Existence, Uniqueness and Stability of Neutral Stochastic Functional Integro differential Evolution Equations with Infinite Delay

Existence, Uniqueness and Stability of Neutral

Stochastic Functional Integro-differential Evolution

Equations with Infinite Delay

C. Parthasarathy

Department of Mathematics,

Karunya University, Karunya Nagar,

Coimbatore- 641 114, Tamil Nadu, India.

A. Vinodkumar

Department of Mathematics, PSG College of Technology,

Coimbatore- 641 004, Tamil Nadu, India.

M. Mallika Arjunan

Department of Mathematics,

C. B. M. College, Kovaipudur,

Coimbatore- 641 042, Tamil Nadu, India.

ABSTRACT

This article presents the results on existence, uniqueness and sta-bility of mild solutions to neutral stochastic functional evolu-tion integro-differential equaevolu-tions with non-Lipschitz condievolu-tion and Lipschitz condition. The existence of mild solutions for the equa-tions are discussed by means of semigroup theory and theory of re-solvent operator. Under some sufficient conditions, the results are obtained by using the method of successive approximation and Bi-hari’s inequality. Moreover, an example is given to illustrate our results.

Keywords:

Resolvent operator, Evolution operator, Existence, Uniqueness, Stability, Successive approximation, Bihari’s inequality

1. INTRODUCTION

Neutral stochastic differential equation occurs in many areas of science and engineering have attain much attention in the past decades. The partial integro-differential equations has wide ap-plications in the field of mechanical, electrical and so on., and refer [12]. For abstract model of partial integro-differential equa-tions with resolvent operators, see for instance [6, 8, 11]. The de-terministic model often fluctuate due to noise. Under this circum-stance, we move the deterministic model problems to stochastic model problems, for more details reader may refer [3, 7, 10, 18]. The existence and uniqueness of the neutral stochastic differen-tial equations with infinite delay have been studied by many au-thors [5, 16]. Recently, the auau-thors have established the prob-lem with Lipschitz and non-Lipschitz condition, we suggest [2, 17, 20, 22] and reference therein.

On other hand, stochastic differential equations are well known problem in many areas of engineering and science. There are only few works on existence, uniqueness and stability of stochas-tic differential systems have been established [1, 16, 17, 22]. The stochastic systems with resolvent operators has occur in differ-ent applications such as heat equation, viscoelasticity and many other physical phenomena, see for instance [15]. The study of existence, uniqueness and stability of stochastic functional dif-ferential with resolvent operator is an unprocessed issue and it is also the motivation of this paper.

In [1] Anguraj et al. studied the impulsive stochastic neutral functional differential equations under non-Lipschitz condition and Lipschitz condition, whereas A. Lin et al. [17] have estab-lished on neutral impulsive stochastic integro-differential

equa-tions with infinite delay via fractional operators and H. Bin Chen [5] have proved the existence and uniqueness for the solution of neutral stochastic functional differential equations with infi-nite delay, then A. Vinodkumar [22] have examine the existence, uniqueness and stability results of impulsive stochastic semilin-ear functional differential equations with infinite delay. Recently, Y. Ren [21] have described the existence, uniqueness and sta-bility of mild solutions for time-dependent stochastic evolution equations with poisson jumps and infinite delay. Moreover, the study was conducted on stability through the continuous depen-dence on the initial values by means of Bihari’s inequality. For more details reader may refer [2, 10, 19].

Inspired by the above mentioned works [5, 8, 22], the purpose of this paper is to study the existence, uniqueness and stability for neutral stochastic functional integro-differential equations of the form

d[x(t) +g(t, xt)] =A(t)

x(t) +g(t, xt)

dt

+

Z t

0

f(t, s)

x(s) +g(s, xs)

ds+h(t, xt)

dt

+σ(t, xt)dw(t), t∈J:= [0, T], (1)

x0=ϕ∈ B. (2)

Here, the statex(·)takes the values in a real separable Hilbert space H with inner product (·,·) and the norm k · k, A(t)

is the linear operators generates a linear evolution systems

{R(t, s), t ≥ 0}on H, andf(t, s), t ∈ J is a bounded lin-ear operator. The historyxt: (−∞,0]→H,xt(θ) =x(t+θ),

fort≥0, belongs to the phase spaceB, which will be described axiomatically in Preliminaries. Suppose{w(t);t≥0}is a given K-valued Brownian motion with a finite trace nuclear covari-ance operatorQ ≥ 0defined on a complete probability space

(Ω,F, P)equipped with a normal filtration{Ft}t≥0, which is generated by Wiener processw. we are also employing the same notationk · kfor the normL(K, H), whereL(K, H)denotes the space of all bounded linear operator fromKintoH. Assume thath:_R+_{× B →H andσ:}

R+× B → LQ(K, H), where

R+ = [0,∞)are Borel measurable andg : R+× B → H is

continuous. Here,LQ(K, H)denotes the space of all

Q-Hilbert-Schmidt operator fromKintoH, which will be defined in Sec-tion 2.

