Stability of a Class of Impulsive Neutral Stochastic Functional Partial Differential Equations

Liu, Yue; Ruan, Dehao

doi:https://doi.org/10.1155/2020/9051396

Discrete Dynamics in Nature and Society

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 9051396 | https://doi.org/10.1155/2020/9051396

Stability of a Class of Impulsive Neutral Stochastic Functional Partial Differential Equations

Yue Liu¹and Dehao Ruan²

Academic Editor: Seenith Sivasundaram

Received28 Jul 2019

Revised20 Sept 2019

Accepted09 Dec 2019

Published21 Jan 2020

Abstract

In this paper, a class of impulsive neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion is investigated. Under some suitable assumptions, the pth moment exponential stability is discussed by means of the fixed-point theorem. Our results also improve and generalize some previous studies. Moreover, one example is given to illustrate our main results.

1. Introduction

In recent years, stochastic differential equations (SDEs) have come to play an important role in many areas such as physics, population dynamics, electrical engineering, medicine biology, ecology, economics and other areas of science, and engineering. Because of their great applications, stochastic differential equations have been developed very fast; see, for example, [1–27].

In observing the process of stock price fluctuations, it is found that the fluctuations of stock prices are not self-similar; on a larger time scale (month or year), these processes are more stable and more stable than on a small time scale (hour or day). One reason is that random noise in the market is a sum of irregular “trading” noise. Therefore, it can be assumed that the stock price is affected by two random phenomena; one is that the incremental process is independent, and the other is that the incremental process is related. Generally speaking, the random perturbation of stock prices consists of two parts: one is the basic part, that is, the overall economic situation of the society, comes from the actual financial background of the stock market and has a long correlation, so it can be expressed by fractional Brownian motion; the other is trading part, that is, the random trading conditions of stockholders in the stock market, is derived from the stochastic inherent factors of stockholders, so it can be expressed by Brownian motion. In addition, similar phenomena have appeared in the research of fluid mechanics, electrical communication, economics, and finance. Therefore, the mixed model has been considered by many authors; see, for example, [7, 10, 12, 16, 20, 23, 24, 27].

On the other hand, impulsive effects are caused by instantaneous perturbations at a certain moment which can be used to model many practical problems that arise in the areas of mechanics, electrical engineering, medicine biology, ecology, and so on. Therefore, there has been increasing interest in the theory of impulsive differential equations (for example, [2, 6, 10, 21, 22]).

Stochastic partial differential equation is one of the most important, active, and rapidly developing key research fields in probability due to its wide and great applications in physics, chemistry, biology, economic, finance, and so on. On the other hand, many dynamical systems not only depend on present and past states but also involve derivatives with delays. Neutral stochastic functional differential partial differential equations are often used to describe such systems. It is well known that the time delay and stochastic perturbations may cause oscillation and instability in systems. It is important to consider the influence of delay and stochastic perturbations in the investigation of these systems. Therefore, the stability of neutral stochastic functional partial differential equations has been studied by many researchers (see, for instance, [2, 4, 6, 10, 12, 13, 15, 27] and the reference therein).

Motivated by the above discussion, this paper is concerned with the exponential stability results for a class of neutral stochastic functional partial differential equations driven by standard Brownian motion and fractional Brownian motion with impulses:under suitable conditions on the operator A; the coefficient functions G, f, , σ, ; and the initial value φ. Here, denotes a Brownian motion and denotes an fBm with the Hurst parameter .

The contents of this paper are as follows. In Section 2, some necessary notions, conceptions, and lemmas are introduced. In Section 3, the pth moment exponential stability of a class of impulsive neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion is investigated by means of the fixed point theorem. In Section 4, one example is given to illustrate our main results. At last, in Section 5, our conclusion is presented.

2. Preliminaries

In this section, we collect some notions, definitions, and lemmas which will be used throughout the whole of this paper.

Definition 1 (see [5]). Given , a continuous centered Gaussian process with the covariance functionis called a two-sided one-dimensional fractional Brownian motion (fBm) and H is the Hurst parameter.
Let stand for a complete probability space equipped with some filtration satisfying normal assumptions. , and X stand for three real Hilbert spaces, respectively. represents the space of all bounded linear operators from to X, . Let be a complete orthonormal basis in . Let be a operator defined by with finite trace , where are nonnegative real numbers. Then there exits a -valued sequence of one-dimensional Brownian motions mutually independent over such thatand the infinite-dimensional cylindrical -valued fBm is defined by the formal sum (see [5]).where the sequence are stochastically independent scalar fBms with Hurst parameter .
Let be the space of all Hilbert–Schmidt operators from to X, . Now we show the following definition.

Definition 2 (see [20]). Let and defineIf , then is called a Hilbert–Schmidt operator and the space is a real separable Hilbert space equipped with the inner product , .
In order to set our problem, we need the following lemmas.

Lemma 1 (see [19]). For any and for arbitrary predictable process ,

Lemma 2 (see [4]). If satisfies then, we havewhere .

Lemma 3 (see [10]). For any and for arbitrary predictable process ,

Now, we turn to state some notations and basic facts about the theory of semigroups and fractional power operators. Let A: be the infinitesimal generator of an analytic semigroup of bounded linear operators on X. It is well known that there exists a pair of constants and such that for every If is a uniformly bounded and analytic semigroup such that , where is the resolvent set of A, then it is possible to define the fractional power for any which is a closed linear operator with its domain . Furthermore, the subspace is dense in X, and the expression defines a norm in . If represents the space equipped with the norm , then the following properties are well known (see [28]).

Lemma 4. Suppose that the preceding conditions are satisfied.

