Existence of mild solutions of random impulsive functional differential equations with almost

Research Article

Existence of mild solutions of random impulsive functional differential equations with almost

sectorial operators

A. Anguraj^∗, M.C. Ranjini

Department of Mathematics, P.S.G. College of Arts and Science, Coimbatore-641 014, Tamil Nadu, India.

This paper is dedicated to Professor Ljubomir ´Ciri´c Communicated by Professor V. Berinde

Abstract

By using the theory of semigroups of growthα, we prove the existence and uniqueness of the mild solution for the random impulsive functional differential equations involving almost sectorial operators. An example is given to illustrate the theory. c2012 NGA. All rights reserved.

Keywords: Impusive differential equations, random impulses, almost sectorial operator, semigroup of growthα, mild solution

2010 MSC: 34A37, 35R10, 46C05

1. Introduction

Sectorial operators, that is, linear operatorsAdefined in Banach spaces, whose spectrum lies in a sector Sw=

λ∈C/{0} | |argλ| ≤w ∪ {0} f or some 0≤w≤π and whose resolvent satisfies an estimate

||(λ−A)⁻¹|| ≤ M|λ|⁻¹, ∀ λ∈C\S_w, (1.1)

∗Corresponding author

Email addresses: [email protected](A. Anguraj),[email protected]( M.C. Ranjini )

Received 2011-6-3

have been studied extensively during the last 40 years, both in abstract settings and for their applications to partial differential equations. Many important elliptic differential operators belong to the class of sectorial operators, especially when they are considered in the Lebesgue spaces or in spaces of continuous functions (see [1] and [[2], chapter 3]). However, if we look at spaces of more regular functions such as the spaces of Holder continuous functions, we find that these elliptic operators do no longer satisfy the estimate 1.1 and therefore are not sectorial as was pointed out by Von Wahl (see [[3], Ex.3.1.33], see [4]).

Neverthless, for these operators estimates such as

||(λ−A)⁻¹|| ≤ M

|λ|^1−α , λ∈X

w,v

=

λ∈C:|arg(λ−w)|< v (1.2)

where α ∈ (0,1), w ∈ R and v ∈ (^π₂, π), can be obtained, (see[4]) which allows to define an associated

”analytic semigroup” by means of the Dunford Integral T(t) = 1

2πi Z

Γθ

e^λt(λ−A)⁻¹dλ, t >0 (1.3)

where Γ_θ =

te^iθ :t∈R\{0} ,θ∈(v,^π₂).

In the literature, a linear operator A :D(A) ⊂ X→ X which satisfy the condition 1.2 is called almost sectorial and the operator family

T(t), T(0) =I, t≥0 is said the ”semigroup of growth α” generated by A. The operator family T(t)t≥0 has properties similar at those of analytic semigroup which allow to study some classes of partial differential equations via the usual methods of semigroup theory. Concerning almost sectorial operators, semigroups of growth α and applications to partial differential equations, we refer the reader to [4, 5, 6, 7, 8] and the references there in.

Also, many evolution processes from fields as diverse as physics, population dynamics, aeronautics, economics, telecommunications and engineering etc., are characterized by the fact that they undergo an abrupt change of state at certain moments of time between intervals of continuous evolution. The duration of these changes are often negligible compared to the total duration of the process acting instantaneously in the form of impulses. The impulses may be deterministic or random. There are lot of papers which investigate the qualitative properties of deterministic impulses see for example [9, 10, 11, 12, 13, 14] and the references there in.

When the impulses exist at random points, solutions of the differential systems are stochastic process.

Random impulsive systems are more realistic than deterministic impulsive systems. The study of random impulsive differential equations is a new area of research. So far there are few results have been discussed in random impulsive systems. In [15, 16], the authors proved the existence and uniqueness of differential system with random impulses. In [17], Wu and Duan discussed the oscillation, stability and boundedness of second-order differential systems with random impulses, and in [18, 19], the authors studied the existence and stability results of random impulsive semilinear differential systems.

