Existence of Solutions for Fractional-Order Neutral Differential Inclusions with Impulsive and Nonlocal Conditions

This paper investigates the existence of solutions for fractional-order neutral impulsive differential inclusions with nonlocal conditions. Utilizing the fractional calculus and fixed point theorem for multivalued maps, new sufficient conditions are derived for ensuring the existence of solutions. The obtained results improve and generalize some existed results. Finally, an illustrative example is given to show the effectiveness of theoretical results.


Introduction
This paper deals with the existence of solutions for the following fractional-order differential inclusions with impulsive nonlocal conditions: c D q x t − g t, x t ∈ F t, x t , t ∈ I 0, T , t / t k , where 0 < q < 1 and c D q is the Caputo fractional derivative.I k : R → P R k 1, . . ., m, and F : I × R → P R are multivalued maps P R is the family of all nonempty subsets of R .Δx| t t k x t k − x t − k , x t k , x t − k represents the right and left limits of x t at t t k , respectively.g, h are given functions to be specified later.
The nonlocal problem was more general and has better effect than the classical Cauchy problems.So it has been studied extensively under various conditions in the literature 1-7 and the references contained therein.In 2 , Byszewski and Lakshmikantham considered the existence and uniqueness of mild solutions when f and g satisfied Lipschitz conditions.In 3 , Ntouyas and Tsamatos studied the case of compactness conditions of g and T t .Liang et al. in 5 discussed the nonlinear nonlocal Cauchy problems when T t was compact and g was Lipschitz and g was not Lipschitz and not compact, respectively.Xue in 6 established existence results of mild solutions for semilinear differential equations under various conditions on f, g, and T t .Subsequently in 7 , the author examined the semilinear differential equations when T t was compact and g failed to be compact.
The theory of impulsive differential equations and differential inclusions has received much attention for the past decades because of its wide applicability in control, electrical engineering, mechanics, biology, and so on.For more details on this theory and applications, we refer to the monograph of Lakshmikantham et al. 8 , Samoilenko and Perestyuk 9 and the references therein.In 10 , Abada et al. discussed the impulsive differential inclusions with delays when A was the infinitesimal generator of a strongly continuous semigroup T t .In the following work 11 , they obtained some existence and controllability results when A was a nondensely defined closed linear operator.
Since fractional-order differential equations have proved to be valuable tools in the modeling of many phenomena in physics and technical sciences, differential equations involving Riemann-Liouville as well as Caputo derivatives have been investigated extensively in the last decades see 12-28 and the references therein .In 13 , Mophou obtained the existence and uniqueness of mild solutions for the semilinear impulsive fractional differential equations by means of the Schauder fixed point theorem in Banach spaces.In 18 , Benchohra and Slimani established sufficient conditions for the existence of solutions for a class of initial value problem for impulsive fractional differential equations.In 21 , Chang and Nieto studied the existence of solutions for a class of fractional differential inclusions with boundary conditions by means of Bohnenblust-Karlin fixed point theorem.Balachandran et al. in 25 investigated fractional-order impulsive integrodifferential equations in Banach spaces.In 27 , Agarwal and Ahmad studied the existence of solutions for nonlinear fractional differential equations and inclusions of order q ∈ 3, 4 with antiperiodic boundary conditions.The existence results were established for convex as well as nonconvex multivalued maps by some fixed point theorems, such as Leray-Schauder degree theory, and nonlinear alternative of Leray-Schauder type.
However, there is little information in the literature on neutral fractional-order impulsive differential inclusions with nonlocal conditions.Motivated by works mentioned above, we consider the existence results of 1.1 by using fixed point theorem for multivalued maps.
This paper is organized as follows.In Section 2, we will recall briefly some basic definitions and preliminary results which will be used throughout the paper.In Section 3, based on the fractional calculus and fixed point theorem for multivalued maps, we will prove the existence of solution to the problem 1.1 .In Section 4, An example is given to show the effectiveness of our abstract results.

Preliminaries
In this section, we introduce some definitions, notes, and preliminary facts which will be used in this paper.
Let C I, R be a Banach space, with the norm x C sup{|x t |, t ∈ I}.Let L p I, R denote the space of Bochner integrable functions on I with the norm x L p T 0 |x t | p dt 1/p .Let X, • be a Banach space.P X is the family of all nonempty subsets of X. P bd X , P cl X , P bd,cl X , P cp,cv X , and P bd,cl,cv X denote, respectively, the family of all nonempty bounded, closed, bounded-closed, compact-convex, and bounded-closed-convex subsets of X.A multivalued map G : ii u → F t, u is upper semicontinuous for almost all t ∈ I; iii for each q > 0, there exists φ q ∈ L 1 I, R such that for all u ≤ q and a.e.t ∈ I.
The key tool in our approach is the following fixed point theorem.
Lemma 2.3 see 29 .Let B 0, r and B 0, r denote, respectively, the open and closed balls in a Banach space X centered at origin and of radius r, and let Φ 1 : X → P bd,cl,cv X and Φ 2 : B 0, r → P cp,cv X be two multivalued operators satisfying the following: Then either ii there exists an u ∈ X with u r such that λu ∈ Φ 1 u Φ 2 u, for some λ > 1.
Lemma 2.4 see 30 .Let I be a compact real interval and X a Banach space.Let F be an L 1 -Caratheodory multivalued mapping with compact convex values with S F,y / ∅, where S F,y {f ∈ L 1 I, X : f t ∈ F t, y t , a.e.t ∈ I} and let Γ : L 1 I, X → C I, X be a linear continuous mapping.Then the operator For more details on multivalued map, see the books of Aubin and Cellina 31 , Deimling 32 .Now, we recall some definitions and facts about fractional derivatives and fractional integrals of arbitrary order, see 33 .
Definition 2.5.The Riemann-Liouville fractional integral operator of order q > 0 of a function h is defined by where Γ is the gamma function.

