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# Fractional nonlocal impulsive quasilinear multi-delay integro-differential systems

- Amar Debbouche
^{1}Email author

**2011**:5

https://doi.org/10.1186/1687-1847-2011-5

© Debbouche; licensee Springer. 2011

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

## 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

in a Banach space *X*, where 0 < *α* ≤ 1, *t* ∈ [0, *a*], *u*
_{0} ∈ *X*, *i* = 1, 2,..., *m* and 0 < *t*
_{1} < *t*
_{2} < ··· < *t*
_{
m
} < *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*,.) generates an evolution operator in the Banach space *X*. The functions *f* : *J* *X*
^{
r+1}→ *X*, *g* : Λ × X^{
k+1}→ *X*, *h* : *PC*(*J*, *X*) → *X*, *u*(*β*) = (*u*(*β*
_{1}),..., *u*(*β*
_{
r
})), *u*(*γ*) = (*u*(*γ*
_{1}),..., *u*(*γ*
_{
k
})), and *β*
_{
p
}, *γ*
_{
q
}: *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 continuous at *t* ≠ *t*
_{
i
}and left continuous at *t* = *t*
_{
i
}and the right limit
exists for *i* = 1, 2,..., *m*. Clearly *PC*(*J*, *X*) is a Banach space with the norm ||*u*||_{
PC
}= sup_{
t∈J
}||*u*(*t*)||, and let
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 of physics and mathematics [6–8]. There has been a significant development in fractional differential and partial differential equations in recent years, see Kilbas et al. [9, 10], also in fractional nonlinear systems with delay and fractional variational principles with delay, see Baleanu et al. [11, 12].

*u*(0) +

*h*(

*u*) =

*u*

_{0}to describe for instance, the diffusion phenomenon of a small amount of gas in a transparent tube can give better result than using the usual local Cauchy problem

*u*(0) =

*u*

_{0}. Let us observe also that since Deng's papers, the function

*h*is considered

where *c*
_{
k
} , *k* = 1, 2,..., *p* are given constants and 0 ≤ *t*
_{1} < ··· < *t*
_{
p
} ≤ *a*. However, among the previous research on nonlocal cauchy problems, few are concerned with mild solutions of fractional semilinear differential equations, 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 monographs 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 integrodifferential 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 solutions to nonlocal and impulsive integrodifferential equations of quasilinear type with delays of arbitrary orders. Section "Preliminaries" is devoted to a review of some essential 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 function

*f*∈

*C*([0, ∞)) is defined as

*α*with the lower limit zero for a function

*f*∈

*C*([0, ∞)) can be written as

*Remark 2.1*

- (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 Definition 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

*R*

_{ u }(

*t*,

*s*) ∈

*B*(

*X*), 0 ≤

*s*≤

*t*≤

*a*, the space of bounded linear operators on

*X*, having the following properties:

- (i)
*R*_{ u }(*t*,*s*) is strongly continuous in*s*and*t*,*R*_{ u }(*s*,*s*) =*I*, 0 ≤*s*≤*a*, ||*R*_{ u }(*t*,*s*)|| ≤*Me*^{ N(t, s)}for some constants*M*and*N*. - (ii)
*R*_{ u }(*t*,*s*)*E*⊂*E*,*R*_{ u }(*t*,*s*) is strongly continuous in*s*and*t*on*E*. - (iii)
- (iv)

with
and
are strongly continuous on 0 ≤ *s* ≤ *t* ≤ *a*. Here *R*
_{
u
} (*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. Let Ω 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

*u*∈

*PC*(

*J*:

*X*) with values in Ω satisfying the integral equation

for all u_{0} ∈ *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*\{*t*_{1},*t*_{2},...,*t*_{ m }} and*u*(*t*) ∈*D*(*A*), - (2)
exists and continuous on

*J*_{0}, 0 <*α*< 1, - (3)
*u*satisfies (1.1) on*J*_{0}, the nonlocal condition (1.2) and the impulsive condition (1.3), where*J*_{0}= (0,*a*]\{*t*_{1},*t*_{2},...,*t*_{ m }}. We assume the following conditions

_{1})

*h*:

*PC*(

*J*: Ω) →

*Y*is Lipschitz continuous in

*X*and bounded in

*Y*, i.e., there exist constants

*k*

_{1}> 0 and

*k*

_{2}> 0 such that

For the conditions (H _{2} ) and (H _{3} ) let Z be taken as both × and Y.

_{2})

*g*: Λ ×

*Z*

^{ k+1}→

*Z*is continuous and there exist constants

*k*

_{3}> 0 and

*k*

_{4}> 0 such that

_{3})

*f*:

*J*× Z

^{ r+1}→

*Z*is continuous and there exist constants

*k*

_{5}> 0 and

*k*

_{6}> 0 such that

(H_{4}) *β*
_{
p
} , *γ*
_{
q
} : *J* → *J* are bijective absolutely continuous and there exist constants *c*
_{
p
} > 0 and *b*
_{
q
} > 0 such that
and
, respectively, for *t* ∈ *J*, *p* = 1,..., *r* and *q* = 1,..., *k*.

_{5})

*I*

_{ i }:

*X*→

*X*are continuous and there exist constants

*l*

_{ i }> 0,

*i*= 1, 2,...,

*m*such that

Let us take *M*
_{0} = max ||*R*
_{
u
}(*t*, *s*)||_{
B(Z)}, 0 ≤ *s* ≤ *t* ≤ *a*, *u* ∈ Ω.

