# Existence and Exponential Stability of Solutions to Stochastic Neutral Functional Differential Equations

Authors: Ling Hu; Zheng Wu; Zhangzhi Wei; Lianglong Wang
Journal: Discrete Dynamics in Nature and Society
 (2017)

---

## Abstract

In this paper we consider the existence and stability of solutions to stochastic neutral functional differential equations with finite delays. Under suitable conditions, the existence and exponential stability of solutions were obtained by using the semigroup approach and Banach fixed point theorem.


---

## Body

## 1. Introduction

In natural world, stochastic phenomena are everywhere. Many stochastic facts exist in biology, chemistry, physics, and economical systems. In addition, delays also appear sometimes to change the results. These facts imply the necessity to study stochastic functional differential equations (SFDEs), although there are a lot of papers on the related topics for deterministic partial functional differential equations [1–4].A large amount of basic knowledge about stochastic differential equations (SDEs) has been given in [5, 6]. Recently, many researchers studied the existence, stability, and other properties of solutions to SFDEs. Some of these topics have been solved by the semigroup approach and others have been solved by the variational one. There are many types of equations worth attentions; say [7] is focused on retarded equations with finite delays while [8] is concerned with neutral ones. In fact the stability of SDEs is most important among all qualitative properties and also determines whether the model is significant. Aside from all kinds of stability, other properties have drawn many attentions. For example, Wu et al. [9, 10] investigated the stochastic delay population dynamics under regime switching by using generalized Itô’s formula, Gronwall inequality, and Young’s inequality. Furthermore, some stochastic systems have added the impulse factors [11].On the other hand, although stochastic partial FDEs with finite delays also seem very important, the corresponding properties of these systems have not been studied in great detail.Taniguchi et al. [12] discussed the stochastic partial FDEs with finite delay as follows:(1)dXt=-AXt+ft,Xtdt+gt,XtdWt,t≥t0,Xt0=ϕ∈LpΩ,Cα,t0≥0.Under some suitable conditions, the existence and asymptotic behavior of solutions were obtained employing the semigroup approach and fixed point theorem.In this paper, we shall discuss the existence and uniqueness of mild solutions to a class of stochastic NFDEs with finite delays,(2)dXt-ft,Xt=-AXt+Ft,Xtdt+Gt,XtdWt,t≥t0,Xt0=ϕ∈LpΩ,Cα,t0≥0,where ϕ is Ft0-measurable. -A is a linear and densely closed operator which generates an analytic semigroup S(t),  t≥0, on a real separable Hilbert space H with the inner product (·,·) and norm ·.Notice that (1) is of retarded type; here we put a neutral type one, (2). When f(t,Xt)≡0, our equation is the same as that in [12]. In this paper we assume 0<α<1,  p>2 and define D(Aα) with the norm xα≔Aαx for x∈D(Aα), where D(Aα) is the domain of the fractional power operator Aα:H→H. Let Hα≔D(Aα) and Cα=C([-r,0],Hα),  0<r<∞. K is another real separable Hilbert space with the inner product (·,·)K and norm ·K, W(t) is a K-valued Wiener process with a finite trace nuclear covariance operator Q≥0. Assume f,F:(-∞,+∞)×Cα→H,  G:(-∞,+∞)×Cα→L20(K,H) are measurable mappings, satisfying that f(t,0),  F(t,0),  G(t,0) are locally bounded in H-norm and L20(K,H)-norm, respectively.This paper is organized as follows. In Section2 some preliminary results are given, which are fundamental for the subsequent developments. The existence and uniqueness of solutions are investigated in Section 3. At last, we obtain the almost sure exponential stability of the solutions.


