- Research Article
- Open access
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Existence and Stability of Solutions for Nonautonomous Stochastic Functional Evolution Equations
Journal of Inequalities and Applications volume 2009, Article number: 785628 (2009)
Abstract
We establish the results on existence and exponent stability of solutions for a semilinear nonautonomous neutral stochastic evolution equation with finite delay; the linear part of this equation is dependent on time and generates a linear evolution system. The obtained results are applied to some neutral stochastic partial differential equations. These kinds of equations arise in systems related to couple oscillators in a noisy environment or in viscoelastic materials under random or stochastic influences.
1. Introduction
In this paper we study the existence and asymptotic behavior of mild solutions for the following neutral non-autonomous stochastic evolution equation with finite delay:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ1_HTML.gif)
where generates a linear evolution system, or say linear evolution operator
on a separable Hilbert space
with the inner product
and norm
.
;
and
are given functions to be specified later.
In recent years, existence, uniqueness, stability, invariant measures, and other quantitative and qualitative properties of solutions to stochastic partial differential equations have been extensively investigated by many authors. One of the important techniques to discuss these topics is the semigroup approach; see, for example, Da Prato and Zabczyk [1], Dawson [2], Ichikawa [3], and Kotelenez [4]. In paper [5] Taniguchi et al have investigated the existence and asymptotic behavior of solutions for the following stochastic functional differential equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ2_HTML.gif)
by using analytic semigroups approach and fractional power operator arguments. In this work as well as other related literatures like [6–9], the linear part of the discussed equation is an operator independent of time and generates a strongly continuous (one-parameter) semigroup or analytic semigroup so that the semigroup approach can be employed. We would also like to mention that some similar topics to the above for stochastic ordinary functional differential equations with finite delays have already been investigated successfully by various authors (cf. [6, 10–13] and references in [14] among others). Related work on functional stochastic evolution equations of McKean-Vlasov type and of second-order are discussed in [15, 16].
However, it occurs very often that the linear part of (1.2) is dependent on time . Indeed, a lot of stochastic partial functional differential equations can be rewritten to semilinear non-autonomous equations having the form of (1.2) with
. There exists much work on existence, asymptotic behavior, and controllability for deterministic non-autonomous partial (functional) differential equations with finite or infinite delays; see, for example, [17–20]. But little is known to us for non-autonomous stochastic differential equations in abstract space, especially for the case that
is a family of unbounded operators.
Our purpose in the present paper is to obtain results concerning existence, uniqueness, and stability of the solutions of the non-autonomous stochastic differential equations (1.1). A motivation example for this class of equations is the following non-autonomous boundary problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ3_HTML.gif)
As stated in paper [21], these problems arise in systems related to couple oscillators in a noisy environment or in viscoelastic materials under random or stochastic influences (see also [22] for the discussion for the corresponding determined systems). Therefore, it is meaningful to deal with (1.1) to acquire some results applicable to problem (1.3). In paper [21], Caraballo et al. have, under coercivity condition in an integral form, investigated the second moment (almost sure) exponential stability and ultimate boundedness of solutions to the following non-autonomous semilinear stochastic delay equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ4_HTML.gif)
on a Hilbert space , where
with
.
As we know, non-autonomous evolution equations are much more complicate than autonomous ones to be dealt with. Our approach here is inspired by the work in paper [5, 18, 19]. That is, we assume that is a family of unbounded linear operators on
with (common) dense domain such that it generates a linear evolution system. Thus we will apply the theory of linear evolution system and fractional power operators methods to discuss existence, uniqueness, and
(
) moment exponential stability of mild solutions to the stochastic partial functional differential equation (1.1). Clearly our work can be regarded as extension and development of that in [5, 21] and other related papers mentioned above.
We will firstly in Section 2 introduce some notations, concepts, and basic results about linear evolution system and stochastic process. The existence and uniqueness of mild solutions are discussed in Section 3 by using Banach fixed point theorem. In Section 4, we investigate the exponential stability for the mild solutions obtained in Section 3, and the conditions for stability are somewhat weaker than in [5]. Finally, in Section 5 we apply the obtained results to (1.3) to illustrate the applications.
