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

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.


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, .
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. erefore, 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. erefore, 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. erefore, 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. erefore, 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, g, σ, I k ; and the initial value φ. Here, W(t) denotes a Brownian motion and B H (t) denotes an fBm with the Hurst parameter H ∈ (1/2, 1). e 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.

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 H ∈ (0, 1), a continuous centered Gaussian process β H (t), t ∈ R with the covariance function is called a two-sided one-dimensional fractional Brownian motion (fBm) and H is the Hurst parameter. Let (Ω, F, F t t≥0 , P) stand for a complete probability space equipped with some filtration F t t≥0 satisfying normal assumptions. Y 1 , Y 2 , and X stand for three real Hilbert spaces, respectively. L(Y i ; X) represents the space of all bounded linear operators from Y i to X, i � 1, 2. Let e (i) n n∈N + be a complete orthonormal basis in Y i . Let en there exits a R-valued sequence ω n (t) n∈N + of one-dimensional Brownian motions mutually independent over (Ω, F, F t t≥0 , P) such that and the infinite-dimensional cylindrical Y 2 -valued fBm B H (t) is defined by the formal sum (see [5]).
be the space of all Q (i) Hilbert-Schmidt operators from Y i to X, i � 1, 2. Now we show the following definition.
Definition 2 (see [20]). Let ξ i ∈ L(Y i ; X) and define In order to set our problem, we need the following lemmas.
Lemma 1 (see [19]). For any r ≥ 1 and for arbitrary Lemma 2 (see [4]). If ψ: 2 Discrete Dynamics in Nature and Society (8) Now, we turn to state some notations and basic facts about the theory of semigroups and fractional power operators. Let A: D(A) ⟶ X be the infinitesimal generator of an analytic semigroup S(t), t ≥ 0 { } of bounded linear operators on X. It is well known that there exists a pair of constants } is a uniformly bounded and analytic semigroup equipped with the norm ‖ · ‖ α , then the following properties are well known (see [28]).

Lemma 4.
Suppose that the preceding conditions are satisfied.

The pth Moment Exponential Stability
Consider (Ω, F, F t t≥0 , P) the complete probability space which was introduced in Section 2.
is continuous for all but at most a finite number of points t ∈ J and at these points t ∈ J, φ(t + ) and and φ(t − ) stand for the right-hand and left-hand limits of the function φ(t), 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 W(t) is the Brownian motion and B H (t)(H ∈ (1/2, 1)) is the fractional Brownian motion which were introduced in the previous section. e mappings G: stands for the jump in the state x at time t k and I k (x(t k )) determines the size of the jump. (1) x(t) is Borel measurable and F t -adapted and has the càldàg path on t ∈ [0, +∞) almost everywhere. (2) For each t ∈ [0, +∞), x(t) satisfies the following integral equation: Definition 4. Equation (10) is said to be exponentially stable in pth (p ≥ 2) moment, if for any initial value φ, there exists a pair of positive constants ] and C such that In order to set the stability problem, we assume that the following conditions hold. ere exist constants L f ≥ 0, a f ≥ 0 and c f > 0 such that for any x, y ∈ PC and for all t ∈ [0, +∞), Condition 3. ere exist constants L g ≥ 0, a g ≥ 0 and c g > 0 such that for any x, y ∈ PC and for all t ∈ [0, +∞), Condition 4. ere exist constants α ∈ (0, 1] and L G ≥ 0, a G ≥ 0 and c G > 0 such that G is X α -valued, for any x, y ∈ PC and for all t ∈ [0, +∞), Condition 5. e mapping σ: ≤ a σ e − c σ t , as for all t ≥ 0, where a σ ≥ 0 and c σ > 0.

Theorem 1. Suppose that conditions 1-6 hold. en equation (10) is exponentially stable in pth moment if
Next, we show that π(S) ⊂ S. It follows from (20) that 4 Discrete Dynamics in Nature and Society Now we estimate the terms on the right-hand side of (21). Firstly, by condition 1, we can obtain Secondly, Hölder's inequality and condition 4 yield For any x(t) ∈ S and any ε 1 > 0, there exists a T 1 > 0, such that e μt E‖x(t)‖ p X < ε 1 for t ≥ T 1 − τ; thus, we can get So k 21 (t) ⟶ 0 as t ⟶ +∞. From condition 4, we get k 22 (t) ⟶ 0 as t ⟶ +∞. at is to say Further, Hölder's inequality, Lemma 4, and condition 4 yield Discrete Dynamics in Nature and Society 5 For any x(t) ∈ S and any ε 2 > 0, there exists a T 2 > 0, such that e μt E‖x(t)‖ p X < ε 2 for t ≥ T 2 − τ. us, we can get e μs sup θ∈ [− τ,0] E‖x(s + θ)‖ p X < e μr ε 2 , s ∈ T 2 , +∞ , (27) then As e − (λ− μ)t ⟶ 0 as t ⟶ +∞, then there exists T 3 ≥ T 2 such that for any t ≥ T 3 , we have So from the above, we obtain for any t ≥ T 3 that is to say, k 31 (t) ⟶ 0 as t ⟶ +∞. Similar computations can be used to show that k 32 (t) ⟶ 0 as t ⟶ +∞. So, we conclude that As for the fourth term on the right-hand side of (21), we have For any x(t) ∈ S and any ε 3 > 0, there exists a T 4 > 0, such that e μt E‖x(t)‖ p X < ε 3 for t ≥ T 4 . us, we can get then As e − (λ− μ)t ⟶ 0 as t ⟶ +∞, there exists T 5 ≥ T 4 such that for any t ≥ T 5 , we have So from the above, we obtain for any t ≥ T 5 at is to say, k 41 (t) ⟶ 0 as t ⟶ +∞. Similar computations can be used to show that k 41 (t) ⟶ 0 as t ⟶ +∞. So, we conclude that As for the fifth term on the right-hand side of (21), from Lemma 1 and Hölder's inequality, we have 6 Discrete Dynamics in Nature and Society ds. (38) Similar to the proof of (32), for (38), we can obtain As for the sixth term on the right-hand side of (21), by Lemmas 2 and 3, we have As for the last term on the right-hand side of (21), we get Discrete Dynamics in Nature and Society