## 2. Preliminaries

Let(Ω,F,P) be a probability space and Xt:Ω→Hα,  t≥t0-r is a continuous Ft-adapted, Hα-valued stochastic process. We give process Xt:Ω→Cα,  t≥t0, by setting Xt(s)(ω)=X(t+s)(ω),  s∈[-r,0].LetMCα(t,p),  p>2, be the space of all Ft-measurable Cα-valued functions ψ:Ω→Cα:(3)EψCαp=Esup-r≤s≤0⁡Aαψsp<∞.Next we give three important assumptions.Assumption A.
(a)H is a separable Hilbert space on which there is an analytic semigroup S(t),  t≥0, and -A is the infinitesimal generator of S(t).
(b) There existM≥1 and a>0 such that S(t)h≤Me-ath,  t≥0,  h∈H.
(c)AαS(t)h≤Mαe-att-αh,  t≥0, for any h∈H, where Mα≥1,  A-α≤C.
(d)S(t)h-h≤NαtαAαh,  h∈D(Aα),  Nα≥1.
The details can be seen in Pazy [13].
Under Assumption A, we will consider the stochastic integral equation next instead of (2) by carrying out a semigroup type argument mentioned above:(4)Xt=St-t0ϕ0-ft0,ϕ+ft,Xt-∫t0tASt-ufu,Xudu+∫t0tSt-uFu,Xudu+∫t0tSt-uGu,XudWu(5)Xt0=ϕ∈MCαt0,p,t0≥0.The function X(t) which satisfies (4) is called the mild solution of (2).Lemma 1 (see [12]).
SupposeΦ(t):Ω→L20(K,H),  t≥0. If Φ(t)k∈Hα,  t≥0,  k∈K, and ∫0tEΦ(u)L202du<∞,  ∫0tEAαΦ(u)L202du<∞ then(6)Aα∫0tΦsdWs=∫0tAαΦsdWs.Assumption B.
For anyγ,ξ∈Cα and t0≤t≤T, there exist three positive real constants N0=N0(T), N1=N1(T), and N2=N2(T)>0 such that(7)ft,γ-ft,ξp≤N0γ-ξCαp,Ft,γ-Ft,ξp≤N1γ-ξCαp,Gt,γ-Gt,ξL20p≤N2γ-ξCαp.Under Assumption B, we get a real number N3=N3(T)>0 such that(8)ft,ξp+Ft,ξp+Gt,ξL20p≤N31+ξCαp,for t≤t≤T, where T is any fixed time.Assumption C.
There existα∈(0,1) and L≥0 such that(9)Aαft,ψ1-Aαfs,ψ2≤Lt-s+ψ1-ψ2Cα,Aαft,ψ≤L1+ψCα,for every t0≤s,t≤T,  ψ1,ψ2,ψ∈Ω, and 4p-1Lp<1.


