Existence and stability of solutions of ψ-Hilfer fractional functional differential inclusions with non-instantaneous impulses

Abstract: In this paper, we prove two existence results of solutions for an ψ-Hilfer fractional noninstantaneous impulsive differential inclusion in the presence of delay in an infinite dimensional Banah spaces. Then, by using the multivalued weakly Picard operator theory, we study the stability of solutions for the considered problem in the sense of ψ-generalized Ulam-Hyers. To achieve our aim, we present a relation between any solution of the considered problem and the corresponding fractional integral equation. The given problem here is new because it contains a delay and non-instantaneous impulses effect. Examples are given to clarify the possibility of applicability our assumptions.


Introduction
A non-instantaneous impulsive differential equation is due to Hernándaz et. al. [1], and is used to describe impulsive action, which stays active on a finite time interval. Hilfer [2] introduced a fractional derivative, which is a generalization for Riemann-Liouville fractional derivative and Caputo fractional derivative. Many works have been appeared studying various models involving fractional differential with instantaneous and non-instantaneous impulses and providing solutions to those models. For example, Saravanakumar et al. [3] analyzed the existence of mild solution of non instantaneous impulsive for Hilfer fractional stochastic differential equations driven by fractional Brownian motion,. Shu et al. [4] presented a right formula of mild solutions to a fractional semilinear evolution equation generated by a sectorial operator, and its order belongs to the intervals (0, 1) and (1,2), Wang et al. [5] studied the global attracting solutions to non-instantaneous impulsive differential inclusions containing Hilfer fractional, and Ngo et al. [6] presented a formula of solution for a non-instantaneous impulsive differential equation containing ψ−Hilfer derivative with lower limit of the fractional derivative at zero. For more works on non-instantaneous impulsive differential equations and inclusions, we refer to [7][8][9][10][11][12][13].
Moreover, Ulam problem [14] has been attracted by many researchers. We highlight some recent works on the existence and Hyers-Ulam stability of solutions for fractional differential equations. Guo et al. [15] investigated the existence and Hyers-Ulam stability of mild solution for an impulsive Riemann-Liouville fractional neutral functional stochastic differential equation with infinite delay of order between one and two, Guo et al. [16] proved the existence and Hyers-Ulam stability of the almost periodic solution to fractional differential equations with impulse involving fractional Brownian, Wang et al. [11] presented the generalized Ulam-Hyers stability for a non-instantaneous impulsive differential inclusions containing the Caputo derivative and Vanterler et al. [17] studied, in finite dimensional Banach spaces, the stability of a Volterra integro-differential equation containing ψ−Hilfer derivative in the sense of Ulam-Hyers. More recently, Vanterler et al. [18] investigated, in finite dimensional Banach spaces, the δ−Ulam-Hyers-Rassias stability for a non-instantaneous impulsive fractional differential equation containing ψ−Hilfer derivative, Benchohra et al. [19] established, in finite dimensional spaces, the existence and stability of solutions for an implicit fractional differential equations with Riemann-Liouville fractional derivative, and Kumar et al. [20] studied the existence and stability of solution for a fractional differential equation with non-instantaneous integral impulses. Very recently, Ben Mahlouf et al. [21] given sufficient conditions to guarantee the existence and stability of solutions for generalized nonlinear fractional differential equations of order α ∈ (1, 2), Elsayed et al. [22] established the existence and stability of boundary value problem for differential equation with Hilfer-Katugampola fractional derivative. For more papers on Ulam-Hyers stability of solutions, we refer to [23][24][25][26][27][28][29][30][31].
It is worth noting that, when the considered problem contains non-instantaneous impulses, there are two approaches in the literature, one by keeping the lower limit of the fractional derivative at zero [6,11,17,18], and the other by switching it at the impulsive points [5,10] Motivated by the above cited work, we prove two existence results of solutions, in infinite dimensional Banach spaces, for a non-instantaneous impulsive fractional differential inclusion involving ψ−Hilfer derivative with delay and we switch the lower limit of the fractional derivative at the impulsive points, and then we study the ψ−generalized Ulam-Hyers stability.
To make a comparison between the present paper objectives and other relevant recent papers, we refer to the following: 1-Abbas et al. [31] proved the existence of solutions and studied Ulam-Hyers-Rassias stability of problem (1) in the absence of both delay and impulses effect, 2-Benchohra [19] investigated the existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville derivative, which is including in Hilfer derivative 3-Ngo et al. [6] presented a formula of solution for a non-instantaneous impulsive differential equation containing ψ−Hilfer derivative with lower limit of the fractional derivative at zero and in the absence of delay.
4-Vanterler el al. [17] established the existence and stability of solutions for Problem (1) in the absence of delay. and when E = R, the lower limit of the fractional derivative at zero F is a singlevalued function 5-Wang et al. [10] considered Problem (1) in the absence of delay, when ψ( ) = and without studying the stability of solutions.
6-Wang et al. [11] consider a non-instantaneous impulsive semilinear differential inclusions containing Caputo derivative and in the absence of delay.
To clarify the novelty and contribution of this study, we refer to, in this paper, we present a relation between a solution of Problem (1) and the corresponding fractional integral equation (Lemma5), provide two methods to demonstrate the existence of solutions for Problem (1), then, investigate the ψ−generalized Ulam-Hyers stability of solutions. Because our considered problem contains ψ−Hilfer fractional derivative, non-instantaneous impulses with the lower limit of the fractional derivative switches at the impulsive points, presence of delay, and the right hand side is a multi-valued function, therefore, this study generalize recent results, as it is shown above, such as [6,10,11,17,19,31]. In addition, there isn't work in the literature, on ψ−Hilfer fractional non-instantaneous impulsive differential inclusions, in infinite dimensional spaces, in the presence of delay, and the lower limit of the fractional derivative switches at the impulsive points. Moreover, the technique presented in this paper can be used to study the existence and Ulam-Hyers stability of solutions or mild solutions for the problems considered in [3,4,15,16,[20][21][22] to the case when, there are impulses and delay on the system, the right hand side is a multi-valued function and involving ψ−Hilfer fractional derivative.
In section 2, we prove some properties for ψ−fractional integral and ψ−fractional derivative, then we present, in Lemma 5, a relation between any solution of problem (1) and the corresponding fractional integral equation. In section 3, we prove an existence result of Problem (1). In section 4, we give another existence result of (1), then we investigate the ψ−generalized Ulam-Hyers stability of solutions. In the last section, examples are given to clarify the possibility of applicability of our assumptions.

