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

Authors: Yue Liu; Dehao Ruan
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## 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, thepth 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.


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## Body

## 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:(1)dxt+Gt,xt=Axt+ft,xtdt+gt,xtdWt+σtdBHt,t≥0,t≠tk,△xtk=xtk+−xtk−=Ikxtk,t=tk,k=1,2,…,xt=φt,−τ≤t≤0,τ≥0,under suitable conditions on the operator A; the coefficient functions G, f, g, σ, Ik; and the initial value φ. Here, Wt denotes a Brownian motion and BHt denotes an fBm with the Hurst parameter H∈1/2,1.The contents of this paper are as follows. In Section2, 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]).
GivenH∈0,1, a continuous centered Gaussian process βHt,t∈ℝ with the covariance function(2)RHt,s=EβHtβHs=12t2H+s2H−t−s2H,t,s∈ℝ,is called a two-sided one-dimensional fractional Brownian motion (fBm) and H is the Hurst parameter.
LetΩ,ℱ,ℱtt≥0,P stand for a complete probability space equipped with some filtration ℱtt≥0 satisfying normal assumptions. Y1,Y2, and X stand for three real Hilbert spaces, respectively. ℒYi;X represents the space of all bounded linear operators from Yi to X, i=1,2. Let enin∈ℕ+ be a complete orthonormal basis in Yi. Let Qi∈ℒYi;X be a operator defined by Qieni=λnieni with finite trace trQi=∑n=1∞λni<∞, where λnin∈ℕ+ are nonnegative real numbers. Then there exits a ℝ-valued sequence ωntn∈ℕ+ of one-dimensional Brownian motions mutually independent over Ω,ℱ,ℱtt≥0,P such that(3)Wt=∑n=1∞λn1en1ωnt,t≥0,and the infinite-dimensional cylindrical Y2-valued fBm BHt is defined by the formal sum (see [5]).(4)BHt=∑n=1∞λn2en2βnHt,t≥0,where the sequence βnHtn∈ℕ+ are stochastically independent scalar fBms with Hurst parameter H∈1/2,1.
LetℒQi0Yi;X be the space of all Qi Hilbert–Schmidt operators from Yi to X, i=1,2. Now we show the following definition.Definition 2 (see [20]).
Letξi∈ℒYi;X and define(5)ξiℒQi02≔trξiQiξi∗=∑n=1∞λniξieniX2.
IfξiℒQi02<∞, then ξi is called a Qi Hilbert–Schmidt operator and the space ℒQi0≔ℒQi0Yi;X is a real separable Hilbert space equipped with the inner product φ,ψℒQi0=∑n=1∞φeni,ψeni, i=1,2.
In order to set our problem, we need the following lemmas.Lemma 1 (see [19]).
For anyr≥1 and for arbitrary ℒQ10Y1;X predictable process ϕ⋅,(6)sups∈0,tE∫0sϕudWuX2r≤r2r−1r∫0tEϕsℒQ102r1/rdsr,t∈0,T.Lemma 2 (see [4]).
Ifψ:0,T⟶ℒQ20Y2;X satisfies ∫0TψsℒQ202ds<+∞ then, we have(7)E∫0tψsdBHsX2≤2Ht2H−1∫0tψsℒQ202ds,t∈0,T,where H∈1/2,1.Lemma 3 (see [10]).
For anyp≥2 and for arbitrary ℒQ20Y2;X predictable process ψ⋅,(8)E∫0tψsdBHsXp≤CpE∫0tψsdBHsX2p/2,t∈0,T.Now, we turn to state some notations and basic facts about the theory of semigroups and fractional power operators. LetA: DA⟶X be the infinitesimal generator of an analytic semigroup St,t≥0 of bounded linear operators on X. It is well known that there exists a pair of constants λ∈ℝ and M≥1 such that StX≤Meλt for every t≥0. If St,t≥0 is a uniformly bounded and analytic semigroup such that 0∈ρA, where ρA is the resolvent set of A, then it is possible to define the fractional power −Aα for any α∈0,1 which is a closed linear operator with its domain D−Aα. Furthermore, the subspace D−Aα is dense in X, and the expression ηα=−AαηX⋅η∈D−Aα defines a norm in D−Aα. If Xα represents the space D−Aα equipped with the norm ⋅α, then the following properties are well known (see [28]).Lemma 4.
