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Fractional nonlocal impulsive quasilinear multidelay integrodifferential systems
Advances in Difference Equations volume 2011, Article number: 5 (2011)
Abstract
In this article, sufficient conditions for the existence result of quasilinear multidelay integrodifferential 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 multidelay integropartial differential system of fractional order is given.
AMS Subject Classifications. 34K05, 34G20, 26A33, 35A05.
Introduction
Many fractional models can be represented by the following system
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].
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 condition 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 GelfandShilov [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
provided the right side is pointwise defined on [0, ∞), where Γ is the gamma function. RiemannLiouville derivative of order α with the lower limit zero for a function f ∈ C([0, ∞)) can be written as
The Caputo derivative of order for a function f ∈ C([0, ∞)) can be written as
Remark 2.1

(1)
If f ∈ C ^{1}([0, ∞)), then

(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)
For x ∈ X, R _{ u } (t, s)x is continuously differentiable in s ∈ [0, a] and

(iv)
For x ∈ X and s ∈ [0, a], R _{ u } (t, s)x is continuously differentiable in t ∈ [s, a] and
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
(H_{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.
(H_{2}) g : Λ × Z ^{k+1}→ Z is continuous and there exist constants k _{3} > 0 and k _{4} > 0 such that
(H_{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.
(H_{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.
(H_{6}) There exist positive constants δ _{1}, δ _{2}, δ _{3} ∈ (0, δ /3] and λ_{1}, λ_{2}, λ_{3} ∈ [0, ) such that
and
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}, Δu(t _{i}) = I _{ i } (u(t _{ i } )), u ≤ δ}, for t ∈ J, δ > 0, u _{0} ∈ X and i = 1,..., m.
Lemma 3.2
where
Proof We have
Using H_{2}, H_{3}, and H_{4}, we get
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(ts)}. If the hypotheses (H_{1})(H_{6}) are satisfied, then the fractional integrodifferential 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.
Proof Consider a mapping P on S _{ δ } defined by
We shall show that P : S _{ δ } → S _{ δ } . For u ∈ S _{ δ } , we have
Using H_{1}, Lemma 3.2 and H_{5}, we get
From assumption H_{6}, one gets (Pu _{ μ } )(t) _{ Y } ≤ δ. Thus, P maps S _{ δ } into itself. Now for u, v ∈ S _{ δ } , we have
where
and
Applying Lemma 3.1 and H_{1}, we get
Also, we apply Lemmas 3.1,3.2, H_{2}, H_{3}, H_{4}, and H_{6}, we obtain
Again, Lemma 3.1, H_{5} and H_{6}, we have
It follows from these estimations that
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
In order to prove the regularity of the mild solution, we use the further assumptions, it is easy to conclude that the function 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).
According to Pazy [30], the late problem has a unique solution v on J intoX given by
Noting that, each term on the righthand 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
Consider the nonlinear integropartial differential equation of fractional order
where 0 < α ≤ 1, 0 ≤ t _{1} < ··· < t _{ p } ≤ a, x ∈ R ^{n} , , , q= (q _{1},...,q _{ n } ) is an ndimensional multiindex, q = q _{1} + ··· + q _{ n } , and w _{ i } , i = 1, 2, is given by
Let L_{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 realvalued 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
It is supposed that

(i)
The operator is uniformly elliptic on R ^{n} . In other words, all the coefficients a _{ q } , q = 2m, are continuous and bounded on R ^{n} and there is a positive number c such that
for all x ∈ R ^{n} and all ξ ≠ 0, ξ ∈ R ^{n} , and .

(ii)
All the coefficients a _{ q } , q = 2m, 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 ≤ 2m  1 satisfy a uniform Hölder condition on R ^{n} and the operators F and G satisfy
There are numbers L _{1}, L _{2} ≥ 0 and λ _{1}, λ _{2} ∈ (0, 1) such that
and
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.
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Keywords
 Fractional integrodifferential systems
 resolvent operators
 nonlocal and impulsive conditions
 fixed point theorem