To the best of our knowledge, the study of the existence of solutions of abstract system as 2.1 for which the operator A is almost sectorial is an untreated topic in the literature. In [20], Hernandez proved the existence of mild solutions for a class of abstract functional differential equations with almost sectorial op- erators and in [19], A. Anguraj et al. proved the existence and exponential stability of semilinear functional differential equations with random impulses under non-uniqueness. By the motivation of the above papers, we present a new idea of research to prove the existence and uniqueness of mild solutions of functional differential equations with random impulses involving almost sectorial operators.

2. Preliminaries

Here, we introduce some notations and technicalities. Let (Z,||.||_z) be a Banach space. In this paper, L(Z, W) represents the space of bounded linear operators fromZintoW endowed with the norm of operators

denoted ||.||_L(Z,W), and we writeL(Z) and ||.||_L(Z) when Z =W. In addition, B_l(z, Z) denotes the closed ball with center at z ∈ Z and radius l > 0 in Z. As usual, C([c, d], Z) represents the space formed by all the continuous functions from [c, d] into Z endowed with the sup-norm denoted by ||.||_C([c,d],Z) and L^p([c, d],X), p≥1, denotes the space formed by all the classes of Lebesgue-integrable functions from [c, d]

intoXendowed with the norm

||h||_Lp([c,d],X) = Z

[c,d]

||h(s)||^pds ¹

p.

Throughout this paper, (X,||.||) is a Banach space, A :D(A)⊂X →X is an almost sectorial operator and (T(t))t≥0 is the semigroup of growth α generated by A. For simplicity, next we assume w = 0. The next lemma consider some properties of the operator family (T(t))t≥0.

Lemma 2.1. ([5, 8]). Under the above conditions, the followings properties are satisfied.

(a) The operator A is closed, T(t+s) = T(t)T(s) and AT(t)x = T(t)Ax for all t, s ∈ [0,∞) and each x∈D(A).

(b) T(.)∈C((0,∞),X)∩C¹((0,∞),X) and _dt^dT(t) =AT(t) for allt >0.

(c) For n ∈ N∪ {0}, AⁿT(.) ∈ C((0,∞),X) and there exists Dn > 0 and a constant γ > 0, which is independent of n, such that ||AⁿT(t)||_L(X)≤Dne^γtt^−(n+α) for all t >0.

LetXbe the Banach space and Ω a non - empty set. Assume thatτ_kis a random variable defined from Ω toD_k^def.= (0, d_k) for all k= 1,2, ...where 0< d_k<∞. Furthermore, assume thatτ_i andτ_j are independent of each other asi6=j fori, j= 1,2, .... For the sake of simplicity, we denoteR⁺ = [0,∞).

We consider the functional differential equations with random impulses of the form,







x⁰(t) =Ax(t) +f(t, x_t), t≥0, t6=ξ_k x(ξ_k) =b_k(τ_k)x(ξ⁻_k), k= 1,2, ...

x0 =φ

(2.1)

whereA:D(A)⊂X→Xis an almost sectorial operator, the functionf :R⁺×Cb→X,Cb=C([−r,0],X) is the set of piecewise continuous functions mapping [−r,0] into Xwith some given r >0; xt is a function where t is fixed, defined by xt(s) = x(t+s) for all s∈ [−r,0]; ξ0 =t0 and ξ_k =ξk−1+τ_k fork = 1,2, ....

Here t₀ = 0 = ξ₀ < ξ₁ < ξ₂ < ... < ξ_k < ..., b_k :D_k → Xfor each k = 1,2, ..., x(ξ_k⁻) = limt→ξ_kx(t) with the norm||x||_t= supt−r<s<t|x(s)|for eachtsatisfying 0≤t≤T andT ∈R⁺ is a given number,||.||is any given norm inX;φis a function defined from [−r,0] toX.