Suppose that the preceding conditions are satisfied.(1)
Let0≤α≤1, then Xα is a Banach space
(2)
If0≤β≤α, then the injection Xα ↪ Xβ is continuous
(3)
For every0≤α≤1, there exists a constant Mα>0 such that(9)−AαSt≤Mαt−αe−λt,t>0λ>0.


## 3. Thepth Moment Exponential Stability

ConsiderΩ,ℱ,ℱtt≥0,P the complete probability space which was introduced in Section 2. Denote ℱt=ℱ0, for all t≤0. We denote by C≔C−τ,0;X the Banach space of all continuous X-value functions ϕ defined on −τ,0 equipped with the norm ϕC=sup−τ≤θ≤0ϕθ, where ⋅denotes the Euclidean norm. Let PCJ,ℝn≔φ:J⟶ℝnis continuous for all but at most a finite number of pointst∈Jand at these pointst∈J,φt+andφt−exist,φt+=φt, where J⊂ℝ is a bounded interval. φt+ and φt− stand for the right-hand and left-hand limits of the function φt, respectively. Define PC≔PC−τ,0;X. Let PCℱ0B−τ,0;XPCℱtB−τ,0;X be the family of all bounded ℱ0ℱt-measurable, PC−τ,0;X-value random variables ϕ, satisfying ϕℒpp=sup−τ≤θ≤0Eϕθp<+∞ for p>0.In this section, we consider thepth moment exponential stability of mild solutions of the following impulsive neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion:(10)dxt+Gt,xt=Axt+ft,xtdt+gt,xtdWt+σtdBHt,t≥0,t≠tk,△xtk=xtk+−xtk−=Ikxtk,t=tk,k=1,2,…,xt=φt∈PCℱ0B−τ,0;X,−τ≤t≤0,where Wt is the Brownian motion and BHtH∈1/2,1 is the fractional Brownian motion which were introduced in the previous section. The mappings G:0,+∞×X⟶X,f:0,+∞×X⟶X,g:0,+∞×X⟶ℒQ10Y1;X and σ:0,+∞ are all Borel measurable functions. For each t∈0,+∞, xt∈PC, the function xt defined by xtθ=xt+θ, for θ∈−τ,0. The fixed tk satisfies 0=t0<t1<t2<…<tn<… and limk⟶+∞tk=+∞. xt at each impulsive point tk is right continuous. Δxtk=xtk+−xtk− stands for the jump in the state x at time tk and Ikxtk determines the size of the jump.Definition 3.
AX-valued stochastic process xt is called a mild solution of equation (10) if xt=φt, t∈−τ,0, and the following assumptions hold:(1)
xt is Borel measurable and ℱt-adapted and has the càldàg path on t∈0,+∞ almost everywhere.(2)
For eacht∈0,+∞, xt satisfies the following integral equation:(11)xt=Stφ0+G0,φ−Gt,xt−∫0tASt−sGs,xsds+∫0tSt−sfs,xsds+∫0tSt−sgs,xsdWs+∫0tSt−sσsdBHs+∑0<tk<tSt−sIkxtk.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 ν and C such that(12)ExtXp≤CφXpe−νt,t≥0.
In order to set the stability problem, we assume that the following conditions hold.Condition 1.
A is the infinitesimal generator of an analytic semigroupSt,t≥0 of bounded linear operators on X satisfies(13)StX≤Me−λt,∀t≥0,whereM≥1andλ>0.Condition 2.
There exist constantsLf≥0, af≥0 and γf>0 such that for any x,y∈PC and for all t∈0,+∞,(14)ft,xt−ft,ytX≤Lfxt−ytX,ft,0Xp≤afe−γft,as.Condition 3.
There exist constantsLg≥0, ag≥0 and γg>0 such that for any x,y∈PC and for all t∈0,+∞,(15)gt,xt−gt,ytX≤Lgxt−ytℒQ10,gt,0Xp≤age−γgt,as.Condition 4.
There exist constantsα∈0,1 and LG≥0, aG≥0 and γG>0 such that G is Xα-valued, for any x,y∈PC and for all t∈0,+∞,(16)−AαGt,xt−−AαGt,ytX≤LGxt−ytX,−AαGt,0Xp≤aGe−γGt.Condition 5.
The mappingσ:0,+∞⟶ℒQ20Y2;X satisfies(17)σtℒQ202≤aσe−γσt,as for allt≥0,whereaσ≥0andγσ>0.Condition 6.