## 3. Existence of Mild Solutions

Definition 2.
If functionX(t) satisfies (4) on [t0,T] and T-t0>0 is small, it is called local solution. If X(t) satisfied (4) on [t0,+∞], it is called global solution.Theorem 3.
Let0<α<(p-1)/2p. Suppose that Assumptions A, B, and C hold. There exists a unique local continuous solution to (4) for any initial value (t0,ϕ) with t0≥0 and ϕ∈MCα(t0,p).In order to prove Theorem3, we need some lemmas. Assume T>t0 is a fixed time and DT is the subspace of all continuous processes Z which belong to the space C([t0-r,T],Lp(Ω,H)) with ZDT<∞, where (10)ZDT≔supt0≤t≤T⁡EZtCp1/p,EZtCp≔Esup-r≤s≤0⁡Ztsp.Introduce the following mapping Φ on DT: (11)ΦZt=AαSt-t0ϕ0-ft0,ϕ+Aαft,A-αZt-∫t0tAα+1St-sfs,A-αZsds+∫t0tAαSt-sFs,A-αZsds+∫t0tAαSt-sGs,A-αZsdWs,t≥t0,ΦZt=Aαϕt,t0≥t≥t0-r.Lemma 4.
ForZ∈DT, (ΦZ)(t) is continuous on [t0,T] in the Lp-sense.Proof.
Lett0<t1<t2<T. For any fixed Z∈DT, (12)EΦZt2-ΦZt1p≤5p-1ESt2-t0-St1-t0Aαϕ0-ft0,ϕp+5p-1EAαft2,A-αZt2-Aαft1,A-αZt1p+5p-1E∫t0t2Aα+1St2-sfs,A-αZs-∫t0t1Aα+1St1-sfs,A-αZsdsp+5p-1E∫t0t2AαSt2-sFs,A-αZs-∫t0t1AαSt1-sFs,A-αZsdsp+5p-1E∫t0t2AαSt2-sGs,A-αZs-∫t0t1AαSt1-sGs,A-αZsdWsp=I1+I2+I3+I4+I5.
From Assumptions A, B, and C we get that(13)I1≤5p-1Nαpt2-t1pαEAαSt1-t0Aαϕ0-ft0,ϕp,I2≤10p-1Lpt2-t1p+10p-1EZt2-Zt1p,because of the continuity of Zt,  I2→0, when t2→t1. (14)I3≤10pE∫t1t2Aα+1St2-sfs,A-αZsdsp+10pE∫t0t1Aα+1St1-sSt2-t1-Ifs,A-αZsdsp≤I31+I32.There exist Q31,Q32>0 and let q=p/(p-1),  ϵ1=1-α+p/q>0 such that(15)I31≤10p-1Mα+1pN3E∫t1t2t2-s-α-1e-at2-s1+ZsCp1/pdsp≤Q31t2-t1ϵ11+ZDTp,I32≤10p-1E∫t0t1Aα+1St1-s2St2-t1-ISt1-s2fs,A-αZsdsp≤Q32t2-t1pα1+ZDTp.In the similar way, there exist Q41,Q42>0 and we have(16)I4≤Q41t2-t11+ϵ11+ZDTp+Q42t2-t1pα1+ZDTp.At last (17)I5≤10pE∫t1t2AαSt2-sGs,A-αZsL202dsp/2+10pE∫t0t1AαSt1-sSt2-t1-IGs,A-αZsL202dsp/2≤I51+I52.Then, there exist Q51,Q52>0 and ϵ2=(p-1-2p2α)/2p>0 such that(18)I51≤10p-1MαpN3pE∫t1t2t2-s-2αe-2at2-s1+ZsC2dsp/2≤Q51t2-t11+ϵ21+ZDTp,I52≤10p-1Mα2pNαpE∫t0t1t1-s2-2αe-2at1-st2-t1αGs,A-αZsL202dsp≤Q52t2-t1pα1+ZDTp.
