On System of Nonlinear Sequential Hybrid Fractional Differential Equations

In this study, the existence and uniqueness of the solution for a system consisting of sequential fractional differential equations that contain Caputo–Hadamard (CH) derivative are verified. To study the existence and uniqueness of these solutions, some of the most important results from the fixed point theorems in Banach space were used. A practical example is also given to support the theoretical side that was obtained.

A fractional di erential equation is an equation that contains fractional derivatives and di erentials of some mathematical functions and appears in the form of variables. e goal of solving these equations is to nd these mathematical functions whose derivatives achieve these equations. Before starting to search for solutions to these equations, studying the conditions of existence and uniqueness is a major matter. To study these conditions, most researchers use the most important xed point theorems in Banach space, such as Banach contraction principle and Leray Schauder's theorem (see [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]).
In 2014, Zhang et al. [19] published a study investigating the existence results for H D p 1

x(t) g(t, x(t))
∈ G(t, x(t)), t ∈ [1, e], x (1) where H D p 1 , p ∈ (1, 2] is the Hadamard fractional derivative, g ∈ C([1, e] × R, R\ 0 { }). In 2016, Algoudi et al. [25] published a study investigating the existence results for the following boundary value problem (sequential Hadamard type): 1 , p, q ∈ (1, 2], r, v ∈ (0, 1) is the Hadamard fractional derivative and H I θ 1 is the Hadamard fractional integral with order Some researchers went deeper into their research and verified the stability of the solutions to these equations (see [26,27]). Furthermore, many specialists in the field have paid attention to hybrid fractional differential equations; the importance of fractional hybrid differential equations is that they have a different dynamic than ordinary differential equations and that the hybrid type describes the nonlinear relationship in the derivative of the hybrid function (see [28][29][30][31]).
Based on what has been studied in the articles mentioned above, existence and uniqueness of the following nonlinear coupled differential equations are investigated. Unlike the previous studies, the main results of this article are different; that is, we generalize the problem mentioned in [36] by converting one single fractional differential equation into a system, using different fractional derivatives; here, we consider the problems in the context of sequential type.
where H D c , c � p, q is the Hadamard fractional derivative of order , and λ, μ ∈ R. In this work, we will follow the steps of researchers and specialists in the field. By organizing our results on existence of a solution to the problem as follows, Section 2 contains some fundamental results of fractional calculus and important result for establishing our main results. In Section 3, we introduce our main results. In Section 4, a practical example shows the applicability of our results is given. In Section 5, conclusion and future work are presented.

Preliminaries
In this section, we introduce some useful definitions, lemmas, and notations of fractional calculus.
e Hadamard fractional integral of order θ for a continuous function ψ: Definition 2 (see [36]). e Hadamard fractional derivative of order θ > 0 for a continuous function ψ: where denotes the integer part of the real number θ.
Let C([a, T], R) denote the Banach space of all real valued continuous functions defined on [a, T] and C n δ ([a, T], R) denote the Banach space of all real valued functions φ such that δ n φ ∈ C([a, T], R) (see [38]).

], R), and h 1 ∈ C([a, T], R), and
en, the solution of problem (7) is given by Proof. Applying H I p a + to (2), we get where b 1 , b 2 , c 1 ∈ R. e condition x(a + ) � 0 implies that b 2 � 0. e first derivative of (4) is calculated as follows: Note that e integrating factor η(t) � e (λ/t)dt ; then, multiplying η(t) by (10), we get then integrating (11) and again using x(a + ) � 0, we conclude that consequently, In a like manner of Lemma 2, one can easily find the solution y(t) as similar logic is applied for the case when μ � 0. Here, these cases will not be taken for consideration in this study.

Main Results
In this section, we will present the main results to be obtained from this study.
is a Banach space with the norm defined as ‖(x, y)‖ H � ‖x‖ + ‖y‖ ∀(x, y) ∈ H. Based on Lemma 1, we define an operator ℵ: H ⟶ H as
First, we show that
Proof. We first prove that the operator ℵ: H ⟶ H is completely continuous; obviously, the operator is continuous as a result that f, g, ψ, and φ are all assumed to be continuous.
Next, we prove equicontinuity for the operator ℵ; for this, we let t 1 , t 2 ∈ [a, T], (t 1 < t 2 ). en, R.H.Ss are both independent on (x, y); in addition, R.H.Ss of both (29) and (30) approach to zero when t 1 ⟶ t 2 and they imply that the operator ℵ(x, y) is equicontinuous; consequently, the operator ℵ(x, y) is completely continuous.

Conclusion and Future Work
In this article, the existence and uniqueness theory of solutions for sequential fractional differential system involving Hadamard fractional derivatives of order 1 < p, q < 2 with initial conditions were investigated. For the future work, the researcher may generalize our system by taking an n × 1 system of sequential fractional differential equations and may apply another type of fractional derivatives such as Psi-Hilfer and Psi-Caputo fractional derivatives.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.