There exists some constantsdk such that for any x,y∈X, and ∑k=1∞dk<+∞,(18)Ikxtk−IkxtkX≤dkxtk−ytkX,Ik0=0,k=1,2,….Theorem 1.
Suppose that conditions1–6 hold. Then equation (10) is exponentially stable in pth moment if(19)k+4p−1M1−αpΓpαLGP1−kp−1λpα+4p−1MpLfp1−kp−1λp+4p−1pp−1/2p/2MpLgp1−kp−12λp/2+4p−1∑k=1+∞dkp1−kp−1<1,where k≔LG−A−αX<1, Γ⋅ is the gamma function, and M1−α is the corresponding constant in Lemma 4.Proof.
Denote byS the Banach space of all ℱ-adapted processes ϕt,w:−τ,+∞×Ω⟶ℝ, which satisfies ϕs,w=φs for s∈−τ,0 and eμtEϕt,wXp⟶0 as t⟶+∞, where µ is a positive constant such that 0<μ<minγG,γf,γg,γσ,λ.
Define an operatorπ:S⟶S by πxt=φt for t∈−τ,0 and for t∈0,+∞,(20)πxt=Stφ0+G0,φ−Gt,xt−∫0tASt−sGs,xsds+∫0tSt−sfs,xsds+∫0tSt−sgs,xsdWs+∫0tSt−sσsdBHs+∑0<tk<tSt−sIkxtk.
Next, we show thatπS⊂S. It follows from (20) that(21)eμtEπxtXp≤7p−1eμtEStφ0+G0,φXp+7p−1eμtEGt,xtXp+7p−1eμtE∫0tASt−sGs,xsdsXp+7p−1eμtE∫0tSt−sfs,xsdsXp+7p−1eμtE∫0tSt−sgs,xsdWsXp+7p−1eμtE∫0tSt−sσsdBHsXp+7p−1eμt∑0<tk<tSt−sIkxtkXp≔7p−1∑i=17EkitXp.
Now we estimate the terms on the right-hand side of (21). Firstly, by condition 1, we can obtain(22)Ek1tXp≤Mpe−pλ−μtφ0+G0,φXp⟶0,ast⟶+∞.
Secondly, Hölder’s inequality and condition4 yield(23)Ek2tXp≤eμtEGt,xt−Gt,0+Gt,0Xp≤−A−αXpeμtE−AαGt,xt−Gt,0+Gt,0Xp≤2p−1LGp−A−αXpeμtsupθ∈−τ,0Ext+θXp+2p−1−A−αXpeμt−AαGt,0Xp≔2p−1LGp−A−αXpk21t+2p−1−A−αXpk22t.
For anyxt∈S and any ε1>0, there exists a T1>0, such that eμtExtXp<ε1 for t≥T1−τ; thus, we can get(24)k21t=eμtsupθ∈−τ,0Ext+θXp≤supθ∈−τ,0e−μθsupθ∈−τ,0eμt+θExt+θXp≤eμτε1,t∈T1,+∞.
Sok21t⟶0 as t⟶+∞. From condition 4, we get k22t⟶0 as t⟶+∞. That is to say(25)Ek2tXp⟶0,ast⟶+∞.
Further, Hölder’s inequality, Lemma4, and condition 4 yield(26)Ek3tXp≤eμtE∫0t−A1−αSt−s−AαGs,xsdsXp≤M1−αpeμtE∫0te−λt−st−s1−α−AαGs,xsdsXp≤M1−αpeμt∫0te−λt−st−s1−αdsp−1∫0te−λt−st−s1−αE−AαGs,xsXpds≤M1−αpΓp−1αλ−αeμt∫0te−λt−st−s1−αE−AαGs,xs−Gs,0+Gs,0Xpds≤2p−1LGpM1−αpΓp−1αλ−αeμt∫0te−λt−st−s1−αsupθ∈−τ,0Exs+θXpds+2p−1M1−αpΓp−1αλ−αeμt∫0te−λt−st−s1−α−AαGs,0Xpds≔2p−1LGpM1−αpΓp−1αλ−αk31t+2p−1M1−αpΓp−1αλ−αk32t.
For anyxt∈S and any ε2>0, there exists a T2>0, such that eμtExtXp<ε2 for t≥T2−τ. Thus, we can get(27)eμssupθ∈−τ,0Exs+θXp<eμrε2,s∈T2,+∞,then(28)eμt∫T2te−λt−st−s1−αe−μseμssupθ∈−τ,0Exs+θXpds<eμrε2∫T2te−λ−μt−st−s1−αds≤Γαλ−μ−αeμrε2.
