Abstract
This work investigates the existence and uniqueness of solutions for a coupled system of fractional differential equations with three-point generalized fractional integral boundary conditions within generalized proportional fractional derivatives of the Riemann-Liouville type. By using the Schauder and Banach fixed point theorems, we study the existence and uniqueness of solutions for the aforesaid system. Finally, we present an example to validate our theoretical outcomes.
1. Introduction
The theory of fractional calculus has become an attractive area of research for mathematicians and physicians because of its fertile aspects in many applications in natural science [1, 2], engineering [3], and many other fields. Moreover, the fractional differential equations have been employed successfully in the modeling of many biological problems, for example, human liver [4], hepatitis B [5–7], mumps virus [8] and methanol detoxification in the human body [9], and other differential models in thermodynamic and physics such as thermostat [10], pantograph [11], diffusion-wave system [12], and dynamical systems [13]. For additional specifics about the theory of fractional calculus and applications, we suggest the books of Kilbas et al. [14], Podlubny [15], and Samko et al. [16]. During the last years, there have exhibited several concepts about fractional derivatives. Here, we point out the most famous kinds including Riemann-Liouville, Liouville-Caputo, generalized Caputo [17], and Hadamard derivatives [18]. This has lead researchers to numerous research papers concerning several fractional operators which were conducted that one can see, for example, in complex plain [19, 20], extended Riemann-Liouville [21], the Mittag-Leer type function [22], the -derivative [23], the local fractional derivative [24], and in stability result [25–27].
More recently, Jarad et al. [28] constructed a new generalized fractional derivative which is called the generalized proportional fractional derivative. This new fractional operator has the advantage of being well-behaved as it is considered to be a generalization of many of the previously known and widely used fractional operators such as Liouville-Caputo and Riemann-Liouville fractional operators. In detail, fractional differential equations with generalized proportional derivatives have seen significant contributions from an interested researcher. For instance, we refer to works of Abbas and Ragusa and Hristova and Abbas [29, 30] and Khaminsou et al. [31, 32], and the references existing therein.
At the same time, coupled systems of differential equations of fractional order with different boundary conditions have been the focus of many mathematicians. The literature on the topic involves the existence, uniqueness, and stability results. Ahmad and Luca [33] studied a system of nonlinear Caputo fractional differential equations with coupled boundary conditions involving Riemann-Liouville fractional integrals. Baitiche et al. [34] discussed the existence and uniqueness of solutions to some nonlinear fractional differential equations involving the -Caputo fractional derivative with multipoint boundary conditions. Mahmudov et al. [35] investigated existence and uniqueness results for a coupled system of Caputo fractional differential equations with integral boundary conditions.
Some existing frameworks mentioned above encourage us to study the following coupled system of fractional differential equations:
equipped with the generalized fractional integral boundary conditions:
where , and denote the generalized proportional fractional derivatives of Riemann-Liouville type of order , and denote the generalized proportional fractional integrals of order , and and are continuous functions. In the current work, we establish the existence and uniqueness of solutions of the coupled system (1) and (2) by means of Schauder’s and Banach’s fixed point theorems.
To the best of our knowledge, there are no contributions considering a coupled system of generalized proportional fractional differential equations with generalized fractional integral boundary conditions.
The paper structure is designed as follows: in Section 2, we collect some essential definitions and lemmas relevant to the generalized proportional fractional derivatives and integrals; in Section 3, we establish the Green function associated with the linear issue of the coupled system (1) and (2), while in Section 4, we prove the main existence and uniqueness results in the current paper; in Section 5, an example is given to validate our theoretical outcomes.
2. Preliminaries
Here, we review some definitions of the generalized proportional fractional derivatives and integrals; see [28, 30, 36].
Definition 1 (see [37]). Take , let the functions be continuous such that for all we have , , , , for , and for . Then, the amended conformable derivative of order is defined by The above amended conformable derivative (3) is said to be a proportional derivative (see [37]). When and , (3) takes the form
Note that, and .
Remark 2. By using (4) for the function and any arbitrary order , it can be easily concluded that .
Example 1. If , then . One can find the graphs of for different value of , in Figure 1. As can be seen from Figure 1, in some points, the value of conformable derivative in this case is independent of , and this can be one of the interesting properties of fractional calculus.
Definition 3 (see [28, 30]). Take , , we define the left generalized proportional fractional integral of the function by and
Definition 4 (see [28, 30]). Take , , we define the left generalized Caputo-proportional fractional derivative of the function by and where , , and .
