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R E S E A R C H Open AccessFractional nonlocal impulsive quasilinear multi-delay integro-differential systems Amar Debbouche Correspondence: amar_debbouche@yahoo.fr Department of Mathema

R E S E A R C H Open Access

Fractional nonlocal impulsive quasilinear

multi-delay integro-differential systems

Amar Debbouche

Correspondence:

amar_debbouche@yahoo.fr

Department of Mathematics,

Faculty of Science, Guelma

University Guelma, Algeria

Abstract

In this article, sufficient conditions for the existence result of quasilinear multi-delay integro-differential equations of fractional orders with nonlocal impulsive conditions

in Banach spaces have been presented using fractional calculus, resolvent operators, and Banach fixed point theorem As an application that illustrates the abstract results,

a nonlocal impulsive quasilinear multi-delay integro-partial differential system of fractional order is given

AMS Subject Classifications 34K05, 34G20, 26A33, 35A05

Keywords: Fractional integrodifferential systems, resolvent operators, nonlocal and impulsive conditions, fixed point theorem

Introduction

Many fractional models can be represented by the following system

dα u(t)

dt α + A(t, u(t))u(t) = f (t, u(t), u( β(t))) +

t

 0

g(t, s, u(s), u( γ (s))) ds, (1:1)

in a Banach space X, where 0 <a≤ 1, t Î [0, a], u0 Î X, i = 1, 2, , m and 0 <t1<t2<

··· <tm<a We assume that -A(t,.) is a closed linear operator defined on a dense domain D(A) in X into X such that D(A) is independent of t It is assumed also that -A(t,.) gen-erates an evolution operator in the Banach space X The functions f : J Xr+1® X, g : Λ

× Xk+1® X, h : PC(J, X) ® X, u(b) = (u(b1), , u(br)), u(g) = (u(g1), , u(gk)), and bp, gq : J® J are given, where p = 1, 2, , r and q = 1, 2, , k Here J = [0, a] and Λ = {(t, s)

0≤ s ≤ t ≤ a} Let PC (J, X) consist of functions u from J into X, such that u(t) is con-tinuous at t≠ ti and left continuous at t = tiand the right limitu(t+i)exists for i = 1, 2, , m Clearly PC(J, X) is a Banach space with the norm ||u||PC= suptÎJ||u(t)||, and letu(ti) = u(t+

i)− u(t−

i )constitutes an impulsive condition Fractional differential equations have proved to be valuable tools in the modelling of many phenomena in various fields of science and engineering Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc (see [1-5]) They involve a wide area of applications by bringing into a broader paradigm concepts

© 2011 Debbouche; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium,

of physics and mathematics [6-8] There has been a significant development in

frac-tional differential and partial differential equations in recent years, see Kilbas et al

[9,10], also in fractional nonlinear systems with delay and fractional variational

princi-ples with delay, see Baleanu et al [11,12]

The existence results to evolution equations with nonlocal conditions in Banach space was studied first by Byszewski [13,14], subsequently, many authors were pointed

in the same field, see reference therein Deng [15] indicated that, using the nonlocal

small amount of gas in a transparent tube can give better result than using the usual

local Cauchy problem u(0) = u0 Let us observe also that since Deng’s papers, the

func-tion h is considered

h(u) =

p



k=1 cku(tk),

where ck, k = 1, 2, , p are given constants and 0 ≤ t1< ··· <tp≤ a However, among the previous research on nonlocal cauchy problems, few are concerned with mild

solu-tions of fractional semilinear differential equasolu-tions, see Mophou and N’Guérékata [16],

and others with fractional nonlocal boundary value problems, for instance, Ahmad et

al [17,18]

The theory of impulsive differential equations has been emerging as an important area of investigation in recent years, because all the structures of its emergence have

deep physical background and realistic mathematical model The theory of impulsive

differential equations appears as a natural description of several real processes subject

to certain perturbations whose duration is negligible in comparison with the duration

of the process It has seen considerable development in the last decade, see the

mono-graphs of Bainov and Simeonov [19], Lakshmikantham et al [20], and Samoilenko and

Perestyuk [21] where numerous properties of their solutions are studied, and detailed

bibliographies are given

Recently, the existence of solutions of fractional abstract differential equations with nonlocal initial condition was investigated by N’Guérékata [22] and Li [23] Much

attention has been paid to existence results for the impulsive differential and

integro-differential equations of fractional order in abstract spaces, see Benchohra et al [2,24]