2. Preliminaries
In this section we collect some notions, conceptions, and lemmas on stochastic process and linear evolution system which will be used throughout the whole paper.
Let be a probability space on which an increasing and right continuous family
of complete sub-
-algebras of
is defined.
and
are two separable Hilbert space. Suppose that
is a given
-valued Wiener Process with a finite trace nuclear covariance operator
. Let
,
, be a sequence of real-valued one-dimensional standard Brownian motions mutually independent over
. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ5_HTML.gif)
where are nonnegative real numbers, and
is a complete orthonormal basis in
. Let
be an operator defined by
with finite trace
. Then the above
-valued stochastic process
is called a
-Wiener process.
Definition 2.1.
Let and define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ6_HTML.gif)
If , then
is called a
-Hilbert-Schmidt operator, and let
denote, the space of all
-Hilbert-Schmidt operators
.
In the next section the following lemma (see [1, Lemma  7.2]) plays an important role.
Lemma 2.2.
For any and for arbitrary
-valued predictable process
,
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ7_HTML.gif)
for some constant .
Now we turn to state some notations and basic facts from the theory of linear evolution system.
Throughout this paper, is a family of linear operators defined on Hilbert space
, and for this family we always impose on the following restrictions.
The domain of
is dense in
and independent of
;
is closed linear operator.
For each , the resolvent
exists for all
with Re
and there exists
so that
.
There exists and
such that
for all
.
Under these assumptions, the family generates a unique linear evolution system, or called linear evolution operators
, and there exists a family of bounded linear operators
with
such that
has the representation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ8_HTML.gif)
where denotes the analytic semigroup having infinitesimal generator
(note that Assumption
guarantees that
generates an analytic semigroup on
).
For the linear evolution system , the following properties are well known:
(a), the space of bounded linear transformations on
, whenever
and maps
into
as
. For each
, the mapping
is continuous jointly in
and
;
(b) for
;
(c);
(d) for
.
We also have the following inequalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ9_HTML.gif)
Furthermore, Assumptions imply that for each
the integral
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ10_HTML.gif)
exists for each . The operator defined by (2.6) is a bounded linear operator and yields
. Thus, we can define the fractional power as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ11_HTML.gif)
which is a closed linear operator with dense in
and
for
.
becomes a Banach space endowed with the norm 
, which is denoted by
.
The following estimates and Lemma 2.3 are from ([23, Part II]):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ12_HTML.gif)
for and
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ13_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ14_HTML.gif)
for some , where
and
indicate their dependence on the constants
,
.
Lemma 2.3.
Assume that hold. If
,
,
, then, for any
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ15_HTML.gif)
For more details about the theory of linear evolution system, operator semigroups, and fraction powers of operators, we can refer to [23–25].
In the sequel, we denote for brevity that for some
, and
, the space of all continuous functions from
into
. Suppose that
,
, is a continuous
-adapted,
-valued stochastic process, we can associate with another process
,
, by setting
,
. Then we say that the process
is generated by the process
. Let
,
, denote the space of all
-measurable functions which belong to
; that is,
,
, is the space of all
-measurable
-valued functions
with the norm
.
Now we end this section by stating the following result which is fundamental to the work of this note and can be proved by the similar method as that of [1, Proposition  4.15].
Lemma 2.4.
Let ,
be a predictable,
-adapted process. If
,
, for arbitrary
, and
,
, then there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ16_HTML.gif)
3. Existence and Uniqueness
In this section we study the existence and uniqueness of mild solutions for (1.1). For this equation we assume that the following conditions hold (let ).
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_IEq158_HTML.gif)
The function satisfies the following Lipschitz conditions: that is, there is a constant
such that, for any
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ17_HTML.gif)
and .
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_IEq165_HTML.gif)
The function satisfies the following Lipschitz conditions: that is, there is a constant
such that, for any
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_IEq170_HTML.gif)
For function , there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ19_HTML.gif)
for any and
Under and
, we may suppose that there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ20_HTML.gif)
for any and
.
Similar to the deterministic situation we give the following definition of mild solutions for (1.1).
Definition 3.1.
A continuous process is said to be a mild solution of (1.1) if
(i) is measurable and
-adapted for each
;
(ii), a.s.;
(iii) verifies the stochastic integral equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ21_HTML.gif)
on interval , and
for
.
Next we prove the existence and uniqueness of mild solutions for (1.1).
Theorem 3.2.
Let and
. Suppose that the assumptions
hold. Then there exists a unique (local) continuous mild solution to (1.1) for any initial value
.
Proof.
Denote by the Banach space of all the continuous processes
which belong to the space
with
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ22_HTML.gif)
Define the operator on
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ23_HTML.gif)
Then it is clear that to prove the existence of mild solutions to (1.1) is equivalent to find a fixed point for the operator . Next we will show by using Banach fixed point theorem that
has a unique fixed point. We divide the subsequent proof into three steps.
Step 1.
For arbitrary ,
is continuous on the interval
in the
-sense.
Let and
be sufficiently small. Then for any fixed
, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ24_HTML.gif)
Thus, by Lemma 2.3 we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ25_HTML.gif)
where satisfies that
and
are constant. From Condition
it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ26_HTML.gif)
And by (2.8)–(2.10) one has that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ27_HTML.gif)
where solves
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ28_HTML.gif)
while
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ29_HTML.gif)
where is very small. Since
is uniformly continuous in
for
,
and
, where
is any positive number (see [23, 25]), we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ30_HTML.gif)
In a similar way, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ31_HTML.gif)
By virtue of Condition and by using Lemma 2.2, we infer that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ32_HTML.gif)
while
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ33_HTML.gif)
Again by the uniform continuity of and (2.9) and (2.10), we can compute that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ34_HTML.gif)
where .
The above arguments show that ,
and
are all tend to
as
and
, and
also clearly tends to
from Condition
. Therefore,
is continuous on the interval
in the
-sense.
Step 2.
We prove that .
To this end, let . Then we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ35_HTML.gif)
Again by Lemma 2.3, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ36_HTML.gif)
By Condition one easily has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ37_HTML.gif)
And (2.9) and (2.10) imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ38_HTML.gif)
From the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ39_HTML.gif)
established in [26], it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ40_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ41_HTML.gif)
and . Thus,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ42_HTML.gif)
where . Hence
and so
.
Step 3.
It remains to verify that is a contraction on
.
Suppose that ,
, then for any fixed
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ43_HTML.gif)
Clearly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ44_HTML.gif)
Next, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ45_HTML.gif)
then there holds that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ46_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ47_HTML.gif)
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ48_HTML.gif)
Then we can take a suitable sufficient small such that
, and hence
is a contraction on
(
denotes
with
substituted by
). Thus, by the well-known Banach fixed point theorem we obtain a unique fixed point
for operator
, and hence
is a mild solution of (1.1). This procedure can be repeated to extend the solution to the entire interval
in finitely many similar steps, thereby completing the proof for the existence and uniqueness of mild solutions on the whole interval
.
For the globe existence of mild solutions for (1.1), it is easy to prove the following result.
Theorem 3.3.
Suppose that the family satisfies
on interval
such that
is defined for all
. Let the functions
,
and
satisfy the Assumptions
respectively. Then there exists a unique, global, continuous solution
to (1.1) for any initial value
.
Proof.
Since is arbitrary in the proof of the previous theorem, this assertion follows immediately.
4. Exponential Stability
Now, we consider the stability result of mild solutions to (1.1). For this purpose we need to assume further that the family verifies additionally the following.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_IEq274_HTML.gif)
, and there exists a closed operator with bounded inverse and domain
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ49_HTML.gif)
as . Then the following inequalities are true:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ50_HTML.gif)
Theorem 4.1.
Let the functions ,
,
satisfy the Lipschitz conditions
, respectively. Furthermore, assume that there exist nonnegative real numbers
and continuous functions
with
(
),
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ51_HTML.gif)
for any mild solution of (1.1). If the constants
and
are small enough such that
with
determined by (4.13) below, then the solution is (the
th moment) exponentially stable. In other words, there exist positive constants
and
such that, for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ52_HTML.gif)
Proof.