Ase−λ−μt⟶0 as t⟶+∞, then there exists T3≥T2 such that for any t≥T3, we have(29)eμ−λt∫0T2eλst−s1−αsupθ∈−τ,0Exs+θXpds<ε2.
So from the above, we obtain for anyt≥T3(30)k31t<Γαλ−μ−αeμr+1ε2.that is to say, k31t⟶0 as t⟶+∞. Similar computations can be used to show that k32t⟶0 as t⟶+∞. So, we conclude that(31)Ek3tXp⟶0,ast⟶+∞.
As for the fourth term on the right-hand side of (21), we have(32)Ek4tXp≤eμtE∫0tMe−λt−sfs,xsdsXp≤Mpeμt∫0te−λt−sdsp−1∫0te−λt−sEfs,xsXpds≤Mpλ−p−1eμt∫0te−λt−sEfs,xsXpds=Mpλ−p−1eμt∫0te−λt−sEfs,xs−fs,0+fs,0Xpds≤2p−1MpLfpλ−p−1eμ−λt∫0teλ−μseμssupθ∈−τ,0Exs+θXpds+2p−1Mpλ−p−1eμ−λt∫0teλ−μseμsfs,0Xpds≔2p−1MpLfpλ−p−1k41t+2p−1Mpλ−p−1k42t.
For anyxt∈S and any ε3>0, there exists a T4>0, such that eμtExtXp<ε3 for t≥T4. Thus, we can get(33)eμssupθ∈−τ,0Exs+θXp<eμrε3,s∈T4,+∞,then(34)eμ−λt∫T4teλ−μseμssupθ∈−τ,0ExsXpds<λ−μ−1ε3.
Ase−λ−μt⟶0 as t⟶+∞, there exists T5≥T4 such that for any t≥T5, we have(35)eμ−λt∫0T4eλssupθ∈−τ,0Exs+θXpds<ε3.
So from the above, we obtain for anyt≥T5(36)k41t<λ−μ−1+1ε3.
That is to say,k41t⟶0 as t⟶+∞. Similar computations can be used to show that k41t⟶0 as t⟶+∞. So, we conclude that(37)Ek4tXp⟶0,ast⟶+∞.
As for the fifth term on the right-hand side of (21), from Lemma 1 and Hölder’s inequality, we have(38)Ek5tXp≤pp−12p/2Mpeμt∫0te−pλt−sEgs,xsℒQ10p2/pdsp/2=pp−12p/2Mpeμt∫0te−2λp−2t−s/pe−2λt−s/pEgs,xsℒQ10p2/pdsp/2≤pp−12p/2Mpeμt∫0te−2λt−sdsp/2−1∫0te−2λt−sEgs,xsℒQ10pds≤pp−12p/2Mp2λ−p/2−1eμt∫0te−2λt−sEgs,xs−gs,0+gs,0ℒQ10pds≤pp−12p/2MpLgp2λ−p/2−12p−1eμ−2λt∫0te2λ−μseμssupθ∈−τ,0Exs+θXpds+pp−12p/2Mp2λ−p/2−12p−1eμ−2λt∫0te2λ−μseμsgs,0ℒQ10pds.
Similar to the proof of (32), for (38), we can obtain(39)Ek5tXp⟶0,ast⟶+∞.
As for the sixth term on the right-hand side of (21), by Lemmas 2 and 3, we have(40)Ek6tXp≤eμtCpE∫0tSt−sσsdBHsX2p/2≤eμtCp2M2Ht2H−1∫0te−2λt−sσsℒQ202dsp/2≤Cp2p/2Hp/2Mpt2H−1p/2eμte−2λt∫0taσe2λ−γσsdsp/2≤Cp2p/2Hp/2Mpaσp/2t2H−1p/2eμte−γσt−e−2λt2λ−γ4p/2≤Cp2pHp/2Mpaσp/2t2H−1p/2eμ−pminγσ,2λ/2t2λ−γσp/2⟶0,ast⟶+∞.
As for the last term on the right-hand side of (21), we get(41)Ek7tXp≤eμtE∑0<tk<tSt−tkIkxtkXp≤eμtE∑0<tk<tSt−tkXIkxtkXp≤MpeμtE∑0<tk<tdke−λt−tkxtk−Xp≤Mp∑k=1+∞dkp−1∑0<tk<tdke−pλt−tkeμtExtk−Xp.
