Mild Solutions for Nonlocal Impulsive Fractional Semilinear Differential Inclusions with Delay in Banach Spaces

In this paper, we give various existence results concerning the existence of mild solutions for nonlocal impulsive differential inclusions with delay and of fractional order in Caputo sense in Banach space. We consider the case when the values of the orient field are convex as well as nonconvex. Our obtained results improve and generalize many results proved in recent papers.


Introduction
During the past two decades, fractional differential equations and fractional differential inclusions have gained considerable importance due to their applications in various fields, such as physics, mechanics and engineering.For some of these applications, one can see [1][2][3][4] and the references therein.El Sayed et al. [5] initiated the study of fractional multi-valued differential inclusions.Recently, some basic theory for initial-value problems for fractional differential equations and inclusions was discussed by [6][7][8][9][10][11][12][13][14].
The theory of impulsive differential equations and impulsive differential inclusions has been an object interest because of its wide applications in physics, biology, engineering, medical fields, industry and technology.The reason for this applicability arises from the fact that impulsive differential problems are an appropriate model for describing process which at certain moments change their state rapidly and which cannot described using the classical differential problems.For some of these applications we refer to [15][16][17].During the last ten years, impulsive differential inclusions with different conditions have intensely student by many mathematicians.At present, the foundations of the general theory of impulsive differential equations and inclusions are already laid, and many of them are investigated in details in the book of Benchohra et al. [18].
Moreover, a strong motivation for investigating the nonlocal Cauchy problems, which is a generalization for the classical Cauchy problems with initial condition, comes from physical problems.For example, it used to determine the unknown physical parameters in some inverse heat condition problems.The nonlocal condition can be applied in physics with better effect than the classical initial condition   0 0 .x x  For example,   g x may be given by are given constants and For the applications of nonlocal conditions problems we refer to [19,20].In the few past years, several papers have been devoted to study the existence of solutions for differential equations or differential inclusions with nonlocal conditions [21][22][23].For impulsive differential equation or inclusions with nonlocal conditions of order one we refer to [22,23].For impulsive differential equation or inclusions of fractional order we refer to [10,[24][25][26][27] and the references therein.
In this paper we are concerned with the existence of mild solution to the following nonlocal impulsive semilinear differential inclusions with delay and of order where and will define in the next section.  To study the theory of abstract impulsive differential inclusions with fractional order, the first step is how to define the mild solution.Mophou [24] firstly introduced a concept on a mild solution which was inspired by Jaradat et al. [25].However, it does not incorporate the memory effects involved in fractional calculus and impulsive conditions.Wang et al. [10] introduced a new concept of PC-mild solutions for (1.1) without delay and derived existence and uniqueness results concerning the PC-mild solutions for (1.1) when F is a Lipschitz single-valued function or continuous and maps bounded sets into bounded sets and is compact.

 
T t ,t  0 In order to do a comparison between our obtained results in this paper and the known recent results in the same domain, we refer to: Ouahab [9] proved a version of Fillippov's theorem for (1.1) without impulse, without delay and A is an almost sectorial operator, Wang et al. [11] proved existence and controllability results for (1.1) without impulse, without delay and with local condition, Zhang et al. [12] considered the problem (1.1) without impulse, without delay, F is a single-valued function and is strongly equicontinuous C 0 -semigroup, Zhou et al. [13,14] introduced a suitable definition of mild solution for (1.1) based on Laplace transformation and probability density functions for (1.1) when    ,t   0 T t F is single-valued function and without impulse, Cardinali et al. [22] proved the existence of mild solutions to the problem (1.1) without delay, when 1   and the multivalued function F satisfies the lower Scorza-Dragoni property and  0 t  is a family of linear operator, generating a strongly continuous evolution operators, Fan [23] studied a nonlocal Cauchy problem in the presence of impulses, governed by autonomous semilinear differential equation, Dads et al. [26] and Henderson et al. [27] considered the problem (1.1) when Among the previous works, little is concerned with nonlocal fractional differential inclusions with impulses and with delay.

A 
In Section 3 in this paper, motivated by the works mentioned above, we derive various existence results of mild solutions for (1.1) when the values of the orient field are convex as well as non-convex.
The paper is organized as follows: In Section 2, we collect some background material and lemmas to be used later.In Section 3, we prove three existence results for (1.1).We adopt the definition of mild solution introduced by Wang et al. [10].Our basic tools are the properties of multi-functions, methods and results for semilinear differential inclusions, and fixed point techniques.