5 is reserved for an example is to illustrate the efficiency of the obtained results.

2. PRELIMINARIES

Let(K,k · kK)and(H,k · kH)be the two real separable Hilbert

space with inner producth·,·iKandh·,·iH, respectively. We

de-noteL(K, H)be the set of all linear bounded operator fromK intoH, equipped with the usual operator normk·k. In this article, we use the symbolk · kto denote norms of operator regardless of the space involved when no confusion possibly arises. Let(Ω,F, P;H) be the complete probability space furnished with a complete family of right continuous increasingσ- algebra

{Ft, t ∈ J}satisfyingFt ⊂ F. AnH- valued random

vari-able is anF- measurable functionx(t) : Ω→H and a collec-tion of random variablesS = {x(t, ω) : Ω → H \t ∈ J}

is called stochastic process. Usually we writex(t) instead of x(t, ω) and x(t) : J → H in the space of S. Let {ei}∞i=1 be a complete orthonormal basis ofK. Suppose that{w(t) :

t ≥ 0}is a cylindricalK-valued wiener process with a finite trace nuclear covariance operator Q ≥ 0, denote Tr(Q) =

P∞

i=1λi=λ <∞, which satisfies thatQei=λiei. So,

actu-ally,ω(t) =P∞

i=1

√

λiωi(t)ei, where{ωi(t)}∞i=1are mutually independent one-dimensional standard Wiener processes. We as-sume thatFt=σ{ω(s) : 0≤s≤t}is theσ-algebra generated

byωandFt=F. LetΨ∈ L(K, H)and define

kΨk2

Q= Tr(ΨQΨ ∗

) = ∞

X

n=1

kpλnΨenk2.

IfkΨkQ <∞, thenΨis called aQ-Hilbert-Schmidt operator.

LetLQ(K, H)denote the space of allQ-Hilbert-Schmidt

op-eratorsΨ :K → H. The completionLQ(K, H)ofL(K, H)

with respect to the topology induced by the normk · kQwhere

kΨk2

Q=hΨ,Ψiis a Hilbert space with the above norm

topol-ogy.

In this work, the axiomatic definition of the phase spaceB is introduced by Hale et al. [13]. To establish the axioms of the phase spaceB, we use the following terminology used in Hinto et al. [14]. The axioms of the spaceBare established forF0 -measurable functions from (−∞,0] intoH, endowed with a seminormk · kBwhich satisfies the following axioms:

(A1) Ifx: (−∞, T]→H, T >0is such thatx0 ∈ Bthen for everyt∈[0, T], the following conditions hold:

(i) xt∈ B;

(ii) kx(t)k ≤LkxtkB;

(iii) kxtkB ≤M(t) sup

0≤s≤t

kx(s)k+N(t)kx0kB, whereL >0

is a constant;M(·),N(·) : [0,+∞)→[1,+∞), is contin-uousN(·)is locally bounded, andL, M(·), N(·)are inde-pendent ofx(·).

(A2) For the functionx(·)in (A1),xtis aB-valued continuous

function on[0, b]. (A3) The spaceBis complete.

TheB- valued stochastic processxt : Ω → B,t ≥ 0, is

de-fined byxt(s) = {x(t+s)(ω) : s ∈ (−∞,0]}The

collec-tion of all strongly measurable, square integrable,H-valued ran-dom variables, denoted byL2(Ω,F, P;H) ≡ L2(Ω;H), is a Banach space equipped with normkx(·)k2

L2 = Ekx(·, w)k

2

H,

whereEdenotes expectation defined byE(h) =R

Ωh(w)dP. LetC(J, L2(Ω;H))be the Banach space of all continuous map fromJintoL2(Ω;H)satisfying the conditionsup

t∈J

Ekx(t)k2_<

∞. An important subspace is given by L0

2(Ω, H) = {f ∈ L2(Ω, H) :fisF0−is measurable}.

