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On the asymptotic stability of a class of jump-diffusions of neutral type with impulses
Journal of Inequalities and Applications volume 2013, Article number: 561 (2013)
This paper is concerned with the asymptotic stability in the p th moment for a class of jump-diffusions of neutral type with impulses. Sufficient conditions ensuring the stability of jump-diffusions of neutral type with impulses are established by means of the Banach fixed point theorem. The results obtained here generalize and improve some well-known results.
Recently, the existence, uniqueness and stability of solutions of stochastic differential equations, especially stochastic partial differential equations [1–5], have been considered by many authors . Besides stochastic effects, impulsive effects also occur in real systems. The study of impulsive systems in a separable Hilbert space is motivated by modeling some evolution phenomena arising in physics, communications, engineering, etc. [7–9].
In addition, many dynamical systems not only depend on present and past states, but also involve derivative with delays, and neutral systems are often used to describe such systems. It should pointed out that there are a few works about the existence and stability of mild solutions of neutral systems [10–16]. Meanwhile, there are also a few works on jump diffusions, and some results on the existence, uniqueness, stability and qualitative properties of solutions have been obtained. For example, Bao et al.  studied almost sure asymptotic stability of stochastic partial differential equations with jumps. Bao et al.  discussed stability in distribution of mild solutions to stochastic partial differential equations with jumps. Cui et al.  discussed exponential stability for neutral stochastic partial differential equations with delays and Poisson jumps. Peszat and Zabczyk  discussed the theory of stochastic partial differential equations with Lévy noise. Motivated by the above papers, in the paper we aim to study the existence and asymptotic stability of a class of jump-diffusions of neutral type with impulses by means of the Banach fixed point theorem, the results obtained here generalize the main results from Mahmudov , Jiang and Shen , Sakthivel .
The organization of the paper is as follows. In the next section, we introduce some notations and definitions of mild solution and asymptotic stability. Then we give sufficient conditions ensuring the stability of jump-diffusions of neutral type with impulses by means of the Banach fixed point theorem.
Throughout this paper, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right-continuous while contains all P-null sets) . Moreover, let X, Y be two real separable Hilbert spaces, and let denote the space of all bounded linear operators from Y into X.
For simplicity, we use the notation to denote the norm in X, Y and to denote the operator norm in and . Let , denote the inner products of X, Y, respectively. Let denote an Y-valued Wiener process defined on the probability space with the covariance operator Q, that is, for all , where Q is a positive, self-adjoint, trace class operator on Y. In particular, we denote by a Y-valued Q-Wiener process with respect to . We assume that there exists a complete orthonormal system in Y, a bounded sequence of nonnegative real numbers such that , , and a sequence of independent Brownian motions such that , , and , where is the σ-algebra generated by . Let be the space of all Hilbert-Schmidt operators from to X with the inner product ; see, for example, .
Let , be a stationary -Poisson point process with characteristic measure λ. Denote by the Poisson counting measure associated with p, that is, with a measurable set , which denotes the Borel σ-field of . Let be the compensated Poisson measure, which is independent of . Denote by the space of all predictable mappings for which
We may then define the X-valued stochastic integral , which is a centered square-integrable martingale . We always assume that and are independent of .
Now consider a class of jump-diffusions of neutral type with impulses of the form
with the initial data , where , , , are all Borel measurable; , , , are continuous; A is the infinitesimal generator of a semigroup of bounded linear operators , , in X; . Furthermore, the fixed moments of times satisfy , and represent the right and left limits of at , respectively. Also represents the jump in the state x at time with determining the size of the jump. and denotes a family of all right-continuous functions with left-hand limits η from to X. Denote the norm of by . Here, is a family of all almost surely bounded, -measurable, continuous random variables from to X.
Suppose that is an analytic semigroup with its infinitesimal generator A. For a basic reference, the reader is referred to Pazy . We always assume , the resolvent set of −A. For any , it is possible to define the fractional power which is a closed linear operator with its domain .
