EXISTENCE OF MILD SOLUTION FOR NONLOCAL IMPULSIVE FRACTIONAL SEMILINEAR DIFFERENTIAL INCLUSION IN BANACH SPACE

In this paper, we deal with the existence of PC-mild solutions of nonlocal impulsive differential inclusions in Banach space when the values of the orient field is convex (P). By using methods and results of semilinear differential inclusions, and techniques of fixed point theorems, we establish sufficient conditions that guarantee the existence of PC-mild solutions of (P). Our results develop and extend various results proved recently.


Introduction
Fractional differential equations and fractional differential inclusions have been an object of interest since two decades due to their wide applications in various fields, such as physics, biology, mechanics and engineering, medical field, industry and technology.Also fractional differential equations are used as an excellent tool for the description of hereditary properties of various materials and processes.For instance, we refer to [18], [9], [11], [15] and the references therein.
In particular, impulsive differential equations and impulsive differential inclusions have gained much more attention because they serve as an appropriate model to describe processes which can not be described by classical differential equations, such as processes which at certain moments change their state rapidly.For some of these applications, one can see, [5], [1] and the references therein.During the last ten years, impulsive differential inclusions with different conditions have been intensely investigated by numerous researchers, we refer readers to [4], [13], [20], [17], [3], [8], [19], [10], [12], [22], [7], [24].Moreover, a strong motivation for studying the nonlocal Cauchy problems comes from physical problems.For instance, using nonlocal Cauchy problems in determining unknown parameters in some inverse heat condition problems, see [4], [13].
In this study, we are concerned with the existence of mild solution for the following impulsive nonlocal Cauchy problem of fractional order α ∈ (0, 1) driven by a semilinear differential inclusion in a real separable Banach space E of the form c D α x(t) ∈ Ax(t) + F(t, x(t)), t ∈ J = [0, b], t = t i , i = 1, ..., m, x(t + i ) = x(t i ) + I i (x(t i )), i = 1, ..., m, x(0) = g(x), where c D α is the Caputo derivative of order α, A : characterize the jump of the solutions at impulse points, g : PC(J, E) → E, is a nonlinear function related to the nonlocal condition at the origin and x(t + i ), x(t − i ) are the right and left limits of x at the point t i respectively and PC(J, E) will be defined later.
The concept of mild solution was firstly introduced by Mophou [19], inspired by Jaradat et.al.[17].Since then, the main goal for many mathematicians has been to establish sufficient conditions regarding the existence of mild solution for differential equations or inclusions problems.In [3], R. Al-Omair and A. Ibrahim studied (P) without impulsive, when α = 1.Also Fan [12] considered a nonlocal Cauchy problem in the presence of impulses, and controlled by autonomous semilinear differential equation.While Wang et.al.[23] introduced a new concept of PC-mild solutions for (P) and obtained existence and uniqueness results concerning the PC-mild solutions for (P) when F is a Lipschitz single-valued function or continuous and maps bounded sets into bounded sets and {T (t)} t>0 is compact.Cardinali et.al.[7] proved the existence of mild solutions to the problem (P) when α = 1 and the multivalued function F satisfies the lower Scorza-Dragoni property and {A(t)} t≥0 is a family of linear operator, generating a strongly continuous evolution operators.Recently, A. G. Ibrahim and N. Almoulhim [16] discussed (p) without impulsive.Motivated by the above works, by using properties of multifunctions, some methods and results semilinear differential inclusions, and fixed point theorems, we develop the results shown in [16] as well as we extend the results in [23] to the case when (P) is taken with impulsive and nonlocal conditions.This paper is organized as: Section 2 recalls some basic foundations related to multifunctions and fractional calculus to be used later.In section 3, the existence of mild solution for (P) is proved.We adopt the definition of mild solution introduced by Wang et.al.[23].We used the properties of multifunctions, methods and results regarding semilinear differential inclusions, and fixed point techniques to obtain the results.

