Solving Coupled Impulsive Fractional Diﬀerential Equations With Caputo-Hadamard Derivatives in Phase Spaces

. In this manuscript, we incorporate Caputo-Hadamard derivatives in impulsive fractional differential equations to obtain a new class of impulsive fractional form. Further, the existence of solutions to the proposed problem has been inferred under a state-dependent delay and suitable hypotheses in phase spaces. Finally, the considered problem has been supported by an illustrative application.


Introduction
The modeling of several events in numerous branches of science and engineering can greatly benefit from the use of fractional differential equations (FDEs) and fractional integral equations (FIEs).
Modeling scientific phenomena has long employed delay differential equations (DEs) or functional DEs with or without impulse. The delay has frequently been thought of as either a fixed constant or as an integral, in which case it is referred to as a distributed delay; for examples, see the books [15][16][17], and the papers [18,19].
In the study of both qualitative and quantitative theory for functional DEs, the idea of the phase space (PS) Ω is crucial. A seminormed space that satisfies the appropriate axioms is a usual choice, as described by Hale and Kato [19]. We recommend reading [11,20,21] for a more in-depth discussion on this subject.
However, in recent years, modeling has been suggested for complex scenarios where the delay depends on the unidentified functions; for example, see [22,23] and the references therein. These equations are usually referred to as state-dependent delay equations. Recently, among other things, existence results for functional differential equations were developed when the solution to impulsive situations depended on the delay over a finite interval. Many papers have addressed this purpose, either by introducing the Caputo functional fractional operators or by introducing other fractional operators with state-dependent delays. For more information, see [24][25][26][27][28][29][30][31][32][33]. According to our knowledge, there are no works in the literature that discuss Caputo-Hadamard fractional (CHF) order functional DEs with state-dependent delay and impulses. The paper's goal is to get that study started.
In this study, we focus on the existence of solutions to the following initial value problems (IVPs) for coupled Impulsive fractional differential equations (IFDEs): where CH D and CH D σ are CHF derivatives with order and σ, respectively, χ, χ : , similarly ∆ξ| τ =τ j and Ω is an abstract PS.

Preliminaries
This part is devoted to present some preliminary facts that will be used in the sequel.
Assume that C (K, R) refers to the space of all continuous functions on the interval K. It is a Banach space (BS) under the norm Definition 2.1. [4] For the function ω : [b, c] → R, the HF integral of order is described as provided that the integral exists.
The CH derivative of fractional order for the function is defined as follows: where [Re( )] is the integer part of the real number Re( ) and log(.) = log e (.).
The above spaces are BSs with norms respectively.
Clearly, the product space = AC × AC , R is a BS with the norm Put Definition 3.1. Suppose that the functions , ξ ∈ Ω Q have th and σ th derivative on K , respectively.
We say that the pair ( , ξ) is a solution to Problem (1.1) if and ξ satisfy (1.1).
The next result will be helpful in the following as a result of Lemma 2.1.
Lemma 3.1. Assume that , σ ∈ (0, 1] and ρ, η ∈ (K, R) . The pair ( , ξ) is a solution to the FIEs if and only if and ξ are a solution to the fractional IVP It is necessary to present the following postulates.
(P 1 ) The functions τ → φ τ and τ → ψ τ are continuous from into Ω and there is a bounded and continuous functions M φ , M ψ :

and a continuous and nondecreasing function
and
Step 3: We carry on with this method while keeping in mind that m = | [m,Q] and ξ m = ξ| [m,Q] are solutions to the problem Following that, the solution to Problem (1.1) is defined by · · · ξ m (τ ), τ ∈ (τ m , Q].

Conclusion
Fractional DEs are considered a fruitful branch of nonlinear analysis. Impulsive DEs appear when, at certain moments, they change their state quickly and have variant applications in medicine, engineering, physics, dynamics, economics, pharmacology, etc. On the other hand, functional DEs via statedependent delay typically arise in applications as models of equations. As a consequence, work on these types of equations has gained a lot of attention in the last few years by using several kinds of fractional derivatives. In this work, by using a fixed point technique, we ensured the existence of solutions to an IVP for a coupled FDE involving Caputo-Hadamard derivatives in PSs with a statedependent delay and an impulsive. Moreover, we illustrated the obtained results with a concrete application where the suitable conditions are well applicable.
Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publication of this paper.