Abstract

In this paper, we study existence and uniqueness of solutions for a system of Caputo-Hadamard fractional differential equations supplemented with multi-point boundary conditions. Our results are based on some classical fixed point theorems such as Banach contraction mapping principle, Leray-Schauder fixed point theorems. At last, we have presented two examples for the illustration of main results.

1. Introduction

In recent years, fractional differential equations (FDE) gain enormous attention among scientists due to the applications which were not possible with ordinary or partial differential equations of integer order. FDEs becomes a very successful tool in modeling anomalous diffusion and fractal-like nature. Agrawal discusses diffusion and heat equations of fractional order in [1–3]. Agrawal et al., Baleanu, and others investigated the boundary value problems for fractional differential equations [4]. Fractional dynamic models, fractional control systems, fractional population dynamics models, and fractional fluid dynamics all involve at least one ordinary or partial fractional derivative.

Fractional differential equations have several kinds of fractional derivatives, such as Riemann-Liouville fractional derivative, Caputo fractional derivative, and Grunwald-Letnikov fractional derivative. Another kind of fractional derivative is Hadamard type which was introduced in 1892 [5]. This derivative differs from various derivatives in the sense that the kernel of the integral in the definition of Hadamard derivative contains logarithmic function of arbitrary exponent. A detailed description of Hadamard fractional derivative and integral can be found in [6]. The readers who are interested in the subject of fractional calculus is referred to the books by Kilbas et al. [7], Podlubny [8], Miller and Ross [9], Samko et al. [10], Diethelm [11], and Zhou [12] and the references therein.

Coupled systems of fractional differential equations play a key role in developing differential models such as the synchronization of chaotic systems [13–15], anomalous diffusion [16, 17], disease models [18, 19], ecological models [20], Lorenz system [21], and nonlocal thermoelectricity systems [22, 23]. For recent theoretical results on the topic, we refer the reader to a series of papers [24–37] and the references cited therein. Ahmad and Ntouyas [32, 33] discussed some fractional integral boundary value problems involving Hadamard fractional differential equations/systems and obtained the existence and uniqueness of solutions by applying the Banach fixed point theorem and Leray–Schauder alternative, respectively.

In [35], the authors investigated the existence and uniqueness of solutions for the coupled system of nonlinear fractional differential equations with three-point boundary conditions where and are the standard Riemann-Liouville fractional derivative and are given continuous functions.

Recently, Alsulami et al. [36] established the existence and uniqueness results for a nonlinear coupled system of Caputo type fractional differential equations supplemented with nonseparated coupled boundary conditions. where denote the Caputo fractional derivatives of order and , respectively, are appropriately chosen functions, and are real constants with

Motivated by the research going on in this direction, in this paper, we study existence and uniqueness of solutions for a coupled system of Caputo-Hadamard fractional differential equations. with multipoint boundary conditions where , for for are real positive constants denotes the Caputo-Hadamard fractional derivatives of order for , are appropriately chosen functions.

The paper is organized as follows. In Sect. 2, we present some preliminary concepts of fractional calculus. Sect. 3 contains main results concerning the existence and uniqueness of solutions for the given problem (3), (4). The Leray-Schauder alternative theorem is applied to prove existence, while the uniqueness result was obtained via the Banach contraction mapping principle. Finally, we also discuss some examples for illustration of the existence-uniqueness results.

2. Preliminaries

For the convenience of the reader, we present some concepts of Hadamard type fractional calculus to facilitate the analysis of system (3), (4).

Definition 1 [7]. The Hadamard fractional integral of order of a function for all is defined by where is the gamma function,s provided the right side is pointwise defined on .

Definition 2 [7]. The Hadamard fractional derivative of order of a function for all is defined by where with denotes the integral part of the real number and .

Definition 3 [38]. Let and If where and The Caputo type modification of the Hadamard fractional derivative of order is defined by

Theorem 4 [38]. Let , and . If where Then exist everywhere on and (i)if , can be represented by (ii)if then

Remark 5. If then

Lemma 6 [38]. Let and . If , then the Caputo-Hadamard fractional differential equation has a solution: and the following formula holds: where .
Now, we present an auxiliary lemma for boundary value problem of linear fractional differential equation with Caputo-Hadamard derivative.