Several authors have studied the existence of solutions of abstract quasilinear evolution

equations in Banach space [25-27]

Regarding this article, it generalizes previous results concerned the existence of solu-tions to nonlocal and impulsive integrodifferential equasolu-tions of quasilinear type with

delays of arbitrary orders Section “Preliminaries” is devoted to a review of some

essen-tial results In next section, we state and prove our main results, the last section deals

to giving an example to illustrate the abstract results

1 Preliminaries

Let X and Y be two Banach spaces such that Y is densely and continuously embedded

in X For any Banach space Z, the norm of Z is denoted by ||·||Z The space of all

bounded linear operators from X to Y is denoted by B(X, Y) and B(X, X) is written as

B(X) We recall some definitions in fractional calculus from Gelfand-Shilov [28] and

Podlubny [29], then some known facts of the theory of semigroups from Pazy [30]

Definition 2.1 The fractional integral of order with the lower limit zero for a func-tion fÎ C([0, ∞)) is defined as

I α f (t) = 1

(α)

t

 0

f (s) (t − s)1−α ds, t > 0, 0 < α < 1,

provided the right side is pointwise defined on [0, ∞), where Γ is the gamma func-tion Riemann-Liouville derivative of order a with the lower limit zero for a function f

Î C([0, ∞)) can be written as

L D α f (t) = 1

(1 − α)

d

dt

t

 0

f (s) (t − s) α ds, t > 0, 0 < α < 1.

The Caputo derivative of order for a function f Î C([0, ∞)) can be written as

C D α f (t)= L D α (f (t) − f (0)), t > 0, 0 < α < 1.

Remark 2.1 (1) If f Î C1

([0,∞)), then

C D α f (t) = 1

(1 − α)

t

 0

f(s) (t − s) α ds = I1−α f(t), t > 0, 0 < α < 1.

(2) The Caputo derivative of a constant is equal to zero

(3) If f is an abstract function with values in X, then integrals which appear in Defini-tion 2.1 are taken in Bochner’s sense

Definition 2.2 A two parameter family of bounded linear operators U(t, s), 0 ≤ s ≤ t

≤ a, on X is called an evolution system if the following two conditions are satisfied

(i) U(t, t) = I, U(t, r)U(r, s) = U(t, s) for 0≤ s ≤ r ≤ t ≤ a, (ii) (t, s) ® U(t, s) is strongly continuous for 0 ≤ s ≤ t ≤ a

More detail about evolution system and quasilinear equation of evolution can be found in [30, Chap 5 and Sect 6.4, respectively]

Let E be the Banach space formed from D(A) with the graph norm Since - A(t) is a closed operator, it follows that - A(t) is in the set of bounded operators from E to X

Definition 2.3 [31-33] A resolvent operators for problem (1.1)-(1.3) is a bounded operators valued function Ru(t, s)Î B(X), 0 ≤ s ≤ t ≤ a, the space of bounded linear

operators on X, having the following properties:

(i) Ru(t, s) is strongly continuous in s and t, Ru(s, s) = I, 0≤ s ≤ a, ||Ru(t, s)||≤ MeN (t, s)

for some constants M and N

(ii) Ru(t, s)E⊂ E, Ru(t, s) is strongly continuous in s and t on E

(iii) For xÎ X, Ru(t, s)x is continuously differentiable in sÎ [0, a] and

∂Ru

∂s (t, s)x = Ru(t, s)A(s, u(s))x.

(iv) For xÎ X and s Î [0, a], Ru(t, s)x is continuously differentiable in t Î [s, a]

and

∂Ru

∂t (t, s)x = −A(t, u(t))Ru (t, s)x,

with ∂Ru

∂s (t, s)xand

∂Ru

∂t (t, s)xare strongly continuous on 0≤ s ≤ t ≤ a Here Ru(t, s)

can be extracted from the evolution operator of the generator - A(t, u) The resolvent

operator is similar to the evolution operator for nonautonomous differential equations

in a Banach space LetΩ be a subset of X

Definition 2.4 (Compare [31] with [7,22,34]) By a mild solution of (1.1)-(1.3) we mean a function uÎ PC(J : X) with values in Ω satisfying the integral equation

u(t) = R u (t, 0)u0− R u (t, 0)h(u)