For anyxt∈S and any ε4>0, there exists a T6>0, such that eμtExtXp<ε3 for t≥T6. Thus, we can get(42)eμtkExtk−Xp<ε4,t∈T6,+∞,then(43)Mp∑k=1+∞dkp−1∑T6<tk<tdke−pλt−tkeμtExtk−Xp≤Mp∑k=1+∞dkp−1∑T6<tk<tdke−pλ−μt−tkε4≤Mp∑k=1+∞dkpε4.
Ase−λ−μt⟶0 as t⟶+∞, then there exists T7≥T6 such that for any t≥T7, we have(44)Mp∑k=1+∞dkp−1∑0<tk≤T6dke−pλt−tkeμtExtk−Xp≤e−pλ−μtMp∑k=1+∞dkp−1∑0<tk≤T6dkepλtkExtk−Xp<ε4.
So from the above, we obtain for anyt≥T7(45)k7t≤Mp∑k=1+∞dkp+1ε4.
So we have(46)Ek7tXp⟶0,ast⟶+∞.
Thus, from (20)–(32), (37), and (39)–(46), we know that eμtEπxtXp⟶0 as t⟶+∞. So, we conclude that πS⊂S.
Thirdly, we will show thatπ is contractive. Let x,y∈S by using the inequality(47)d1+d2+d3+d4+d5p≤d1pkp−1+4p−1d2p1−kp−1+4p−1d3p1−kp−1+4p−1d4p1−kp−1+4p−1d5p1−kp−1,where d1,d2,d3,d3,d5 are nonnegative constants and k=LG−A−αX<1. For any fixed t∈0,T, we have(48)Eπxt−πxtXp≤1kp−1EGt,xt−Gt,ytXp+4p−11−kp−1E∫0t−A1−αSt−s−AαGs,xs−Gs,ysdsXp+4p−11−kp−1E∫0tSt−sfs,xs−fs,ysdsXp+4p−11−kp−1E∫0tSt−sgs,xs−gs,ysdWsXp+4p−11−kp−1E∑0<tk<tSt−sIkxtk−IkytkXp.
Hence,(49)sups∈−τ,TEπxs−πysXp≤sups∈−τ,TExs−ysXp×k+4p−1M1−αpΓpαLGP1−kp−1λpα+4p−1MpLfp1−kp−1λp+4p−1pp−1/2p/2MpLgp1−kp−12λp/2+4p−1∑k=1+∞dkp1−kp−1.
Thus, by (19), we know that π is a contraction mapping.
Hence, by the Contraction Mapping Theorem,π has a unique fixed point xt in S, which is a solution of equation (10) with xs=φs on −τ,0 and eμtExtXp⟶0 as t⟶+∞. 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 (p≥2) moment exponential stability: 1/p<α≤1 and(50)4p−1LGp−A−αXp+4p−1M1−αpΓp−1pα−1/p−1LGpλpα+4p−1MpLfpλp+4p−1MpLgppp−1/2p/2p−2/2p−1p/2−1λp/2<1,where 00≔1. Our condition is 0<α≤1 and(51)k+3p−1M1−αpΓpαLGP1−kp−1λpα+3p−1MpLfp1−kp−1λp+3p−1pp−1/2p/2MpLgp1−kp−12λp/2<1.
Now we show that why (51) is weaker than (50). Obviously,(52)Γpα=∫0+∞e−xxα−1dsp=∫0+∞e−x/pe−p−1x/pxα−1dsp+∫0+∞e−xds∫0+∞e−xxpα−1/p−1−1dsp−1≤Γp−1pα−1p−112p/2≤p−22p−1p/2−1,p>2.
Then we can transfer to prove (51) is weaker than following condition:(53)4p−1LGp−A−αXp+4p−1M1−αpΓpαLGPλpα+4p−1MpLfpλp+4p−1pp−1/2p/2MpLgp2λp/2<1.
Obviously, (53) is not hold for k=LG−A−αX∈41−p/p,1 and (51) may hold for k∈41−p/p,1. On the other hand, when k=LG−A−αX∈0,41−p/p, conditions (51) and (53) can be transferred into the following conditions, respectively.(54)3p−1M1−αpΓpαLGP1−kpλpα+3p−1MpLfp1−kpλp+3p−1pp−1/2p/2MpLgp1−kp2λp/2<14p−1M1−αpΓpαLGP1−4p−1LGp−A−αXpλpα+4p−1MpLfp1−4p−1LGp−A−αXpλp+4p−1pp−1/2p/2MpLgp1−4p−1LGp−A−αXp2λp/2<1.
