Fixed Point Theory and Applications for Function Spaces 2021View this Special Issue
On System of Mixed Fractional Hybrid Differential Equations
In this article, we find 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.
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 .
Fractional derivatives have played a very important role in mathematical modeling in many diverse applied sciences, see [2, 3]. For example, the authors in  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  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  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 [7–14].
In 2012, the authors in  studied a nonlinear three-point 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.
Here, is the ordinary derivative, is a real number such that .
In 2019, Ahmad et al.  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  in which fractional hybrid modeling of a thermostat is simulated, for some recent results on hybrid.
For FDEs, we refer to [19, 20]. Freshly, some authors have studied different characteristics of hybrid FDEs including the existence of solutions, see for some detail [21–29], and some go further and studied Hyers-Ulam stability for FDEs by different mathematical theories, see for some detail .
Zhao et al.  investigated the existence result for the fractional hybrid differential equations with Riemann–Liouville fractional derivatives given by
where is Riemann–Liouville fractional derivative, are assumed to be continuous.
Hilal and Kajouni  studied the Caputo hybrid BVP of the form
in which , are assumed to be continuous and .
In , 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
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.
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 ). The Riemann-Liouville (RL) fractional integral is defined by
Definition 2 (see ). The Caputo fractional derivative of order of a function is given by
Theorem 3 (see , Banach’s contraction mapping principle). Let be a complete metric space; is a contraction then (i) has a unique fixed point ; that, is (ii), we have
Theorem 4 (see , nonlinear alternative of Leray-Schauder type). Assume that is an open subset of a Banach space , , and be a contraction such that is bounded then either (i) has a fixed point in , or(ii) and such that holds
Theorem 5 (see , Arzela-Ascoli theorem). is compact if and only if it is closed, bounded, and equicontinuous.
3. Main Results
Lemma 6. If , and then the solution to the problem mentioned above is given by
Proof. Taking to , then take to the resulting equation, we get Substitution of and in Equation (11) gives and , respectively, and consequently, Equation (6) becomes Use of the condition in Equation (12) yields Inserting in Equation (12) gives Alternatively, we have Equation (15) is equivalent to Equation (10), which makes the proof done.
Denote the Banach space by with the norm . Then, the product space with the norm , is indeed a Banach space too. We define an operator as where
To construct the necessary conditions for the results of uniqueness and existence of the problem (6), let us consider the following hypotheses.
(C1) Let the functions and are assumed to be continuous and bounded; that is, such that
(C2) Let the functions and are assumed to be continuous, and such that
(C3) There is positive constants , , and such that
(C4) Let be a bounded set, then such that and
To facilitate the calculations below, let us say
Theorem 7. If both (C1) and (C2) are satisfied, and assume that . Then, the system in Equation (6) has a unique solution.
Proof. Define a closed ball with , where .
Observe that .
First, we show that . For any , we have similar to what was done above, we get From Equation (25) and Equation (26), we deduce that .
Next, for , we have Similarly, we can find Combining Equation (27) and Equation (28) yields Equation (29) becomes . That is, is a contraction; consequently, Banach fixed point theorem applies; thus, the uniqueness of solutions for Equation (6) holds on .
Theorem 8. If (C1), (C3), and (C4) are satisfied, and if and , then Equation (6) has at least one solution.
Proof. In the first step, we verify that the operator is completely continuous; obviously, the operator is continuous as a result that , , , and are all assumed to be continuous.
With the aid of (C4), , we have Similarly, Combining the inequalities (30) and (31) yields , implying that is uniformly bounded.
Next, to verify the equicontinuity for the operator , we let then The R.H.S for both inequalities (33) and (34) tend to zero as , and they are both independent on . So, operator is equicontinuous and yields; is completely continuous.
Finally, we establish the bounded set given by ; then, ; the equation gives Using the hypothesis (C3), we get Consequently, we have Inequality (37) can be written as follows: where .
Inequality (38) shows that is bounded. Hence, Leray-Schauder alternative applies, implying the existence of the solution for Equation (6).
In this part, we address the issue of stability of solutions to the system of equations defined by Equation (6) via U-H definition.
Definition 9. The system of the coupled sequential fractional differential BVPs Equation (6) is stable in U–H sense if a real number exists so that, for any and for any satisfying there exists a unique solution of (6) with It is clear that satisfies the inequalities (39) if there exists a function (which depends on ), such that (i)(ii)For
Theorem 10. Suppose that (C2) is fulfilled. Moreover Then, the system of coupled sequential fractional differential BVPs (6) is U–H stable.