LetZbe the closed subspace of all continuously differentiable processxthat belongs to the spaceC(J, L2(Ω;H))consisting

ofFt- adapted measurable process such that theF0-adapted pro-cessϕ∈L0

2(Ω,B). Letk · kZbe a seminorm inZdefined by

kxkZ=

sup t∈J

kxtk2B

1₂

,

where

kxtkB≤NTEkϕkB+MTsup{Ekx(s)k: 0≤s≤T},

MT = supt∈J{M(t)},NT = supt∈J{N(t)}. It is easy to

ver-ify thatZfurnished with the norm topology as defined above, is a Banach space.

The resolvent operator plays an important role in the study of the existence of solutions and to give a variation of constant formula for linear systems. However, need to know when the linear sys-tem (3) has a resolvent operator. For more details on resolvent operator, reader may refer [11].

The following assumptions are:

(H1) A(t)generates a strongly continuous semigroup of evolu-tion operators..

(H2) SupposeY is a Banach space formed fromD(A)with the graph norm. A(t) and f(t, s) are closed operators it fol-lows thatA(t) and f(t, s) are in the set of bounded lin-ear operators fromY toH,f(Y, H), for0≤ t ≤ T and

0≤s≤t≤T, respectively.A(t)andf(t, s)are continu-ous on0≤t≤ T and0≤s ≤ t≤T, respectively, into B(Y, H).

To obtain the results, consider the integro-differential abstract Cauchy problem

dx(t) =

A(t)x(t) +

Z t

0

f(t, s)x(s)ds

dt,

0≤s≤t≤T, (3)

x(0) =x0∈H.

DEFINITION 1. [11]A family of bounded linear operator R(t, s)∈ P(H),0≤s≤t≤T is called a resolvent operator for

dx dt =A(t)

x(t) +

Z t

0

f(t, s)x(s)ds

,

if

(i) R(t, s)is strongly continuous insandt.R(t, t) = I, the identity operator onH.kR(t, s)k ≤M eβ(t−s)_{t, s∈Jand} M, βare constants;

(ii) R(t, s)Y ⊂Y,R(t, s)is strongly continuous insandton Y;

(iii) Fory∈Y,R(t, s)yis continuously differentiable insand t, and for0≤s≤t≤T,

∂

∂tR(t, s)y=A(t)R(t, s)y+ Z t

s

f(t, r)R(r, s)ydr,

∂

∂sR(t, s)y=−R(t, s)A(s)y− Z t

s

R(t, r)f(r, s)ydr,

with ∂

∂tR(t, s)yand ∂

∂sR(t, s)yare strongly continuous on0≤

s ≤ t ≤ T. HereR(t, s)can be extracted from the evolution operator of the generator A(t).

For the family{A(t) : 0 ≤ t ≤ T}of linear operators, the following restrictions are imposed:

(B1) The domainD(A)of{A(t) : 0≤t≤T}is dense inX and independent oft,A(t)is closed linear operator;

(B2) For eacht ∈ [0, T], the resolventR(λ, A(t))exists for all λ with Reλ ≤ 0 and there exists K > 0 so that

(B3) There exists0< δ ≤ 1andK > 0such thatk(A(t)−

A(s))A−1_{(τ)k ≤K|t−s|}δ_{for allt, s, τ∈[0, T];}

(B4) For eacht∈[0, T]and someλ∈ρ(A(t)), the resolvent set ofA(t), the resolventR(λ, A(t)), is a compact operator.

Under these assumptions, the family{A(t) : 0≤t≤T} gener-ates a unique linear evolution system, or called linear evolution operator.

DEFINITION 2. [19]A two parameter family of bounded lin-ear operatorsR(t, s),0≤s≤t≤T , onHis called an evolu-tion system if the following two condievolu-tions hold

(i) R(s, s) =I,R(t, r)R(r, s) =R(t, s), for0 ≤ s≤ r ≤

t≤T.

