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Existence and Stability Results for Second-Order Neutral Stochastic Differential Equations

With Random Impulses and Poisson Jumps

K. Ravikumar1, K. Ramkumar1, Dimplekumar Chalishajar2,∗
1Department of Mathematics, PSG College of Arts and Science, Coimbatore, 641 046, India;

ravikumarkpsg@gmail.com, ramkumarkpsg@gmail.com2Department of Applied Mathematics, Mallory Hall, Virginia Military Institute,
Lexington, VA 24450, USA

∗Correspondence: chalishajardn@vmi.edu

Abstract. The objective of this paper is to investigate the existence and stability results of second-order neutral stochastic functional differential equations (NSFDEs) in Hilbert space. Initially, weestablish the existence results of mild solutions of the aforementioned system using the Banachcontraction principle. The results are formulated using stochastic analysis techniques. In the laterpart, we investigate the stability results through the continuous dependence of solutions on initialconditions.

1. Introduction
Stochastic differential equations (SDEs) captures disturbances from random factors. Mathe-matical models obtained by integrating stochastic process provide a better understanding of thereal-world system [12]. For elementary study of stochastic differential equations, the reader mayrefer to [7, 12, 14, 24].Impulsive differential equations also attracted the attention of researchers (see [4, 11, 13, 21, 22]etc.). Impulse in general occurs as deterministic or random models. Nevertheless by naturalphenomena, the impulses often occur at random time points. Many researches have been undergonesolving various differential equations with fixed time impulses [1, 9, 16, 23]. Random impulsivedifferential equations involving fractional derivative are also studied see [20, 25].It is known that impulsive stochastic differential equations play a vital role in modelling practicalprocesses. Not only from Guassian white noise there are certain other factors that results in therise of random effects. Random impulsive stochastic differential equations (ISDEs) are widely used
Received: 20 Aug 2021.
Key words and phrases. existence; stability; Banach contraction principle; second-order neutral stochastic functionaldifferential equations; random impulse; stochastic differential system.1

https://adac.ee
https://doi.org/10.28924/ada/ma.1.1
https://orcid.org/0000-0002-6146-5544


Eur. J. Math. Anal. 1 (2021) 2
in the fields of medicine, biology, economy, finance and so on. For example, the classical stockprice model [28].

d[S(t)] = fS(t)dt + σS(t)dw(t), t ≥ 0, t 6= τk,
S(τk) = akS(τ−k ), k = 1, 2, ...,
S(0) = S0,

is described using an ISDEs. Here wt is a Brownian motion or Wiener process, S(t) representsthe price of the stock at time t, and {τk} represents the release time of the important informationrelating to the stock. S(τ−k ) = limt→τk−0 S(t) and S0 ∈ R. In reality, {τk} is a sequence ofrandom variables, which satisfies 0 < τ2 < τ3 < · · · . Recently, in [10] the authors have contributedthe existence and Hyers-Ulam stability of mild solutions for random impulsive stochastic functionalordinary differential equations which are studied using Krasnoselskii’s fixed point theorem.Solving second-order differential equations has been observed by many scholars. Many authorssolved second-order stochastic differential equations see [5, 6, 8, 19]. However, there are not manypapers considering the existence and stability results on stochastic differential equations withrandom impulse. Anguraj et.al [3], considered the SDEs with random impulses and Poisson jumpsof the form
d[x(t)] = f(t,xt) + g(t,xt)dw(t) + ∫

U
h(t,xt,u)Ñ(dt,du), t ≥ t0, t 6= τk,

x(ζk) = bk(τk)x(ζ−k ), k = 1, 2, ...,
xt0 = ζ = {ζ(θ) : −τ ≤ θ ≤ 0} .

The authors studied the existence, uniqueness, and stability through continuous dependence oninitial conditions for SDEs with random impulses and Poisson jumps by using Banach fixed pointtheorem. Very recently, Anguraj et.al [2] investigated the Existence and Hyers Ulam stability ofrandom impulsive stochastic functional integrodifferential equations with finite delays.Motivated by the above discussion, here we consider the following second-order NSFDEs withrandom impulses and Poisson jumps.
d
[
x′(t) −h(t,xt)] = [Ax(t) + f(t,xt)] dt + g(t,xt)dω(t) + ∫

U
σ(t,xt,u)Ñ(dt,du), , t ≥ t0, t 6= ξk,

x(ξk) = bk(δk)x(ξ−k ), x′(ξk) = bk(δk)x′(ξ−k ), k = 1, 2, ..., (1.1)
xt0 = φ, x′(t0) = φ,

where A : D(A) ⊂ H → H is the infinitesimal generator of a strongly continuous cosine family
{C(t), t ≥ 0}. W(t) is a given Q-Wiener process with a finite trace nuclear covariance operator
Q > 0. δk is a random variable defined from Ω to D ≡ (0,dk) for k = 1, 2 · · · . Suppose that δi and
δj are independent of each other as i 6= j, (i, j = 1, 2, · · · ). The impulsive moments ξk are randomvariables and satisfy ξk = ξk−1 + δk, k = 1, 2, · · · . Obviously, {ξk} is a process with independentincrements. 0 < t0 = ξ0 < ξ2 < ξ3 < · · · < limk→∞ξk = ∞, and x(ξ−k ) = limt→ξk−0 x(t). bk : Dk → H,



Eur. J. Math. Anal. 1 (2021) 3
for each k = 1, 2, · · · . The time history xt(θ) = {x(t + θ) : −δ ≤ θ ≤ 0} with some given δ > 0.Moreover, h, f,g,σ, and φ,φ will be specified later.To the best of authors knowledge, up to now, no work has been reported to derive the second-order NSFDEs with random impulses and Poisson jumps. The main contributions are summarizedas follows:(1) second-order NSFDEs with random impulses and Poisson jumps is formulated.(2) Initially, we establish the existence results of mild solutions of the aforementioned system usingBanach contraction principle.(3) Next, we investigate the stability results through continuous dependence of solutions on initialconditions.(4) An example is provided to illustrate the obtained theoretical results.The rest of the paper is organised as follows. Section 2 is devoted to basic definitions, notions andlemma. In section 3, existence of mild solutions of the aforementioned system (1.1) is investigatedusing Banach contraction principle. Eventually in section 4, the stability of mild solution is obtainedthrough continuous dependence of solutions on initial conditions.

