IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES

This paper is devoted to study the existence of solutions for a class of initial value problems for impulsive fractional differential equations involving the Caputo fractional derivative in a Banach space. The arguments are based upon Mönch’s fixed point theorem and the technique of measures of noncompactness.

On the other hand, the theory of impulsive differential equations has undergone rapid development over the years and played a very important role in modern applied mathematical models of real processes rising in phenomena studied in physics, population dynamics, chemical technology, biotechnology and economics; see for instance the monographs by Bainov and Simeonov [10], Benchohra et al [14], Lakshmikantham et al [34], and Samoilenko and Perestyuk [43] and references therein.Moreover, impulsive differential equations present a natural framework for mathematical modeling of several real-world problems.However, the theory for fractional differential equations in Banach spaces has yet been sufficiently developed.Recently, Benchohra et al [16] applied the measure of noncompactness to a class of Caputo fractional differential equations of order r ∈ (0, 1] in a Banach space.Lakshmikantham and Devi [33] discussed the uniqueness and continuous dependence of the solutions of a class of fractional differential equations using the Riemann-Liouville derivative of order r ∈ (0, 1] on Banach spaces.Let E be a Banach space with norm • .In this paper, we study the following initial value problem (IVP for short), for fractional order differential equations where c D r is the Caputo fractional derivative, f : ) represent the right and left limits of y(t) at t = t k , k = 1, . . ., m.To our knowledge no paper has considered impulsive fractional differential equations in abstract spaces.This paper fills the gap in the literature.To investigate the existence of solutions of the problem above, we use Mönch's fixed point theorem combined with the technique of measures of noncompactness, which is an important method for seeking solutions of differential equations.See Akhmerov et al. [4], Alvàrez [5], Banas et al. [6,7,8,9], El-Sayed and Rzepka [24], Guo et al. [30], Mönch [40], Mönch and Von Harten [41] and Szufla [44].

Preliminaries
In what follows, we first state the following definitions, lemmas and some notation.Denote by C(J, E) the Banach space of continuous functions J → E, with the usual supremum norm y ∞ = sup{ y(t) , t ∈ J}.
Let L 1 (J, E) be the Banach space of measurable functions y : J → E which are Bochner integrable, equipped with the norm Moreover, for a given set V of functions v : J → E let us denote by Now let us recall some fundamental facts of the notion of Kuratowski measure of noncompactness.

Definition 2.1 ([7]
) Let E be a Banach space and Ω E the bounded subsets of E. The Kuratowski measure of noncompactness is the map α : Properties: The Kuratowski measure of noncompactness satisfies some properties (for more details see [7]) For completeness we recall the definition of Caputo derivative of fractional order.Let ϕ r (t) = t r−1 Γ(r) for t > 0 and ϕ r (t) = 0 for t ≤ 0, and ϕ r → δ(t) as r → 0, where δ is the delta function.

Definition 2.2 ([32]) The fractional order integral of the function
where Γ is the gamma function.When a = 0, we write I r h(t) = [h * ϕ r ](t).

Definition 2.3 ([32] For a function h given on the interval [a, b],
the Caputo fractionalorder derivative of h, of order r > 0 is defined by Here n = [r] + 1 and [r] denotes the integer part of r.
For example for 0 < r ≤ 1 and h : [a, b] → E an absolutely continuous function, then the fractional derivative of order r of h exists.
From the definition of Caputo derivative, we can obtain the following auxiliary results [45].
For our purpose we will only need the following fixed point theorem, and the important Lemma.
Theorem 2.1 ( [3,40]) Let D be a bounded, closed and convex subset of a Banach space such that 0 ∈ D, and let N be a continuous mapping of holds for every subset V of D, then N has a fixed point.