Preliminaries and Notations
Let   , C J E the space of -valued continuous functions on the space of E-valued Bochner integrable functions on J with the norm is called the infinitesimal generator of the semigroup is the domain of ([29-33]).Let X and Y be two topological spaces.A multifunction is said to be upper semicontinuous (u.s.c.) if Then the multivalued function In particular for every measurable singlevalued function the multivalued function is measurable and for every Caratheodory single-valued function , is said to be decomposable provided for every f g M  and each Lebesgue measurable set Z in , E is said to be semi-compact if: 1) It is integrably bounded, i.e. there is 2)The set . a e t J  We recall one fundamental result which follows from Dunford-Pettis Theorem.
Lemma 2 ([33]).Every semi-compact sequence in L J E For more about multifunctions we refer to [29][30][31][32][33]. Lemma 3 ([11], lemma 2.10).For Definition 6 According to the Riemann -Liouville approach, the fractional integral of order provided the right side is defined on J , where  is the Euler gamma function defined by   Note that the integrals appear in the two previous definitions are taken in Bochner' sense and For more informations about the fractional calculus we refer to [2,4].Definition 8 ([14], Lemma 3.1 and Definition 3.1, see also [11][12][13]).Let A function is said to be a mild solution of the following system: if it satisfies the following integral equation For any fixed are linear bounded operators.
then for any function must be chosen such that the integral appears in (2.2) is well be defined.
. , .K K are associated with the numbrer ,  there are no analogue of the semigroup property, i.e.
In the following we recall the properties of   1 ., K .
For any x   and any the element of   , t J t x    defined by It is easy to check that are are Banach spaces endowed with the norms Here   t x  represents the history of the state time t r  up the present time For any subset and for any , and , , 0,1, , .
Let us recall the concept of mild solutions, introduced by Wang et al. [10], for the impulsive fractional evolution equation: , , , , , .
Indeed, by adding together (2.5) with (2.6), it follows (2.4).Note is continuous, so v  .On the other hand, any solution of (2.4) can be decomposed to (2.5) and (2.6).By Definition 9, a mild solution of (2.5) is given by Now we rewrite system (2.6) in the equivalent integral equation The above equation can be expressed as We apply the Laplace transform for (2.8) to get (see, [25]) Note that the Laplace transform for   Thus we can derive the mild solution of (2.6) as By (2.7) and (2.10), the mild solution of (2.4) is given by By using the above results, we can write the following definition of mild solution of the system (2.3).
Definition 9 ([10], Definition 3.1).By a mild solution of the system (2.3) we mean a function Now we can give the concept of mild solution for our considered problem (1.1).
Definition 10 By a mild solution for (1.1), we mean a function x   which satisfies the following integral equation where and Remark 2 It is easily to see that the solution given by (2.11) satisfies the relation

Remark 3
If for all and if there is no delay then Formula (2.11) will take the form This means that when there is no neither impulse nor delay in the problem (1.1), its solution is equal to the formula given in (2.2).

Theorem 1 ([34]). Let be a nonempty subset of a Banach space
, which is bounded, closed and convex.Suppose The following fixed point theorem for contraction multivalued is proved by Govitz and Nadler [35].

Existence Results for the Problem (1.1)
In this section, we give the main results of mild solutions of (1.1).

Convex Case
In the following Theorem we derive the first existence result concerning the mild solution for the problem (1.1).

Theorem 4 Let
  E be a multifunction.Assume the following conditions: .
Then, for a given continuous function where, such that 0 is nonempty.So, we can define a multifunction as follows: .
Obviously, every fixed point for G is a mild solution for the problem (1.1).So, our goal is to apply Theorem 1.The proof will be given in several steps. Step Then, according to the definition of there is a se- .
Not that, from (3.1), for any for almost 1, This show that the set . is integrably bounded.Moreover, because Therefore, the set is semi-compact and then, by Lemma 2 it is weakly compact in So, without loss of generality we can assume that n and n g converges strongly to f .Since, the values of F are convex, and hence, by the compactness of .
We consider the following cases: In view of Holder's inequality we get

4(v)), and hence, 1  
K h is uniformly continuous on J (see [28]).Therefore, the last inequality tends to zero as 0,   independently of .
x Case 2. Let , where We only need to check as 0 By the uniform continuity of , we conlude that indepen Arguing as in the first case we get According the definition of we get Arguing as in the first case we can see th From (3.9)  (3.15) we conclude that | i J B is equicontinuous for every Step 4. Our aim in this step is to show that for any is relatively compact in Let us introduce the following maps: , where is a bounded subset in and  g is compact, the set relatively compact in .Also, since the functions 2, ,  I k m are compact, the set is relatively compact in It remains to show that the set . Moreover, by using (H 3 ) and (H 4 ) we get Obviously, by Lemma 4(2), the right hand side of the previous inequality tend to zero as ., 0 h   Hence, there exists a relatively compact set that can be arbitrary close to the set     Let us show that the seque micompact.From the un conver there exists a atural numb such that fo y , mpact values, then for every 0, .
. a e .In addition, assumption (H 3 ) implies is semicompact, hence weakly compact.Ar Step 1 from Mazur's theorem, there is a such that  , and converges strongly to This proves that the graph of is closed.Now, as a consequence of Step 1 to Step 5, we conclude that the multifunction of is a compact multivalued function, u.s.c with c pact values.By pplying Theorem 1, we can deduce that has a where 1 , : , In view and for a.e. of (H 9 ), for every L J E So, we may n Then, by Lemma 2, it is weakly compact.
pass to a subsequence if necessary to get that f converges weakly to a function Therefore, by means of the Lebesgue dominated convergence Theorem and the continuity of By interchanging the role of and , we obtain rom (3.21), (3.22) and (H ) Therefore, on and thus by Theorem R has a fixed point w ich is a mild solution for (1.1 ).

Nonconvex Case
In )

Conclusion
this paper, existence problems of nonlocal fractional-

4.
In order impulsive semi-linear differential inclusions with delay have been considered.We have been considered the case when the values of the orient field are convex as well as non-convex.Some sufficient conditions have been obtained, as pointed in the first section, theses conditions are strictly weaker than the most of the existing on ue al o discuss som es.In addition, our techniq lows us t e fractional differential inclusions with delay.

Definition 7
The Caputo derivative of order
R R let x   and  , 1 n and z converges strongly to . 21) the following Theorem we give nonconv rsion for Theorem 4. Our hypothesis on the orient field is the [37]s closed values, 1 FS is closed ([37]).Because F is integrably bounded, fixe sfies all the conditions of Th nt theorem).Thus, there is x       .