2. Preliminaries
Let (Ω,=,P) be a complete probability space equipped with the normal filtration {=}t≥t0 . =t0containing all P-null sets. H and K be two real Hilbert spaces. L(H, K) denotes the space of allbounded linear operators from K to H.We may assume that, {N(t), t ≥ t0} be a counting process generated by {ξk, k ≥ 0}. =(1)t denotethe minimal σ algebra denoted by {N(r), r ≤ t} and denote =(2)t the σ-algebra generated by

{ω(s),s ≤ t}. We assume that =(1)∞ ,=(2)∞ and ξ are mutually independent and =t = =(1)t ∨=(2)t .We assume that there exist a complete orthonormal system {en}∞n=1 in K, a bounded sequenceof non-negative real numbers λn such that, Qen = λnen, n = 1, 2, · · · . Let {βn(t)}(n = 1, 2, 3...) bea sequence of real valued one dimensional standard Brownian motion mutually independent over
(Ω,=,P). A Q-Wiener process can be defined by ω(t) = ∞∑

n=1
√
λnβn(t)en, (t ≥ 0). Set Φ ∈L(K, H)

we define, ∥∥Φ∥∥2Q = Tr(ΦQΦ∗) = ∞∑
n=1
∥∥∥√λnΦen∥∥∥2

If ∥∥Φ∥∥2Q < ∞, then Φ is called a Q-Hilbert-Schmidt operator. Let LQ(K, H) denote the space ofall Q-Hilbert-Schmidt operator Φ : K → H. The completion LQ(K, H) of L(K, H) with respect tothe topology induced by the norm ∥.∥Q, where ∥∥Φ∥∥2Q = 〈Φ, Φ〉 is a Hilbert space.Let T ∈ (t0, +∞), J := [t0,T ], Jk = [ξk,ξk+1) , k = 0, 1, · · · , J̃ = {t : t ∈ J, t 6= ξk, k = 1, 2, · · ·}.
L2(Ω, H) be the collection of square integrable =t-measurable, H-valued random variables definedby the norm ∥x∥L2 = (E∥x∥2)12 , the expectation being expressed by the form E∥x∥2 = ∫Ω ∥x∥2 dP.Let PC (J,L2(Ω, H)) = {x : J → L2(Ω, H)}, x is continuous on every Jk, and the left limits
x(ξ−k ),x′(ξ−k ) exist k = 1, 2, · · · be a piecewise continuous space.



Eur. J. Math. Anal. 1 (2021) 4
We may define the space C = C ([−δ, 0], H) which contains all piecewise continuous functionsmapping from [−δ, 0] to H with the norm ∥x∥t = sup

t−δ≤s≤t

∥∥x(s)∥∥ for each t ≥ t0. B be the Banachspace, B([t0−δ,T ],L2(Ω, H)) consists of continuous, =t-measurable, C-valued processes. The normis defined by
∥x∥B = (sup

t∈J E
∥x∥2t)12 .

In (1.1), Ñ(dt,du) = N(dt,du)−dtv(du) denotes the compensated Poisson measure independentof ω(t) and N(dt,du) represents the Poisson counting measure associated with a characteristicmeasure v. For a basic study on the Poisson jumps we refer to the book by [27].Subsequently, we introduce certain definitions of sine and cosine operators.A bounded linear operators family {C(t), t ∈ R} is called a strongly continuous cosine family ifand only if(i) C(0) = I (I is the identity operator in H);(ii) C(t)x is continuous in t, for all x ∈ H;(iii) C(t + s) + C(t −s) = 2C(t)C(s) for all t,s ∈ R.The corresponding strongly continuous sine family {S(t), t ∈ R} is defined by
S(t)x = ∫ t0 C(s)xds, x ∈ H, t ∈ RThen the following property holds:
A

∫ t
t0
S(s)xds = [C(t) −C(t0)] x

Lemma 2.1. [18] Let {C(t), t ∈ R} be a strongly continuous cosine family in H, then for all s,t ∈ R,
the following results are true:
(i) C(t) = C(−t);
(ii) S(s + t) + S(s− t) = 2S(s)C(t);
(iii) S(s + t) = S(s)C(t) + S(t)C(s);
(iv) S(t) = −S(−t);
(v) C(t + s) + C(s− t) = 2C(s)C(t);
(vi) C(t + s) −C(t −s) = 2AS(t)S(s).

Before investigating mild solution (1.1), we consider the second-order neutral functional differ-ential equation, which is given byd[u′(t) −g(t,u(t))] = Autdt + f(t,ut)dt, t ≥ 0,u0 = φ ∈ C,u′(0) = φ ∈ H, t ∈ (−r, 0], (2.1)
where A is the infinitesimal generator of a strongly continuous cosine family {C(t), t ∈ R+} andthe functions g, f ∈L1(0,T ; H).



Eur. J. Math. Anal. 1 (2021) 5
Lemma 2.2. [15] A continuously differentiable function u(t) : [0,T ] → H is called the mild solution
for the Cauchy problem (2.1), if it satisfies,

u(t) = C(t)φ(0) + S(t)[φ −g(0,φ)] + ∫ t0 C(t −s)g(s,us)ds +
∫ t

0 S(t −s)f(s,xs)ds, t ≥ 0,
where

S(t) = 12πi
∫

Γ eλtR(λ2; A)dλ;
C(t) = 12πi

∫
Γ eλtλR(λ2; A)dλ,

and Γ is a suitable path.
Consider the linear second-order linear differential equation with impulse conditions,

u′′(t) = Au(t) + f(t), t ≥ 0, t 6= tk,
u(0) = u0,u′(0) = v0,
u(tk) = bku(t−k ),u′(tk) = bku′(t−k ), k = 1, 2, · · · ,