SinceZ∈DT, it follows that I1,  I2,  I3,  I4,  I5 tend to zero, as t2→t1. Therefore, the proof of the lemma is complete.Lemma 5.
Suppose the operator mappingΦ and the corresponding domain DT are defined as above; then Φ(DT)⊂DT.The proof is similar to that of Lemma4.Proof of Theorem3.
LetX,Y∈DT, and then for any fixed t∈[t0,T], (19)EΦXt-ΦYtCp≤Esup-r≤θ≤0⁡ΦXt+θ-ΦYt+θp≤4p-1Esup-r≤θ≤0⁡Aαft+θ,A-αXt+θ-Aαft+θ,A-αYt+θp+4p-1Esup-r≤θ≤0⁡∫t0t+θAα+1St+θ-sfs,A-αXs-fs,A-αYsdsp+4p-1Esup-r≤θ≤0⁡∫t0t+θAαSt+θ-sFs,A-αXs-Fs,A-αYsdsp+4p-1Esup-r≤θ≤0⁡∫t0t+θAαSt+θ-sGs,A-αXs-Gs,A-αYsdWsp=I11+I12+I13+I14,I11≤4p-1LpEsup-r≤θ≤0⁡A-αXt+θ-A-αYt+θp≤4p-1LpX-YDT,I12≤4p-1N0Mα+1pT-t0Γ1-αqaqqα-1p/qX-YDTp,I13≤4p-1N1MαpT-t0Γ1-αqaqqα-1p/qX-YDTp.
Next, let1/p+α<ρ<1/2 and(20)Δs=∫t0ss-σ-ρSs-σGσ,A-αXσ-Gσ,A-αYσdWσ,and then by the use of the stochastic Fubini Theorem again we have(21)I14≤4p-1Esup-r≤θ≤0⁡Aα∫t0t+θt+θ-sρ-1St+θ-sΔsdsp≤4p-1N2MαpT-t0Γ1-1+α-ρqaqq1+α-ρ-1p/q·cpMpT-pρ+p/21-2ρp/2X-YDTp.
So we can choose a suitableT>t0 to make T-t0>0 sufficiently small; we obtain B(T)∈(0,1) such that(22)ΦX-ΦYDT≤BTX-YDT,for any X,Y∈DT. Thus, by Banach fixed point theorem we get a unique fixed point U∈DT. Setting X(t)=A-aU(t), we obtain(23)Xt=St-t0ϕ0-ft0,ϕ+ft,Xt-∫t0tASt-sfs,Xsds+∫t0tSt-sFs,Xsds+∫t0tSt-sGs,XsdWs,t≥t0,Xt0=ϕt,t0≥t≥t0-r.The proof is complete.Theorem 6.
Assume0<α<(p-2)/2p and let f,F:(-∞,+∞)×Ca→H,  g:(-∞,+∞)×Ca→L20(K,H) satisfy Assumptions A, B, and C. If there exists a constant B2>0 such that(24)ft,ψp+Ft,ψp+Gt,ψL20p≤B21+ψCαpfor all ψ∈Ca,  t≥0, then there exists a unique, global continuous solution X(t):Ω→Hα to (4) for any initial value (t0,ϕ) with ϕ∈MCα(t0,p).Proof.
IfF and G satisfy the global Lipschitz conditions, the proof of the theorem can be given similarly as a corollary of Theorem 3. If F and G satisfy the local Lipschitz conditions, the proof can be given similarly by the truncation method in [14]. Hence, we omit the proof.