Denote {B_t, t ≥ 0} the simple counting process generated by ξ_n, that is, {B_t ≥ n} = {ξ_n ≤ t}, and denote F_t theσ - algebra generated by{B_t, t≥0}. Then (Ω, P,{F_t}) is a probability space.

Definition 2.2. For a given T ∈(0,∞), a stochastic process {x(t),−r ≤t ≤T} is called a mild solution to the equation 2.1 in (Ω, P,{F_t}) , if

(i) x(t) isF_t-adapted.

(ii)x(s) =φ(s) whens∈[−r,0], and

x(t) =

∞

X

k=0

hY^k

i=1

bi(τi)T(t)φ(0) +

k

X

i=1 k

Y

j=i

bj(τj) Z ξi

ξi−1

T(t−s) f(s, xs)ds +

Z t ξ_k

T(t−s)f(s, x_s)dsi

I_[ξk,ξk+1)(t), t∈[0, T] whereQn

j=m(.) = 1 as m > n,Qk

j=ibj(τj) =b_k(τ_k)bk−1(τk−1)...bi(τi), andIA(.) is the index function, i.e., IA(t) =

(1, if t∈A 0, if t /∈A

3. Existence Results

In this section, we discuss the existence and uniqueness of the solution of the system 2.1.

Remark 3.1. In the remainder of this paper, φ: [−r,0]→X is a given function and y : [−r, T]→ X is the function defined byy(θ) =φ(θ) forθ≤0 and y(t) =P∞

k=0

hQk

i=1b_i(τ_i)T(t)φ(0)i

I_[ξk,ξk+1)(t) fort >0.

In addition, Cn, n∈N, are positive constants such that

||AⁿT(t)||_L(X) ≤C_nt^−(n+α),∀ t∈(0, T],

and for a bounded set B⊂X, we use the notationDiam_X(B) for Diam_X(B) = sup

a,b∈B

||a−b||.

To prove our results, we introduce the following hypotheses. In the next assumptions, q∈ _1−α¹ ,∞ or q=∞ and q⁰ = _p−1^p forq <∞ and q⁰ = 1 if q=∞.

(H₁) The function f(. , ψ) is strongly measurable on [0, T] for all ψ ∈ Cb and f(t , .) ∈ C(C,b X) for each t∈[0, T]. There exists a non-decreasing function W_f ∈C(R⁺,(0,∞)) and m_f ∈L^q([0, T], R⁺) such that

E||f(t, ψ_t)||² ≤mf(t)Wf(E||ψ||²_t), ∀ (t, ψ)∈[0, T]×Cb

(H₂) The function f is continuous and for all l >0 with [0, l]×B_l(φ,C)b ⊂[0, T]×C, there existsb L_f,l ∈ L^q([0, T], R⁺) such that

E||f(s, ψ₁)−f(s, ψ2)||² ≤L_f,l(s) E||ψ₁−ψ2||², ∀ (s, ψi)∈[0, l]×B_l(φ,C)b (H₃) The condition max

i,k

nQk

j=i||b_j(τ_j)||o

is uniformly bounded if there is a constantB >0 such that

maxi,k

nY^k

j=i

||b_j(τ_j)||o

≤ B, ∀τ_j ∈D_j, j= 1,2, ...

Theorem 3.2. If the hypotheses (H₁), (H₃) are satisfied, T(.)φ(0)∈C([0, T],X) and T(t) is compact for allt >0, then there exists a mild solution of 2.1 on [−r, T].