(ii) (t, s)→R(t, s)is strongly continuous for0≤s≤t≤T. LEMMA 1. [19]Assume that (B1)-(B3) hold. Then, there exist a unique evolution systemU(t, s),0≤s≤ t≤T and a constantK >0such that

(i) R(t, s)≤Kfor0≤s≤t≤T;

(ii) for0 ≤ s ≤ t ≤ T,R(t, s) :H → Y andt → R(t, s)

is strongly differentiable inH.The derivative ∂

∂tR(t, s)

be-longs toL(H)and it is strongly continuous on0≤s≤t≤

T. Moreover, for all0≤s≤t≤T, it holds ∂

∂tR(t, s) +A(t)R(t, s) = 0;

k∂

∂tR(t, s)k=kA(t)R(t, s)k ≤ K t−s;

kA(t)R(t, s)A(s−1)k ≤K,

(iii) for eachy ∈ Y and t ∈ J , R(t, s)y is differentiable with respect toson 0 ≤ s ≤ t ≤ T and _∂t∂R(t, s)y =

R(t, s)A(s)y.

LEMMA 2. [9]Let{A(t), t ∈J}be a family of linear op-erators satisfying(B1)−(B4). If{R(t, s),0 ≤ s ≤ t≤ T}

is the linear evolution system generated by{A(t), t∈J}, then

{R(t, s),0 ≤ s ≤ t ≤ T}is a compact operator whenever t−s >0.

LEMMA 3. [4]LetT > 0and u0 ≥ 0,u(t), v(t)be the continuous function on[0, T]. LetK:R+→R+be a concave

continuous and nondecreasing function such thatK(r)>0for r >0. If

u(t)≤u0+ Zt

0

v(s)K(u(s))ds for all 0≤t≤T, then

u(t)≤G−1 G(u0) + Z t

0

v(s)dsfor allt∈[0, T]such that

G(u0) + Z t

0

v(s)ds∈Dom(G−1),

whereG(r) =Rr

1

ds

K(s) forr≥0andG

−1_{is the inverse}

func-tion ofG. In particular, moreover if,u0= 0and R

0+_Kds_(s)=∞, thenu(t) = 0for allt∈[0, T].

In order to obtain the stability of the solutions, the following extended Bihari’s inequality is used.

LEMMA 4. [20]Let the assumption of Lemma 3 holds. If u(t)≤u0+

Zt

0

v(s)K(u(s))ds for all 0≤t≤T,

then

u(t)≤G−1 G(u0) + Z t

0

v(s)ds

for allt∈[0, T]such that

G(u0) + Z t

0

v(s)ds∈Dom(G−1_),

whereG(r) =Rr

1

ds

K(s) forr≥0andG

−1_{is the inverse}

func-tion ofG.

COROLLARY 1. [20]Let the assumption of Lemma 3 hold andv(t)≥0fort∈[0, T]. If for all >0, there existst1≥0 such that for0≤u0≤,

RT

t1v(s)ds≤ R

u0

ds

K(s)holds, then for everyt∈[t1, T], the estimateu(t)≤holds.

LEMMA 5. [7]For anyr ≥ 1and for arbitraryL0 2-valued predictable processΨ(·)

sup s∈[0,t]

Ek

Z s 0

Ψ(u)dw(u)k2r X =

(r(2r−1))r

Z t

0

(EkΨ(s)k2r L0

2)ds r

.

The following is the definition of the mild solution of the system (1)-(2).

DEFINITION 3. A stochastic process {x(t) ∈

C(J, L2(Ω; H)), t ∈ (−∞, T]}, (0 < T < ∞), is said to be a mild solution of the equation (1)-(2) if

(i) x(t)∈HisFt-adapted;

(ii) for eacht∈J,x(t)satisfies the following integral equation

x(t) =



        

        

ϕ(t), for t∈(−∞,0], R(t,0)

ϕ(0) +g(0, ϕ)

−g(t, xt) +

Z t

0

R(t, s)h(s, xs)ds

+

Z t

0

R(t, s)σ(s, xs)dw(s)

for a.s t∈[0, T].

(4)

3. EXISTENCE AND UNIQUENESS

In this section, the existence and uniqueness of mild solution of the system (1)-(2) are discussed and worked under the following assumptions:

(H3) There exists a resolvent operatorR(t, s) which is com-pact and continuous in the uniform operator topology for t > s. Further, there exists a constantM1 > 0such that

kR(t, s)k2_≤M

1, for allt∈J.

(H4) For eachx, y∈ Band for allt∈[0, T]such that

kh(t, xt)−h(t, yt)k2∨ kσ(t, xt)−σ(t, yt)k2

≤K(kxt−ytk2B),

whereK(·)is a concave non-decreasing function fromR+

toR+,K(0) = 0,K(u)>0, foru >0andR0+Kdu(u) =

∞.

