Abstract
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.
1. 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 wide-ranging 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 [5–9]. 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 [12–18].
In 2016, Aljoudi et al. [19] published a study investigating the existence results for the following boundary value problem (sequential Hadamard type). where is the Hadamard fractional derivative, and is the Hadamard fractional integral with order , .
In 2017, Ahmad and Ntouyas [20] published a study investigating the existence results for the following initial value problem where is the Hadamard fractional derivative,
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 [23–26].
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. where are the Caputo and Caputo-Hadamard fractional derivatives of order .
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.
2. 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).
Definition 1 (see [7]). The Hadamard fractional integral of order for a continuous function is defined as
Definition 2 (see [7]). The Hadamard fractional derivative of order for a continuous function is defined as , where denotes the integer part of the real number .
Definition 3 (see [5]). The Caputo-Hadamard fractional derivative of order for at least differentiable function is defined as
Lemma 4 (see [20]). Let , then and
Denote the Banach space of all continuous functions from into by ,
accompanied by the norm: .
Definition 5 (see [28]). The Kuratowski measure of noncompactness .
Defined on bounded set of Banach space is
Lemma 6 (see [28]). Given the Banach space with U, V are two bounded proper subsets of , then the following properties hold true (I)(II)(III)(IV)(V)(VI)(VII)
Lemma 7 (see [29]). Given an equicontinuous and bounded set then the function.
is continuous on , and
Definition 8 (see [3]). Given the function satisfy the Carathéodory conditions, if the following conditions applies:
(I) is measurable in for
(II) is continuous in for
Theorem 9 (Mönch’s fixed point theorem [4]). Given a bounded, closed, and convex subset such that , let also be a continuous mapping of into itself.
If satisfied then has a fixed point.
3. Existence Results
Let Obviously, the defined set is a Banach space with .
The measurable functions 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 , then the solution of With is given by
Proof. Apply to Equation (10), respectively, implies
Now, apply to Equation (13) and Equation (14), respectively, implies
Using the conditions in Equation (15) and Equation (16), respectively, yields are both zeros. Again the conditions in Equation (15) and Equation (16), respectively, give
Back substituting obtained above in equations Equation (15) and Equation (16), we get
and . The proof is completed.
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.
(C1). Assume the functions satisfy Carathéodory conditions.
(C2). , and there exist a nondecreasing continuous function , such that we have
(C3). Let be a bounded set, and , then
Also, one can use the fact that to deduce that
Theorem 11. Assume that the conditions (C1), (C2), and (C3) are satisfied. If then there exist at least one solution for the boundary value problem Equation (3) on .
Proof. Beginning with introducing the following continuous operator as, where
According to the conditions (C1) and (C2), the operator is well defined. Then, the following operator equation can be an equivalent equation to the fractional equations given by Equation (11) and Equation (12)
Subsequently, proving the existence of the solution to Equation (22) is equivalent to proving the existence of a solution to Equation (3).
Let be a closed bounded convex ball in B with where .
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).
Firstly, we show that for this, we let and for any we have
Based on (C2), observe that
then
Similarly,
Equation (25) and Equation (26) imply that
This proves that .
Secondly, we need to show the continuity for to see this, we take the sequence such that .
Owing to the Carathéodory continuity of , 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
is the Lebsegue integrable on , we have
Yields to we get
that is the operator is continuous.
In a like manner, we have
Combining (31) and (32), we obtain
From equation (33), we conclude that the operator is continuous.
Third, to verify the equicontinuity for the operator , let , and for any then
Similarly, we get
Note that the R.H.S’s of the above inequalities of Equation (34) and Equation (35) are free of which implies that is equicontinuous and bounded.
Fourth and finally, we need to satisfy Mönch’s hypothesis, so we let.
where Moreover, are assumed to be bounded and equicontinuous, such that
Thus, the functions are continuous on .
Based on lemma Equation(10), lemma Equation (11), and (C3), we get
That is
but it is assumed that which implies that , i.e.,
In a like manner, we have So and.
, which implies that is relatively compact in Now, Arzela-Ascoli is applicable, which means that is relatively compact in , and therefore, using theorem 9, we deduce that the operator has a fixed point (solution of the problem Equation (3)) on And that ends the proof.
4. Example
In this section, we provide an applied example that supports the theoretical results reached through this study.
Define , it is obvious that is a Banach space with For this, we consider the following boundary value problem:
Here, , , , , , and
Now, let us take for example
, assumption (C1) of theorem 11 is satisfied. Furthermore, Similarly,
That is (C2) of theorem 11 is satisfied as well.
Next, if we consider the bounded subset we obtain where in our case, we have ; the latter two inequalities show that the condition (C2) of the theorem 11 is satisfied.
Finally, we calculate
Then, So all conditions of theorem 11 satisfied, that is the problem Equation (39) has at least one solution.
5. Conclusion
In the current paper, we studied the existence and uniqueness of solution for a coupled system of a mixed fractional differential equations. The main results are established by the aid “Mönch’s fixed point theorem.” In addition, an applied example that supports the theoretical results reached through this study is included. For future work, more investigations can be performed for such a system by applying another type of fractional derivatives to verify the existence and uniqueness issue, stability via Ulam-Hyeres technique is also possible to be verified.
Data Availability
No data sets were used in this study.
Conflicts of Interest
The author declares that he has no conflict of interest.
Acknowledgments
This work was supported through the Annual Funding track by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Project No. GRANT115], King Faisal University (KFU), Ahsa, Saudi Arabia. The author, therefore, acknowledges technical and financial support of DSR at KFU.