(α)

t

 0

(t − s) α−1 R

u (t, s)[f (s, u(s), u( β(s))) +

s

 0

g(s, η, u(η), u(γ (η)))dη]ds

0<t i <t

R u (t, t i )I i (u(t i )), t ∈ J

(2:1)

for all u0Î X

Definition 2.5 (Compare [35,36] with [2]) By a classical solution of (1.1)-(1.3) on J,

we mean a function u with values in X such that:

(1) u is continuous function on J \{t1, t2, , tm} and u(t)Î D(A), (2)dα u

dt α exists and continuous on J0, 0 <a < 1,

(3) u satisfies (1.1) on J0, the nonlocal condition (1.2) and the impulsive condition (1.3), where J0 = (0, a]\{t1, t2, , tm} We assume the following conditions

(H1) h : PC(J :Ω) ® Y is Lipschitz continuous in X and bounded in Y , i.e., there exist constants k1> 0 and k2> 0 such that

||h(u)||Y ≤ k1,

||h(u) − h(v)||Y ≤ k2max

t ∈J ||u − v||PC, u, v ∈ PC(J : X).

For the conditions (H 2) and (H3) let Z be taken as both × and Y

(H2) g :Λ × Zk+1® Z is continuous and there exist constants k3> 0 and k4> 0 such that

t

 0

||g(t, s, u1 , , u k+1)− g(t, s, v1 , , v k+1) ||Z ds ≤ k3

k+1

q=1

||u q − v q||Z, u q , v q ∈ X, q = 1, , k + 1,

k4= max

t 0

||g(t, s, 0, , 0)|| Z ds : (t, s) ∈



(H3) f : J × Zr+1® Z is continuous and there exist constants k5 > 0 and k6 > 0 such that

||f (t, u1 , , u r+1)− f (t, v1 , , v r+1) ||Z ≤ k5

r+1



p=1 ||u p − v p||Z, u p , v p ∈ X, p = 1, , r + 1,

k6 = max

t ∈J ||f (t, 0, , 0)|| Z.

(H4) bp, gq : J® J are bijective absolutely continuous and there exist constants cp> 0 and bq> 0 such thatβ

p (t) ≥ cpandγ

q (t) ≥ bq, respectively, for tÎ J, p = 1, , r and q = 1, , k

(H5) Ii: X ® X are continuous and there exist constants li> 0, i = 1, 2, , m such that

||Ii (u) − Ii (v) || ≤ li||u − v||, u, v ∈ X.

Let us take M0= max ||Ru(t, s)||B(Z), 0≤ s ≤ t ≤ a, u Î Ω.

(H6) There exist positive constants δ1,δ2,δ3 Î (0, δ /3] and l1, l2, l3 Î [0,1

3) such that

δ1= M0||u0||Y+ M0k1, δ2= M0θ, δ3= M0ξ,

and

λ1 = Ka||u0||Y+ k1Ka + M0k2,

λ2 = Ka θ + M0σ [k5(1 + 1/c1+· · · + 1/cr) + k3(1 + 1/b1+· · · + 1/bk)],

λ3 = Kaξ + M0

m



i=1 li,

where r = s [k5(1/c1 + ··· +1/cr)+ k3(1/b1+ ··· +1/bk)],θ = sδ (k3 + k5)+ rδ + s (k4 +

k6),σ = (1 + α) a α andξ =m

i=1 (liδ + ||Ii(0)||)

Main results

Lemma 3.1 Let Ru(t, s) the resolvent operators for the fractional problem (1.1)-(1.3).

There exists a constant K > 0 such that

||Ru (t, s) ω − Rv(t, s) ω|| ≤ K||ω||Y

t



s

||u(τ) − v(τ)||dτ,

for every u, v Î PC(J : X) with values in Ω and every ω Î Y , see [30, lemma 4.4, p

202]

Let Sδ= {u : uÎ PC(J : X), u(0) + h(u) = u0,Δu(ti) = Ii(u(ti)), ||u||≤ δ}, for t Î J, δ >

0, u0Î X and i = 1, , m

Lemma 3.2

||ϕ(t)||Y ≤ θ,

where

(α)

t

 0

(t − s) α−1

⎡

⎣f(s, u(s), u(β(s))) +

s

 0

g(s, τ, u(τ), u(γ (τ)))dτ

⎤

⎦ ds.