(H5) Assuming that there exists a positive numberMg, such that

Mg<18, for anyx, y∈ Band fort∈[0, T], we have

kg(t, xt)−g(t, yt)k2≤Mgkxt−ytk2B.

(H6) For allt∈[0, T], it follows thatσ(t,0), h(t,0), g(t, yt)∈

L2_{such that}

kσ(t,0)k2_{∨ k}

h(t,0)k2_{∨ k}

g(t,0)k2_≤ κ0,

Let us now introduce the successive approximation to equation (4) as follows

x0(t) =

(

ϕ(t)fort∈(−∞,0],

R(t,0)ϕ(0)fort∈[0, T]. (5) and, forn= 1,2, . . . ,

xn_{(t) =}

                  

ϕ(t),fort∈(−∞,0], R(t,0)

ϕ(0) +g(0, ϕ)

−g(t, xn t) +

Z t

0

R(t, s)h(s, xns−1)ds

+

Z t

0

R(t, s)σ(s, xn−1

s )dw(s),

for a.s, t∈J,

(6)

with an arbitrary non-negative initial approximationx0 _∈ C(J, L2(Ω;H)).

THEOREM 1. Assume that(H3)−(H6)hold. Then the sys-tem (1)-(2) has unique mild solutionx(t)inC(J, L2(Ω;H))and

E{sup 0≤t≤T

kxn_(t)−x(t)k2_{} →0, n→ ∞,}

where{xn_(t)}

n≥1are successive approximations (6). PROOF. Letx0_{∈C(J, L}

2(Ω;H))be a fixed initial approxi-mation to (6). To begin with the assumptions(H3)−(H6)and observing thatkR(t, s)k2 _≤M

1for someM1 ≥ 1and for all t∈[0, T]. Then for anyn≥1, we have

kxn_(t)k2_≤4M 1k

ϕ(0)−g(0, ϕ)

k2 + 8

kg(t, xn

t)−g(t,0)k

2_+kg(t,0)k2

+ 8M1T Z t

0

[kh(s, xn−1

s )−h(s,0)k

2_+kh(s,0)k2_]ds

+ 8M1 Z t

0

[kσ(s, xns−1)−σ(s,0)k

2

+kσ(s,0)k2 ]ds.

Thus,

Ekxntk

2

B≤

N1 1−8Mg

+8M1(T+ 1) 1−8Mg

E Z t

0

K(kxns−1k

2

B)ds,

whereN1= 8M1

Ekϕ(0)k+MgEkϕk20

+ 8

1 +M1T(T+ 1)

κ0.

Given thatK(·)is concave andK(0) = 0, the following pair of positive constantsaandbare found such that

K(u)≤a+bu,for allu≥0. Then, we have

Ekxntk

2

B≤N2+

8M1(T+ 1)b 1−8Mg

Z t

0

Ekxns−1k

2

Bds

≤N2+

8M1(T+ 1)b 1−8Mg

Z t

0

N(t)Ekϕk2

B+M(t) sup

0≤s≤T

Ekxn−1_(s)k2 ds

≤N2+

8M1T(T+ 1)b 1−8Mg

NTEkϕk2B

+8M1(T+ 1)b 1−8Mg

Z t 0

sup 0≤s≤T

Ekxn−1_(s)k2_ds,

whereN2=₁₋N_8M1_g +8M₁1₋T₈(T_M+1)_g a. Therefore,

Ekxntk

2

B≤N3+

8M1(T+ 1)b 1−8Mg

Z t

0 sup 0≤s≤T

Ekxn−1(s)k2 ds,

(7)

where,N3=N2+

8M1T(T+1)b

1−8Mg NTEkϕk

2

B.

Since

Ekx0

tk

2

B≤M1Ekϕ(0)k2=N4<∞. (8) Thus,

Ekxn

tk2B≤N5<∞, (9) for alln= 0,1,2, . . ., andt∈[0, T]. This proves the bounded-ness of{xn_{(t), n∈}

N}.

Let us next show that {xn_{(t)} is a Cauchy sequence in}

C(J, L2(Ω;H)). Forn, m≥1, we have Ekxn+1(t)−xm+1(t)k2

≤3MgEkxn+1(t)−xm+1(t)k2B + 3M1(T+ 1)

Z t

0

K(Ekxn_(s)−xm_(s)k2

B)ds

≤ 3M1(T+ 1)

1−3Mg

Z t

0

K(Ekxn(s)−xm(s)k2

B)ds.

