Existence of mild solutions for impulsive neutral Hilfer fractional evolution equations

*Correspondence: tabdeljawad@psu.edu.sa 2Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia 3Department of Medical Research, China Medical University, Taichung, Taiwan Full list of author information is available at the end of the article Abstract In this paper, we investigate the existence of mild solutions for neutral Hilfer fractional evolution equations with noninstantaneous impulsive conditions in a Banach space. We obtain the existence results by applying the theory of resolvent operator functions, Hausdorff measure of noncompactness, and Sadovskii’s fixed point theorem. We also present an example to show the validity of obtained results.


Introduction
Fractional calculus primarily involves the description of fractional-order derivatives and integral operators [28]. In the last few decades, it has gained significant importance because of its wide range of applicability in diverse scientific domains. Fractional differential equations (FDEs) are among the strongest tools of mathematical modeling and are successfully employed to model complex physical and biological phenomena like anomalous diffusion, viscoelastic behavior, power laws, and automatic remote control systems. In the available literature, notable definitions of fractional derivatives were given by famous mathematicians, but the most commonly used are the Riemann-Liouville (RL) and Caputo derivatives [1,2,21,22,24,26,29,39]. Thus FDEs involving the RL fractional derivative or Caputo derivative have considered frequently for investigating the existence of mild solutions. However, little attention has been devoted to FDEs with generalized fractional derivatives. The Hilfer fractional derivative (HFD), a generalization of the RL fractional derivative was first introduced by Hilfer [4,11,15,16]. The existence and uniqueness of general initial and boundary value problems involving HFD were first examined by Fu-rati and Kassim [10] and Wang and Zhang [36], respectively. Thereafter, by means of the controllability and existence of solutions. Recently, Subashini et al. [34] obtained mild solutions for Hilfer integro-differential equations of fractional order by means of Monch's fixed point technique and noncompact measure. FDEs involving HFD are widely applicable in biomedical research. These equations are successfully employed to model the irregular boundaries of biological cells and microscopic fluctuations of biomedical matters [37].
Hernandez et al. [14] first introduced the concept of noninstantaneous impulsive conditions. These conditions appeared in the mathematical description of real-world dynamical processes experiencing a sudden change over a short interval of time like DNA sequences, heart beat intervals, optimal control models, and so on [19]. In the published work, FDEs involving either RL or Caputo derivative are commonly considered with impulsive conditions for obtaining mild solutions [5,7,20,27,30]. However, Sousa et al. [31] for the first time obtained the mild solutions for Hilfer FDEs with noninstantaneous impulsive conditions. Similalrly, no existence results have been established for neutral Hilfer FDEs in contrast to neutral FDEs with RL or Caputo derivative [9,25,35,40]. Such equations have a bundle of applications in physics, biology, and electrical engineering. Thus, to make a little contribution to existing works, we consider the neutral Hilfer FDEs with impulsive conditions of the mentioned form for obtaining mild solutions. We obtain an existence result with the help of fixed point theory, which is proven to be an authoritative modeling tool for obtaining exact or approximate solutions of FDEs t ∈ (s l , t l+1 ] ⊂ J = [0, a], l = 0, 1, 2, . . . , m; t ∈ (t l , s l ], l = 1, 2, . . . , m; where I 1-θ 0 + is the RL fractional integral, H D p,q 0 + is the HFD of order (p, q) with 0 ≤ p < 1, 0 ≤ q ≤ 1, and 0 ≤ θ = p + qpq ≤ 1, the linear operator A : D(A) ⊂ Z − → Z is the infinitesimal generator of a strongly continuous semigroup {T (t)} t≥0 in a Banach space Z, This manuscript is structured as follows. In Sect. 2, we discuss the Hilfer fractional derivative, Hausdorff measure of noncompactness, and mild solutions of equation (1.1) along with some basic results and lemmas. In later section, we obtain existence results by means of fixed point technique, measure of noncompactness, and Lebesgue dominated convergence theorem. We confirm the validity of obtained results by offering an example in the last section.

Preliminaries
Let C(J , Z) denote the complete normed linear space of continuous functions z(t) defined on the interval J = [0, a] with z = sup t ∈ J z(t) . We define the Banach We will establish the existence results in the Banach space for l = 1, 2, . . . , m with the corresponding norm By L(Z) we denotes the family of bounded linear operators defined on Z, and by {W p,q (t)} t≥0 the (pq)-resolvent operator or the q-times integrated p-resolvent operator generated by A.
The Hilfer fractional derivative of order n -1 ≤ p < n, n ∈ N; 0 ≤ q ≤ 1, with lower limit c is defined as where I p(n-q) c + is the RL integral, and D q+pn-pq a + is the RL derivative.

Lemma 2.3 ([3, 23])
The measure of noncompactness ζ defined on bounded subsets P and Q of a Banach space Z has following properties:

Lemma 2.7 ([31])
After applying Lemmas 2.1 and 2.2, the system of fractional nonlinear differential equations (1.1) reduces to the following integral equation: where M p (ν) is the Wright function defined as and satisfying the equality This definition of a mild solution is obtained by means of the Laplace transform of the Hilfer fractional derivative.
Remark 2.1 The Laplace transform of the Hilfer derivative of a function f (t) of order 0 < p < 1 and 0 < q < 1 is as follows [15]: 2 The norm continuity of family {T (t)} for t > 0.

Existence result
In the beginning of this section, we introduce some assumptions required to obtain the desired result: for t ∈ J , and there exist ψ 1 ∈ L 1 r (J , R + ), 1 r > 1, and a continuous function ψ 2 such that Q (·, z 1 , z 2 ) ≤ ψ 1 (t) z 1 + ψ 2 (t) z 2 for almost all t ∈ J .
The operator Θ is well defined.
We establish our results in six steps.
Since P and Q are continuous functions with respect to the second and third variables, it follows that lim n− →∞ P t, z n (t) = P t, z(t) and By assumptions (H 1 ) and (H 3 ) we have P t, z n (t) -P t, z(t) ≤ 2M P Q t, z n (t), Since ψ 1 ∈ L 1 r [0,a] and ψ 2 (t) is continuous, both functions on the right-hand side are integrable.

Conclusion
We established the existence of mild solutions for neutral fractional-order system involving the Hilfer fractional derivative in a Banach space by converting it into an integral form and hence applying Sadovskii's fixed point technique. For future research work, for the proposed problem, we suggest stability analysis, multiple solutions, and singular solutions.