(1)Let , then is a Banach space(2)If , then the injection X_α ↪ X_β is continuous(3)For every , there exists a constant such that

3. The pth Moment Exponential Stability

Consider the complete probability space which was introduced in Section 2. Denote , for all . We denote by the Banach space of all continuous -value functions ϕ defined on equipped with the norm , where denotes the Euclidean norm. Let , where is a bounded interval. and stand for the right-hand and left-hand limits of the function , respectively. Define . Let be the family of all bounded -measurable, -value random variables ϕ, satisfying for .

In this section, we consider the pth moment exponential stability of mild solutions of the following impulsive neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion:where is the Brownian motion and is the fractional Brownian motion which were introduced in the previous section. The mappings and are all Borel measurable functions. For each , , the function defined by , for The fixed satisfies and . at each impulsive point is right continuous. stands for the jump in the state x at time and determines the size of the jump.

Definition 3. A X-valued stochastic process is called a mild solution of equation (10) if , , and the following assumptions hold:(1) is Borel measurable and -adapted and has the càldàg path on almost everywhere.(2)For each , satisfies the following integral equation:

Definition 4. Equation (10) is said to be exponentially stable in pth moment, if for any initial value φ, there exists a pair of positive constants ν and C such thatIn order to set the stability problem, we assume that the following conditions hold.

Condition 1. A is the infinitesimal generator of an analytic semigroup of bounded linear operators on X satisfies

Condition 2. There exist constants , and such that for any and for all ,

Condition 3. There exist constants , and such that for any and for all ,

Condition 4. There exist constants and , and such that G is -valued, for any and for all ,

Condition 5. The mapping satisfies

Condition 6. There exists some constants such that for any , and ,

Theorem 1. Suppose that conditions 1–6 hold. Then equation (10) is exponentially stable in pth moment ifwhere , is the gamma function, and is the corresponding constant in Lemma 4.

Proof. Denote by the Banach space of all -adapted processes , which satisfies for and as , where µ is a positive constant such that .
Define an operator by for and for ,Next, we show that . It follows from (20) thatNow we estimate the terms on the right-hand side of (21). Firstly, by condition 1, we can obtainSecondly, Hölder’s inequality and condition 4 yieldFor any and any , there exists a , such that for ; thus, we can getSo as From condition 4, we get as That is to sayFurther, Hölder’s inequality, Lemma 4, and condition 4 yieldFor any and any , there exists a , such that for . Thus, we can getthenAs as , then there exists such that for any , we haveSo from the above, we obtain for any that is to say, as . Similar computations can be used to show that as . So, we conclude thatAs for the fourth term on the right-hand side of (21), we haveFor any and any , there exists a , such that for . Thus, we can getthenAs as , there exists such that for any , we haveSo from the above, we obtain for any That is to say, as . Similar computations can be used to show that as . So, we conclude thatAs for the fifth term on the right-hand side of (21), from Lemma 1 and Hölder’s inequality, we haveSimilar to the proof of (32), for (38), we can obtainAs for the sixth term on the right-hand side of (21), by Lemmas 2 and 3, we haveAs for the last term on the right-hand side of (21), we getFor any and any , there exists a , such that for . Thus, we can getthenAs as , then there exists such that for any , we haveSo from the above, we obtain for any So we haveThus, from (20)–(32), (37), and (39)–(46), we know that as . So, we conclude that .
Thirdly, we will show that π is contractive. Let by using the inequalitywhere are nonnegative constants and . For any fixed , we haveHence,Thus, by (19), we know that π is a contraction mapping.
Hence, by the Contraction Mapping Theorem, π has a unique fixed point in , which is a solution of equation (10) with on and as . This completes the proof.

Remark 1. Zhang and Ruan in [27] considered (10) without impulsive effects and required the following conditions to ensure the pth () moment exponential stability: andwhere . Our condition is andNow we show that why (51) is weaker than (50). Obviously,Then we can transfer to prove (51) is weaker than following condition:Obviously, (53) is not hold for and (51) may hold for . On the other hand, when , conditions (51) and (53) can be transferred into the following conditions, respectively.Obviously,Then we conclude that (51) is weaker than (50); in this sense, this paper improves and generalizes the results in [27].

Remark 2. Theorem 1 does not ask for , , and , which are imposed in [10]. Even in this special case, our results also improve the result in [10]. Our conditions are andHowever, the corresponding conditions in [15] are andIn this sense, this paper improves the results in [10].

Remark 3. A similar discussion can be used to show that this paper improves and generalizes the results in [6, 15].

4. Example

Consider the following neutral stochastic functional partial differential equation driven by Brownian motion and fractional Brownian motion with impluses:where is the Brownian motion and is the fractional Brownian motion with Hurst parameter , , and . Let and , and define the operator A: given by with domain

It is well known that an analytic semigroup generated by the operator A on X satisfies . Furthermore, has a discrete spectrum, the eigenvalues are , , with the corresponding normalized eigenvectors . Then, for , we have

Let

Obviously,

From the definition of , we get

It is obvious that all the assumptions are satisfied with

Then, the mild solution of (58) is exponentially stable in pth moment provided thatwhere and .

5. Conclusion

In this paper, a class of neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion with impulses is investigated by means of the fixed point theorem. Some sufficient conditions to ensure that the mild solution is exponentially stable in pth moment are established. The obtained results improve and generalize the results in [6, 10, 15, 27].

In our next paper, we will investigate the global attracting set and quasi-invariant set of a class of neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion with impulses by means of the fixed point theorem.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was supported by the National Natural Science Foundation of Hunan Province under grant no. 2019JJ50111 and the National Natural Science Foundation of China under grant no. 11701115.

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Copyright © 2020 Yue Liu and Dehao Ruan. 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.

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