Preliminaries
During this section, we state some previous known results so that we can use them later throw out this paper.Let C(J, E) be the Banach space of all E−valued continuous functions from J into E with the uniform norm x = sup{ x(t) ,t ∈ J}, L 1 (J, E) the space of E−valued Bochner integrable functions on J with the norm We also introduce the set of functions It is easy to check that PC(J, E) is a Banach space endowed with the norm x PC(J,E) = max{ x(t) : t ∈ J} For any subset B ⊆ PC(J, E) and for any i = 0, 1, ..., m, let space X is said to be (1) uniformly continuous if lim t→0 + T (t) − I = 0, where I is the identity operator; (2) strongly continuous if lim t→0 + T (t)x = x, for every x ∈ X.
A strongly continuous semigroup of bounded linear operators on X will be called a semigroup of

be a semigroup of bounded linear operators an a
Banach space X.The linear operator A defined by is called the infinitesimal generator of the semigroup T (t), D(A) is the domain of A.
Definition 2.3.([2], [14]) Let X and Y be two topological spaces.A multifunction F : (iii) completely continuous if F(B) is relatively compact for every bounded subset B of X.
Note that if the multifunction F is completely continuous with non empty compact values, then Let X,Y be (not necessarily separable) Banach spaces, and let F : J × X → P k (Y ) be such that (i) for every x ∈ X the multifunction F(•, x) has a strongly measurable selection; (ii) for a.e.t ∈ J the multifunction F(t, •) is upper semicontinuous.
Then for every strongly measurable function z : J → X there exists a strongly measurable func- Remark 2.1.[14] For single-valued or compact valued multifunction acting on a separable Banach space the notions measurability and strongly measurable coincide.So, if X,Y be separable Banach spaces, we can replace strongly measurable with measurable in the previous lemma.E) is said to be semi-compact if: (i) It is integrably bounded i.e. there is q ∈ L 1 (J, R + ) such that Definition 2.5.According to the Riemann-Liouville approach, the fractional integral of order α ∈ (0, 1) of a function f ∈ L 1 (J, E) is defined by provided the right side is defined on J, where Γ is the Euler gamma function defined by Definition 2.6.The Caputo derivative of order α ∈ (0, 1) of continuously differentiable function (1) .
Note that the integral appear in the two previous definitions are taken in Bochner'sense and For more informations about the fractional calculus we refer to ( [11], [18]).
Definition 2.7.A function x ∈ PC(J, E) is an impulsive mild solution for (p) if where where θ ∈ (0, ∞) and ξ is a probability density function defined on (0, ∞), that is are associated with the number α, there are no analogue of the semigroup property, i.e.
In the following we recall the properties of Lemma 2.4.[25] (i) For any fixed t ≥ 0, K 1 (t), K 2 (t) are linear bounded operators.