Proof 1. Let T be an arbitrary number 0< T < ∞ where T < b₁ <∞ and forC >0 let W_f(||ψ||²) ≤C, for all ψ∈Bb1(φ,C).b

Also, let sup

s∈[0,T]

||y_s−φ||² ≤ b²₁ 4 and

CC₀² ||m_f||_Lq([0,T]) max{1, B²}T

1 q0−2α+1

(1−2q⁰α)^q10

≤ b²₁ 4 For the simplification, on the space

Bb1 2

(0, S(T)) =

u∈C([−r, T],X) :u₀= 0, ||u||²_C([0,T],

X)≤ b²₁ 4 we define the map

Γ :Bb1 2

(0, S(T))→C([−r, T],X) by (Γu)₀= 0 and

Γu(t) =











φ(t) t∈[−r,0]

P∞ k=0

hPk i=1

Qk

j=ib_j(τ_j)Rξi

ξi−1T(t−s)f(s, u_s+y_s)ds +Rt

ξ_kT(t−s)f(s, us+ys)ds i

I_[ξk,ξk+1)(t), t∈[0, T] with the norm defined as

||χ||²_Γ= sup

0≤t≤T

E||χ||²_t

Now, we prove that Γ is completely continuous from Bb1 2

(0, S(T)) intoBb1 2

(0, S(T)).

For that, (s, u)∈[0, T]×B^b1 2

(0, S(T)),

||u_s+y_s−φ||² ≤ 2h sup

θ∈[0,s]

||u(θ)||²+||y_s−φ||²i

≤ 2 hb²₁

4 +b²₁ 4 i

≤ b²₁

which implies that u_s+y_s∈B_b1(φ,C) andb W_f(||u_s+y_s||²)≤C.

Now, from the properties of (T(t))t≥0 and f, the Bochner’s criterion for integrable functions and the inequality,

||T(t−s)f(s, u_s+y_s)||² ≤ C₀² mf(s)Wf(||u_s+ys||²) (t−s)^2α

≤ C₀²Cm_f(s) (t−s)^2α

Therefore, the function s→ T(t−s)f(s, u_s+y_s) is integrable on [0, t] for all t∈ [0, T], which implies that Γu∈C([−r, T], X) and Γ is well defined.

Next, we show that ΓBb1 2

(0, S(T))⊂Bb1 2

(0, S(T)).

Consider,

||Γu(t)||² = ||

∞

X

k=0

hX^k

i=1 k

Y

j=i

b_j(τ_j) Z ξi

ξi−1

T(t−s) f(s, u_s+y_s)ds +

Z t ξk

T(t−s) f(s, us+ys)ds i

I_[ξk,ξk+1)(t)||²

≤ h maxi,k

n 1,

k

Y

j=i

||b_j(τ_j)||oi2

Z t 0

||T(t−s)|| ||f(s, u_s+ys)||ds I_[ξk,ξk+1)(t) 2

E||Γu(t)||² ≤ maxn 1, B²o

tZ t 0

||T(t−s)||²E||f(s, u_s+y_s)||²ds

Taking supremum overt, we get sup

t∈[0,T]

E||Γu(t)||² ≤ max n

1, B² o

T C₀² Z t

0

m_f(s)W_f(sup_t∈[0,T]E||u_s+y_s||²)

(t−s)^2α ds

≤ CC₀²Tmax n

1, B² oZ t

0

mf(s) (t−s)^2αds

≤ CC₀²max n

1, B² o

||m_f||_Lq([0,T])

T

1 q0−2α+1

(1−2q⁰α)

1 q0

Thus,

||Γu(t)||² ≤ b²₁ 4

which implies that Γu ∈ Bb1 2

(0, S(T)) and therefore ΓBb1 2

(0, S(T))⊂Bb1 2

(0, S(T)). Moreover, a stan- dard application of the Lebesgue dominated convergence theorem proves that Γ is continuous.

Now, we prove the compactness of the operator Γ.

Step 1: The set ΓB^b1 2

(0, S(T)) = n

Γu(t) :u∈B^b1 2

(0, S(T)) o

is relatively compact for allt∈[−r, T].

