Existence of solutions of non-autonomous fractional differential equations with integral impulse condition

*Correspondence: n.sooppy@psau.edu.sa 4Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawasir 11991, Saudi Arabia Full list of author information is available at the end of the article Abstract In this paper, we investigate the existence of solution of non-autonomous fractional differential equations with integral impulse condition by the measure of non-compactness (MNC), fixed point theorems, and k-set contraction. The obtained results are verified via a supporting example.


Introduction
Fractional calculus is a generalization of the standard integer calculus. The advantage of fractional calculus over integer-order calculus is that it provides a great deal for the kind of thought and hereditary characteristics of diversified materials and methods. From the past two decades, fractional calculus has attracted research attention towards itself due to its importance in several parts of science, like physics, fluid mechanics, heat conduction [1,19,21,24,26,27,32,33,[37][38][39][40][41][42]. We can relate to the monographs [2,22,30,36] for the fundamentals and to [4,44,45] for the current developments in the field of fractional calculus.
In recent years, non-autonomous differential equations of integer order, as well as fractional order, have been studied by many researchers. One can see the references [5-8, 10-13, 15-17, 20, 25, 35] for more details. In [14], Chen et al. discussed the existence of mild solutions as well as approximate controllability for a class of non-autonomous evolution systems of parabolic type with nonlocal conditions in Banach spaces by using the Schauder's fixed-point theorem as well as the theory of an evolution family. In the same year, Chen et al. [9] explored the existence of mild solutions for the initial value problem to a new class of abstract evolution equations with non-instantaneous impulses on ordered Banach spaces by using a perturbation technique and by dropping the compactness condition on the semigroup. Malik et al. [28] used the Rothe's fixed point theorem to study the controllability of non-autonomous nonlinear differential system with non-instantaneous impulses in the space R n . By using Krasnoselskii's fixed point theorem, Wang et al. [43] formed a set of sufficient conditions for the existence and stability for a class of impulsive non-autonomous differential equations. Kucche [23] investigated the existence and uniqueness of mild solutions for impulsive delay integrodifferential equations with integral impulses in Banach spaces by using Krasnoselskii-Schaefer fixed point theorem.
Motivated by the above, we discuss the non-autonomous fractional differential system with integral impulses in the following form: Suppose that F is a Banach space and {A(s)} s∈J is a family of closed linear operator from

Preliminaries
Let F be a Banach space with norm · .

Definition 2.2 ([22])
We define the derivative of u of the fractional order q > 0 in the Caputo sense as here 0 < q ≤ 1 and u (t) = du(t) dt .
A measurable function f : [0, ∞) → F is called Bochner integrable if f is Lebesgue integrable. The integrals which appear in (2.1) and (2.2) are taken in Bochner's sense. Let the operator -A(s) satisfies the following conditions: (H 1 ) A(s) is a closed operator, the domain of A(s) is independent of s, and dense in F. (H 2 ) For any λ ≥ 0, the operator λI + A(s) has a bounded inverse operator [λI + A(s)] -1 in L(F) and where C is a positive constant independent of s and λ. (H 3 ) For any s, τ , t ∈ J, there is a constant p ∈ (0, 1] such that where the constants p and C > 0 are independent of s, τ , and t. Following Pazy [34], (H 1 ) means that for each t ∈ J, -A(t) generates an analytic semigroup e -sA(t) (s > 0), and there is a C > 0 independent of both s and t such that A n (t)e -sA(t) ≤ C s n , where n = 0, 1, s > 0, t ∈ J. By [18], we can give the definition of operatorsΨ (s, t),Φ(s, σ ), andŨ(s): where ξ q is probability density function defined on [0, ∞) such that it's Laplace transform is given by Using the above facts, we define the mild solution of problem (1.1)-(1.3).
The following lemma gives some properties ofΨ (s, t),Φ(s, σ ), andŨ(s) that are required to prove our main result.
where C > 0 is independent of both s and σ . Moreover, and Next, we define the MNC, which is required in our results.
Some properties of μ(·) are given in the following lemma.

Lemma 2.2 ([3]) Let Z, W be bounded subsets of F and
In this article, MNC on the set F and C(J, F) is denoted by μ(·) and μ C (·), respectively. For any s ∈ J and The prove our main result, the following lemmas are required.

3)
and Proof: Let the operator P : C(J, F) → C(J, F) be defined by First, we show that P maps G R to G R which is a bounded, closed and convex set, where R is a positive constant such that G R = {u ∈ C(J, F) : u(s) ≤ R for ∀s ∈ J}. If this were not true, then there would exist s r ∈ J and u r ∈ G R such that (Pu r )(s r ) > r for each r > 0. Now by using Hölder inequality, (F 1 ) and Lemma 2.1, we get Dividing both sides of (3.6) by r(1 -C * S q+1 ), using (F 1 ), and taking the limit as r → ∞, we get which is a contradiction. Therefore P : G R → G R . Now, we prove that P : Since the function f is continuous in the second variable, for any s ∈ J, we get which gives that, for every s ∈ J, (Pu n )(s) -(Pu)(s) → 0 as n → ∞, Therefore, P : G R :→ G R is a continuous operator. It remains to prove that {Pu : u ∈ G R } is an equicontinuous function set. For any u ∈ S R and s 1 , s 2 ∈ [0, S], s 1 < s 2 , we get (Pu(s 2 ) -(Pu)(s 1 )
Also I t = 0 as s 2 → s 1 . Hence (Pu)(s 2 ) -(Pu)(s 1 ) tends to 0 independently of u ∈ G R as s 2 → s 1 , which means that the operator P : G R → G R is equicontinuous.
Let D = co P(G R ), where co denotes the closure of the convex hull. Then it can be easily seen that the operator P : D → D (D ⊂ C(J, F)) is equicontinuous. Now, we show that P : D → D is a condensing operator. For any B ⊂ D, by Lemma 2.3, there exists a countable set B 0 = {u n } ⊂ B such that From the equicontinuity of B, B 0 ⊂ B is also equicontinuous. Consequently, from Lemma 2.4 and (F 2 ), we get Therefore, by (2.5)-(2.6), we have (3.10) By combining (3.10), (3.4) and Definition 2.5, we know that P : G R → G R is a k-set contractive operator. By Lemma 2.6, P has at least one fixed point u ∈ G R . Therefore P is a mild solution of (1.1)-(1.3).

Example
Consider the following nonlinear reaction-diffusion equation with integral impulse condition: