastic integrodifferential equations. Our paper expands the usefulness of stochastic integrodifferential equations since the literature shows results for existence and stability for such equations under semigroup theory.
The remaining of the paper is organized as follows. Section 2, presents notations and preliminary results. We study also the existence of the mild solutions of Equation (1). Section 3, shows the stability of the mild solutions. Finally, Section 4, presents an example that illustrates our results.
2. Stochastic processes and integrodifferential equations
Let and
be Banach spaces.
denotes the space of bounded linear operator from
to
, simply
when
. We will assume that
is a complete filtered probability space. We are given a
-Wiener process the probability space and having value in
a separable Hilbert space, one can construct
as follows,
where
is a sequence of real-valued standard Brownian motions mutually independent of
,
are positive real numbers such that
is a complete orthonormal basis in
, and
is the incremental covariance operator of the
which is a symmetric nonnegative trace class operator defined by
For this analysis, we recall the definition of -valued stochastic integral with respect to the
-valued
-Wiener process
. Let
denote the space of all Hilbert–Schmidt operator from
to
which is a separable Hilbert space, equipped with the following norm:
Clearly, for any bounded operators , this norm is given by
Let be a predictable
-adapted process such
Then, define the -valued stochastic integral
which is a continuous square integrable martingale. For more details on stochastic integrals, we refer to Da Prato and Zabczyk (1992a).
Next, we recall conditions that guarantee the existence of solution for the deterministic, integrodifferential equation (2)
(2)
Definition 2.1. (Grimmer, 1982) A resolvent operator for Equation (2) is a bounded linear operator-valued function for
, having the following properties:
(i) (the identity map of
) and
for some constants
and
(ii) For each ,
is strongly continuous for
.
(iii) for
. For
and
In the whole of this work, we assume that
(A1) The operator is the infinitesimal generator of a
-semigroup
on
.
(A2) For all ,
is closed linear operator from
to
and
. For any
, the map
is bounded, differentiable and the derivative
is bounded uniformly continuous on
.
The following theorem gives a satisfactory answer to the problem of existence of solutions.
Theorem 2.1. (Grimmer, 1982) Assume that hold. Then there exists a unique resolvent operator for the Cauchy problem (2).
In the following, we give some results for the existence of solutions for the following integro-differential equation: (3)
(3)
where is a continuous function.
Definition 2.2. (Grimmer, 1982) A continuous function is said to be a strict solution of Equation (3) if
and
satisfies Equation (3).
Theorem 2.2. (Grimmer, 1982) Assume that hold. If
is a strict solution of Equation (3), then
In order to set our problem, we make the following assumptions
(A3) There exist and
such that the resolvent operator
of Equation (2) satisfies
(4)
(4)
The exponential stability of the resolvent operator will play a crucial role to prove the main results of this work. It has been studied in Grimmer (1982).
(A4) The functions and
are Lipschitz continuous. Let
be such that for every
and
the following conditions are satisfied:
(5)
(5)
(6)
(6)
(7)
(7)
2.1. Existence of the mild solution of Equation (1)
The next definition introduces the concept of solution for the stochastic system (1).
Definition 2.3. A mild solution of the integrodifferential Equation (1) on is a stochastic process
defined on the probability space
such that
is
-adapted predictable on
satisfies with probability one,
and
Let and
denote the space of
-adapted predictable random process with values in the Hilbert space
satisfying
We define the following norm on by
is a norm. Then, we have
Lemma 2.3 (Dieye et al., 2016c) is a Banach space.
Theorem 2.4. If hypotheses and
hold, then for each initial datum
-measurable
-valued square-integrable random variable, the integrodifferential Equation (1) has a unique mild solution on
.
. Let
. We define for
the following applications on
by
(9)
(9)
(10)
(10)
Since , then
, therefore
becomes a complete metric space. Now, we define the following map:
(11)
(11)
Note that a fixed point of is a mild solution of Equation (1). Using the same arguments developed in Dieye et al. (2016c), we obtain that
applied
to itself. Moreover, we have the following estimation:
(12)
(12)
where with
. Taking the supremum over
, we get that
(13)
(13)
By iterative process involving repeated substitution of the expression (13) into itself, we obtain after iterations the following inequality:
(14)
(14)
where and
denotes the
-fold composition of the operator
. Hence, by taking
the above inequality gives that
(15)
(15)
The constants and
being finite, then for
large enough:
and hence the
-th iterate
of the operator
is a contraction on the metric space
. Since this is a complete metric space it follows from Banach fixed point Theorem that for
sufficiently large,
has a unique fixed point, that is a unique fixed point also of
. We deduce that the mild solution exists on
, this is true for any
which means, we have a global existence.
3. Exponential stability of the mild solutions Equation (1)
We are mainly interested in the stability properties of the mild solutions, we consider the following equation: (16)
(16)
instead of Equation (1).
3.1. Exponential stability in the -th mean
In the next, we discuss the exponential asymptotic stability in the -th mean of mild solutions of Equation (1). From now on, let
denote the solution of Equation (16) with an initial value
random variable and we always assume that
is
measurable with
and
is independent of
.
Definition 3.1. Let be an integer. The mild solution
of Equation (1) is said to be globally exponentially asymptotically stable in the
-th mean if there exist
and
such that, for any mild solution of Equation (1),
corresponding to an initial value
with
, the following inequality holds:
(17)
(17)
Theorem 3.1. Let be an integer and let
and
be solutions of Equation (16) with initial values
and
respectively. Supposse that
hold true. Then, the following inequality holds:
where
. Let
and
be solutions of Equation (16) with initial values
and
respectively. Then, we have
Thus it follows that (19)
(19)
Now, we compute the terms on the right-hand side of the above inequality. From assumption (A3), we have (20)
(20)
From the assumptions (A3) and (A3), we obtain (21)
(21)
where . To estimate
, we recall the following result
Lemma 3.2. (Da Prato & Zabczyk, 1992a) For any and for an
-predictable process
we have the following inquality
By using Hölder’s inequality we obtain that (23)
(23)
where and
We remark if
, the inequality (23) holds with convention
. From inequalities (20), (21) and (23), one can see that the inequality (19) becomes
that is, (24)
(24)
Hence Gronwall’s inequality yields (25)
(25)
that is, (26)
(26)
which completes the proof.
Consequently, we have the following result as corollary.
Corollary 3.3. Suppose that all hypotheses of Theorem 3.1 hold and let . Then mild solutions of Equation (1) are globally exponentially asymptotically stable in the
-th mean.
Corollary 3.4. Assume that all hypotheses of Theorem 3.1 hold, and
and
. Then
(27)
(27)
3.2. Almost sure asymptotic stability
In this subsection, we state the pathwise asymptotic stability for the mild solutions of equation (1). Due to the properties of the stochastic convolution, we study the case . At first, we need the following lemma
Lemma 3.5. (Dieye et al., 2016c) Suppose that the hypothesis and
are satisfied. Let
be a predictable,
-adapted process with
for some integer
and any
. Then there exists a constant
such that for any
, the following holds
(28)
(28)
Theorem 3.6. Supposse that hold. Let
be an integer,
and
be solutions of Equation (16) with initial values
and
respectively. If
then there exists
such that for
, we have
(29)
(29)
. Let
be a sufficiently large integer and
denote the interval
. Then for
, we have
It follows that
Thus, for any fixed , we obtain that
By Theorem 3.1 and Holder’s inequality, we have
Moreover,
Using Lemma 3.5, we get that
It follows that (30)
(30)
where
Now, for each integer , we set
. Then, it follows
(31)
(31)
where .
We consider the subset of given by
.
By using inequality (31) and applying Borel-Cantelli’s Lemma it follows that
Then, the set of all such that there exists an infinite number index
with
is a negligible set. Hence, for almost sure, there exists infinite number index
such that
i.e
Let be the lower bound of index
almost surely.
For any , there exists a positive integer
such that
. For
, we have
It follows that
Hence, for , we have
(32)
(32)
Corollary 3.7. Under the hypotheses of Theorem 3.6, if , then the zero solution is almost sure asymtotically stable.
4. Application
Consider the following stochastic partial functional integrodifferential equation: (33)
(33)
where denotes the standard
-valued Brownian motion,
is continuous.
Let with the norm
and
denote the completed orthonormal basis in
.
are bounded Lipschitz functions and
is a given continuous function.
Let be a complete filtered probability space where the Brownian motion
is defined. Let
, where
are one-dimensional standard Brownian motion mutually independent of
. Let
Then
is a
valued
-Brownian motion.
Define by
, with domain
.
Then , where
, is, also the orthonormal set of eigenvectors of
. It is well known that
is the infinitesimal generator of a strongly continuous semigroup on
; thus,
is true. Moreover,
Let be the operator defined by
for
and
.
Let
Then, Equation (33) takes the following form (34)
(34)
Clearly and
satisfy the assumption
with
and
where
and
are the Lipschitz constants of the functions
and
, respectively.
Moreover, if is bounded and
function such that
is bounded and uniformly continuous, then
and
are satisfied, and hence, by Theorem 2.1, Equation (2) has a resolvent operator
on
.
Proposition 4.1. (Dieye et al., 2016c) Suppose that is bounded and
function such that
is bounded and uniformly continuous and
for all
where
. Then the resolvent operator of the abstract form of Equation (34) decays exponentially to zero. Specifically
where
.
In the next, we assume that is bounded and
function such that
is bounded and uniformly continuous and
for all
where
.
Therefore, by Theorem 2.4 the existence and uniqueness of the mild solution of the stochastic partial functional integrodifferential Equation (33) is true.
Considering the case , we have the following constants:
(35)
(35)
(36)
(36)
(37)
(37)
Hence . Then, by Corollary 3.3, we have
Proposition 4.2. The system (33) is mean square globally exponentially aymtotically stable provided that:
For , we have
Therefore .
Moreover, by Theorem 3.6, we have also that
Proposition 4.3 If
Then the mild solution of the system (33) is exponentially aymtotically stable. If the pathwise exponential stabilitye is true.
Theorem 4.1. If and condition (39) holds, then zero solutions of the stochastic partial functional integrodifferential Equation (33), is almost sure asymtotically stable.
5. Conclusion
In this paper, we have initiated a study on stochastic neutral partial functional differential equations in a real separable Hilbert space. By using stochastic convolution, existence and uniqueness have been discussed. Further, the exponential stability of the moments of the mild solution as well its sample paths have been studied.