The case t ≤ 0 is trivial. Let 0 < t ≤ T be fixed and let be a real number with 0 < < t. For u∈B^b1

2

(0, S(T)), we define

Γu(t) =

∞

X

k=0

hX^k

i=1 k

Y

j=i

bj(τj) Z ξi

ξi−1

T(t−s) f(s, us+ys)ds +

Z t−

ξk

T(t−s) f(s, u_s+y_s)dsi

I_[ξk,ξk+1)(t) Since, T(t) is compact, the set Γu(t) =n

u(t) : u∈Bb1 2

(0, S(T))o

is relatively compact inX for every ∈(0, t).

Moreover, for every u∈Bb1 2

(0, S(T)), we have

Γu(t)−Γu(t) =

∞

X

k=0

hX^k

i=1 k

Y

j=i

b_j(τ_j) Z ξi

ξi−1

T(t−s) f(s, u_s+y_s)ds +

Z t ξk

T(t−s) f(s, us+ys)ds i

I_[ξk,ξk+1)(t)

−

∞

X

k=0

hX^k

i=1 k

Y

j=i

b_j(τ_j) Z ξi

ξi−1

T(t−s)f(s, u_s+y_s)ds +

Z t−

ξk

T(t−s)f(s, us+ys)ds i

I_[ξk,ξk+1)(t)

||Γu(t)−Γu(t)||² ≤ maxn 1, B²o

t Z _t

t−

||T(t−s)||²E||f(s, us+ys)||²ds E||Γu(t)−Γu(t)||² ≤ maxn

1, B²o T C₀²

Z _t

t−

mf(s)Wf(E||u_s+ys||²) (t−s)^2α ds sup

t∈[0,T]

E||Γu(t)−Γu(t)||² ≤ CC₀²Tmaxn

1, B²oZ _t

t−

mf(s) (t−s)^2αds

Thus,

||Γu(t)−Γu(t)||² ≤ CC₀²Tmaxn

1, B²oZ t t−

mf(s) (t−s)^2αds

Therefore, letting→0, we see that there are relatively compact sets arbitrary close to the set n

Γu(t) :u∈ B^b1

2

(0, S(T)) o

. Hence, the set n

Γu(t) :u∈B^b1 2

(0, S(T)) o

is relatively compact inX.

Step 2: Γ is equicontinuous.

For any 0≤t1 < t2 ≤T and for u∈Bb1 2

(0, S(T)), we have

Γu(t₂)−Γu(t₁) =

∞

X

k=0

hX^k

i=1 k

Y

j=i

b_j(τ_j) Z ξi

ξi−1

T(t₂−s) f(s, u_s+y_s)ds +

Z t2

ξk

T(t2−s) f(s, us+ys)ds i

I_[ξk,ξk+1)(t2)

−

∞

X

k=0

hX^k

i=1 k

Y

j=i

b_j(τ_j) Z ξi

ξi−1

T(t₁−s) f(s, u_s+y_s)ds +

Z t1

ξk

T(t1−s) f(s, us+ys)ds i

I_[ξk,ξk+1)(t1)

=

∞

X

k=0

hX^k

i=1 k

Y

j=i

b_j(τ_j) Z ξi

ξi−1

T(t₂−s) f(s, u_s+y_s)ds +

Z t2

ξk

T(t2−s) f(s, us+ys)ds i

I_[ξk,ξk+1)(t2)−I_[ξk,ξk+1)(t1)

