Impulsive fractional differential inclusions with state-dependent delay

In this paper, we prove the existence of mild solution for impulsive fractional inclusions with state-dependent delay in Banach spaces. Our study is based on the nonlinear alternative of LeraySchauder type for multivalued maps due to Martelli. An example is provided to illustrate the main result.


Introduction
Fractional differential equations have been proved to be one of the most effective tools in the modeling of many phenomena in various fields of physics, mechanics, chemistry, engineering, etc. For more details, see [1,2,31,34,41,42,48]. In order to describe various real-world problems in physical and engineering science subject to abrupt changes at certain instants during the evolution process, impulsive differential equations have been used to model the systems. The theory of impulsive differential equations is an important branch of differential equations, which has an extensive physical background [9,12,33].
On the other hand, functional differential equations with state-dependent delay appear frequently in applications as model of equations and for this reason the study of this type of equations has received great attention in the last few years, see for instance [4,10,23,24,25,26,27,28,44,47] and the references therein. The literature devoted to this subject is concerned fundamentally with first-order functional differential equations for which the state belongs to some finite dimensional space, see among other works, [13,15,17,22,32,36,46]. The problem of the existence and uniqueness of solutions for fractional differential equations with delay was recently studied by Maraaba et al. in [37,38]. In [5], the authors provide sufficient conditions for the existence of mild solutions for a class of fractional integro-differential equations with state-dependent delay, where as in [7,40] we investigate the existence and controllability results for a class of impulsive fractional evolution equations with state-dependent delay.
El-Sayed and Ibrahim initiated the study of fractional differential inclusions in [19]. Recently several qualitative results for fractional differential inclusion several results were obtained in [14]. In [6] we establish sufficient conditions for the existence of mild solutions for fractional integro-differential inclusions with state-dependent delay in Banach spaces.
In this work we establish the existence of mild solutions for the class of impulsive fractional inclusions with state-dependent delay described by the form is the family of all nonempty subsets of the separable Banach space (E, · ) and J = [0, T ], T > 0, and φ ∈ B with φ(0) = 0. Here, 0 = t 0 < t 1 < . . . < t m < t m+1 = T, I k : E → E, k = 1, 2, . . . , m, are maps, x(t k − h) represent the right and the left limit of x(t) at t = t k , respectively. We denote by x t the element of B defined by x t (θ) = x(t + θ), θ ∈ (−∞, 0]. Here x t represents the history of the state from −∞ up to the present time t. We assume that the histories x t belongs to some abstract phase space B, to be specified later, and φ ∈ B.

Preliminaries
We will briefly recall some basic definitions and facts from multivalued analysis that we will use in the sequel. C = C(J, E) denotes the Banach space of continuous functions from J into E with the norm Let L(E) be the Banach space of all linear and bounded operators on E. Let L 1 (J, E) be the space of E−valued Bochner integrable functions on J with the norm G is called upper semi-continuous (u.s.c.) on E if for each x 0 ∈ E the set G(x 0 ) is a nonempty, closed subset of E, and if for each open set U of E containing G(x 0 ), there exists an open neighborhood V of x 0 such that G(V ) ⊆ U.
G is said to be completely continuous if G(B) is relatively compact for every B ∈ P b (E). If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph (i.e. x n −→ x * , y n −→ y * , y n ∈ G(x n ) imply y * ∈ G(x * )). For more details on multivalued maps see the books of Deimling [18], Górniewicz [20] and Hu and Papageorgiou [30].
x, y) is upper semicontinuous for almost all t ∈ J.
Definition 2.2. Let α > 0 and f ∈ L 1 (J, E). The Riemann-Liouville integral is defined by For more details on the Riemann-Liouville fractional derivative, we refer the reader to [16]. Definition 2.3. [42]. The Caputo derivative of order α for a function f : J → E is defined by Obviously, The Caputo derivative of a constant is equal to zero.
In order to defined the mild solution of the problems (1) we recall the following definition.
Definition 2.5. [8] if A is a closed linear operator with domain D(A) defined on a Banach space E and α > 0, then we say that A is the generator of an α-resolvent family if there exists ω ≥ 0 and a strongly continuous function S α : R + →L(E) such that {λ α : Re(λ) > ω} ⊂ ρ(A)) and In this case, S α (t) is called the α-resolvent family generated by A.
Definition 2.6. (see Definition 2.1 in [3]) if A is a closed linear operator with domain D(A) defined on a Banach space E and α > 0, then we say that A is the generator of a solution operator if there exist ω ≥ 0 and a strongly continuous in this case, S α (t) is called the solution operator generated by A. For more details see [36,43].
In this paper, we will employ an axiomatic definition for the phase space B which is similar to those introduced by Hale and Kato [21]. Specifically, B will be a linear space of functions mapping (−∞, 0] into E endowed with a seminorm · B , and satisfies the following axioms: where C > 0 is a constant. (A2): There exist a continuous function C 1 (t) > 0 and a locally bounded Let r ≥ 0, 1 ≤ p < ∞, and let g : (−∞, −r) → R be a nonnegative measurable function which satisfies the conditions (g − 5), (g − 6) in the terminology of [29]. Briefly, this means that g is locally integrable and there exists a nonnegative, locally consists of all classes of functions ϕ : (−∞, 0] → X, such that ϕ is continuous on [−r, 0], Lebesgue-measurable, and g ϕ p on (−∞, −r). The seminorm in . B is defined by The space B = C r × L p (g, X) satisfies axioms (A1), (A2), (A3). Moreover, for r = 0 and p = 2, this space coincides with C 0 × L 2 (g, X), H = 1, [29], Theorem 1.3.8 for details).
Let S F •x be a set defined by Lemma 2.1. [35]. Let F : J × B × E −→ P cp,c (E) be an L 1 -Carathéodory multivalued map and let Ψ be a linear continuous mapping from L 1 (J, E) to C(J, E), then the operator The following is the multivalued version of the fixed-point theorem due to Martelli [39].
Lemma 2.2. Let X be a Banach space, and N : X → P cp,cv (X) be an upper semicontinuous and completely continuous multivalued map. If the set is bounded, then N has a fixed point.