Preliminaries and notations
Let P ck (E) be the family of non-empty convex and compact subsets of E . Since the given problem containing Hilfer derivative we need to the the spaces: Obviously C 1−µ,ψ (J, E) and C n 1−µ,ψ (J, E) are Banach spaces with norms Because Problem (1) involving impulses effect we recall the Banach space:

given by by
and In the sequel, I q,ψ a+ denotes to the ψ−Riemann-Liouville fractional integral operator of order q with the lower limit at a , D ϑ,ψ a+ f to the ψ−Riemann-Liouville fractional derivative operator of order ϑ with the lower limit at a and c D ϑ,ψ a+ f to the ψ−Caputo fractional derivative of order ϑ with the lower limit at a for f ∈ AC 1,ψ ([a, b], E), where If ψ( ) = , we obtain the Caputo fractional derivative, and if ψ( ) = ln , we obtain the Caputo-Hadamard fractional derivative. The following remark and more information about ψ−fractional integral and derivative can be found in [32][33][34][35] Remark 2.1. If q = 1, then I 1,ψ a+ f ( ) = a ψ (s) f (s) ds, and hence In the following lemma we give an important for I ϑ,ψ a+ , which we need later.
], E). The ψ−Hilfer fractional derivative of order 0 < ϑ < 1and type 0 ≤ ν ≤ 1 and with lower limit at a for a function f : [a, b] → E is defined by Notice that, the operator D ϑ,ν,ψ a+ can be written as: Let

2-The metric space (the space of solutions)
where the metric function is given by:

3-The Banach space
endowed with the norm: , then x will be continuous at zero.
It is easily seen that the function: define a measure of noncompactness on H, where B is a bounded subset of H. We need to the following fixed point for multi-valued functions.
, Theorem 3.1) Let W be a closed convex subset of a Banach space X and : W → P c (W). Suppose that is closed, (D) is relatively compact, whenever D is compact, and that, for some Then, there is a fixed point for .