+

∞

X

k=0

hX^k

i=1 k

Y

j=i

b_j(τ_j) Z ξi

ξi−1

[T(t₂−s)−T(t₁−s)]f(s, u_s+y_s)ds +

Z t1

ξk

[T(t2−s)−T(t1−s)]f(s, us+ys)ds+ Z t2

t1

T(t2−s) f(s, us+ys)ds i

I_[ξk,ξk+1)(t1) Then

EkΓu(t₂)−Γu(t1)k² ≤ 2EkI₁k²+ 2EkI₂k² (3.1)

where, I1 =

∞

X

k=0

hX^k

i=1 k

Y

j=i

bj(τj) Z ξi

ξi−1

T(t2−s) f(s, us+ys)ds +

Z t2

ξk

T(t₂−s)f(s, u_s+y_s)dsi

I_[ξk,ξk+1)(t₂)−I_[ξk,ξk+1)(t₁) and

I2 =

∞

X

k=0

hX^k

i=1 k

Y

j=i

bj(τj) Z ξi

ξi−1

[T(t2−s)−T(t1−s)]f(s, us+ys)ds +

Z _t1

ξk

[T(t2−s)−T(t1−s)]f(s, us+ys)ds +

Z _t2

t1

T(t₂−s)f(s, u_s+y_s)dsi

I_[ξk,ξk+1)(t₁)

Consider,

E||I₁||² ≤ h maxn

1,||

k

Y

j=i

||b_j(τ_j)||oi2

E hZ t2

0

||T(t2−s)|| ||f(s, us+ys)||dsI_[ξk,ξk+1)(t2)−I_[ξk,ξk+1)(t1) i2

≤ max n

1, B² o

T hZ t2

0

||T(t₂−s)||² E||f(s, u_s+ys)||²ds i

E

I_[ξk,ξk+1)(t₂)−I_[ξk,ξk+1)(t₁)

≤ max n

1, B² o

T C₀² hZ t2

0

m_f(s)W_f(E||u_s+y_s||²) (t−s)^2α ds

i

E

I_[ξk,ξk+1)(t2)−I_[ξk,ξk+1)(t1)

≤ CC₀²Tmaxn

1, B²ohZ t2

0

m_f(s) (t−s)^2αdsi

E

I_[ξk,ξk+1)(t₂)−I_[ξk,ξk+1)(t₁)

→0 as t₂ →t₁ (3.2)

E||I₂||² ≤ hX^∞

k=0

hX^k

i=1 k

Y

j=i

b_j(τ_j) Z ξi

ξi−1

[T(t₂−s)−T(t₁−s)]f(s, u_s+y_s)ds +

Z t1

ξk

[T(t2−s)−T(t1−s)]f(s, us+ys)ds +

Z t2

t1

T(t2−s) f(s, us+ys)ds i

I_[ξk,ξk+1)(t1) i2

≤ 2 maxn 1, B²o

t₁ Z _t1

0

||T(t₂−s)−T(t₁−s)||² E||f(s, u_s+y_s)||²ds +2(t2−t1)

Z t2

t1

||T(t2−s)||² E||f(s, us+ys)||²ds

≤ 2 max n

1, B² o

T Z t1

0

||T(t2−s)−T(t1−s)||² mf(s) Wf(E||f(s, us+ys)||²)ds +2(t₂−t₁)

Z t2

t1

||T(t₂−s)||² m_f(s) W_f(E||f(s, u_s+y_s)||²)ds

≤ 2 max n

1, B² o

T C Z t1

0

||T(t2−s)−T(t1−s)||² mf(s)ds +2(t₂−t₁)C

Z t2

t1

||T(t₂−s)||² m_f(s)ds

→0 as t₂ →t₁ (3.3)

From the equations (3.2) and (3.3), it follows that the right hand side of 3.1 tends to zero as t2 → t1. Thus Γ mapsBb1

2

(0, S(T)) into an equicontinuous family of functions.

Finally, from the Schauder’s fixed point theorem, Γ has a fixed point x=u+y which is a mild solution of the problem 2.1.

Theorem 3.3. Let the hypothesis (H₂) and (H₃) hold. If the following inequality C₀² ||L_f,b1||_Lq([0,T]) max{1, B²} T^q10−2α+1

(1−2q⁰α)^q10

<1

is satisfied, then the system 2.1 has a unique mild solution on [−r, T].