Let be a mild solution of (1.1), then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ53_HTML.gif)
For we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ54_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ55_HTML.gif)
with satisfying
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ56_HTML.gif)
Since, by Lemma 2.2,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ57_HTML.gif)
and the Young inequality enables us to get immediately that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ58_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ59_HTML.gif)
Therefore, combining the above estimates yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ60_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ61_HTML.gif)
Now, Taking arbitrarily with
and
large enough, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ62_HTML.gif)
While,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ63_HTML.gif)
Substituting (4.15) into (4.14) gives that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ64_HTML.gif)
Since can be small enough by assumption, it is possible to choose a suitable
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ65_HTML.gif)
Hence letting in (4.16) yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ66_HTML.gif)
On the other hand, from the deduction of (4.12) it is not difficult to see that it also holds with substituted by
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ67_HTML.gif)
then it follows from (4.18) and (4.19) that (note the conditions for )
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ68_HTML.gif)
which is our desired inequality. Then the proof is completed.
We also have the following result for almost surely exponential stability.
Theorem 4.2.
Suppose that all the conditions of Theorem 4.1 are satisfied. Then the solution is almost surely exponentially stable. Moreover, there exists a positive constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ69_HTML.gif)
Proof.
The proof is similar to that of [5, Theorem  3.3] and we omit it.
5. Examples
Now we apply the results obtained above to consider the following non-autonomous stochastic functional differential equation with finite delay (i.e. , (1.3).
Example 5.1.
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ70_HTML.gif)
where ,
is a continuous function and is uniformly Hölder continuous in
(with exponent
) and satisfies that
as
. Let
and
,
denote a one-dimensional standard Brownian motion.
We define the operators by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ71_HTML.gif)
with the domain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ72_HTML.gif)
Then generates an evolution operator
satisfying assumptions
(see [23]). Set
for some
and
.
In order to discuss the system (5.1), we also need the following assumptions on functions and
.
The functions ,
are continuous and global Lipschitz continuous in the second variable.
There exist real numbers and continuous functions
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ73_HTML.gif)
where ,
satisfy that
, for some
small enough (
).
Now we can define ,
and
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ74_HTML.gif)
Then it is not difficult to verify that and
satisfy the conditions
,
and
, respectively, due to Assumptions
,
, since, by the embody property of
(also see [25, Corollary  2.6.11]),
for some constant
. Hence we have, by Theorems 3.3 and 4.1, the following.
Theorem 5.2.
Let and
. Suppose that all the above assumptions are satisfied. Then for the stochastic system (5.1) there exists a global mild solution
, and it is exponentially stable provided that
and
are small enough.
We present another system for which the linear evolution is given explicitly, and so all the coefficients for the conditions of the obtained results can be estimated properly.
Example 5.3.
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ75_HTML.gif)
where is a positive function and is Hölder continuous in
with parameter
.
,
and
are as in Example 5.1,
and
.
Let ,
.
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ76_HTML.gif)
with the domain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ77_HTML.gif)
Then it is not difficult to verify that generates an evolution operator
satisfying assumptions
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ78_HTML.gif)
where is the compact analytic semigroup generated by the operator
with
for
. It is easy to compute that
has a discrete spectrum, and the eigenvalues are
, with the corresponding normalized eigenvectors
. Thus for
, there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ79_HTML.gif)
and clearly the common domain coincides with that of the operator . Furthermore, We may define
(
) for self-adjoint operator
by the classical spectral theorem, and it is easy to deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ80_HTML.gif)
on the domain . Particularly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ81_HTML.gif)
Therefore, we have that, for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ82_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ83_HTML.gif)
for ,
. And
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ84_HTML.gif)
which shows that for
.
Now we define as above and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F785628/MediaObjects/13660_2009_Article_2007_Equ85_HTML.gif)
for all and any
. Thus (5.6) has the form (1.1). Thus we can easily obtain its existence and stability of mild solutions for (5.6) by Theorems 3.3 and 4.1 under some proper conditions.
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Acknowledgments
The author would like to thank the referees very much for their valuable suggestions to this paper. This work is supported by the NNSF of China (no. 10671069), NSF of Shanghai (no. 09ZR1408900), and Shanghai Leading Academic Discipline Project (no. B407).
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Fu, X. Existence and Stability of Solutions for Nonautonomous Stochastic Functional Evolution Equations. J Inequal Appl 2009, 785628 (2009). https://doi.org/10.1155/2009/785628
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DOI: https://doi.org/10.1155/2009/785628