Obviously,(55)3p−11−kp≤4p−11−4p−1kp,k∈0,41−p/p.
Then we conclude that (51) is weaker than (50); in this sense, this paper improves and generalizes the results in [27].Remark 2.
Theorem1 does not ask for Gt,0≡0, ft,0≡0, and gt,0≡0, which are imposed in [10]. Even in this special case, our results also improve the result in [10]. Our conditions are 0<α≤1 and(56)k+4p−1M1−αpΓpαLGP1−kp−1λpα+4p−1MpLfp1−kp−1λp+4p−1pp−1/2p/2MpLgp1−kp−12λp/2+4p−1∑k=1+∞dkp1−kp−1<1.
However, the corresponding conditions in [15] are 1/p<α≤1 and(57)ρ≔7p−1−A−αXpLGp+7p−1M1−αpΓp−1pα−1/p−1LGp1−kp−1λpα+14p−1MpLfpλp+14p−1MpLgppp−1/2p/2p−2/2p−1p/2−1λp/2+7p−1Mp∑k=1+∞dkp<1.
In 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:(58)dxt,ξ+L1xt−τGt,ξ+a1e−γ1tM1−α−Aα=∂2xt,ξ∂ξ2+L2xt−τft,ξ+a2e−γ2tdt+L3xt−τgt,ξ+a3e−γ3tdWt+a4e−γ4tdBHt,t≥0,t≠tk,Δxtk,ξ=Ikxtk,ξ=CIxtk−kk+1,t=tk,k=1,2,…,xt,0=xt,π=0,t≥0,xt,ξ=φt,ξ,t∈−τ,0,0≤ξ≤π,where Wt is the Brownian motion and BHt is the fractional Brownian motion with Hurst parameter H∈1/2,1, γ1,γ2,γ3,γ4>0, and 0≤τGt,τft,τgt≤τ. Let X=ℒ20,π and Y=ℝ1, and define the operator A: DA⊂X⟶X given by A=∂2/∂ξ2 with domain(59)DA≔v⋅∈X,v,v′are absolutely continuous,v″∈X,v0=vπ=0.It is well known that an analytic semigroupSt,t≥0 generated by the operator A on X satisfies StX≤e−π2t. Furthermore, −A has a discrete spectrum, the eigenvalues are n2, n∈ℕ, with the corresponding normalized eigenvectors enx=2/πsinnx. Then, for y∈DA, we have(60)Ay=∑n=1∞n2y,enen,Sty=∑n=1∞e−π2ty,enen.Let(61)Gt,xt−τGt,ξ=L1xt−τut,ξ+a1e−γ1tM1−α−Aα,ft,xt−τft,ξ=L2xt−τft,ξ+a2e−γ2t,gt,xt−τgt,ξ=L3xt−τgt,ξ+a3e−γ3t,σt=a4e−γ4t.Obviously,(62)−AαGt,xt−τGt,ξ−−AαGt,yt−τGt,ξX≤L1M1−αxt−τGt,ξ−yt−τGt,ξX,ft,xt−τft,ξ−ft,yt−τft,ξX≤L2xt−τft,ξ−yt−τft,ξX,gt,xt−τgt,ξ−gt,yt−τgt,ξℒQ10≤L3xt−τgt,ξ−yt−τgt,ξX,−AαGt,0,ξX≤a1e−γ1t,ft,0X≤a2e−γ2t,gt,0ℒQ10≤a3e−γ3t,σtℒQ20=a4e−γ4t.From the definition of−A−α, we get(63)−A−αX≤1Γα∫0+∞tα−1StXdt≤1Γα∫0+∞tα−1e−π2tdt≤1π2α.It is obvious that all the assumptions are satisfied with(64)λ=π2,LG=L1M1−απ2α,Lf=L2,Lg=L3,∑k=1∞dk=1.Then, the mild solution of (58) is exponentially stable in pth moment provided that(65)k+4p−1ΓpαL1p1−kp−1π4pα+4p−1L2p1−kp−1π2p+4p−1pp−1/2p/2L3p1−kp−12π2p/2+5p−1CIp1−kp−1<1,where k=L1/M1−απ4α and α∈0,1.


## 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 inpth 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.



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*Source: 9051396-2020-01-21.xml*