Definition 5 (see [28, 30]). Take , , we define the left generalized proportional fractional derivative of Riemann-Liouville type of the function by and where .
Lemma 6 (see [36]). If , and with and , we have the following statements: where
Theorem 7 (Schauder’s fixed point theorem) [38]. Let be a closed, convex, and nonempty subset of a Banach space ; let be a continuous mapping such that is a relatively compact subset of . Then, has at least one fixed point in .
3. The Equivalent Integral Equations
Let be the Banach space of all continuous functions from into with the norm
and be the product Banach space with the norm
Definition 8. By a solution of the coupled system (1) and (2), we mean a coupled ordered pair of continuous functions that satisfy (1) and (2).
Lemma 9. Let , , and . Then, the solution of is equivalent to the integral equation where
Proof. By applying the generalized fractional proportional integral to both sides of the first equation (16) and using (13) in Lemma 6, one has Using the boundary condition and (19), one has Using (18), the above equation becomes In the light of (8) and (10) in Lemma 6, the boundary condition and (19) give Again, using (18), the above equation takes the form Therefore, by merging equations (21) and (23), using (18), we get Thus, by inserting the values of and in (19), we obtain (17). The proof is finished.
By hint of Lemma 9, the solution of the system (1) and (2) is given by
where
4. Existence and Uniqueness Results
Define the operator by
where
According to Lemma 9, the solution of the coupled system (1) and (2) conforms with the fixed point operator .
For fulfillment the main results, the following assumptions will be imposed.
(A1) The functions are continuous
(A2) There exist nonnegative constants and such that for each and
Further, we set .
The following notations will be introduced:
Theorem 10. Assume that the assumptions (A1) and (A2) are satisfied. Then, the coupled system (1) and (2) has at least one solution on .
Proof. Consider the operator as defined in (27). Let us introduce the ball
where is a positive real number such that
It is obvious that is a closed, bounded, and convex subset of the Banach space . We shall show that achieves the hypothesis of Schauder’s fixed point theorem in four steps.
Step 1. maps bounded sets into bounded sets in .
By virtue of (A2) and since , then for each and , one has
Thus, by using (31), we get
Similarly, we obtain that
where is defined in (32). Hence, we conclude that
which implies that
Step 2. is continuous.
In view of the assumption (A1), we conclude that and are continuous on . Thus, the operator is also continuous
Step 3. is equicontinuous.
Set . For , with and , we have
where the mean value theorem is used on the function with .
Thus, we get
In an identical way, we obtain that
As , the R.H.S. of the last two inequalities independently of . As a consequence of Steps 1 to 3 and in the light of the Arzelá-Ascoli theorem, we conclude that the operator is relatively compact in . Hence, in accordance with Schauder’s fixed point theorem (Theorem 7), the operator has a fixed point and so the coupled system (1) and (2) possesses at least one solution on . The proof is completed.
Theorem 11. Assume that the assumptions (A1) and (A2) are satisfied. If , then the coupled system (1) and (2) has a unique solution on .
Proof. Consider the operator as defined in (27). We have to show that is a contraction mapping.
For each and , one has
Thus, by (31), we get
In a similar way, using (32), we get
From (43) and (44), we get
By virtue of the condition , we conclude that is a contraction mapping.
Hence, with the aid of Banach’s fixed point theorem, we deduce that has a unique fixed point, and so the coupled system (1) and (2) possesses a solution on uniquely. The proof is finished.
Example 2. Consider the following coupled system of fractional differential equations with the generalized fractional integral boundary conditions:
Here, , and . Set and that their graphs show in Figures 2 and 3.
For each and , one has
which implies that the assumption (A2) holds true with and . We calculate functions in (18), (26), (31), and (32) for , and and present their numerical results in Table 1. We have in all three cases:
By virtue of the above discussion, we infer that all the assumptions of Theorems 10 and 11 are satisfied. Consequently, we deduce that the coupled system (46) and (47) has a solution on uniquely.
5. Conclusion
As you know, there are many events in nature which we know nothing about those. One of the best ways for better understanding these types of phenomena is studying new notions in the fractional calculus field. In this work, we investigated the existence and uniqueness of solutions for a coupled system of fractional differential equations with three-point generalized fractional integral boundary conditions in the frame of the generalized proportional fractional derivatives of the Riemann-Liouville type which was introduced in 2017 by Jarad et al. In this way, we provided some results under some conditions. To better explain the notion, we gave some figures of some functions. Finally, we provided an illustrated example for our main result.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.
Acknowledgments
The second and third authors were supported by the Azarbaijan Shahid Madani University. The fourth author was supported by the University of Aden.