Research On System of Mixed Fractional Hybrid Differential Equations

In this article, we ﬁ nd the necessary conditions for the existence and uniqueness of solutions to a system of hybrid equations that contain mixed fractional derivatives (Caputo and Riemann-Liouville). We also verify the stability of these solutions using the Ulam-Hyers (U-H) technique. Finally, this study ends with applied examples that show how to proceed and verify the conditions of our theoretical results.


Introduction
Although the concept of fractional calculus was established 300 years ago, interest in this type of derivative appeared for a short period. So that it is no secret to anyone that the most important use of fractional derivatives is to find analytical solutions to differential equations if possible, or by using numerical analysis methods to find an approximation to these solutions. In this study, we will focus on the idea of studying theories that investigate the existence of a solution to a system of hybrid fractional equations that contain mixed fractional derivatives with boundary conditions attached to them.
As mentioned before, fractional calculus as a concept is not very recent. It is worth mentioning here the great names who have given a lot to this science, such as A.V. Letnikov, J. Hadamard, J. Liouville, B. Riemann M., and Caputo L. worked in this field. These names must be mentioned by way of example. To get acquainted with some of the names of scientists who have made great contributions to fractional calculus in the modern world, we ask the reader to look at [1].
Fractional derivatives have played a very important role in mathematical modeling in many diverse applied sciences, see [2,3]. For example, the authors in [4] employed the fractional derivative of the Psi-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 [5] employed the fractional derivative of the Psi-Caputo type and used the kernel Rayleigh, to improve the model again in modeling the logistic population equation.
As a final example, the authors in [6] employed the fractional derivatives of the Caputo and Caputo-Fabrizio type by modeling the equation that gives the relationship between atmospheric pressure and altitude, and they were also able to show that the fractional equation gave less error in estimating atmospheric pressure at a certain altitude. There are many scientific papers in the literature that prove the superiority of fractional derivatives over classical ones.
There are a large number of manuscripts published in the literature that investigate the issue of the existence of a solution to fractional differential equations, whether they are sequential equations of type or nonsequential equations [7][8][9][10][11][12][13][14].
In 2012, the authors in [7] studied a nonlinear threepoint boundary value problem of sequential fractional differential equations. Green's function of the associated problem involving the classical gamma function is obtained. Existence results are obtained using Banach's contraction mapping principle and Krasnoselskii's fixed point theorem.
In 2019, Ahmad et al. [15] developed the existence theory for a new kind of nonlocal three-point boundary value problems for differential equations involving both Caputo and Riemann-Liouville fractional derivatives. The existence of solutions for the multivalued problem concerning the upper semicontinuous and Lipschitz cases is proved by applying nonlinear alternative for Kakutani maps and Covitz and Nadler fixed point theorem.
It is known that fractional calculus and FDEs are used in different fields such as physics, signal and image processing, control theory, robotics, economics, biology, and metallurgy, see for example [16,17] and references therein. On the other hand, recently, many researchers have paid much attention to hybrid differential equations of fractional order. This is because of the development and new advanced applications of fractional calculus. The fractional hybrid modeling is of great significance in different engineering fields, and it can be a unique idea for future combined research between various applied sciences, for example, see [18] in which fractional hybrid modeling of a thermostat is simulated, for some recent results on hybrid.
In [31], the authors have considered the following coupled hybrid system. A new generalization of Darbo's theorem associated with measures of noncompactness is the main tool in their approach: supplemented with nonlocal hybrid boundary conditions. Inspired by the aforementioned studies, the following sequential hybrid BVP is considered for investigating the existence of the solution and for the stability of its solution via the U-H sense

Journal of Function Spaces
After this introductory section of this work, the manuscript is organized as the following hierarchical structure: Section 2 delivers the basic elements of fractional calculus definitions, Section 3 introduces the main results of the work, Section 4 introduces the (U-H) stability result for our problem, and the last section is arranged for a numerical example to support the theoretical results.

Preliminaries
In this part, we present some basic elements and definitions needed to find solutions to the main mathematical problem presented in this study.
Definition 1 (see [3]). The Riemann-Liouville (RL) fractional integral is defined by Definition 2 (see [3]). The Caputo fractional derivative of order ν of a function ϑ : ℝ + ⟶ ℝ is given by Theorem 3 (see [3], Banach's contraction mapping principle). Let ðS, dÞ be a complete metric space; H : S ⟶ S is a contraction then Theorem 4 (see [3], nonlinear alternative of Leray-Schauder type). Assume that V is an open subset of a Banach space U, 0 ∈ V, and F : V ⟶ U be a contraction such that Fð VÞ is bounded then either and v ∈ ∂V such that v = μFðvÞ holds Theorem 5 (see [2], Arzela-Ascoli theorem). F ⊂ CðU, ℝÞ is compact if and only if it is closed, bounded, and equicontinuous.

Lemma 6.
If h ∈ Cð½0, 1, ℝÞ, and then the solution to the problem mentioned above is given by Proof. Taking Substitution of χð0Þ = 0 and χ ′ ð0Þ = 0 in Equation (11) gives a 2 = 0 and a 0 = 0, respectively, and consequently, 3 Journal of Function Spaces Equation (6) becomes Use of the condition χð1Þ = δχðζÞ in Equation (12) yields Inserting a 1 in Equation (12) gives Alternatively, we have Equation (15) is equivalent to Equation (10), which makes the proof done.
Denote the Banach space by C = C½0, 1 with the norm k hk = sup 0≤t≤1 jhðtÞj. Then, the product space ðC × C, kðχ, ϑÞkÞ with the norm kðχ, ϑÞk = kχk + kϑk, ∀ðx, yÞ ∈ C × C is indeed a Banach space too. We define an operator ϒ : where To construct the necessary conditions for the results of uniqueness and existence of the problem (6), let us consider the following hypotheses.
Proof. In the first step, we verify that the operator ϒ : C × C ⟶ C × C is completely continuous; obviously, the operator is continuous as a result that ℏ, ƛ, ψ, and φ are all assumed to be continuous.

Stability
In this part, we address the issue of stability of solutions to the system of equations defined by Equation (6) via U-H definition.