Thus

sup 0≤s≤t

Ekxn+1

s −x m+1

s k

2

B

≤N6 Zt

0 K

sup 0≤r≤s

Ekxn r−x

m r k 2 B ds, (10)

whereN6=3M₁−13(TM+1)g .

Integrating both sides of equation (10) and applying Jensen’s in-equality gives that

Z t

0 sup 0≤l≤s

Ekxn+1

l −x m+1

l k

2

Bds

≤N6 Z t

0 Z s

0

K sup

0≤r≤l

Ekxn r−x

m r k 2 B dlds

≤N6 Z t 0 s Zs 0 K sup

0≤r≤l

Ekxnr −x m r k 2 B 1 sdlds

≤N6t Z t 0 K Z s 0 sup 0≤r≤l

Ekxnr−x m r k 2 B 1 sdl ds. Then

Φn+1,m+1(t)≤N6 Z t

0

K Φn,m(s)

ds, (11)

where

Φn,m(t) =

Z t

0 sup 0≤r≤l

Ekxn

r−xmr k2Bds

t .

From (9), it is easy see that

sup n,m

Φn,m(t)<∞.

So lettingΦ(t) = lim supn,m→∞Φn,m(t)and enchanting the

relation Fatou’s lemma, it capitulate that

Φ(t) =N6 Zt

0 K

Φ(s)

ds.

Now, applying the Lemma 3, instantaneously exposeΦ(t) = 0

for any t ∈ [0, T]. This further means {xn_{(t), n ∈}

N} is

a Cauchy sequence inC(J, L2(Ω;H)). So there is an x ∈ C(J, L2(Ω;H))such that

lim n→∞

ZT

0 sup 0≤s≤t

In addition, by (9), it is easy to follow thatEkxtk2B≤N5. Thus, x(t)is a mild solution to(1)−(2). On the other hand, by (H4) then, lettingn→ ∞, we can also claim that fort∈[0, T]

Ek

Z t

0 R(t, s)

h(t, xn−1

s )−h(t, xs)

dsk2

B→0,

Ek

Zt 0

R(t, s)

σ(t, xn−1

s )−σ(t, xs)

dw(s)k2

B→0.

On further, by applying (H5) then, fort∈[0, T], that Ekg(s, xn

s)−g(s, xs)k2≤MgEkxn(s)−x(s)k2B→0.

At this instant, taking limits in both sides of (6) leads, fort≥0, to

x(t) =R(t,0)

ϕ(0)−g(0, ϕ)

+g(t, xt)

+

Z t

0

R(t, s)h(s, xs)ds+

Z t

0

R(t, s)σ(s, xs)dw(s).

This certainly exhibit by the Definition 3 thatx(t)is a mild so-lution of (1)-(2) on the interval[0, T].

Now, the uniqueness of the solutions of (4) is to be proved by the following: Letx, y ∈ C(J, L2(Ω;H))be the two solution of (1)-(2) on some interval(−∞, T]. Then, fort∈(−∞,0], the uniqueness is obvious and for0≤t≤T, we have

Ekx(t)−y(t)k2_≤3M

gEkxt−ytk2B + 3M1(T+ 1)

Z t

0

K(Ekxs−ysk2B)ds.

Thus,

Ekxt−ytk2B≤

3M1(T+ 1) 1−3Mg

Z t

0

K(Ekxs−ysk2B)ds

≤N6 Z t

0

K(Ekxs−ysk2B)ds.

Thus, Bihari’s inequality yield that

sup t∈[0,T]

Ekxt−ytk2B= 0,0≤t≤T.

Thus,x(t) =y(t), for all0≤t≤T. Therefore, for all−∞ ≤

t≤T,x(t) =y(t). This completes the proof.

4. STABILITY

In this section, the stability through the continuous dependence on initial values is studied by the following definition.

DEFINITION 4. A mild solutionx(t) of the system (1)-(2) with initial valueϕis said to be stable in the mean square if for all >0, there existsδ >0such that

Ekxt−bxtk 2

B≤,wheneverEkϕ−ϕbk 2_≤

δ,for allt∈[0, T], wherexb(t)is another mild solution of the system (1)-(2) with initial dataϕ.b

THEOREM 2. Letx(t)andy(t)be the mild solution of the system (1)-(2) with initial valuesϕ1 andϕ2respectively. If the assumption of Theorem 1 are satisfied, then the mild solution of the system (1)-(2) is stable in the mean square.