Main results
Now, after the preliminaries are laid, we encounter our main problem (P).The following theorem proves the existence of mild solution for (P).
Theorem 3.1.Let F : J × E → P ck (E) be a multifunction.Assume the following conditions: (C 1 ) A is the infinitesimal generator of a C 0 −semigroup T (t) : t ≥ 0 and T (t),t > 0 is compact.
(C 4 ) g : PC(J, E) → E is continuous, compact and there exist two positive numbers a, d such that g(x) ≤ a x + d, ∀x ∈ PC(J, E).
(C 5 ) For every i = 1, 2, ..., m, I i is continuous, compact and there exists a positive constant h i such that Then the problem (P) has a mild solution provided that there is r > 0 such that where M > 0 such that sup t∈J T (t) ≤ M, and , Proof.From C 2 , Lemma 2.1 and Remark 2.1 the set is nonempty, for any x ∈ PC(J, E).Thus, we can consider the operator R : PC(J, E) → 2 PC(J,E) , which defined by y ∈ R(x) if and only if Where f ∈ S 1 F(•,x(•)) .It can be easily to see that any fixed point for R is a mild solution for the problem (P).So, our goal is to prove that R satisfies the conditions of Theorem 2.1 in the preliminaries.The proof will be given in six steps.
Step 1.We show that the values of R are convex subset in PC(J, E).
Step 2. We prove that R(x) is closed for every x ∈ PC(J, E).To prove that the values of R are closed, let x ∈ PC(J, E) and (z n ) n≥1 be a sequence in R(x) such that z n → z in PC(J, E).Then, we need to prove that z ∈ R(x).According to the definition of R there is a sequence We can assume that f n converges weakly to a function f ∈ L 1 (J, E).From Mazur's lemma, there is a sequence (g n ), n ≥ 1 such that {g n (t) : n ≥ 1} ⊆ Conv{ f n (t) : n ≥ 1};t ∈ J and g n converges strongly to f .Since, the values of F are convex, g n ∈ S 1 F(t,x(t)) and hence, by the compactness of F(t, x(t)), f ∈ S 1 F(t,x(t)) .Moreover, for every t, s ∈ J and for every n ≥ 1, Therefore, based on the Lebesgue dominated convergence theorem, taking n → ∞ on both sides of (3.3) we get for every i = 0, 1, .., Which means that z ∈ R(x).
We claim that R(B r ) ⊆ B r .To prove that, let x ∈ B r , y ∈ R(x) and let t ∈ J 0 we have Similarly, by using Lemma 2.4, C 3 ,C 4 ,C 5 and (3.1) we have for every t ∈ J i , i = 1, 2, ..., m, Therefore, we conclude that R(B r ) ⊆ B r .
step 4. R sends bounded sets into equicontinuous sets of PC(E, J).Put B = R(B r ).We claim that B is equicontinuous, let x ∈ B r and y ∈ R(B r ).According to (3.2) we have where f ∈ S 1 F(•,x(•)) .To show that B |J is equicontinuous it suffices to verify that B |J i is equicontinuous for every i = 0, 1, ..., m, where We study the following cases: case 1.If i = 0. Let t,t + λ be two points in J 0 = J 0 , then where We only need to check R i → 0 as λ → 0 for every i = 1, 2, 3, 4. Since K 1 (t),t > 0, is uniformly continuous on J. So, R 1 → 0 as λ → 0 independently of x ∈ B r .For R 2 , by the Holder inequality we have lim By applying Lemma 2.3 and taking into account 1 − q ∈ (0, 1) we get lim For R 4 , by using the Lebesgue dominated convergence theorem and the condition C 1 , we obtain lim Since K 2 (t),t > 0, is uniformly continuous, so, we conclude that R 4 → 0 as λ → 0, independently of x ∈ B r .
Case 2. If i ∈ {1, 2, ..., m}, let t,t + λ be two points in J i .Recalling (3.2) we have Arguing as in the first case we get lim ., m let λ > 0 be such that t i + λ ∈ J i and δ > 0 such that t i < t i + λ ≤ t i+1, then we have Based on (3.2) we get Arguing as in the first case we can see that Now we can say that, B |J i is equicontinuous for every i = 0, 1, 2, ..., m.Thus, B is equicontinuous on J.
Step 5. We prove that for any t ∈ [0, b] the set ∆(t) = {y(t) : Clearly, by Lemma 2.4, the right hand side of the previous inequality tend to zero as ε, l → 0. Therefore, we can say that there exists a relatively compact set which can be arbitrary close to the set ∆ 2 (t),t ∈ (0, b].Hence, this set is relatively compact in E. So, ∆(t),t ∈ J is relatively compact.
From step 4 and step 5 we conclude that B is relatively compact.
Step 6. R is closed i.e. its graph is closed.Let x n ∈ B r , x n → x in PC(J, E) and y n ∈ R(x n )∀n ≥ 1 with y n → y in B. We will prove that y ∈ R(x).By the definition of R, for any n ≥ 1 there exists + t 0 (t − s) α−1 K 2 (t − s) f n (s)ds,t ∈ J i , i = 1, ..., m. (3.4) We will prove that the sequence ( f n ), n ≥ 1 is semicompact.The assumption C 3 implies f n (t) ≤ ϕ(t) x n ≤ rϕ(t), a.e.t ∈ J.
nonempty and bounded }, P cl (E) = {B ⊆ E : Bis nonempty and closed }, P k (E) = {B ⊆ E : B is nonempty and compact }, P cl,cv (E) = {B ⊆ E : B is nonempty, closed and convex }, P ck (E) = {B ⊆ E : B is nonempty, convex and compact }, conv(B), conv(B) be the convex hull and convex closed hull in E of subset B. Considering a division of [0, b], i.e a finite set {t 0

Theorem 2 . 1 .
[6] Let B be a nonempty subset of a Banach space E, which is bounded, closed and convex.Suppose G : B → 2 E is u.s.c. with closed and convex value such that G(B) ⊆ B and G(B) is compact.Then G has a fixed point.