_{6}) There exist positive constants

*δ*

_{1},

*δ*

_{2},

*δ*

_{3}∈ (0,

*δ*/3] and λ

_{1}, λ

_{2}, λ

_{3}∈ [0, ) such that

where *ρ* = *σ* [*k*
_{5}(1/*c*
_{1} + ··· +1/*c*
_{
r
} )+ *k*
_{3}(1/*b*
_{1} + ··· +1/*b*
_{
k
} )], *θ* = *σδ* (*k*
_{3} + *k*
_{5})+ *ρδ* + *σ* (*k*
_{4} + *k*
_{6}),
and
.

## Main results

*Lemma 3.1*Let

*R*

_{ u }(

*t*,

*s*) the resolvent operators for the fractional problem (1.1)-(1.3). There exists a constant

*K*> 0 such that

for every *u*, *v* ∈ *PC*(*J* : *X*) with values in Ω and every *ω* ∈ *Y* , see [30, lemma 4.4, p. 202].

Let *S*
_{
δ
} = {*u* : *u* ∈ *PC*(*J* : *X*), *u*(0) + *h*(*u*) = *u*
_{0}, Δ*u*(*t*
_{i}) = *I*
_{
i
} (*u*(*t*
_{
i
} )), ||*u*|| ≤ *δ*}, for *t* ∈ *J*, *δ* > 0, *u*
_{0} ∈ *X* and *i* = 1,..., *m*.

Hence the required result.

**Theorem 3.3** Suppose that the operator -*A*(*t*, *u*) generates the resolvent operator *R*
_{
u
}(*t*, *s*) with ||*R*
_{
u
}(*t*, *s*)||≤ *Me*
^{
N(t-s)}. If the hypotheses (H_{1})-(H_{6}) are satisfied, then the fractional integro-differential equation (1.1) with nonlocal condition (1.2) and impulsive condition (1.3) has a unique mild solution on *J* for all *u*
_{0} ∈ *X*.

_{6}, one gets ||(

*Pu*

_{ μ })(

*t*)||

_{ Y }≤

*δ*. Thus,

*P*maps

*S*

_{ δ }into itself. Now for

*u*,

*v*∈

*S*

_{ δ }, we have

where 0 ≤ *λ* < 1. Thus *P* is a contraction on *S*
_{
δ
} . From the contraction mapping theorem, *P* has a unique fixed point *u* ∈ *S*
_{
δ
} which is the mild solution of (1.1)-(1.3) on *J*.

**Theorem 3.4**Assume that

- (i)
Conditions (H

_{1})-(H_{6}) hold, - (ii)
Y is a reflexive Banach space with norm ||·||,

- (iii)
The functions

*f*and*g*are uniformly Hölder continuous in*t*∈*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 solution

*u*∈

*S*

_{ δ }.. Set

*ω*(

*t*) is also uniformly Hölder continuous in

*t*∈

*J*. Consider the following fractional differential equation

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

Noting that, each term on the right-hand side belongs to *D*(*A*), using the uniqueness of v(t), we have that *u*(*t*) ∈ *D*(*A*). It follows that *u* is a unique classical solution of (1.1)-(1.3) on *J*.

## Application

*α*≤ 1, 0 ≤

*t*

_{1}< ··· <

*t*

_{ p }≤

*a*,

*x*∈

*R*

^{ n }, , ,

*q*= (

*q*

_{1},...,

*q*

_{ n }) is an

*n*-dimensional multi-index, |

*q*| =

*q*

_{1}+ ··· +

*q*

_{ n }, and

*w*

_{ i },

*i*= 1, 2, is given by

_{2}(

*R*

^{ n }) be the set of all square integrable functions on

*R*

^{ n }. We denote by

*C*

^{ m }(

*R*

^{ n }) the set of all continuous real-valued functions defined on

*R*

^{ n }which have continuous partial derivatives of order less than or equal to

*m*. By we denote the set of all functions

*f*∈

*C*

^{ m }(

*R*

^{ n }) with compact supports. Let

*H*

^{ m }(

*R*

^{ n }) be the completion of with respect to the norm

- (i)The operator is uniformly elliptic on
*R*^{ n }. In other words, all the coefficients*a*_{ q }, |*q*| = 2*m*, are continuous and bounded on*R*^{ n }and there is a positive number*c*such thatfor all

*x**∈**R*^{ n }and all*ξ*≠ 0, ξ ∈*R*^{ n }, and . - (ii)
All the coefficients

*a*_{ q }, |*q*| = 2*m*, satisfy a uniform Hölder condition on*R*^{ n }. Under these conditions the operator*A*with domain of definition*D*(*A*) =*H*^{2m }(*R*^{ n }) generates an evolution operator defined on*L*_{2}(*R*^{ n }), and it is well known that*H*^{2m }(*R*^{ n }) is dense in*X*=*L*_{2}(*R*^{ n }) and the initial function*g*(*x*) is an element in Hilbert space*H*^{2m }(*R*^{ n }), see [14, 15, 35]. Applying Theorem 3.3, this achieves the proof of the existence of mild solutions of the system (4.1)-(4.3). In addition, - (iii)
If the coefficients

*b*_{ q },*c*_{ q }, |*q*| ≤ 2*m*- 1 satisfy a uniform Hölder condition on*R*^{ n }and the operators*F*and*G*satisfy

for all *t*, *s* ∈ *I*, (*t*, *η*), (*s*, *η*) *∈* Δ, and all *x* ∈ *R*
^{
n
} . Applying Theorem 3.4, we deduce that (4.1)-(4.3) has a unique strong solution.

## Declarations

## Authors’ Affiliations

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