Proof We have

||ϕ(t)|| Y

≤(α)1

t



0

(t − s) α−1[||f (s, u(s), u(β 1(s)), , u(β r (s)) − f (s, 0, , 0)|| + ||f (s, 0, , 0)||

+

s



0

||g(s, τ, u(τ), u(γ1 (τ)), , u(γ k(τ)) − g(s, τ, 0, , 0)||dτ +

s



0

||g(s, τ, 0, , 0)||dτ

⎤

⎦ ds.

Using H2, H3, and H4, we get

||ϕ(t)|| Y

≤(α)1

t



0

(t − s) α−1 [k

5 (||u(s)|| + ||u(β 1(s))|| + · · · + ||u(β r (s))||) + k6

+ k3 (||u(s)|| + ||u(γ1(s)) || + · · · + ||u(γ k (s)) ||) + k4]ds

≤(α)1

t



0

(t − s) α−1 [k

5{δ + ||u(β1(s))||(β

1(s) / c1 ) +· · · + ||u(β r (s))||(β

r (s) / c r)} + k 6

+ k3{δ + ||u(γ1(s))||(γ

1(s) / b1 ) +· · · + ||u(γ k (s))||(γ

k (s) / b k)} + k 4] ds

≤ σ δ(k3+ k5 ) +σ (k4+ k6 ) + k5

c1(α)

β 1(t)

β1 (0)

(t − β−1

1 (τ)) α−1 ||u(τ)||dτ + · · · + k5

c r (α)

β r (t)



β r(0)

(t − β−1

r (τ)) α−1 ||u(τ)||dτ

+ k3

b1(α)

γ 1(t)

γ1 (0)

(t − γ−1

1 (η)) α−1 ||u(η)||dη + · · · + k3

b k (α)

γ k (t)



γ k(0)

(t − γ−1

k (η)) α−1 ||u(η)||dη.

Hence the required result

Theorem 3.3 Suppose that the operator -A(t, u) generates the resolvent operator Ru (t, s) with ||Ru(t, s)||≤ MeN(t-s) If the hypotheses (H1)-(H6) are satisfied, then the

frac-tional integro-differential equation (1.1) with nonlocal condition (1.2) and impulsive

condition (1.3) has a unique mild solution on J for all u0Î X

Proof Consider a mapping P on Sδdefined by

(Pu) (t) = R u (t, 0)u0− R u (t, 0)h(u)

+ 1

(α)

t

 0

(t − s) α−1 R

u (t, s)

⎡

⎣f(s, u(s), u(β(s))) +

s

 0

g(s,η, u(η), u(γ (η)))dη

⎤

⎦ ds

0<ti<t

R u (t, t i )I i (u(t i)).

We shall show that P : Sδ® Sδ For uÎ Sδ, we have

||Pu(t)|| Y ≤ ||R u (t, 0)u0|| + ||R u (t, 0)h(u)||

+

(α)1

t

 0

(t − s) α−1 R

u (t, s)

⎡

⎣f(s, u(s), u(β(s))) +

s

 0

g(s, η, u(η), u(γ (η)))dη

⎤

⎦ ds

0<ti<t

||R u (t, t i)||(||Ii (u(t i))− I i(0)|| + ||Ii(0)||).

Using H1, Lemma 3.2 and H5, we get

||Pu(t) Y || ≤ M0 ||u0|| + k1 +θ +

m



i=1

(l i δ + ||I i(0)||)



From assumption H6, one gets ||(Puμ)(t)||Y≤ δ Thus, P maps Sδinto itself Now for

u, v Î Sδ, we have

||Pu(t) − Pv(t) || ≤ I1+ I2+ I3, where

I1 =||R u (t, 0)u0− R v (t, 0)u0|| + ||R u (t, 0)h(u) − R v (t, 0)h(v)||,

I2 = 1

(α)

t

 0

(t − s) α−1 ||R u (t, s)

⎡

⎣f(s, u(s), u(β(s))) +

s

 0

g(s, η, u(η), u(γ (η)))dη

⎤

⎦

− R v (t, s)[f (s, v(s), v( β(s))) +

s



g(s, η, v(η), v(γ (η)))dη]||ds

I3=

m



i=1

||Ru(t, ti)Ii(u(ti)) − Rv(t, ti)Ii(v(ti))||

Applying Lemma 3.1 and H1, we get

I1≤ ||Ru (t, 0)u0− Rv (t, 0)u0|| + ||Ru(t, 0)h(u) − Rv(t, 0)h(u)||

+||Rv(t, 0)h(u) − Rv(t, 0)h(v)||

≤ {Ka||u0||Y+ k1Ka + M0k2} max

τ∈J ||u(τ) − v(τ)||.