(2.2)
where 0 = t0 < t1 < t2 < · · · < tk < · · · ,{tk, k ≥ 1} is a sequence of fixed impulsive points,
f(t) : [0,T) → H is an integrable function.
Lemma 2.3. The piecewise continuous differentiable function u(t) : [0,T ] → H is a mild solution of
(2.2), if and only if x(t) satisfies the integral equation

u(t) = k∏
i=1 biC(t)u0 +

k∏
i=1 biS(t)v0 +

k∑
i=1

k∏
j=i bj

∫ ti
ti−1
S(t −s)f(s)ds

×
∫ t
tk
S(t −s)f(s)ds, t ∈ [tk, tk+1), k = 0, 1, · · · . (2.3)

Proof. (i)For t ∈ [0, t1), the mild solution is studied in [17],
u(t) = C(t)u0 + S(t)v0 + ∫ t0 S(t −s)f(s)ds, t ∈ [0, t1).(ii) For t ∈ [t1, t2), we set

u(t) = C(t − t1)u(t1) + S(t − t1)u′(t1) + ∫ t
t1
S(t −s)f(s)ds, t ∈ [t1, t2). (2.4)

Since,
u(t1) = b1u(t−1 ), u′(t1) = b1u′(t−1 ),and from (i) we know

u(t−1 ) = C(t1)u0 + S(t1)v0 + ∫ t10 S(t1 −s)f(s)ds; (2.5)
u′(t−1 ) = AS(t1)u0 + C(t1)v0 + ∫ t10 C(t1 −s)f(s)ds. (2.6)



Eur. J. Math. Anal. 1 (2021) 6
Thus,
u(t) = b1C(t − t1)C(t1)u0 + b1S(t − t1)AS(t1)u0 + b1C(t − t1)S(t1)v0 + b1S(t − t1)C(t1)v0

+ b1C(t − t1)∫ t10 S(t1 −s)f(s)ds + b1S(t − t1)
∫ t1

0 S(t1 −s)f(s)ds
+ ∫ t

t1
S(t −s)f(s)ds, t ∈ [t1, t2).

Applying Lemma 2.1, we get
u(t) = b1C(t)u0 + b1S(t)v0 + b1 ∫ t10 S(t1 −s)f(s)ds +

∫ t
t1
S(t −s)f(s)ds, t ∈ [t1, t2).

(iii) For t ∈ [t2, t3),
u(t) = C(t − t2)u(t2) + S(t − t2)u′(t2) + ∫ t

t2
S(t −s)f(s)ds

= C(t − t2)b2u(t−2 ) + S(t − t2)b2u′(t−2 ) + ∫ t
t2
S(t −s)f(s)ds. (2.7)

From the conclusions of (ii), it is known that,
u(t−2 ) = b1C(t2)u0 + b1S(t2)v0 + b1 ∫ t20 S(t2 −s)f(s)ds +

∫ t2
t1
S(t2 −s)f(s)ds; (2.8)

u′(t−2 ) = b1AS(t2)u0 + b1C(t2)v0 + b1 ∫ t20 C(t2 −s)f(s)ds +
∫ t2
t1
C(t2 −s)f(s)ds (2.9)

Along with (2.7) and using Lemma 2.1, we have
u(t) = b2b1C(t)u0 + b2b1S(t)v0 + b2b1 ∫ t10 S(t −s)f(s)ds

+ b2 ∫ t2
t1
S(t −s)f(s)ds + ∫ t2

t1
S(t −s)f(s)ds, t ∈ [t2, t3)

Similarly, for all t ∈ [tk, tk−1).
x(t) = k∏

i=1 biC(t)u0 +
k∏
i=1 biS(t)v0 +

k∑
i=1

k∏
j=i bj

∫ ti
ti−1
S(t −s)f(s)ds + ∫ t

ξk
S(t −s)f(s)ds.

�

By Lemma 2.2, Lemma 2.3 the mild solution of the system (1.1) applying index function for t ∈ J.
Definition 2.1. For a given T ∈ (t0, +∞), a =-adapted process function {x ∈B, t0 −δ ≤ t ≤ T}
is called a mild solution of system (1.1), if (i) xt0 (s) = φ(s) ∈L02(Ω,B) for δ ≤ s ≤ 0;
(ii) x′(t0) = φ(t) ∈L02(Ω, H) for t ∈ J;
(iii) The functions f(s,xt),g(s,xt),h(s,xt) and σ(s,xs,u) are integrable, and for a.e. t ∈ J, the



Eur. J. Math. Anal. 1 (2021) 7
following integral equation is satisfied.

x(t) = +∞∑k=0
[ k∏
i=1 bi(δi)C(t − t0)φ(0) +

k∏
i=1 bi(δi)S(t − t0)[φ −h(0,φ)] +

k∑
i=1

k∏
j=i bj(δj)

×
∫ ξi
ξi−1
C(t −s)h(s,xs)ds + ∫ t

ξk
C(t −s)h(s,xs)ds + k∑

i=1
k∏
j=i bj(δj)

∫ ξi
ξi−1
S(t −s)

× f(s,xs)ds + ∫ t
ξk
S(t −s)f(s,xs)ds + k∑

i=1
k∏
j=i bj(δj)

∫ ξi
ξi−1
S(t −s)g(s,xs)dω(s)

+ ∫ t
ξk
S(t −s)g(s,xs)dω(s) + k∑

i=1
k∏
j=i bj(δj)

∫ ξi
ξi−1
∫
U
S(t −s)σ(s,xs,u)N(ds,du)

+ ∫ t
ξk
∫
U
S(t −s)σ(s,xs,u)N(ds,du)]I[ξk,ξk+1)(t), t ∈ [t0,T ]. (2.10)

where,
k∏
j=i bj(δj) = bk(δk)bk−1(δk−1) · · ·bi(δi),

and I(A)(.) is the index function, i.e.,
IA(t) =

1, if t ∈ A,0, if t /∈ A.
Lemma 2.4. For any p ≥ 1, and for LQ(K, H)-valued predictable process u(.) such that,

sup
s∈[0,T ] E

∥∥∥∥∫ s0 u(η)dω(η)
∥∥∥∥2p ≤ (p(2p− 1))p (∫ t0

(
E
∥∥u(s)∥∥2pQ )1/p ds)p , t ∈ J.

