Existence of Mild Solutions for a Class of Impulsive Hilfer Fractional Coupled Systems

The aim of this paper is to give existence results for a class of coupled systems of fractional integrodifferential equations with Hilfer fractional derivative in Banach spaces. We first give some definitions, namely the Hilfer fractional derivative and the Hausdorff’s measure of noncompactness and the Sadovskii’s fixed point theorem.


Introduction
Fractional differential equations have been a good tool in many research areas in the last decade, such as engineering, mathematics, physics, and many other sciences [1,2]. For some basic results on this theory, we refer the readers to the papers [3,4] and the references therein.
There are many different definitions of fractional derivatives, each one with its importance and application, which helped justify the importance of fractional calculus. We mention here a few of the most notable definitions of fractional derivatives: Hadamard, Caputo-Hadamard, Hilfer, ψ-Hilfer, Caputo-Riesz, Grünwald-Letnikov, for more details we refer the readers to [5][6][7][8][9].
Recently, a lot of attention has been devoted to the existence of fractional differential problems with Hilfer fractional derivative, see [10][11][12]. The Hilfer fractional derivative, which is a generalization of the Riemann-Liouville fractional derivative, was introduced by nonother than Hilfer [1,13].
The first results on the existence of general value problems involving Hilfer fractional derivative were investigated in [14] and after that in [12]. Following these results, Gu and Trujilo [15] gave the existence of solutions for fractional differential equations with Hilfer fractional derivative using the notion of measure of noncompactness. These equations are widely employed in the biomedical field.
On the other hand, the concept of noninstantaneous impulses was first introduced in [16] by Hernandez; these conditions appeared in the mathematical description of problems that experience abrupt changes during their evolution in time. In the established works, fractional differential equations (FDEs) involving Caputo's fractional derivative are commonly considered with impulsive conditions for obtaining mild solutions [17][18][19][20]. However, in [21], Sousa obtained for the first time the mild solutions for Hilfer fractional differential equations with noninstantaneous impulses. To the best of our knowledge, there are few papers dealing with coupled systems, and on top of that, even fewer existence results for neutral Hilfer fractional differential equations. That is why, to make a little contribution to the already existing results, we consider in this paper a class of coupled systems of Hilfer fractional differential equations with not instantaneous impulses in a Banach space as follows: where for j = 1, 2, I 1−α j 0+ are the RL fractional integrals, H D β j ,γ j are the Hilfer fractional derivatives of order (β j , γ j ) with 0 ≤ β j < 1, 0 ≤ γ j ≤ 1, and 0 ≤ α j = β j + γ j − β j γ j ≤ 1, the linear operators A j : DðA j Þ ⊂ X− → X are the infinitesimal generators of strongly continuous semigroups fT j ðtÞg t≥0 in a Banach space X, The functions f j : ½0, T × X × X − → X and g j : ½0, T × X × X × X − → X are satisfying some assumptions that will be given later, and the functions m i : ðt i , s i × X − → X and n i : ðt i , s i × X − → X characterize the impulsive conditions and x 0j ∈ X. The conditions on u j : ½0, T × ½0, T → ℝ and φ j : ½0, T × ½0, T × X × X − → X are given in a later part. This paper is organized as follows: we first give some preliminaries and notions that will be used throughout the work; after that, we will establish the existence results by means of the fixed point theory; last but not least, we will give an example that illustrates the results.

Preliminaries and Notations
Let CðJ, XÞ be the complete normed linear space of all continuous functions xðtÞ defined on the interval J = ½0, T with kxðtÞk = sup t∈j kxðtÞk. We define the Banach space We also define the Banach space for i = 1, 2, ⋯, m with the norm kxk PC 1−α = max fsup t∈J By LðXÞ, we denote the family of bounded linear operators defined on X, and for j = 1, 2, fR β j ,γ j ðtÞg t≥0 are the (β j − γ j )-resolvent operators generated by A j .
Definition 4. (see [22]). The Hausdorff measure of noncompactness on a bounded subset Ω X of Banach space X is the mapping μ : B ⊂ Ω X → ½0,∞Þ defined by We are going to look back on some properties of the measure of noncompactness.
Lemma 8 (see [23]). Let D = fx n g ⊂ CðJ, XÞ be a bounded countable subset of X. Then, μðDðtÞÞ is Lebesque integrable on X, and Lemma 9. (see [21]). We apply lemmas 2 and 3; then, we obtain an equivalent system of equations to the system 1 as follows: Remark 10. The Laplace transform of the Hilfer fractional derivative of a function f ðtÞ of order 0 < β < 1 and 0 < γ < 1 is given in [13] by where I Now, we give a definition of a pair of mild solution to the problem 1, which is obtained by applying the Laplace transform of the Hilfer fractional derivative.
Definition 11 (see [15,21]). A pair ðx, yÞ ∈ PC 1−α1 ðJ, XÞ × P C 1−α2 ðJ, XÞ = X is said to be a pair of mild solutions of the system (1) if the couple (x, y) satisfies the following coupled system: 3 Advances in Mathematical Physics where W β j ðv j Þ are the Wright functions defined as follows: and satisfying From [15,25], we can assume that for j = 1, 2 and fS β ðtÞg t>0 are strongly continuous and verify: (ii) The norm continuity of the family fT j ðtÞg for t > 0
(H 4 ) The impulsive functions m i , n i : ½t i , s i × X → X are Lipschitz continuous, that is, there exist K mi , K ni > 0, i = 1, 2, ⋯, m, such that for all x, y ∈ X, we have: (H 5 ) For j = 1, 2, ϕ j ðt, s, ·, · Þ: X → X are caratheodory functions, and there exist ξ j : Note that K 1 = K mi and K 2 = K ni: Proof. To prove the existence of solutions for system (1), we only have to prove the existence of solutions for the system (11) and (12) because they are equivalent. Let us define Ω r = fðx 1 , x 2 Þ ∈ PC 1−α 1 ðJ, XÞ × PC 1−α 2 ðJ, XÞ = X : kðx 1 , x 2 Þk ≤ rg with fix radius r, Ω r is a nonempty closed convex bounded subset of X.