Also, we apply Lemmas 3.1,3.2, H2, H3, H4, and H6, we obtain

I2≤ 1

(α)

t

 0

(t − s) α−1

⎧

⎨

⎩

R u (t, s)

⎡

⎣f(s, u(s), u(β(s))) +

s

 0

g(s, η, u(η), u(γ (η)))dη

⎤

⎦

− R v (t, s)

⎡

⎣f(s, u(s), u(β(s))) +

s

 0

g(s, η, u(η), u(γ (η)))dη

⎤

⎦ +

R v (t, s)

⎡

⎣f(s, u(s), u(β(s))) +

s

 0

g(s, η, u(η), u(γ (η)))dη

⎤

⎦

− R v (t, s)

⎡

⎣f(s, v(s), v(β(s))) +

s

 0

g(s, η, v(η), v(γ (η)))dη

⎤

⎦ 

⎫

⎬

⎭ds

≤ Kaθmax

τ∈J ||u(τ) − v(τ)||

+ M0 1

(α)

t

 0

(t − s) α−1

⎧

⎨

⎩k5

⎡

⎣||u(s) − v(s)|| +r

p=1

||u(β p (s)) − v(β p (s))||(β

p (s)/c p)

⎤

⎦

+ k3

⎡

⎣||u(s) − v(s)|| +k

q=1

||u(γ q (s)) − v(γ q (s))||(γ

q (s)/b q)

⎤

⎦

⎫

⎬

⎭ds

≤ Kaθmax

τ∈J ||u(τ) − v(τ)||

+ M0σ [k5(1 + 1/c1+· · · + 1/c r ) + k3(1 + 1/b1+· · · + 1/b k)]max

τ∈J ||u(τ) − v(τ)||.

I3≤

m



i=1 {||R u (t, t i )I i (u(t i))− R v (t, t i )I i (u(t i))|| + ||Rv (t, t i )I i (u(t i))− R v (t, t i )I i (v(t i))||}

m



i=1 (l i δ + ||I i(0)||) a + M0

m



i=1

l i

 max

τ∈J ||u(τ) − v(τ)||.

It follows from these estimations that

||Pu(t) − Pv(t) || ≤ λ max

τ∈J ||u(τ) − v(τ)||,

where 0 ≤ l < 1 Thus P is a contraction on Sδ From the contraction mapping theo-rem, P has a unique fixed point u Î Sδwhich is the mild solution of (1.1)-(1.3) on J

Theorem 3.4 Assume that (i) Conditions (H1)-(H6) hold, (ii) Y is a reflexive Banach space with norm ||·||, (iii) The functions f and g are uniformly Hölder continuous in tÎ J

Then the problem (1.1)-(1.3) has a unique classical solution on J.

Proof From (i), applying Theorem 3.3, the problem (1.1)-(1.3) has a unique mild solu-tion u Î Sδ Set

ω(t) = f (t, u(t), u(β(t))) +

t

 0

g(t, s, u(s), u(γ (s))) ds.

In order to prove the regularity of the mild solution, we use the further assumptions,

it is easy to conclude that the functionω(t) is also uniformly Hölder continuous in t Î

J Consider the following fractional differential equation

dα v(t)

with the nonlocal condition (1.2) and impulsive condition (1.3)

According to Pazy [30], the late problem has a unique solution v on J intoX given by

v(t) = Ru (t, 0)u0− Ru (t, 0)h(u) + 1

(α)

t

 0

(t − s) α−1 Ru(t, s) ω(s)ds

0<t i <t Ru(t, ti)Ii(u(ti)).

Noting that, each term on the right-hand side belongs to D(A), using the uniqueness

of v(t), we have that u(t) Î D(A) It follows that u is a unique classical solution of

(1.1)-(1.3) on J

Application

Consider the nonlinear integro-partial differential equation of fractional order

∂ α u(x, t)

∂t α +



|q|≤2m

a q (x, t)u(x, t)D q x u(x, t) = F(x, t, u, w1) +

t

 0

G(x, t, s, u(x, s), w2(s))ds,(4:1)

u(x, 0) +

p



k=1

u(x, tk) =



R n

where 0 <a≤ 1, 0 ≤ t1< ··· <tp≤ a, x Î Rn,D q x = D q1

x1 D q n

x n,Dx i = ∂

∂xi, q= (q1, ,qn) is

an n-dimensional multi-index, |q| = q1 + ··· + qn, and wi, i = 1, 2, is given by

|q|≤2m−1 bqi(x, t)D q xu(x, sint) +







|q|≤2m−1

cq i (x, t)D q y u(y, sint)dy.