Definition 1 A process , , is called a mild solution of Eq. (1) if
is adapted to , with a.s.;
has càdlàg paths on a.s. and for each , satisfies the integral equation(2)
Moreover, for the purposes of stability, we always assume that , , , , (). Hence Eq. (1) has a trivial solution when .
Definition 3 Let be an integer. The trivial solution of Eq. (1) or Eq. (1) itself is said to be asymptotically stable (or globally asymptotically stable) in the p th moment if it is stable in the p th moment and for any ,
When , we say Eq. (1) is mean square asymptotically stable (or mean square globally asymptotically stable).
To establish the stability of Eq. (1), we employ the following assumptions.
A is the infinitesimal generator of a semigroup of bounded linear operators , , in X satisfying , , for some constants and .
The functions f, g and h satisfy the following conditions: there exists a constant K such that for any and ,
There exist a number and a positive constant such that for any and , and
There exists a constant such that for each ().
In this section, we consider the asymptotic stability of Eq. (1) by means of the fixed point theory. Let H be the space of all -adapted processes which is almost certainly continuous in t for fixed . Moreover, for and as .
Now let us state the following well-known lemma , which will be used in the sequel in the proof of the main result.
Lemma 1 If (H1) holds and , then for any ,
for each , ;
there exist constants and such that , .
We can now state our main result of this paper.
Theorem 1 If (H1)-(H4) hold for some , then Eq. (1) is mean square globally asymptotically stable provided
Proof Define an operator by for and for ,
We divide the proof into three steps.
Step 1. We claim that π is mean square continuous on . Let , , and be sufficiently small, then
Consequently, π is mean square continuous on .
Step 2. We claim that . From (4), we have
Note that . By (H1), (H3), (H4) and Lemma 1, we have
By Lemma 1, (H3) and the Hölder inequality, we obtain
For any and , there exists such that for . We thus obtain
We can see as . By (3), there exists such that for any we obtain
This, together with (11), yields for any , . That is,
Further, similar to the proof of (12), from (13), (14) and (15), we then have as . Therefore, we have as . That is, .
Step 3. We claim that π is a contraction mapping. Let , we have
where . Thus π is a contraction mapping. Hence there exists a unique fixed point in H which is the solution of Eq. (1) and as . The proof is complete. □
Similarly, we can easily generalize the above result to global asymptotic stability in the p th moment.
Theorem 2 If (H1)-(H4) hold for some , , and the inequality
also holds, then Eq. (1) is globally asymptotically stable in the pth moment, where and .
Remark Without the impulsive and Poisson jumps, Eq. (1) reduces to a stochastic partial differential equation, which is investigated in . If without Poisson jumps, then Eq. (1) reduces to impulsive stochastic neutral partial differential equations, which is studied in . If without the neutral term and Poisson jumps, then Eq. (1) reduces to an impulsive stochastic partial differential equation, which is studied in . In the sense, the results of this paper are generalized.
This paper discusses the globally asymptotic stability of the mild solutions to jump-diffusions of neutral type with impulses by the fixed point theory. Globally asymptotic stability of the mild solutions to jump-diffusions of neutral type with impulses are derived. Some earlier results are generalized and improved.
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We wish to thank the referees for their valuable suggestions which have considerably improved the presentation of this article. The work is supported by the Fundamental Research Funds for the Central Universities, the National Natural Science Foundation of China under Grant 61304067 and the Natural Science Foundation of Hubei Province of China under Grant 2013CFB443.
The authors declare that they have no competing interests.
FJ carried out the study of asymptotic stability of this paper and drafted the manuscript. HY participated in the proof of the main result of the paper. XZ provided some constructive suggestions for the improvement of the paper. All authors read and approved the final manuscript.
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Jiang, F., Yang, H. & Zhao, X. On the asymptotic stability of a class of jump-diffusions of neutral type with impulses. J Inequal Appl 2013, 561 (2013). https://doi.org/10.1186/1029-242X-2013-561
- Poisson jump
- mild solution
- asymptotic stability