Advances in Mathematical Physics
The upcoming part of the proof needs us to rewrite the operator S as follows: where Step 1. We first show that Sðx 1 , x 2 Þ ∈ X for ðx 1 , x 2 Þ ∈ X, that is, t 1−α 1 Pðx 1 , x 2 ÞðtÞ and t 1−α 2 Qðx 1 , x 2 ÞðtÞ are continuous functions for t ∈ ðt i t i+1 , i = 0, 1, 2, ⋯, m. For 0 ≤ t 2 ≤ t ≤ t 1 , we have: We make the substitution t 2 − s = s 1 in the third and fifth terms, we get This proves the continuity of t 1−α1 Pðx 1 , x 2 ÞðtÞ for t ∈ ½0, t 1 , and similarly, we prove the continuity of t 1−α2 Q ðx 1 , x 2 ÞðtÞ.
To show that the operator S is continuous on the intervals ðt i s i and ðs i , t i+1 for i = 1, 2, ⋯, m, we use the continuity of noninstantaneous impulsive functions m i ðt, x 1 ðtÞÞ and n i ðt, x 2 ðtÞÞ. Thus, we conclude that Sðx 1 , x 2 Þ ∈ X.
Step 2. We show that S : Ω r → Ω r , that is, Sðx 1 , x 2 Þ ∈ Ω r , for We first show that the operator P is bounded, which means, Pðx 1 , x 2 Þ ∈ Ω r , for ðx 1 , x 2 Þ ∈ Ω r . Suppose the opposite, so there exist ðx 1 , x 2 Þ ∈ Ω r and t ∈ J such that kPðx 1 , For t ∈ ½0, t 1 , we have which implies that For t ∈ ðt i , s i , i = 1, 2, ⋯, m, we have Advances in Mathematical Physics which implies that For t ∈ ðs i , t i+1 , which implies that Combining the expressions (30), (32) and (34), we obtain By our assumptions, we have kPðx 1 , x 2 ÞðtÞkC 1−α 1 > r which implies that Dividing both sides by r and taking δ → ∞, we obtain which is a contradiction. Hence, kPðx 1 , Similarly, we show that kQðx 1 , That shows that the operator S maps bounded sets to bounded sets.
Step 3. We prove that the operator S 1 is Lipschitz continuous.
For t ∈ ½0, t 1 , we have For t ∈ ðt i , s i , i = 1, 2, ⋯, m, we have so we get and for t ∈ ðt i , s i , we have From (38), (40) and (41), we can say that the operator P 1 is Lipschitz continuous with constant Since the operator S 1 is a Lipschitz operator with constant max As the operator S = S 1 + S 2 , we obtain μðSðDÞÞ ÞgÞμðDÞ < μðDÞ Thus, S : Ω r → Ω r is a condensing operator. Hence, by Sadovskii's fixed point theorem [26], the operator S has at least a pair of solutions ðx 1 ðtÞ, x 2 ðtÞÞ. Therefore, the problem For these values, the condition (1) of theorem 12 is satisfied: we have for j = 1, 2: The second condition is also verified: Consequently, both conditions are satisfied, which means that the problem (1) has a couple of solutions ðx 1 ðtÞ, x 2 ðtÞÞ in the space X = PC 1−α1 × PC 1−α2 .

Conclusion
In this paper, we achieved the existence of solutions for a class of impulsive Hilfer fractional coupled systems by converting the problem to an integral form and then using the Sadovskii's fixed point theorem. For future works, we can consider other fractional operators for example the ψ-Hilfer fractional operator for its new results and applications.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.