Let L2(Rn) be the set of all square integrable functions on Rn We denote by Cm(Rn) the set of all continuous real-valued functions defined on Rn which have continuous

partial derivatives of order less than or equal to m ByC m0(R n)we denote the set of all

functions f Î Cm(Rn) with compact supports Let Hm(Rn) be the completion ofC m0(R n)

with respect to the norm

||f ||2

|q|≤m



R n

|D q

xf (x)|2dx.

It is supposed that (i) The operator A(t, u) =−|q|≤2m aq(x, t)u(x, t)D q x is uniformly elliptic on Rn In other words, all the coefficients aq, |q| = 2m, are continuous and bounded on Rnand

there is a positive number c such that

|q|=2m aq(x, t)u(x, t)ξ q ≥ c|ξ| 2m,

for all x Î Rnand allξ ≠ 0, ξ Î Rn,ξ q=ξ q1

1 ξ q

n nand|ξ|2=ξ2

1+ + ξ2

n (ii) All the coefficients aq, |q| = 2m, satisfy a uniform Hölder condition on Rn Under these conditions the operator A with domain of definition D(A) = H2m(Rn) generates

an evolution operator defined on L2(Rn), and it is well known that H2m(Rn) is dense in

X = L2(Rn) and the initial function g(x) is an element in Hilbert space H2m(Rn), see

[14,15,35] Applying Theorem 3.3, this achieves the proof of the existence of mild

solu-tions of the system (4.1)-(4.3) In addition,

(iii) If the coefficients bq, cq, |q|≤ 2m - 1 satisfy a uniform Hölder condition on Rn and the operators F and G satisfy

There are numbers L1, L2≥ 0 and l1, l2 Î (0, 1) such that



|q|≤2m−1



R n

|F(x, t, u, D q

x w1)− F(x, s, u, D q

x w∗1)|2dx ≤ L1(|t − s| λ1+|w1− w∗

1|2dx).

and



|q|≤2m−1



R n

| G(x, t, η, u, D q

x w2)− G(x, s, η, u, D q

xw2)|2dx ≤ L2|t − s| λ2

for all t, sÎ I, (t, h), (s, h) Î Δ, and all x Î Rn

Applying Theorem 3.4, we deduce that (4.1)-(4.3) has a unique strong solution

Competing interests

The author declare that he has no competing interests.

Received: 15 December 2010 Accepted: 24 May 2011 Published: 24 May 2011

References

1 Agrawal OP, Defterli O, Baleanu D: Fractional optimal control problems with several state and control variables.

J Vibr Control 2010, 16(13):1967-1976.

2 Benchohra M, Slimani BA: Existence and uniqueness of solutions to impulsive fractional differential equations.

Electron J Diff Eqns 2009, 10:1-11.

3 Hilfer R: Applications of fractional calculus in physics World Scientific, Singapore 2000.

4 Li F, N ’Guerekata GM: Existence and uniqueness of mild solution for fractional integrodifferential equations Adv

Differ Equ 2010, 10, Article ID 158789.

5 Oldham KB, Spanier J: The fractional calculus Academic Press, New York, London; 1974.

6 Agarwal RP, Lakshmikantham V, Nieto JJ: On the concept of solution for fractional differential equations with

uncertainty Nonlinear Anal 2010, 72:2859-2862.

7 Balachandran K, Trujillo JJ: The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in

Banach spaces Nonlinear Anal 2010, 72:4587-4593.

8 Baleanu D, Trujillo JI: A new method of finding the fractional Euler-Lagrange and Hamilton equations within

9 Kilbas AA, Srivastava HM, Trujillo JJ: Theory and applications of fractional differential Equations In North-Holland

Mathematics Studies Volume 204 Elsevier, Amsterdam; 2006.

10 Samko SG, Kilbas AA, Marichev OI: Fractional integrals and derivatives: Theory and applications, Gordon and Breach,

Yverdon 1993.