3. Existence Results of Mild Solution
To prove the existence of mild solutions of random impulsive stochastic differential equations,the following assumptions are to be made.(H1) C(t), S(t)(t ∈ J) are equicontinuous and there exist positive constants M, M̃ such that

sup
t∈J
∥∥C(t)∥∥ ≤M, sup

t∈J
∥∥S(t)∥∥ ≤M̃. (3.1)

(H2) The functions f : J × C → H; h : J × C → H; g : J × C →LQ(K, H) and σ : J × C ×U → H
E
∥∥f(t,xt) − f(t,yt)∥∥2 ≤Lf ∥∥x −y∥∥2t ,

E
∥∥g(t,xt) −g(t,yt)∥∥2 ≤Lg ∥∥x −y∥∥2t ,

E
∥∥h(t,xt) −h(t,yt)∥∥2 ≤Lh ∥∥x −y∥∥2t ,∫

U
E
∥∥σ(t,xt,u) −σ(t,yt,u)∥∥2 v(du)ds∨∫

U

(
E
∥∥σ(t,xt,u) −σ(t,yt,u)∥∥4 v(du)ds)12 ≤Lσ ∥∥x −y∥∥2t ,∫

U

(
E
∥∥σ(t,xt,u) −σ(t,yt,u)∥∥4 v(du)ds)12 ≤Lσ ∥x∥2t .



Eur. J. Math. Anal. 1 (2021) 8
(H3) For all t ∈ J, there exist constants κf,κg,κh,κσ ∈L′(J,R+) such that,

E
∥∥f(t, 0)∥∥2 ≤ κf, E∥∥g(t, 0)∥∥2 ≤ κg,

E
∥∥h(t, 0)∥∥2 ≤ κh, E∥∥σ(t, 0,u)∥∥2 ≤ κσ.

(H4) E
maxi,k {

k∏
j=i
∥∥bj(δj)∥∥}

 is uniformly bounded then there exist constant N for all δj ∈ Djsuch that
E

maxi,k {
k∏
j=i
∥∥bj(δj)∥∥}

 ≤N .
Theorem 3.1. If assumptions (H1)-(H4) gets satisfied then there exist a unique continuous mild
solution of the system (1.1).

Proof. We define an operator φ : B →B by φx such that,

(φx)(t) =



φ(t), t ∈ [t0 −δ,t0],+∞∑
k=0
[ k∏
i=1 bi(δi)C(t − t0)φ(0) +

k∏
i=1 bi(δi)S(t − t0)[φ −h(0,φ)] +

k∑
i=1

k∏
j=i bj(δj)

×
∫ ξi
ξi−1
C(t −s)h(s,xs)ds + ∫ t

ξk
C(t −s)h(s,xs)ds + k∑

i=1
k∏
j=i bj(δj)

∫ ξi
ξi−1
S(t −s)

× f(s,xs)ds + ∫ tξk S(t −s)f(s,xs)ds + k∑
i=1

k∏
j=i bj(δj)

∫ ξi
ξi−1
S(t −s)g(s,xs)dω(s)

+∫ t
ξk
S(t −s)g(s,xs)dω(s) + k∑

i=1
k∏
j=i bj(δj)

∫ ξi
ξi−1
∫
U
S(t −s)σ(s,xs,u)N(ds,du)

+∫ t
ξk
∫
U
S(t −s)σ(s,xs,u)N(ds,du)]I[ξk,ξk+1)(t), t ∈ [t0,T ].

We need to prove that φ maps B into itself.
E
∥∥(φx)(t)∥∥2 ≤ E∥∥∥∥ +∞∑k=0

[ k∏
i=1 bi(δi)C(t − t0)φ(0) +

k∏
i=1 bi(δi)S(t − t0)[φ −h(0,φ)]

+ k∑
i=1

k∏
j=i bj(δj)

∫ ξi
ξi−1
C(t −s)h(s,xs)ds + ∫ t

ξk
C(t −s)h(s,xs)ds



Eur. J. Math. Anal. 1 (2021) 9
+ k∑

i=1
k∏
j=i bj(δj)

∫ ξi
ξi−1
S(t −s)f(s,xs)ds + ∫ t

ξk
S(t −s)f(s,xs)ds

+ k∑
i=1

k∏
j=i bj(δj)

∫ ξi
ξi−1
S(t −s)g(s,xs)dω(s) + ∫ t

ξk
S(t −s)g(s,xs)dω(s)

+ k∑
i=1

k∏
j=i bj(δj)

∫ ξi
ξi−1
∫
U
S(t −s)σ(s,xs,u)N(ds,du)

+ ∫ t
ξk
∫
U
S(t −s)σ(s,xs,u)N(ds,du)]I[ξk,ξk+1)(t)∥∥∥∥2

≤ 6E[+∞∑k=0
[ k∏
i=1
∥∥bi(δi)∥∥∥∥C(t − t0)∥∥∥∥φ(0)∥∥

]
I[ξk,ξk+1)(t)

]2 + 6E[ +∞∑k=0
[ k∏
i=1
∥∥bi(δi)∥∥∥∥S(t − t0)∥∥

×
∥∥φ −h(0,φ)∥∥]I[ξk,ξk+1)(t)]2 + 6E[ +∞∑k=0

[ k∏
j=i
∥∥bj(δj)∥∥∫ ξi

ξi−1
∥∥C(t −s)∥∥∥∥h(s,xs)∥∥ds

+ ∫ t
ξk
∥∥C(t −s)∥∥∥∥h(s,xs)∥∥ds]I[ξk,ξk+1)(t)]2 + 6E[ +∞∑k=0

[ k∏
j=i
∥∥bj(δj)∥∥∫ ξi

ξi−1
∥∥S(t −s)∥∥

×
∥∥f(s,xs)∥∥ds + ∫ t

ξk
∥∥S(t −s)∥∥∥∥f(s,xs)∥∥ds]I[ξk,ξk+1)(t)]2 + 6E[ +∞∑k=0

[ k∏
j=i
∥∥bj(δj)∥∥

×
∫ ξi
ξi−1
∥∥S(t −s)∥∥∥∥g(s,xs)∥∥dω(s) + ∫ t

ξk
∥∥S(t −s)∥∥∥∥g(s,xs)∥∥dω(s)]I[ξk,ξk+1)(t)