Proof 2. LetT < b1<∞ andC >0 such that||f(t, ψ)|| ≤C, ∀(t, ψ)∈[0, b1]×Bb1(φ,C).b Also, let

sup

s∈[0,T]

||y_s−φ||² ≤ b²₁ 4 and

2C₀²max{1, B²}h

b²₁ ||L_f,b1||_Lq([0,T])

T

1 q0−2α+1

(1−2q⁰α)

1 q0

+||f(s, φ)||² T^2(1−α) 1−2α

i

≤ b²₁ 4 Let Γ :B^b1

2

(0, S(T))→C([−r, T],X) be the operator introduced as in the Theorem 3.2. Proceeding as in the proof of the previous theorem, we can prove that Γ is well-defined.

Next, we have to prove that Γ is a contraction mapping.

Before that, for (s, u)∈[0, T]×Bb1 2

(0, S(T)),

||u_s+ys−φ||² ≤ 2 h

sup

θ∈[0,s]

||u(θ)||²+||y_s−φ||²i

≤ 2 hb²₁

4 +b²₁ 4 i

≤ b²₁

which implies that us+ys∈B_b1(φ,C) and thereforeb ||f(s, us+ys)|| ≤C.

Using this fact, for u∈Bb1 2

(0, S(T)),

||Γu(t)||² = ||

∞

X

k=0

hX^k

i=1 k

Y

j=i

b_j(τ_j) Z ξi

ξi−1

T(t−s) f(s, u_s+y_s)ds +

Z t ξ_k

T(t−s) f(s, us+ys)ds i

I_[ξk,ξk+1)(t)||²

≤ h maxi,k

n 1,

k

Y

j=i

||b_j(τ_j)||oi2

Z t 0

||T(t−s)|| ||f(s, u_s+ys)||ds I_[ξk,ξk+1)(t) 2

≤ maxn 1, B²o

tZ t 0

||T(t−s)||² ||f(s, u_s+y_s)−f(s, φ) +f(s, φ)||²ds E||Γu(t)||² ≤ 2 maxn

1, B²o

T C₀² hZ _t

0

E||f(s, us+ys)−f(s, φ)||² (t−s)^2α ds +

Z t 0

E||f(s, φ)||² (t−s)^2α dsi

Taking supremum, we get

||Γu(t)||² ≤ 2C₀²max{1, B²}h

b²₁ ||L_f,b1||_Lq([0,T])

T

1 q0−2α+1

(1−2q⁰α)

1 q0

+||f(s, φ)||² T^2(1−α) 1−2α

i

≤ b²₁ 4 which implies that Γu∈B^b1

2

(0, S(T)) and ΓB^b1 2

(0, S(T))⊂B^b1 2

(0, S(T)).

Moreover,

kΓu(t)−Γv(t)k² ≤ hX^∞

k=0

hX^k

i=1 k

Y

j=i

||b_j(τj)||

Z ξi

ξi−1

||T(t−s)|| ||f(s, u_s+ys)−f(s, vs+ys)||ds +

Z t ξk

||T(t−s)|| ||f(s, u_s+y_s)ds−f(s, v_s+y_s)||dsi

I_[ξk,ξk+1)(t)i2

EkΓu(t)−Γv(t)k² ≤ maxn 1, B²o

T Z t

0

C₀²

(t−s)^2α E||f(s, u_s+y_s)−f(s, v_s+y_s)||²ds

≤ max n

1, B² o

T C₀² Z t

0

L_f,b1 E||u_s−v_s||² (t−s)^2α ds Taking supremum overt,

kΓu(t)−Γv(t)k² ≤ C₀² ||L_f,b1||_Lq([0,T]) max{1, B²} T

1 q0−2α+1

(1−2q⁰α)

1 q0

||u−v||_C([0,T],X) It follows that Γ is contraction onB^b1

2

(0, S(T)) and there exists a unique fixed pointxof Γ which is defined asx=u+y and is a mild solution of 2.1.