PROOF. By the assumption,x(t)andy(t)are two mild solu-tions of equasolu-tions (1)-(2) with initial valuesϕ1 andϕ2

respec-tively, then for0≤t≤T, x(t)−y(t)

=R(t,0)

ϕ1(0)−ϕ2(0)

+

g(0, ϕ1)−g(0, ϕ2)

+

g(t, xt)−g(t, yt)

+

Z t

0

R(t, s)

h(s, xs)−h(s, ys)

ds

+

Z t

0

R(t, s)

σ(s, xs)−σ(s, ys)

dw(s).

So, estimating as before, we get

Ekx(t)−y(t)k2

≤5M1

1 +Mg

Ekϕ1−ϕ2k2+ 5MgEkxt−ytk2B + 5M1(T+ 1)

Z t

0

K(Ekxs−ysk2B)ds.

Thus,

Ekxt−ytk2B≤ 5M1

1 +Mg

1−5Mg

Ekϕ1−ϕ2k2

+5M1(T+ 1) 1−5Mg

Z t

0

K(Ekxs−ysk2B)ds.

LetK1(u) = 5M₁₋1(₅T_M+1)_g K(u), whereKis concave increasing function fromR+ toR+ such thatK(0) = 0,K(u) > 0for

u >0andR

0+Kdu(u) = +∞. So,K1(u)is obviously, a concave function fromR+toR+such thatK1(0) = 0,K1(u)≥K(u), for0 ≤ u ≤ 1andR

0+

du

K1(u) = +∞. Now for any > 0, 1 = 12, thenlim_s→0

Z 1

s

du K1(u)

= ∞. So, there is a positive constantδ < 1, such that

R1

δ du K1(u) ≥T. Let

u0= 5M1

1 +Mg

1−5Mg

Ekϕ1−ϕ2k2, u(t) =Ekxt−ytk2B, v(t) = 1,

whenu0≤δ≤1. From Corollary 1 we have Z 1

u0 du K1(u)

≥

Z 1

δ

du K1(u)

≥T =

Z T

0

v(s)ds.

So, for anyt∈[0, T], we estimateu(t)≤1holds. This com-pletes the proof.

REMARK 1. Consider the following stochastic functional integro-differential systems of the form

d[x(t) +g(t, xt)] =A

x(t) +g(t, xt)

dt

+

Z t

0

f(t, s)

x(s) +g(s, xs)

ds+h(t, xt)

dt

+σ(t, xt)dw(t), t∈J:= [0, T], (12)

x0=ϕ∈ B, (13)

whereAis the infinitesimal generator of a strongly continuous semigroupR(t), t≥0defined on H. Moreover,g, h, σare same as defined in (1)-(2). HereR(t, s) = R(t−s), t > s. The mild solution of (12)-(13) is as follows

x(t) =



        

        

ϕ(t), for t∈(−∞,0], R(t)

ϕ(0) +g(0, ϕ)

−g(t, xt) +

Z t

0

R(t−s)h(s, xs)ds +

Z t

0

R(t−s)σ(s, xs)dw(s)

for a.s t∈[0, T].

REMARK 2. In [8] the author has studied existence and uniqueness of the stochastic integro-differential systems for (12)-(13) with finite delay. Here, the nature of work is to study the existence, uniqueness and stability results of the equation (12)-(13) with infinite delay. The result of equation (12)-(13) as obtained by the following hypothesis.

(H7) There exists a resolvent operatorR(t−s)which is compact and continuous in the uniform operator topology fort > s. Further, there exists a constantM1>0such thatkR(t)k2≤ M1, for allt∈J.

THEOREM 3. Assume that(H4)−(H6)and (H7)hold. Then the system (12)-(13) has unique mild solution x(t) in C(J, L2(Ω;H))and

E{sup 0≤t≤T

kxn(t)−x(t)k2_{} →}

0, n→ ∞, where{xn_(t)}

n≥1are successive approximations (14). PROOF. The proof is very similar to Theorem 1. Hence, it is omitted.

THEOREM 4. Letx(t)andy(t)be the mild solution of the system (12)-(13) with initial valuesϕ1andϕ2respectively. If the assumption of Theorem 3 are satisfied, then the mild solution of the system (12)-(13) is stable in the mean square.