+ 6E[ +∞∑k=0
[ k∏
j=i
∥∥bj(δj)∥∥×∫ ξi

ξi−1
∫
U

∥∥S(t −s)∥∥∥∥σ(s,xs,u)∥∥Ñ(ds,du)
+ ∫ t

ξk
∫
U

∥∥S(t −s)∥∥∥∥σ(s,xs,u)∥∥Ñ(ds,du)]I[ξk,ξk+1)(t)]2,
= 6 6∑

i=1 Gi.
where,

G1 ≤ E
[+∞∑

k=0
[ k∏
i=1
∥∥bi(δi)∥∥∥∥C(t − t0)∥∥∥∥φ(0)∥∥

]
I[ξk,ξk+1)(t)

]2

≤ M2E
maxi,k {

k∏
j=i
∥∥bj(δj)∥∥}


2
E
∥∥φ(0)∥∥2

≤ M2N2E∥∥φ(0)∥∥2 ,
G2 ≤ E

[ +∞∑
k=0
[ k∏
i=1
∥∥bi(δi)∥∥∥∥S(t − t0)∥∥∥∥φ −h(0,φ)∥∥]I[ξk,ξk+1)(t)]2



Eur. J. Math. Anal. 1 (2021) 10
≤ M̃2E

maxi,k {
k∏
j=i
∥∥bj(δj)∥∥}


2
E
∥∥φ −h(0,φ)∥∥2

≤ M̃2N2E∥∥φ −h(0,φ)∥∥2 ,
G3 ≤ E

[ +∞∑
k=0
[ k∏
j=i
∥∥bj(δj)∥∥∫ ξi

ξi−1
∥∥C(t −s)∥∥∥∥h(s,xs)∥∥ds + ∫ t

ξk
∥∥C(t −s)∥∥∥∥h(s,xs)∥∥ds]I[ξk,ξk+1)(t)]2

≤ M2E
maxi,k {1,

k∏
j=i
∥∥bj(δj)∥∥}


2 (T − t0)∫ t

t0
E
∥∥h(s,xs)∥∥2 ds

≤ 2M2 max{1,N2}(T − t0)[∫ t
t0
E
∥∥h(s,xs) −h(s, 0)∥∥2 ds + ∫ t

t0
∥∥h(s, 0)∥∥2 ds]

≤ 2M2 max{1,N2}(T − t0)∫ t
t0
[
LhE

∥x∥2s + κh]ds
≤ 2M2 max{1,N2}(T − t0)∫ t

t0
LhE

∥x∥2s ds + 2M2 max{1,N2}(T − t0)2κh,
G4 ≤ E

[ +∞∑
k=0
[ k∏
j=i
∥∥bj(δj)∥∥∫ ξi

ξi−1
∥∥S(t −s)∥∥∥∥f(s,xs)∥∥ds + ∫ t

ξk
∥∥S(t −s)∥∥∥∥f(s,xs)∥∥ds]I[ξk,ξk+1)(t)]2

≤ M̃2E
maxi,k {1,

k∏
j=i
∥∥bj(δj)∥∥}


2 (T − t0)∫ t

t0
E
∥∥f(s,xs)∥∥2 ds

≤ 2M̃2 max{1,N2}(T − t0)[∫ t
t0
E
∥∥f(s,xs) − f(s, 0)∥∥2 ds + ∫ t

t0
∥∥f(s, 0)∥∥2 ds]

≤ 2M̃2 max{1,N2}(T − t0)∫ t
t0
[
LfE

∥x∥2s + κf]ds
≤ 2M̃2 max{1,N2}(T − t0)∫ t

t0
LfE

∥x∥2s ds + 2M̃2 max{1,N2}(T − t0)2κf,
G5 ≤ E

[ +∞∑
k=0
[ k∏
j=i
∥∥bj(δj)∥∥∫ ξi

ξi−1
∥∥S(t −s)∥∥∥∥g(s,xs)∥∥dω(s) + ∫ t

ξk
∥∥S(t −s)∥∥∥∥g(s,xs)∥∥dω(s)]I[ξk,ξk+1)(t)]2

≤ M̃2E
maxi,k {1,

k∏
j=i
∥∥bj(δj)∥∥}


2 ∫ t

t0
E
∥∥g(s,xs)∥∥2 ds

≤ 2M̃2 max{1,N2}Tr(Q)[∫ t
t0
E
∥∥g(s,xs) −g(s, 0)∥∥2 ds + ∫ t

t0
∥∥g(s, 0)∥∥2 ds]

≤ 2M̃2 max{1,N2}Tr(Q)∫ t
t0
[
LgE

∥x∥2s + κg]ds
≤ 2M̃2 max{1,N2}Tr(Q)∫ t

t0
LgE

∥x∥2s ds + 2M̃2 max{1,N2}(T − t0)Tr(Q)κg.



Eur. J. Math. Anal. 1 (2021) 11

G6 ≤ E
[ +∞∑

k=0
[ k∏
j=i
∥∥bj(δj)∥∥×∫ ξi

ξi−1
∫
U

∥∥S(t −s)∥∥∥∥σ(s,xs,u)∥∥Ñ(ds,du)
+ ∫ t

ξk
∫
U

∥∥S(t −s)∥∥∥∥σ(s,xs,u)∥∥Ñ(ds,du)]I[ξk,ξk+1)(t)]2
≤ 2M̃2 max{1,N2}∫ t

t0
∫
U

[
E
∥∥σ(s,xs,u) −σ(s, 0,u)∥∥2 + ∥∥σ(s, 0,u)∥∥2]ds

+ 2M̃2 max{1,N2}(∫ t
t0
∫
U
E
∥∥σ(s,xs,u)∥∥4 v(du)ds)12

≤ 4M̃2 max{1,N2}∫ t
t0
LσE

∥x∥2s ds + 2M̃2 max{1,N2}(T − t0)κσ,
Thus we would obtain,
E
∥∥(φx)(t)∥∥2t ≤ 6M2N2E∥∥φ(0)∥∥2 + 6M̃2N2E∥∥φ −h(0,φ)∥∥2 + 12M2 max{1,N2}(T − t0)