Main results
In this section, main results are presented.
Let PC I, R {x : For the study of the system 1.1 , the following hypotheses are given: H1 F is a Carathéodory map with compact convex values, and for each fixed x ∈ R, S F,x is nonempty; H2 there exist a function a • ∈ L p I, R , where p > 1/q, p ∈ N and a nondecreasing continuous function Ω : R → R such that for t ∈ I and each x ∈ R; The existence of solutions is now presented.
Theorem 3.2.Assume that (H1)-(H6) are satisfied; then the problem 1.1 has at least one solution on I provided that there exists a real number r > 0 such that Proof.We transform the problem 1.1 into a fixed point problem by considering the multivalued operator N x : PC I, R → P PC I, R : where f ∈ S F,x , J k ∈ I k x t − k , k 1, . . ., m.Consider the multivalued operators α, β :

3.7
It is clear that N α β.This problem of finding a solution of 1.1 is reduced to find a solution of the operator inclusions x ∈ α x β x .In what follows, we aim to show that the operator α β has a fixed point, which is a solution of the system 1.1 .For this purpose, We will show that α β satisfies all the conditions of Lemma 2.3.For better readability, the proof will be given in several steps.
Step 1.A priori bounded on solution: we show that the second assertion of Lemma 2.3 is not true.
Define an open ball B 0, r ∈ PC I, R , where the real number r satisfies the inequality given in condition 3.5 .Let x be a possible solution of λx ∈ α x β x , for some real number λ > 1 with x r.Then we have t − s q−1 f s ds.

3.8
Hence by H2 -H5 , we have 3.9 Taking the supremum over t, we get Substituting x r in the above inequality yields which is a contradiction to 3.5 .
Step 2. α : PC I, R → P PC I, R is a contraction mapping.Indeed, let x 1 , x 2 ∈ PC I, R , and from H5 , we have

3.12
Hence by 3.4 , α is a contraction mapping.
Step 3. β has compact, convex values, and it is completely continuous.
Claim 1. β is convex for each x ∈ PC I, R .In fact, if ϕ 1 and ϕ 2 belong to β x , then there exist f 1 , f 2 ∈ S F,x such that

3.14
Since S F,x is convex because F has convex values , we have

3.15
Claim 2. β maps bounded sets into bounded sets in PC I, R .Let r > 0 , B r {x ∈ PC I, R : x ≤ r} be a bounded set in PC I, R .For x ∈ B r , ϕ ∈ β x , then there exists f ∈ S F,x such that Therefore, 3.17 Thus

3.18
Claim 3. β maps bounded sets into equicontinuous sets.Let τ 1 , τ 2 ∈ I, τ 1 < τ 2 , B r be a bounded set of PC I, R as in Claim 2 and x ∈ B r .Then there exists f ∈ S F,x such that

3.19
By elementary computation, we have

3.20
As τ 1 → τ 2 , the right-hand side of the above inequality tends to zero.As a consequence of Claims 1 to 3 together with the Arzelá-Ascoli theorem, we conclude that β is completely continuous.
Claim 4. β has a closed graph.Let x n → x * , ϕ n ∈ β x n , and ϕ n → ϕ * ; we prove that ϕ * ∈ β x * .Now ϕ n ∈ β x n means that there exists f n ∈ S F,x n such that 3.21 We will prove that there exists f * ∈ S F,x * such that Consider the continuous operator κ : Therefore, β has a closed graph.As a consequence of Lemma 2.3, the operator inclusion x ∈ α x β x has a solution which implies that the problem 1.1 has a solution on I. Remark 3.3.If we take F t, x {f t, x }, where f is a continuous operator, then the result corresponds to a single-value problem.In this case, our result is also new in the present configuration.
Corollary 3.4.Assume that (H1)-(H6) are satisfied; then the problem 1.1 has at least one solution on I provided that Proof.From 3.25 , we know that there exists a constant r > 0, such that Since the proof of Corollary 3.4 follows the very same lines as the proof of Theorem 3.2, we do not give details.

Corollary 3.5. Assume that (H1), (H3)-(H6), and the following condition are satisfied, (H2 )
Proof.In this case, we take a t ρ t , Ω x x in Corollary 3.4.At last we would like to discuss the impulsive condition. 3.29

An Illustrative Example
In this section, an example is given to show the effectiveness of our theoretical result.Consider the following fractional-order differential inclusions with impulsive and nonlocal conditions:

Corollary 3 . 6 .
H5 I k : R → R k 1, . . ., m are single continuous operator, and there exists c k > 0, k 1, . . ., m, such that |I k y − I k x | ≤ c k |y − x|, for all x, y ∈ R. Assume that (H1)-(H4), H5 , (H6) are satisfied; then the problem 1.1 has at least one solution on I provided that This set is a Banach space with the normx PC sup t∈I |x t |.As a consequence of Lemmas 2.7 and 2.8, we can define the solution of problem 1.1 .
Definition 3.1.The function x ∈ PC I, R is said to be a solution of the problem 1.1 if x 0 x 0 h x , and there exist f t ∈ S F,x , J k There exists a function ρ • ∈ L p I, R , such that ∞ , and F : I × R → P R is a multivalued map given by x −→ F t, x e −2t |x t | 9 e t 1 |x t | , e −t |x t | 9 e t 1 |x t | .