## 4.Pth Moment Exponential Stability

Lemma 7 (see [12]).
Let0<θ<(p-2)/2p and assume β is any fixed real number such that 1/p+θ<β<1/2. Then for all t∈(t0+r,T),(25)Esup-r≤s≤0⁡∫t0t+sSt+s-uΦudWuθp≤cpearMθpMpCp,θ,βΓ1-2β2a-2ap2β-1·∫t0te-at-σEΦσL20pdσ,where C(p,θ,β)≔Γ1-q1+θ-βaq1+θ-βp/q.Now, we are in a position to present the stability.Theorem 8.
Assume0<θ<(p-1)/2p. Let f,F:(-∞,+∞)×Cθ→H,  G:(-∞,+∞)×Cθ→L20(K,H) satisfy the local Lipschitz condition B. Furthermore assume that Assumption A is satisfied and there exist nonnegative real numbers Q1,Q2≥0 and continuous functions ξ1,ξ2:[0,∞)→R+ such that(26)EFt,Xtp≤Q1EXtCθp+ξ1t,t≥t0,EGt,XtL20p≤Q2EXtCθp+ξ2t,t≥t0,for any solution X(t) to (4). Suppose a>L0, where S(t)≤Me-at,  t≥0, and there exist nonnegative real numbers P1,P2≥0 such that ξjt≤Pje-(a-L0)t  (j=1,2), and then there exist positive constants ε>0 and K(p,ε,ϕ)>0 such that for each t≥t0+2r(27)EXtCθp≤Kp,ε,ϕ·e-εt-t0.In other words, the solution is the pth moment exponentially stable.Proof.
Without loss of generality, we supposet0=0. Let -r≤s≤0; then for each t>2r, (28)EXt+sθp≤5p-1ESt+sϕ0-f0,ϕθp+5p-1Eft+s,Xt+sθp+5p-1E∫0t+sASt+s-ufu,Xuduθp+5p-1E∫0t+sSt+s-uFu,Xuduθp+5p-1E∫0t+sSt+s-uGu,XudWuθp=J1+J2+J3+J4+J5,J1≤5p-1Mθpe-pat-rt-r-pθEϕ0-f0,ϕp,J2≤5p-1LpE1+Xt+sCθp.
On the other hand, by the Hölder inequality we can obtain that(29)J3≤5p-1Mθ+1pe-asΓ1-qθaqθ-1p/q∫0t+se-at-uQ1EXuCθp+ξ1du,J4≤5p-1Mθpe-asΓ1-qθaqθ-1p/q∫0t+se-at-uQ1EXuCθp+ξ1udu.
By virtue of Lemma7, we can deduce(30)J5≤5p-1MθpMpe-ascpCp,θ,βΓ1-2β2a-2ap2β-1·∫0t+se-at-uQ2EXuCθp+ξ2udu.Then we have(31)EXt+sθp≤M0·e-at+L0e-at+s∫0t+seauEXuCθpdu+e-at∫0t+seauL1ξ1u+L2ξ2udu,for each t≥2r, where(32)M0=5p-1Mθpr-pβeparEϕ0-f0,ϕp,L1=5p-1Mθ+1p+MθpΓ1-qθaqθ-1p/q,L2=5p-1MθpMpcpCp,θ,βΓ1-2β2a-2ap2β-1L0=L1Q1+L2Q2.Therefore, we have for arbitrary ε∈R+,  0<ε<a and T>0 large enough,(33)∫0TeεtEXt+sθpdt≤M0∫0Te-at+εtdt+L0∫0Teεt-at+s∫0t+seauEXuCθpdudt+∫0Te-at+εt∫0t+seauL1ξ1u+L2ξ2ududt.
On the other hand, for-r≤s≤0 and t>2r, (34)∫0Teεt-at+s∫0t+seauEXuCθpdudt≤1a-ε∫s0eau-sEXuCθpdu+1a-ε∫0Teau-s·e-a-εu-sEXuCθpdu=1a-ε∫s0eau-sEXuCθpdu+1a-ε∫0Teεu-sEXuCθpdu,and therefore,(35)∫0TeεtEXt+sθpdt≤M0∫0Te-at+εtdt+L0eara-ε∫s0eauEXuCθpdu+L0eεra-ε∫0TeεtEXuCθpdt+∫0Te-at+εt∫0t+seauL1ξ1u+L2ξ2ududt,which, by virtue of the continuity of X(t),  t≥0, immediately implies (36)∫0TeεtEXtCθpdt≤M0∫0Te-at+εtdt+L0eara-ε∫s0eauEXuCθpdu+L0eεra-ε∫0TeεtEXuCθpdt+∫0Te-at+εt∫0t+seauL1ξ1u+L2ξ2ududt.
On the other hand, sincea<L0 by assumption, it is possible to choose a suitable ε∈R+ with 0<ε<a-L0 small enough so that L0eεr/(a-ε)<1, which, letting T>0 tend to infinity and using (36), immediately yields (37)∫0∞eεtEXtCθpdt≤11-L0eεr/a-εM0∫0∞e-at+εtdt+L0eara-ε∫-r0eauEXuCθpdu+∫0Te-at+εt∫0t+seauL1ξ1u+L2ξ2ududt≔K′p,ε,ϕ<∞.So we can easily deduce (note 0<ε<a-L0)(38)EXtCθp≤M0·e-εt+M1ε·e-εt+L0·eεre-εt∫0∞eεuEXuCθpdu≤M0+M1ε+L0·eεrK′p,ε,ϕe-εt≔Kp,ε,ϕe-εt,where(39)M1ε=∫0∞eεtL1ξ1u+L2ξ2udu<∞.So(40)EXtCθp≤Kp,ε,ϕe-εt.The proof is complete.



---

*Source: 2623035-2017-02-22.xml*