×
∫ t
t0
LhE

∥x∥2s ds + 12M2 max{1,N2}(T − t0)2κh + 12M̃2 max{1,N2}(T − t0)
×

∫ t
t0
LfE

∥x∥2s ds + 12M̃2 max{1,N2}(T − t0)2κf + 12M̃2 max{1,N2}Tr(Q)
×

∫ t
t0
LgE

∥x∥2s ds + 12M̃2 max{1,N2}(T − t0)Tr(Q)κg
+ 24M̃2 max{1,N2}∫ t

t0
LσE

∥x∥2s ds + 12M̃2 max{1,N2}(T − t0)κσ.
Taking supremum over t,

sup
t0≤t≤T

E
∥∥(φx)(t)∥∥2t

≤ 6M2N2E∥∥φ(0)∥∥2 + 6M̃2N2E∥∥φ −h(0,φ)∥∥2 + 12M2 max{1,N2}(T − t0)
×

∫ t
t0
Lh sup

t0≤t≤T
E∥x∥2s ds + 12M2 max{1,N2}(T − t0)2κh + 12M̃2max{1,N2}

× (T − t0)∫ t
t0
Lf sup

t0≤t≤T
E∥x∥2s ds + 12M̃2 max{1,N2}(T − t0)2κf + 12M̃2

× max{1,N2}Tr(Q)∫ t
t0
Lg sup

t0≤t≤T
E∥x∥2s ds + 12M̃2max{1,N2}(T − t0)Tr(Q)κg

+ 24M̃2 max{1,N2}∫ t
t0
Lσ sup

t0≤t≤T
E∥x∥2s ds + 12M̃2 max{1,N2}(T − t0)κσ



Eur. J. Math. Anal. 1 (2021) 12
≤ 6M2N2E∥∥φ(0)∥∥2 + 6M̃2N2E∥∥φ −h(0,φ)∥∥2 + 12M2 max{1,N2}(T − t0)2
× Lh sup

t0≤t≤T
E∥x∥2t + 12M2 max{1,N2}(T − t0)2κh + 12M̃2 max{1,N2}

× (T − t0)2Lf sup
t0≤t≤T

E∥x∥2t + 12M̃2 max{1,N2}(T − t0)2κf
+ 12M̃2 max{1,N2}Tr(Q)(T − t0)Lg sup

t0≤t≤T
E∥x∥2t

+ 12M̃2 max{1,N2}(T − t0)Tr(Q)κg+ 24M̃2 max{1,N2}(T − t0)Lσ sup
t0≤t≤T

E∥x∥2s ds + 12M̃2 max{1,N2}(T − t0)κσ
≤ 6[N2 [M2E∥∥φ(0)∥∥2 + M̃2E∥∥φ −h(0,φ)∥∥2]] + 12 max{1,N2}(T − t0)
×

[
M2(T − t0)κh + M̃2(T − t0)κf + M̃2Tr(Q)κg + M̃2κσ] + 12 max{1,N2}

× (T − t0)[M2(T − t0)Lh + M̃2(T − t0)Lf + M̃2Tr(Q)Lg + 2M̃2Lσ]∥x∥2t∥∥φx∥∥2B ≤ c1 + c2 ∥x∥2B .
where,
c1 = 6[N2 [M2E∥∥φ(0)∥∥2 + M̃2E∥∥φ −h(0,φ)∥∥2]] + 12 max{1,N2}(T − t0)

×
[
M2(T − t0)κh + M̃2(T − t0)κf + M̃2Tr(Q)κg + M̃2κσ],

c2 = 12 max{1,N2}(T − t0)[M2(T − t0)Lh + M̃2(T − t0)Lf + M̃2Tr(Q)Lg + 2M̃2Lσ] .
where c1 and c2 are constants.Hence φ is bounded.Now we need to prove that φ is a contraction mapping. For any x,y ∈B we have,∥∥(φx)(t) − (φy)(t)∥∥2

≤
∥∥∥∥ +∞∑k=0

[ k∑
i=1

k∏
j=i bj(δj)

∫ ξi
ξi−1
C(t −s)h(s,xs)ds + ∫ t

ξk
C(t −s)h(s,xs)ds

+ k∑
i=1

k∏
j=i bj(δj)

∫ ξi
ξi−1
S(t −s)f(s,xs)ds + ∫ t

ξk
S(t −s)f(s,xs)ds + k∑

i=1
k∏
j=i bj(δj)

×
∫ ξi
ξi−1
S(t −s)g(s,xs)dω(s) + ∫ t

ξk
S(t −s)g(s,xs)dω(s) + k∑

i=1
k∏
j=i bj(δj)

×
∫ ξi
ξi−1
∫
U
S(t −s)σ(s,xs,u)Ñ(ds,dt) + ∫ t

ξk
∫
U
S(t −s)σ(s,xs,u)Ñ(ds,dt)]I[ξk,ξk+1)(t)∥∥∥∥2

−
∥∥∥∥ +∞∑k=0

[ k∑
i=1

k∏
j=i bj(δj)

∫ ξi
ξi−1
C(t −s)h(s,ys)ds + ∫ t

ξk
C(t −s)h(s,ys)ds



Eur. J. Math. Anal. 1 (2021) 13
+ k∑

i=1
k∏
j=i bj(δj)

∫ ξi
ξi−1
S(t −s)f(s,ys)ds + ∫ t

ξk
S(t −s)f(s,ys)ds + k∑

i=1
k∏
j=i bj(δj)

×
∫ ξi
ξi−1
S(t −s)g(s,ys)dω(s) + ∫ t

ξk
S(t −s)g(s,ys)dω(s) + k∑

i=1
k∏
j=i bj(δj)

×
∫ ξi
ξi−1
∫
U
S(t −s)σ(s,ys,u)Ñ(ds,dt) + ∫ t

ξk
∫
U
S(t −s)σ(s,ys,u)Ñ(ds,dt)]I[ξk,ξk+1)(t)∥∥∥∥2

≤ 4 max{1,N2}M2(T − t0)∫ t
t0
∥∥h(t,xs) −h(t,ys)∥∥2 ds + 4 max{1,N2}

× M̃2(T − t0)∫ t
t0
∥∥f(t,xs) − f(t,ys)∥∥2 ds + 4 max{1,N2}M̃2

×
∫ t
t0
∥∥g(t,xs) −g(t,ys)∥∥2 ds + 4 max{1,N2}M̃2

×
[∫ t

t0
∫
U

∥∥σ(t,xs,u) −σ(t,ys,u)∥∥2 v(du)ds
+ (∫ t

t0
∫
U

∥∥σ(t,xs,u) −σ(t,ys,u)∥∥4 v(du)ds)12 ]