PROOF. The proof of this theorem is similar to Theorem 2.

5. EXAMPLE

Consider the following stochastic partial integro-differential equation of the form

d

u(t, ξ) +

Z π

0

a(y, ξ)u(tsint, y)dy

= ∂ 2

∂ξ2

u(t, ξ) +

Z π

0

a(y, ξ)u(tsint, y)dy

dt

+

Z t

0

f(t−s)u(s, ξ) +

Z π

0

a(y, ξ)u(tsint, y)dyds

+H(t, u(tsint, ξ))

dt+G(t, u(tsint, ξ))dβ(t),

0≤ξ≤π, τ >0, t∈J= [0, T], (15) u(t,0) =u(t, π) = 0, t∈J, (16) u(θ, ξ) =ϕ(θ, ξ), θ∈(−∞,0], 0≤ξ≤π, (17) whereβ(t)denotes a standard cylindrical Wiener process inH defined on a stochastic process(Ω,F, P)andH=L2_{([0, π]).} To rewrite (15)-(17) into the form (1)-(2), defineA:H →Hby Az=z00_{with domain}

D(A) =

z ∈ H, z, z0_{are absolutely continuousz}00 _∈

H, z(0) =z(π) = 0

.

Then,Agenerates a strongly continuous semigroupR(t)onH, thus (H1) is true. Moreover, the operatorAcan be expressed as

Az= ∞

X

n=1

n2< z, zn> zn, z∈D(A),

wherezn(s) =

q 2

πsin(ns),n = 1,2, . . ., is orthonormal set

of eigenvectors ofA.

In addition, it follows thatR(t)is compact for everyt >0and

kR(t)k ≤e−t_{for everyt≥0.}

Now, we define an operatorA(t) :D(A)⊂H→Hby A(t)x(ξ) =Ax(ξ) +b(t, ξ)x(ξ).

Letb(·)be continuous andb(t, ξ)≤ −γ(γ >0), for everyt∈

R. Then, the system

(

u0_{(t) =A(t)u(t), t≥s,}

u(s) =x∈H

has an associated evolution family, given by

R(t, s)x(ξ) = [R(t−s)eRstb(ς,ξ)dς_x](ξ)

From the above expression, it follows thatR(t, s)is a compact operator and for everyt, s∈Jwitht > s

kR(t, x)k=e−(1+γ)(t−s)_. The assumption of the following conditions hold:

(i) The functionbis measurable and

Z π 0

a2_{(y, ξ)dydξ <∞.}

(ii) The function ∂

∂tb(y, ξ)is measurableb(y,0) =b(y, π) = 0

and let

Mg=

Z π

0 Z π

0 ∂ ∂ta(y, ξ)

2 dydξ

12 <∞. Letα <0, define the phase space

B=

φ∈C((−∞,0], H) : lim θ→−re

αθ

φ(θ) exists in H

,

and letkφkB= sup θ∈(−∞,0]

eαθkφ(θ)kL2 . Then,(B,k · kB)is a Banach space and satisfied axioms(1)-(2) withL = 1,N(t) =

e−αt_{,M(t) = max{1, e}−αt_{}. Thus for(t, φ)∈J× B, where}

φ(θ)(ξ) =ϕ(θ, ξ),(θ, ξ)∈(−∞,0]×[0, π].

Suppose that conditions (i) and (ii) are verified, then the prob-lem (5.1)-(5.3) can be represent as the abstract neutral stochastic integro-differential equation of the form (1)-(2), as follows

g(t, xt) =

Zπ

0

a(y, ξ)u(tsint, y)dy, h(t, xt) =H(t, u(tsint, ξ)),

σ(t, xt) =G(t, u(tsint, ξ))

The below results are consequence of Theorem 1 and Theorem 2 respectively.

PROPOSITION 1. If the hypothesis (H1)-(H6) hold, then there exists a unique mild solution ofuof the system (15)-(17).

PROPOSITION 2. If all the hypothesis of Proposition 1 hold, then the mild solutionuof the system (15)-(17) is stable in the mean square.

6. CONCLUSION

In this paper, the existence, uniqueness and stability of neutral stochastic integro-differential evolution equations with infinite delay are discussed by using phase space axioms. The results are obtained by using the method of successive approximation and Bihari’s inequality. Finally, an example is illustrated for the effectiveness of the results.

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