11 Baleanu D, Maaraba T, Jarad F: Fractional variational principles with delay J Phys A 2008, 41(31), Article Number

315403.

12 Sadati SJ, Baleanu D, Ranjbar A, Ghaderi R, Abdeljawad T: Mittag-Leffler stability theorem for fractional nonlinear

systems with delay Abstr Appl Anal 2010, 7, Article ID 108651.

13 Byszewski L: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy

problem J Math Anal Appl 1991, 162:494-505.

14 Byszewski L: Theorems about the existence and uniqueness of continuous solutions of nonlocal problem for

nonlinear hyperbolic equation Appl Anal 1991, 40:173-180.

15 Deng K: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions J Math

Annal Appl 1993, 179:630-637.

16 Mophou GM, N ’Guérékata GM: A note on a semilinear fractional differential equation of neutral type with infinite

delay Adv Differ Equ 2010, 8, Article ID 674630.

17 Ahmad B, Sivasundaram S: On four-point nonlocal boundary value problems of non-linear integro-differential

equations of fractional order Appl Math Comput 2010, 217:480-487.

18 Ahmad B, Nieto JJ: Existence of solutions for nonlocal boundary value problems of higher order nonlinear

fractional differential equations Abstr Appl Anal 2009, 9, Article ID 494720.

19 Bainov DD, Simeonov PS: Systems with impulse effect Ellis Horwood Ltd., Chichister; 1989.

20 Lakshmikantham V, Bainov DD, Simeonov PS: Theory of impulsive differential equations World Scientific, Singapore

1989.

21 Samoilenko AM, Perestyuk NA: Impulsive differential equations World Scientific, Singapore 1995.

22 N ’Guerekata GM: A Cauchy problem for some fractional abstract differential equation with non local conditions.

Nonlinear Anal 2009, 70:1873-1876.

23 Li F: Mild solutions for fractional differential equations with nonlocal conditions Adv Differ Equ 2010, 9, Article ID

287861.

24 Benchohra M, Gatsori EP, G ’Orniewicz L, Ntouyas SK: Nondensely defined evolution impulsive differential equations

with nonlocal conditions Fixed Point Theory 2003, 4(2):185-204.

25 Amann H: Quasilinear evolution equations and parabolic systems Trans Am Math Soc 1986, 29:191-227.

26 Dong Q, Li G, Zhang J: Quasilinear nonlocal integrodifferential equations in Banach spaces Electron J Diff Equ 2008,

19:1-8.

27 Sanekata N: Abstract quasilinear equations of evolution in nonreflexive Banach spaces Hiroshima Math J 1989,

19:109-139.

28 Gelfand IM, Shilov GE: Generalized functions Moscow, Nauka; 19591.

29 Podlubny I: Fractional differential equations, Mathematics in Science and Engineering Technical University of Kosice,

Slovak Republic; 1999198.

30 Pazy A: Semigroups of linear operators and applications to partial differential equations Springer, Berlin; 1983.

31 Debbouche A: Fractional evolution integro-differential systems with nonlocal conditions Adv Dyn Syst Appl 2010,

5(1):49-60.

32 Sakthivel R, Choi QH, Anthoni SM: Controllability result for nonlinear evolution integrodifferential systems Appl Math

Lett 2004, 17:1015-1023.

33 Sakthivel R, Anthoni SM, Kim JH: Existence and controllability result for semilinear evolution integrodifferential

systems Math Comput Model 2005, 41:1005-1011.

34 Yan Z: Existence of solutions for nonlocal impulsive partial functional integrodifferential equations via fractional

operators Journal of Computational and Applied Math-ematics 2011, 235(8):2252-2262.

35 Debbouche A, El-Borai MM: Weak almost periodic and optimal mild solutions of fractional evolution equations.

Electron J Diff EqU 2009, 46:1-8.

36 El-Borai MM, Debbouche A: On some fractional integro-differential equations with analytic semigroups Int J C Math

Sci 2009, 4(28):1361-1371.

doi:10.1186/1687-1847-2011-5 Cite this article as: Debbouche: Fractional nonlocal impulsive quasilinear multi-delay integro-differential systems.

Advances in Difference Equations 2011 2011:5.

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9 Kilbas... uniformly Hưlder continuous in tỴ J

Then the problem (1.1)-(1.3) has a unique classical solution... n)we denote the set of all

functions f Ỵ Cm(Rn) with compact