Moreover,
sup

t0≤t≤T
E
∥∥(φx)(t) − (φy)(t)∥∥2

≤ 4 max{1,N2}M2(T − t0)2Lh sup
t0≤t≤T

E
∥∥x −y∥∥2s ds + 4 max{1,N2}

× M̃2(T − t0)2Lf sup
t0≤t≤T

E
∥∥x −y∥∥2s ds + 4 max{1,N2}M̃2Tr(Q)

× (T − t0)Lg sup
t0≤t≤T

E
∥∥x −y∥∥2s ds + 4 max{1,N2}M̃2

× (T − t0)Lσ sup
t0≤t≤T

E
∥∥x −y∥∥2s ds

≤
[4 max{1,N2}M2(T − t0)2Lh + 4 max{1,N2}M̃2(T − t0)

× [(T − t0)Lf + Tr(Q)Lg + Lσ ]] sup
t0≤t≤T

E
∥∥x −y∥∥2t

Hence, ∥∥(φx) − (φy)∥∥2B ≤ γ(T)∥∥x −y∥∥2B .where,
γ(T) = 3 max{1,N2}M2(T − t0)2Lh + 3 max{1,N2}M̃2(T − t0) [(T − t0)Lf + Tr(Q)Lg + Lσ ] .



Eur. J. Math. Anal. 1 (2021) 14
By taking suitable 0 < T1 < T sufficiently small such that, γ(T1) < 1.Hence φ is a contraction on B . By Banach contraction principle, a unique fixed point x ∈ B isobtained for the operator φ and therefore φx = x is a mild solution of the system.The solution can be extended to the entire interval (−δ,T ] in finitely many steps. Thus the existenceand uniqueness of the mild solution on (−δ,T ] is proved.

�

4. Stability
The stability through continuous dependence of solutions on initial conditions are established.

Definition 4.1. A mild solution xξ,x(t) of the system (1.1) with the initial value (ξ,x) is said to be
stable in mean square if for all ε > 0 such that

E

( sup0≤s≤T
∥∥∥xξ,x(s) −yξ,x(t)∥∥∥2) ≤ ε, when E∥∥ξ −ζ∥∥2 + E∥∥x −y∥∥2 < δ,

where xζ,y(t) is another solution of the system (1.1) with initial value (ζ,y).
Theorem 4.1. Let x(t) and x(t) be mild solution of the system (1.1) with the initial condition φ1
and φ2 respectively. If the assumptions of Theorem 3.1 gets satisfied, the mean solution of the
system (1.1) is stable in the mean square.

Proof. We may assume that x(t) and x(t) be the mild solutions of the system (1.1) with initialvalues φ1 and φ2 respectively.
x(t) −x(t) = +∞∑k=0

[ k∏
i=1 bi(δi)C(t − t0)[φ1 −φ2] +

k∏
i=1 bi(δi)S(t − t0)[(φ1 −φ2) − [(h(0,φ1) − (h(0,φ2))]]

+ k∑
i=1

k∏
j=i bj(δj)

∫ ξi
ξi−1
C(t −s) [h(s,xs) −h(s,x(s))] ds + ∫ t

ξk
C(t −s) [h(s,xs) −h(s,xs)] ds

+ k∑
i=1

k∏
j=i bj(δj)

∫ ξi
ξi−1
S(t −s) [f(s,xs) − f(s,xs)] ds + ∫ t

ξk
S(t −s) [f(s,xs) − f(s,xs)] ds

+ k∑
i=1

k∏
j=i bj(δj)

∫ ξi
ξi−1
S(t −s) [g(s,xs) −g(s,xs)] dω(s) + ∫ t

ξk
S(t −s) [g(s,xs) −g(s,xs)] dω(s)

+ k∑
i=1

k∏
j=i bj(δj)

∫ ξi
ξi−1
∫
U
S(t −s) [σ(s,xs,u) −σ(s,xs,u)] Ñ(ds,du)

+ ∫ t
ξk
∫
U
S(t −s) [σ(s,xs,u) −σ(s,xs,u)] Ñ(ds,du)]I[ξk,ξk+1)(t)



Eur. J. Math. Anal. 1 (2021) 15
E
∥∥x(t) −x(t)∥∥2

≤ 6N2M2E∥∥φ1 −φ2∥∥2 + 12N2M̃2E∥∥φ1 −φ2∥∥2 + 12N2M̃2E∥∥h(0,φ1) −h(0,φ2)∥∥2
+ 6M2 max{1,N2}∫ t

t0
E
∥∥h(s,xs) −h(s,xs)∥∥2 ds + 6M̃2 max{1,N2}

×
∫ t
t0
E
∥∥f(s,xs) − f(s,xs)∥∥2 ds + 6M̃2 max{1,N2}∫ t

t0
E
∥∥g(s,xs) −g(s,xs)∥∥2 ds

+ 6 max{1,N2}M̃2 ×[∫ t
t0
∫
U

∥∥σ(t,xs,u) −σ(t,xs,u)∥∥2 v(du)ds
+ (∫ t

t0
∫
U

∥∥σ(t,xs,u) −σ(t,xs,u)∥∥4 v(du)ds)12 ]
≤ 6N2M2E∥∥φ1 −φ2∥∥2 + 10N2M̃2E∥∥φ1 −φ2∥∥2 + 12N2M̃2LhE∥∥φ1 −φ2∥∥2
+ 6M2 max{1,N2}∫ t

t0
LhE

∥∥x −x∥∥2s ds + 6M̃2 max{1,N2}∫ t
t0
LfE

∥∥x −x∥∥2s ds
+ 6M̃2 max{1,N2}Tr(Q)∫ t

t0
LgE

∥∥x −x∥∥2s ds
+ 6M̃2 max{1,N2}∫ t

t0
LσE

∥∥x −x∥∥2s ds
Furthermore,
sup

t0≤t≤T
E
∥∥x −x∥∥2t ≤ 6N2M2E∥∥φ1 −φ2∥∥2 + 12N2M̃2E∥∥φ1 −φ2∥∥2 + 12N2M̃2LhE∥∥φ1 −φ2∥∥2