Main results
Before going further we need the following lemma ( [45]).
if F satisfies the uniform Holder condition with exponent β ∈ (0, 1] and A is a sectorial operator, then the unique solution of the Cauchy problem (2) is given by is called the α-resolvent family and T α (t) is the solution operator, generated by A.
Theorem 3.1. [11,45] If α ∈ (0, 1) and A ∈ A α (θ 0 , ω 0 ), then for any x ∈ E and t > 0, we have Let us consider the set of functions From Lemma 3.1, we can define the mild solution of system (1) as follows: , and x satisfies the following integral equation: Set . We always assume that ρ : J × B → (−∞, T ] is continuous. Additionally, we introduce following hypothesis: (H ϕ ) The function t → ϕ t is continuous from R(ρ − ) into B and there exists a continuous and bounded function L φ : Remark 3.1. The condition (H ϕ ), is frequently verified by continuous and bounded functions. For more details, see for instance [29].
Remark 3.2. In the rest of this section, C * 1 and C * 2 are the constants C * 1 = sup s∈J C 1 (s) and C * 2 = sup s∈J C 2 (s).
Let us introduce the following hypotheses: (H1) The semigroup S(t) is compact for t > 0.
where Set where B is a bounded set of E. Proof. We transform the problem (1) into a fixed-point problem. Consider the multivalued operator N : It is clear that the fixed points of the operator N are mild solutions of the problem (1). Let us define y(·) : (−∞, T ] −→ E as Then y 0 = φ. For each z ∈ C(J, E) with z(0) = 0, we denote by z the function defined by Let x t = y t + z t , t ∈ J. It is easy to see that x(·) satisfies (3) if and only if z 0 = 0 and for t ∈ J, we have where v(s) ∈ S F,y ρ(s,ys+zs) +z ρ(s,ys +zs ) . Let For any z ∈ B 2 , we have Thus (B 2 , · B2 ) is a Banach space. We define the operator P : where v(s) ∈ S F,y ρ(s,ys+zs) +z ρ(s,ys +zs ) . It is clear that the operator N has a fixed point if and only if P has a fixed point. So let us prove that P has a fixed point. We shall show that the operators P satisfy all conditions of Lemma 2.2. For better readability, we break the proof into a sequence of steps.
Step 1: P is convex for each z ∈ B 2 . Indeed, if h 1 and h 2 belong to P (z), then there exist v 1 , v 2 ∈ S F,y+z such that, for t ∈ J, we have t ∈ (t 1 , t 2 ]; (i = 1, 2) . . . 1]. Then for each t ∈ [0, t 1 ], we get Similarly, for any t ∈ (t i , t i+1 ], i = 1, . . . , m, we have Since S F,y ρ(s,ys+zs) +z ρ(s,ys +zs ) is convex (because F has convex values), we have Step 2: P maps bounded sets into bounded sets in B 2 . Indeed, it is enough to show that for any r > 0, there exists a positive constant such that for each z ∈ B r = {z ∈ B 2 : z B2 ≤ r}, we have P (z) B2 ≤ . Let h ∈ P (z), and using (H2) we have for each t ∈ [0, t 1 ], Moreover, when t ∈ (t i , t i+1 ], i = 1, . . . , m, we have the estimate Hence P (B r ) is bounded.
Step 4: The set (P B r )(t) is relatively compact for each t ∈ J, where For all t ∈ [0, t 1 ], by the strong continuity of S α (·) and conditions (H1), (H2), that the set {S α (t − s)v(s), t, s ∈ [0, t 1 ]} is relatively compact in E. Moreover, from the mean value theorem for the Bochner integral, we obtain On the other hand, for t ∈ (t i , t i+1 ], i = 1, . . . , m, using the continuity of the operator T α (·), it follows that (P B r )(t) is relatively compact in E, for every t ∈ [0, T ]. As a consequence of Step 2 to 3 together with Arzelá-Ascoli theorem we can conclude that P is completely continuous.
Since z n → z * , for some v * ∈ S F,y * ρ(s,y * s+z * s)+z * ρ(s,y * s+z * s) it follows that, for every t ∈ (t i , t i+1 ], we have Hence the multivalued operator P is upper semi-continuous.
Integrating from 0 to t we get Thus, for every t ∈ J, there exists a constant Λ such that v(t) ≤ Λ and hence m(t) ≤ Λ. Since z B2 ≤ m(t), we have z B2 ≤ Λ. This shows that the set E is bounded. As a consequence of the Lemma 2.2, we deduce that the operator P has a fixed point which gives rise to a mild solution of the problem (1).

An Example
We consider the impulsive fractional integro-differential problem:  It is well known that A is the infinitesimal generator of an analytic semigroup (S(t)) t≥0 on E. For the phase space, we choose B = B γ defined by Notice that the phase space B γ satisfies axioms (A1) − (A3).
We can show that problem (5) is an abstract formulation of problem (1). The following result is a direct consequence of Theorem 3.2.
Proposition 4.1. Let ϕ ∈ B be such that (H ϕ ) holds, and let t → ϕ t be continuous on R(ρ − ). Then there exists a mild solution of (5).