Applicability of Mönch’s Fixed Point Theorem on Existence of a Solution to a System of Mixed Sequential Fractional Differential Equation

In this paper, we study the existence and uniqueness of the solution for a coupled system of mixed fractional differential equations. The main results are established with the aid of “Mönch’s fixed point theorem.” In addition, an applied example that supports the theoretical results reached through this study is included.


Introduction
Fractional calculus has an extended history, going all the way back to Leibniz's 17th-century explanation of the derivative order in 1965. Mathematicians use fractional calculus to study how derivatives and integrals of noninteger order work and how they change over time. Subsequently, the subject attracted the interest of numerous famous mathematicians, including Fourier, Laplace, Abel, Liouville, Riemann, and Letnikov. For current and wideranging analyses of fractional derivatives and their applications, we recommend the monographs [1,2], and the recently mentioned papers [3,4].
Many problems in various scientific branches can be successfully studied using partial differential equations, such as theoretical physics, biology, viscosity, electrochemistry, and other physical processes see [5][6][7][8][9]. For example, but not limited, the authors in [10] employed the fractional derivative of the ψ-Caputo type in modeling the logistic population equation, through which they were able to show that the model with the fractional derivative led to a better approximation of the variables than the classical model. In addition, the authors in [11] employed the fractional derivative of the ψ-Caputo type and used the kernel Rayleigh, to improve the model again in modeling the logistic population equation.
The obvious difference between the ordinary differential equation and the fractional differential equation is that the latter is an equation that contains fractional derivatives and also comes in a relationship so that the definition of the fractional derivative is an integral equation on the other side of this equation. Fractional derivatives have drawn the attention of researchers in various fields of research. One of the main goals of solving these equations is to investigate whether these derivatives will help in the future in improving the accuracy of predicting the values of variables in various mathematical models in all sciences, whether in scientific or human aspects.
Before starting this research for solutions to these problems, which are recently considered in the applied sciences, verifying the issue of the existence and uniqueness of such equations is an indispensable thing. To study these conditions, most of the researchers use the most important fixed point theorems in the Banach space, such as the Banach contraction principle and Leray-Schauder theorem see [12][13][14][15][16][17][18].
In 2016, Aljoudi et al. [19] published a study investigating the existence results for the following boundary value problem (sequential Hadamard type).
where H D ð:Þ 1 , ν 1 , ν 2 ∈ ð1, 2, r 1 , r 2 ∈ ð0, 1Þ is the Hadamard fractional derivative, and H I θ 1 is the Hadamard fractional integral with order θ 1 , In 2017, Ahmad and Ntouyas [20] published a study investigating the existence results for the following initial value problem ℝÞ,x t ∈ Cð½−τ, 0, ℝÞ, where x t ðγÞ = xðt + γÞ, γ ∈ ½−τ, 0: Many researchers went deeper in their research beyond the issue of verifying the issue of the existence of a solution to such equations and studied the issue of the stability of these solutions, it can be seen in [21,22]. Furthermore, many specialists in this field have taken an interest in hybrid partial differential equations see [23][24][25][26].
Newly, interest in fractional calculus has increased from a purely mathematical theory and from an applied point of view in various sciences. Focusing on the theory, there are many experts in this field who have studied the existence of solutions for many types' fractional differential equations (FDEs) using the most famous fixed-point theories such as Banach's principle and nonlinear Leary-Schauder alternative. While a few of them tried other theories to examine the existence of solutions to these problems, Derbazi and Baitiche [27] publish one of these scientific papers.
The aim of this paper is to investigate the existence of solutions for the following nonlinear sequential fractional differential equation subject to the Dirichlet boundary conditions.
In this work, we will try to follow the researchers and specialists in this field, by working to prove the existence of a solution to the problem presented above. In which the work will be presented in this format: Section 2 contains some basic results for fractional calculus. Section 3 shows an important result for the establishment of our main findings, and after that, we present our main findings. In Section 4, an applied example is obtained illustrating what has been obtained in the theoretical aspect of this manuscript. In Section 5, a conclusion and future work section is introduced.

Preliminaries
This part is dedicated to presenting some definitions, postulates, and theorems related to the fixed point concept of solutions of differential equations, which will be used to verify the existence of a solution to the system of equations given by Equation (3).
The measurable functions ðψ, φÞ ∈ Cð½a, T, ℝÞ × Cð½a, T, ℝÞ are said to be solutions of problem Equation (3) if they satisfy problem (3) associated with the given boundary conditions, our next lemma will introduce the solutions of Equation (3), which indeed needed to investigate the existence results.

Lemma 10. If p, q ∈ Cð½a, T, ℝÞ, then the solution of
With 0 < α i , β i ≤ 1, i = 1, 2:a ≤ t ≤ T, is given by Proof. Apply RL I α i , i = 1, 2 to Equation (10), respectively, Now, apply H I β i , i = 1, 2 to Equation (13) and Equation (14), respectively, implies Using the conditions ψðaÞ = 0, φðaÞ = 0 in Equation (15) and Equation (16), respectively, yields c 1 , d 1 are both zeros. Again the conditions ψðTÞ = 0, φðTÞ = 0 in Equation (15) 3 Journal of Function Spaces and Equation (16), respectively, give Back substituting c i , d i , i = 1, 2 obtained above in equations Equation (15) and Equation (16) To begin formulating theoretical results regarding the problem of having a solution to the system of fractional differential equations given by Equation (3). We will force the following conditions to be hold true.
Let Θ ε = fðψ, φÞ ∈ B : kðψ, φÞk ≤ ε, ε > 0g be a closed bounded convex ball in B with ε ≥ l ς Ξ 1 ϑ ς ðεÞ + l ξ Ξ 2 ϑ ξ ðεÞ, where l ς = sup a≤t≤T l ς ðtÞ,. For the possibility of applying Mönch's fixed point theorem, we will proceed in the proof in the form of four steps, and thus, we achieve the desired goal by proving the existence of a solution to the equation given in Equation (3).
Owing to the Carathéodory continuity of ς, it is obvious that Keeping in mind was given in (C2), one can deduce that Together with the Lebesgue dominated convergence theorem and the fact that the function r ↦ l ς ϑ ς ðεÞððln ðt/rÞÞ β 1 −1 ðr − xÞ α 1 −1 Þ is the Lebsegue integrable on ½a, T, we have Yields to kϒ 1 ðψ n , φ n ÞðtÞ − ϒ 1 ðψ, φÞðtÞk ∞ ⟶ 0 as n ⟶ ∞:∀t ∈ ½a, T, we get that is the operator ϒ 1 is continuous. In a like manner, we have