Lemma 2.3 ([44]
) Let D be a bounded, closed and convex subset of the Banach space C(J, E), G a continuous function on J × J and f a function from J × E → E which satisfies the Carathéodory conditions and assume there exists p ∈ L 1 (J, R + ) such that for each t ∈ J and each bounded set B ⊂ E we have

) if and only if y is a solution of the fractional IVP
Proof: Assume y satisfies ( 5)- (7).
Conversely, assume that y satisfies the impulsive fractional integral equation ( 4).If t ∈ [0, t 1 ], then y(0) = y 0 and, using the fact that c D r is the left inverse of I r , we get . ., m, and using the fact that c D r C = 0, where C is a constant, we get c D r y(t) = h(t), for each t ∈ [t k , t k+1 ).
Also, we can easily show that Let us list some conditions on the functions involved in the IVP (1)- (3).Assume that (H1) f : J × E → E satisfies the Carathéodory conditions.(H2) There exists p ∈ L 1 (J, R + ) ∩ C(J, R + ), such that, f (t, y) ≤ p(t) y , for t ∈ J and each y ∈ E. (H5) For each t ∈ J and each bounded set B ⊂ E we have then the IVP ( 1)-( 3) has at least one solution.
Proof.We shall reduce the existence of solutions of ( 1)-( 3) to a fixed point problem.To this end we consider the operator N : P C(J, E) −→ P C(J, E) defined by Clearly, the fixed points of the operator N are solution of the problem ( 1)-( 3).Let and consider the set D r 0 = {y ∈ P C(J, E) : y ∞ ≤ r 0 }.
Clearly, the subset D r 0 is closed, bounded and convex.We shall show that N satisfies the assumptions of Theorem 2.1.The proof will be given in three steps.
Step 1: N is continuous.
Let {y n } be a sequence such that y n → y in P C(J, E).Then for each t ∈ J Since I k is continuous and f is of Carathéodory type, then by the Lebesgue dominated convergence theorem we have EJQTDE Spec.Ed.I, 2009 No. 8 Step 2: N maps D r 0 into itself.
For each y ∈ D r 0 , by (H2) and ( 8) we have for each t ∈ J N(y Step 3: N(D 0 ) is bounded and equicontinuous.
By Step2, it is obvious that N(D r 0 ) ⊂ P C(J, E) is bounded.
By (8) it follows that v ∞ = 0; that is, v(t) = 0 for each t ∈ J, and then V (t) is relatively compact in P C(J, E).In view of the Ascoli-Arzelà theorem, V is relatively compact in D r 0 .Applying now Theorem 2.1 we conclude that N has a fixed point which is a solution of the problem (1)-(3).

Nonlocal impulsive differential equations
This section is concerned with a generalization of the results presented in the previous section to nonlocal impulsive fractional differential equations.More precisely we shall present some existence and uniqueness results for the following nonlocal problem where f, I k , k = 1, . . ., m are as in Section 3 and g : P C(J, E) → E is a continuous function.Nonlocal conditions were initiated by Byszewski [19] when he proved the As remarked by Byszewski [17,18], the nonlocal condition can be more useful than the standard initial condition to describe some physical phenomena.For example, in [21], the author used where c i , i = 1, . . ., p, are given constants and 0 < τ 1 < ... < τ p ≤ T , to describe the diffusion phenomenon of a small amount of gas in a transparent tube.In this case, (13) allows the additional measurements at t i , i = 1, . . ., p.
Let us introduce the following set of conditions.Clearly, the fixed points of the operator F are solution of the problem (10)- (12).We can easily show the conditions of Theorem 2.1 are satisfied by F .

An Example
In this section we give an example to illustrate the usefulness of our main results.Let us consider the following impulsive fractional initial value problem, Set f (t, x) = 1 10 + e t x, (t, x) ∈ J × E, and I k (x) = 1 5 x, x ∈ E.

8 3Lemma 3 . 1
s))ds.EJQTDE Spec.Ed.I, 2009 No. Existence of Solutions First of all, we define what we mean by a solution of the IVP (1)-(3).Definition 3.1 A function y ∈ P C(J, E) is said to be a solution of (1)-(3) if y satisfies the equation c D r y(t) = f (t, y(t)) on J ′ , and conditions∆y| t=t k = I k (y(t − k )), k = 1, .. ., m, and y(0) = y 0 .Let 0 < r ≤ 1 and let h : J → E be continuous.A function y is a solution of the fractional integral equation