This completes the proof.

4. Application

In this section, we apply our abstract results to random impulsive partial differential equation. To apply our results, we need to introduce the required technical tools. Let U ⊂ Rⁿ is a open bounded set with smooth boundary ∂U, η ∈ (0,1) and X = C^η(U,Rⁿ) is the space formed by all the η - Hlder continuous functions fromU into Rⁿ endowed with the norm

||ξ||_Cη(U,Rⁿ) = ||ξ||_C(U,Rn)+ [|ξ|]_Cη(U,Rⁿ)

where||.||_C(U,

Rⁿ) is the sup-norm onU, [|ξ|]_Cη(U,Rⁿ) = sup

x,y∈U,x6=y

|ξ(x)−ξ(y)|

|x−y|^η and |.|is the Euclidean norm inRⁿ.

On the spaceX, we consider the operatorA:D(A)⊂X→X givenAu= ∆u with domain

D(A) =

u∈C^2+η(U,Rⁿ) :u_|∂U = 0

From[4], we know thatA is an almost sectorial operator which verifies 1.2 with α= ^η₂ and A is not secto- rial. In the remainder of this section, (T(t))t≥0represents the analytic semigroup of growthαgenerated byA.

Let Ω⊂Xbe a bounded domain with smooth boundary ∂Ω.











∂

∂tu(t, ξ) = ∆u(t, ξ) +a1u(x, t−r)u³(x, t), t6=ξ_k, t≥0, u(t, ξ_k) = q(k)τ_ku(t, ξ_k⁻) a.s. x∈Ω,

u(t, ξ) = Φ(t, ξ) a.s. ξ∈Ω, −r ≤t≤0,

u(t, ξ) = 0 a.s. ξ∈∂Ω.

(4.1)

Letτ_k be a random variable defined inD_k≡(0, d_k) for allk= 1,2, ...where 0< d_k <∞. Furthermore, assume thatτ_i and τ_j be independent with each other as i6=j fori, j= 1,2, ...and

Eh maxi,k

( k

Q

j=i

kq(j)(τ_j)k² )

i

<∞.

(H1) The function f(. , ψ) is strongly measurable on [0, T] for all ψ ∈ Cb and f(t , .) ∈ C(C,b X) for each t∈[0, T]. There exists a non-decreasing function W_1f ∈C(R⁺,(0,∞)) andm_1f ∈L^q([0, T], R⁺) such that

E||f(t, ψ_t)||² ≤m_1f(t)W_1f(E||ψ||²_t),∀(t, ψ)∈[0, T]×Cb

(H₂) The function f is continuous and for all l >0 with [0, l]×B_l(φ,C)b ⊂[0, T]×C, there existsb L_1f,l ∈ L^q([0, T], R⁺) such that

E||f(s, ψ₁)−f(s, ψ2)||² ≤L1f,l(s) E||ψ₁−ψ2||²,∀ (s, ψi)∈[0, l]×Bl(φ,C)b (H3) The condition max

i,k

nQk

j=i||b_j(τj)||o

is uniformly bounded if there is a constantB1>0 such that

maxi,k

nY^k

j=i

||b_j(τj)||o

≤ B1,∀τ_j ∈Dj, j= 1,2, ...

If the inequality,

C C₀² ||m_1f||_Lq([0,T]) max{1, B₁²} T

1 q0−2α+1

(1−2q⁰α)^q10

≤ b²₁ 4

holds, then it is easy to check that all the hypotheses of the Theorem 3.2are satisfied and therefore, Theorem 3.2 guarantees the existence of mild solution of the partial differential equation 4.1.

Furthermore, if the following condition holds C₀² ||L_1f,b1||_Lq([0,T]) max{1, B₁²} T^q10−2α+1

(1−2q⁰α)

1 q0

<1

then by Theorem 3.3 we know that the solution to 4.1 is unique.

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