Existence results of solution for (1).
In this section, we demonstrate the existence of solutions of Problem (1). For any x ∈ H let and and for any x ∈ H let In order to derive the relation between any solution for Problem (1) and the corresponding fractional integral equation, we need to the following essential Lemmas.

Now,
By integration by parts, we get It follows from Remark 1, that .
Now, we give in the following lemma the relation between any solution for Problem (1) and the corresponding fractional integral equation.
such that x satisfies the integral equation In the following, we present our first existence result of solutions for Problem (1).
We assume the following conditions with the property that for any bounded subset D ⊆ H, any k = 0, 1, 2, .., n, and a.e., for ∈ J k where Then Problem (1) has a mild solution provided that where, h * = i=n i=0 h i .
Proof. We define a multioperator Φ : H → P(H) as follows: let x ∈ H, then due to (F 1 ) there is f ∈ S 1 F(.,τ(.)x) , and hence we can define y ∈ Φ(x) if and only if Let us clarify that a point x is a fixed point for Φ if and only if x is a solution for (1). Let x be a fixed point to Φ. Then where f ∈ S 1 F(.,τ(.)x) . Therefore, This means x satisfies (4), and hence it is a solution for (1). Similarly, it is easy to see that if x satisfies (4), then x is a fixed point for Φ. So we prove, by application Lemma 5, that Φ has a fixed point.
Obviously the values of Φ are convex.
Step1. We demonstrate that there is a n ∈ N with Φ(B n ) ⊆ B n ,where B n = {x ∈ H : x H ≤ n}. Suppose that for any natural number n, there are x n , y n ∈ H with y n ∈ Φ(x n ), x n H ≤ n and y n H > n.
Step2. Φ is closed on B n 0 .
Step5. Φ maps compact sets into relatively compact sets. Let B be a compact subset of B n 0 ,{y n , : n ≥ 1} ⊆ Φ(B) Then, there is x n ∈ B, n ≥ 1, such that y n ∈ Φ(x n ). So, there is f n ∈ S 1 F(.,x n (.)) such that (15) holds. We have to show that the set Z = {y n : n ≥ 1} is relatively compact in H. Note that, since B is compact in H, then from (F 3 ) we get for a.e. s ∈ J 0 , By the same reasons, one can show that for a.e. s ∈ J k , k = 1, 2, .., n By arguing as in the previous step one can show that Z is relatively compact, and hence Φ(B) is relatively compact. Now, by applying Lemma 8, there is x ∈ H and f ∈ S 1 F(.,τ(.)x) such that Next, in view of (F 1 ), there is h ∈ PC 1−µ,ψ (J, E) such that f ( ) = I ν(1−ϑ),ψ s i + h( ), ∈ J k , k = 0, .., n, and hence, from Lemma 5, This yields that f ∈ PC ν(1−ϑ) 1−µ,ψ (J, E). Then, the function belongs to H and in view of Lemma(6) it is a solution for (1). This completes the proof.

ψ−generalized Ulam-Hyers stability of Problem (1).
In this section, we give another version for the existence of solutions and investigate the generalized ψ−generalized Ulam-Hyers stability of problem (1). For basic information about multivalued weakly Picard operators we refer to [38].

Examples
In this section we give examples to clarify the possibility of applicability our assumptions.
By choosing ρ small enough this inequality becomes realized.

Conclusions
A relation between a solution of the considered problem and the corresponding fractional integral equation is given, then two existence results of solutions for an ψ-Hilfer fractional non-instantaneous impulsive differential inclusion in the presence of delay in an infinite dimensional Banah spaces are obtained. Moreover, by using the multivalued weakly Picard operator theory, the stability of solutions for the considered problem in the sense of generalized Ulam-Hyers is studied. This work generalizes many recent results in the literature, for example [6,10,11,17,19,31]. Moreover, our technique can be used to study the existence and Ulam-Hyers stability of solutions or mild solutions for the problems considered in [3,4,15,16,[20][21][22] to the case when, there are impulses and delay on the system, the right hand side is a multi-valued function and involving ψ−Hilfer fractional derivative. There are many directions for future work, for example: Generalize the obtained results in [3,4,15,16] when, the considered problems in these works involving ψ-Hilfer fractional derivative.