+ 6M2 max{1,N2}(T − t0)Lh sup
t0≤t≤T

E
∥∥x −x∥∥2t + 6M̃2 max{1,N2}(T − t0)

× Lf sup
t0≤t≤T

E
∥∥x −x∥∥2t + 6M̃2 max{1,N2}(T − t0)Tr(Q)Lg sup

t0≤t≤T
E
∥∥x −x∥∥2t

+ 6M̃2 max{1,N2}(T − t0)Lσ sup
t0≤t≤T

E
∥∥x −x∥∥2t

sup
t0≤t≤T

E
∥∥x −x∥∥2t ≤ 6N2

[
M2 + M̃2Lh]

1 − 6 max{1,N2}(T − t0)[M2Lh + M̃2 [Lf + Tr(Q)Lg + Lσ ]]E
∥∥φ1 −φ2∥∥2

+ 12N2M̃21 − 6 max{1,N2}(T − t0)[M2Lh + M̃2 [Lf + Tr(Q)Lg + Lσ ]]E
∥∥φ1 −φ2∥∥2

≤ ρE
∥∥φ1 −φ2∥∥2 + ΥE∥∥φ1 −φ2∥∥2

where,
ρ = 5N2

[
M2 + M̃2Lh]

1 − 5 max{1,N2}(T − t0)[M2Lh + M̃2 [Lf + Tr(Q)Lg + Lσ ]]
Υ = 10N2M̃21 − 5 max{1,N2}(T − t0)[M2Lh + M̃2 [Lf + Tr(Q)Lg + Lσ ]]



Eur. J. Math. Anal. 1 (2021) 16
Given ε > 0,µ > 0 choose, λ = ερ,µ = σΥ such that,

E
∥∥φ1 −φ2∥∥2 ≤ λ and E∥∥φ1 −φ2∥∥2 ≤ µ

Therefore, ∥∥x −y∥∥2B ≤ ε.Thus the proof is complete.
5. Illustration

In this section, the results obtained are applied to a stochastic partial differential equations withrandom impulses. Let us consider a space H = L2([0,π]). The infinitesimal generator A is definedto be A : D(A) ⊂ H → H by A = ∂2∂x2 , with the domain,
D(A) = {z ∈ H | z and ∂z

∂x
are absolutely continuous,

∂2z
∂x2 ∈ H,z(0) = z(π) = 0}

. For z ∈ D(A),Az = − ∞∑
n=1 n

2 < z,zn > zn, where {zn : n ∈ Z} is an orthonormal basis
of H, zn(x) := 1√2πeinx,n ∈ Z+,x ∈ [0,π]. It is known that A generates strongly continuous
operators C(t) and S(t) in a Hilbert space H, such that C(t)z = ∞∑

n=1 cos(nt) < z,zn > zn, and
S(t)z = ∞∑

n=1 sin(nt)/n < z,zn > zn, for t ∈ R. And we assume that S(t) is not a compact semigroupand θ(S(t)D) ≤ θ(D), where D ∈ H denotes a bounded set, θ is the Hausdroff measure of non-compactness.In the sequel, we may consider second-order neutral stochastic functional differential equation ofthe form,
∂
∂t

[
∂
∂t
z(t,x) − m15

∫ 0
−r
ε1(s)z(t + s,x)ds

] (5.1)
= [ ∂2

∂x2 z(t,x) + m25
∫ 0
−r
ε1(s)z(t + s)ds

]
dt

+ m35
∫ 0
−r
ε3(s)z(t + s)dω(t)

+ m45
∫
U

∫ 0
−r
ε4(s)z(t + s)Ñ(dt,du), t ≥ t0, t 6= ξk, x ∈ [0,π],

z(ξk,x) = ρ(k)δkz(ξ−k ,x), k = 1, 2, 3..., (5.2)
∂
∂t
z(ξk,x) = ρ(k)δk ∂∂tz(ξ−k ,x),
z(t0,x) = φ(θ,x),θ ∈ [−r, 0], x ∈ [0,π], r > 0,

∂
∂t
z(t0,x) = φ(x), x ∈ [0,π],
z(t, 0) = z(t,π) = 0.



Eur. J. Math. Anal. 1 (2021) 17
Let δk be a random variable defined on Dk ≡ (0,dk) where, 0 < dk < +∞, for k = 1, 2, · · · . ξ0 =
t0 > 0 and ξk = ξk−1 + δk for k = 1, 2, · · · . ω(t) denotes a standard cylindrical Weiner process inH. Furthermore, let ρ be a function of k. εi : [−r, 0] → R are positive functions and mi > 0 for
i = 1, 2, 3, 4. ∥∥C(t)∥∥ ,∥∥S(t)∥∥ are bounded on R. ∥∥C(t)∥∥ ≤ e−π2t and ∥∥S(t)∥∥ ≤ e−π2t(t ≥ 0).We may assume that,
(i)The function ε(θ) ≥ 0 is continuous on [−r, 0],∫ 0

−r
ε2i (θ)dθ < ∞(i = 1, 2, 3, 4.)

(ii)max
i,k

= { k∏
j=i E[

∥∥ρ(j)δj∥∥2]} < N .
Using above assumptions and functions ε1,ε2,ε3,ρ we can show that Lg = rm125 ∫ 0−r ε21(θ)dθ,Lf =
rm225 ∫ 0−r ε21(θ)dθ, Lh = rm325 ∫ 0−r ε21(θ)dθ and Lσ = rm425 ∫ 0−r ε21(θ)dθ. Hence stability in mean squareof mild solution (5.1) is obtained.

�

6. Conclusion
In this paper, the existence and stability results of second-order neutral stochastic functionalsystems with random impulse is presented. The existence results of aforementioned system is estab-lished using Banach contraction principle. Then the stability of mild solutions through continuousdependence of solutions on initial conditions are calculated.

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	1. Introduction
	2. Preliminaries
	3. Existence Results of Mild Solution
	4. Stability
	5. Illustration
	6. Conclusion
	References

