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.
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 [5–9].
Recently, a lot of attention has been devoted to the existence of fractional differential problems with Hilfer fractional derivative, see [10–12]. 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  and after that in . Following these results, Gu and Trujilo  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  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 [17–20]. However, in , 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 are the RL fractional integrals, are the Hilfer fractional derivatives of order () with , and , the linear operators are the infinitesimal generators of strongly continuous semigroups in a Banach space is a partition of .
The functions and are satisfying some assumptions that will be given later, and the functions : and characterize the impulsive conditions and . The conditions on and 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.
2. Preliminaries and Notations
Let be the complete normed linear space of all continuous functions defined on the interval with . We define the Banach space introduced in  by with norm .
We also define the Banach space for with the norm .
The space equipped with the norm is also a Banach space.
By , we denote the family of bounded linear operators defined on , and for are the ()-resolvent operators generated by .
Definition 1 (see ). The Hilfer fractional derivative of order , ; , with lower limit is defined as follows: where is the RL integral, and is the RL derivative.
Lemma 2. (see ). Let , , and . If is such that D, then
Lemma 3. (see ). Let and . If and , then
Definition 4. (see ). The Hausdorff measure of noncompactness on a bounded subset of Banach space is the mapping defined by We are going to look back on some properties of the measure of noncompactness.
Lemma 7 (see ). For a bounded and equicontinuous function , the Hausdorff measure of noncompactness is continuous on J, and .
Lemma 8 (see ). Let be a bounded countable subset of . Then, is Lebesque integrable on X, and
Remark 10. The Laplace transform of the Hilfer fractional derivative of a function of order and is given in  by
where is the Riemann-Liouville derivative of order .
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 is said to be a pair of mild solutions of the system (1) if the couple () satisfies the following coupled system: for , we have where are the Wright functions defined as follows: and satisfying From [15, 25], we can assume that for (i)The linear operators and are strongly continuous and verify:(ii)The norm continuity of the family for
3. Existence Results
In this section, we make some assumptions that are necessary to obtain our results:
(H1) For , the functions are bounded and Lipschitz continuous, that is, there exist and such that
(H2) For , the functions are Caratheodory, that is, is measurable for all is continuous a.e for , and there exist , and a continuous function such that for almost all .
(H3) For , there exist functions and constants such that for any bounded, equicontinuous, and countable sets .
(H4) The impulsive functions are Lipschitz continuous, that is, there exist , such that for all , we have:
(H5) For are caratheodory functions, and there exist with
(H6) For , and for any bounded set and , there exist functions such that
(H7) For is bounded and measurable on along with the continuity of defined by .
Theorem 12. The system (1) has a pair of solutions in the space if the assumptions (H1)−(H7) hold and the following conditions are verified, for : Note that and
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 with fix radius is a nonempty closed convex bounded subset of .
Define the operator such that , where by splitting both (23) and (24), we have: The upcoming part of the proof needs us to rewrite the operator as follows: where
Step 1. We first show that for , that is, and are continuous functions for .
For , we have: We make the substitution in the third and fifth terms, we get
This proves the continuity of and similarly, we prove the continuity of .
To show that the operator is continuous on the intervals and for , we use the continuity of noninstantaneous impulsive functions and . Thus, we conclude that .
Step 2. We show that , that is, , for .
We first show that the operator is bounded, which means, , for . Suppose the opposite, so there exist and such that .
For , we have which implies that