Lemma 7. Let and . Let . Then, the solution of the linear Caputo-Hadamard fractional differential system is equivalent to the system of integral equations where

Proof. We apply Lemma [6] that the general solution of the Caputo-Hadamard fractional differential equation in (13) can be written as: where are arbitrary real constants. From (17) and (18) we have Using the boundary conditions and from (17) and (18), we have Using the boundary conditions and from (19) to (22), we have Solving the resulting equations for and , we find that substituting and in (23) and (24), we have and Inserting the values of in (17) and (18), which leads to the solution system (14), (15). The converse follows by direct computation. The proof is completed.

3. Existence and Uniqueness Results

This section is concerned with the main results of the paper. First of all, we fix our terminology. Let be the Banach space of all continuous functions from to . Space endowed with the norm is a Banach space. In addition, let with the norm It is obvious that product space is a Banach space with the norm

In view of Lemma 7, we introduce an operator as follows: where

and

Here,

For computational convenience, we set

Now, we are in a position to present our main results. The methods used to prove the existence and uniqueness solutions of boundary value problem (3), (4) via Banach’s contraction principle.

Theorem 8. Suppose that are continuous functions. In addition, we assume that:
(H1) there exist constants and such that for all and we have Then, the system (3), (4) has a unique solution on , if

Proof. Define and and such that Now, we show that , where .
By assumption for we have that which leads to Hence, In the same way, we can obtain that Consequently, it follows that which implies . Next, we show that operator is contraction mapping.
For any and for any we obtain Therefore, we get the following inequality Similarly, From inequalities (43) and (44), it yields Since therefore, is a contraction operator. So, by applying Banach’s fixed point theorem, the operator has a unique fixed point in . Hence, there exists a unique solution of problem (3), (4) on .
Now, we prove our second existence result via the Leray-Schauder alternative.

Lemma 9 (Leray-Schauder alternative [39]). Let be a completely continuous operator (i.e., a map restricted to any bounded set in is compact). Let Then, either the set is unbounded or has at least one fixed point.

Theorem 10. Assume that:
(H2) are continuous functions and there exist real constants and such that we have If and then system (3), (4) has at least one solution on .

Proof. By the continuity of functions on the operator is continuous. Now, we show that the operator is completely continuous. Let be bounded. Then, there exist two positive constants, and , such that Then, for any we have which yields, In the same way, we can obtain that . Hence, from the above inequalities, we get that the operator is uniformly bounded, since
Next, we show that is equicontinuous. For any , and with Then, we have Therefore, we obtain Analogously, we can get the following inequality: Then we can easily show that the operator is equicontinuous. As a consequence of steps together with the Arzel-Ascoli theorem, we get that the operator is completely continuous.
Finally, it will be verified that the set is bounded. Let with For any we have Then, we have which implies that Consequently, where which proves that is bounded. Therefore, by applying Lemma 9, the operator has at least one fixed point in . Therefore, we deduce that the boundary value problem (3), (4) has at least one solution on .

4. Some Examples

In this section, we give an example to illustrate our main results.

Example 11. Consider the following system of Caputo-Hadamard boundary value problem: Here, By simple calculation, we found that . (i)Let two nonlinear functions be given byNote that we obtain Thus, all the conditions of Theorem 8 are satisfied. Problem (59) with (60) and (61)has a unique solution on . (ii)Let two nonlinear functions be given byNote that We get By simple calculation, we have and By Theorem 10, the coupled boundary value problem (59) with (63) and (64) has at least one positive solution on [1,e].

5. Conclusions

In this paper, we studied existence and uniqueness of solutions for the system of Caputo-Hadamard fractional differential equations with multipoint boundary conditions. The existence theory of solutions of a Caputo-Hadamard system using a variety of fixed point theorems. The Leray-Schauder alternative was applied to prove existence, while the uniqueness result was obtained via the Banach contradiction mapping principle. Finally, we have given two examples to demonstrate our result.

Data Availability

No data were used to support this study.

Conflicts of Interest

There is no competing interest among the authors regarding the publication of the article.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.