Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations

In this manuscript, we examine the existence and the Ulam stability of solutions for a class of boundary value problems for nonlinear implicit fractional differential equations with instantaneous impulses in Banach spaces. The results are based on fixed point theorems of Darbo and Mönch associated with the technique of measure of noncompactness. We provide some examples to indicate the applicability of our results.


Introduction
As it is known very well, the roots of fixed point theory go to the method of successive approximations (or Picard's iterative method) that is used to solve certain differential equations. Roughly speaking, Banach derived the fixed point theorem from the method of successive approximations. In the last decades fixed point theory has been enormously and independently from the differential equations. But, recently, fixed point results turn to be the tools for the solutions of the differential equation. In this paper, we shall involve two interesting fixed point theorems (Darbo's fixed point theorem and Mönch's fixed point theorem) in the setting of "measure of noncompactness" to solve the boundary value problem for nonlinear implicit fractional differential equations with instantaneous impulses.
Differential equations of fractional order have been recently proved to be a powerful tool to study many phenomena in various fields of science and engineering such as electrochemistry, electromagnetics, viscoelasticity, finance, and so on. In the literature, it is very common to propose a solution for fractional differential equations by involving different kinds of fractional derivatives, see e.g. [1-10, 12, 13, 21, 22, 36]. On the other hand, there a few results that deal with the boundary value problems for fractional differential equations. The aim of the present paper is to underline the importance of the theory of impulsive differential equations. Further, by the help of these observations, we aim to un-derstand several phenomena that are not clarified by the non-impulsive equations (see e.g. [15,16,18,32]).
In 1940, Ulam [34,35] raised the following problem of the stability of the functional equation (of group homomorphisms): "Under what conditions does it exist an additive mapping near an approximately additive mapping?" Let G 1 be a group, and let G 2 be a metric group with a metric d(·, ·). Given any > 0, does there exist δ > 0 such that if a function h : G 1 → G 2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G 1 , then there exists a homomorphism H : G 1 → G 2 with d(h(x), H(x)) < for all x ∈ G 1 ?
A partial answer was given by Hyers [20] in 1941, and between 1982 and 1998 Rassias [28,29] established the Hyers-Ulam stability of linear and nonlinear mappings. Subsequently, many works have been published in order to generalize Hyers results in various directions, see for example [24,25,30,31,33].

Preliminaries
In this section, we recall and recollect the basic notion, notations together with some fundamental results that will be necessary in the main results. Throughout the paper, (E, · ) represents a Banach space. Set J = [a, b] where 0 < a < b. The letter C is reserved to represent the Banach space which consists of all continuous functions u : J → E where the norm is In what follows, we pay attention to the weighted spaces of continuous functions with the norms and Consider the Banach space Also, we consider the weighted space and PC n γ ,ρ (J) = u ∈ PC n-1 : u (n) ∈ PC γ ,ρ (J) , n ∈ N, equipped with the norm The letter L 1 (J) indicates the space of Bochner-integrable functions f : J − → E with the norm
Since 0 < α < 1, we shall focus only on the case n = 1.

Property 2.10 ([27])
The operator ρ D α,β a + can be written as Property 2.11 ([27]) The fractional derivative ρ D α,β α + is an interpolator of the following fractional derivatives: Definition 2.13 ([14]) Let X be a Banach space, and let X be the family of bounded subsets of X. The Kuratowski measure of noncompactness is the map μ : For all M, M 1 , M 2 ∈ X , the map μ satisfies the following properties:

Lemma 2.14 ([19]) Let D ⊂ PC γ ,ρ (J) be a bounded and equicontinuous set, then
if and only if u satisfies the following Volterra integral equation: Theorem 2.16 (Mönch's theorem [26]) Suppose that D is a closed, bounded, and convex subset of a Banach space X such that 0 ∈ D. Let T be a continuous mapping of D into itself. If the implication holds for every subset V of D, then T has a fixed point. [17]) Suppose that D is a nonempty, closed, bounded, and convex subset of a Banach space X. Let T be a continuous mapping of D into itself such that, for any nonempty subset C of D, where 0 ≤ k < 1, and μ is the Kuratowski measure of noncompactness. Then T has a fixed point in D. and The following theorem shows that problem (6)-(8) has a unique solution given by Proof Assume that u satisfies (6)- (8). If t ∈ J 0 , then Lemma 2.15 implies we have the solution that can be written as If t ∈ J 1 , then Lemma 2.15 implies If t ∈ J 2 , then Lemma 2.15 implies Repeating the process in this way, the solution u(t) for t ∈ J k , k = 1, . . . , m, can be written as on both sides of (11), using Lemma 2.6, and taking t = b, we obtain Multiplying both sides of (12) by c 2 and using condition (8), we obtain Substituting (13) into (11) and (10), we obtain (9).

Also, we can easily show that
This completes the proof.
As a consequence of Theorem 3.1, we have the following result.

Lemma 3.2 Let
where h : (a, b] → R is defined as
In the sequel, we shall use the following hypotheses efficiently: (Ax1) The mapping t → f (t, u, w) is measurable on (a, b] for each u, w ∈ E, and the functions u → f (t, u, w) and w → f (t, u, w) are continuous on E for a.e. t ∈ (a, b], and f ·, u(·), w(·) ∈ PC β(1-α) γ ,ρ for any u, w ∈ PC γ ,ρ (J).
(Ax2) There exists a continuous function p : ∈ (a, b] and for each u, w ∈ E.
(Ax3) For each bounded set B ⊂ E and for each t ∈ (a, b], we have  ∈ (a, b], we find Set p * = sup t∈J p(t).
We are now in a position to investigate the existence result for problem (1) then problem (1)

-(3) has at least one solution in PC
Proof Consider the operator : PC γ ,ρ (J) → PC γ ,ρ (J) defined in (20) and the ball B R := For any u ∈ B R and any t ∈ (a, b], we find By Lemma 2.6, we have Hence, for all u ∈ PC γ ,ρ (J) and each t ∈ (a, b], we get Hence, we conclude that transforms the ball B R into itself. Now, we prove that the operator : B R → B R fulfills all conditions of Theorem 2.16. For the sake of transparency, we shall divide the proof in four steps. Step 1: We shall prove that the operator : B R → B R is continuous. Suppose that the sequence {u n } converges to u in PC γ ,ρ (J). Then we get for each t ∈ (a, b], where h n , h ∈ C γ ,ρ (J) in a way that h n (t) = f t, u n (t), h n (t) .
Since u n → u, then we get h n (t) → h(t) as n → ∞ for each t ∈ (a, b]. So we find that u nu PC γ ,ρ → 0 as n → ∞ by the Lebesgue dominated convergence theorem. Step 2: We shall indicate that (B R ) is equicontinuous and bounded. Indeed, (B R ) is bounded since (B R ) ⊂ B R and B R is bounded. Next, let 1 , 2 ∈ J, 1 < 2 , and let u ∈ B R . Then Consequently, we conclude that (B R ) is bounded and equicontinuous.
By Lemma, we have Therefore So, by (21), the operator is a L-contraction.