A Nonlinear Implicit Fractional Equation with Caputo Derivative

Abstract

In this paper, we study a nonlinear implicit differential equation with initial conditions. The considered problem involves the fractional Caputo derivatives under some conditions on the order. We prove an existence and uniqueness analytic result by application of Banach principle. Then, another result that deals with the existence of at least one solution is delivered and some sufficient conditions for this result are established by means of the fixed point theorem of Schaefer. At the end, we discuss two examples to illustrate the applicability of the main results.

1. Introduction

The theory of differential equations of fractional order and fractional calculus is very important since they can be used in analyzing and modeling real word phenomena. Recently, several researchers are interested in the important progress of differential equations of fractional order. For more information on these works and their applications, one can consult the references [1–9]. In particular, research on the existence of unique solutions for fractional differential equations is of big importance since it helps physicians to better understand the behaviour of real phenomena. See, for more details, the references [10–14].

The motivation for this work arises from both the development of the theory of fractional calculus itself and its wide applications to various fields of science, such as physics, chemistry, biology, electromagnetism of complex media, robotics, and economics.

Much attention has been paid to the existence and uniqueness of solutions of fractional dynamical systems [15–18] due to the fact that existence is the fundamental problem and a necessary condition for considering some other properties for fractional dynamical systems, such as controllability and stability. Chai [19] provided sufficient conditions for the existence of solutions to a class of antiperiodic boundary value problems for fractional differential equations, while Sheng and Jiang [20] considered a class of initial value problems for fractional differential systems. There are several operators studied in the field of fractional calculus, for example, see [21–26], but the difference in this work is that the operator considered is in the sense of Caputo derivative.

Motivated by the works of Benchohra et al. [27], we will establish in this paper existence and uniqueness results of the solutions of the fractional dynamical system with Caputo fractional derivativewhere is in the sense of Caputo, is a given function, , is an matrix, and , , with .

Rest of the paper is organised as follows: in Section 2, we recall some results and definitions which we use for the proof of our main results. In Section 3, we give and prove the main theorems of this paper, and we discuss some illustrative examples.

2. Preliminaries

In this section, we introduce some definitions, lemmas, and preliminaries facts which are used throughout this paper. See [7] for more information. Let be a suitable norm in and be the matrix norm. We denote by the Banach space of continuous functions from to with the norm . We denote by the space of Lebesgue-integrable function with the norm

Letwith the norm

Definition 1. The Riemann–Liouville integral of order for a continuous function is given bywith .

Definition 2. If and , then the Caputo fractional derivative is given by

Lemma 1. Let and , then the general solution of is given bysuch that .

Lemma 2. Taking and , we havewith , .

Lemma 3. Let and . Then, it holds

Proof. For this proof, we use the same method in [28]. We haveWith the change of variable , we haveNow, we get

Definition 3. Let be a Banach space. Then, a map is called a contraction mapping on if there exists such thatfor all .

Theorem 1. (Banach’s fixed point theorem, see [29]). Let be a nonempty closed subset of a Banach space . Then, any contraction mapping of into itself has a unique fixed point.

Theorem 2. (Schaefer’s fixed point theorem, see [29]). Let be a Banach space, and let be a completely continuous operator. If the set is bounded, then has fixed points.

3. Main Results

We begin this section by some results that help us for solving the problem considered in (1).

Lemma 4. For any and , we have

Proof. By the definition of the operator , we have

Lemma 5. Let ,, and . Then, we can state that the problem,has for solution the following function

Proof. By applying to both sides of equation (16), we haveand using the property established in Lemmas 2 and 3, we find thatSome of the initial conditions allow us to have the result.
Conversely, assume that satisfy the equation (16), then we see easily the initial conditions.
We use the fact and , where is a constant; we getLet us now transform the above problem to a fixed point one. Consider the nonlinear operator defined byTo prove the main results, we need to work with the following hypotheses:(H1)The function defined on is continuous.(H2)There exist nonnegative constants , and such that, for any , ,Also, we consider the quantitiesThe first main result deals with the existence of a unique solution for (1). It is based on the application of Banach fixed point theorem for contraction mappings.

Theorem 3. If the conditions (H1) and (H2) are satisfied and , then problem (1) has a unique solution on .

Proof. It is sufficient for us to prove that is a contraction mapping.
Let . Then, we can writewhere defined by and .
From (H2) for each , we haveand using Lemma 4, we haveTherefore, we have for each ,On the other hand, we havewhich is clear in .
Then, with the same arguments as before, we haveThus, we haveSince , then the operator is contraction. Hence, by Banach’s contraction principle, has a unique fixed point which is the unique solution of problem (1).
The following main result deals with the existence of at least one solution of the studied problem.

Theorem 4. Under the hypotheses (H1) and (H2), problem (1) has at least one solution .

Proof. Let us prove the result by considering the following steps: Continuous of : if the proof is trivial, then it is omitted (we just apply the fact that is continuous. Uniform boundness of : let us take and consider the (bounded) ball . For , we can write With a simple calculus, we get where . Then, we have and also we have The above two inequalities show that  Consequently, is uniformly bounded. Equicontinuity of : we prove that, for any bounded set for instance, we obtain that is an equicontinuous set of . Take and consider the above (bounded) ball of . So, by considering , we can state that where . As , the right-hand side of the above inequality tends to zero, and we have also As , the right-hand side of the above inequality tends to zero. From a consequence of the Ascoli-Arzela's theorem, we conclude that is completely continuous. Boundness of : the set is bounded. Let . Then, we have for some . Hence, we can writeFrom (37) and (38), we state that . The set is thus bounded.
Consequently, thanks to Schaefer fixed point theorem, we deduce that has at least one fixed point. Thus, problem (1) has a solution.

Example 1. Let us consider the following example:wherewith , . We take , and .
We can see clearly that the function is continuous.
For any and ,which givewhere .
Hence, the hypotheses (H1) and (H2) are satisfied.
With a simple computation, we get and , which imply .
Thus, all the assumptions from (H1)–(H3) are satisfied. From Theorem 3, we conclude that equation (1) has a unique solution.

4. Conclusion

In this work, we consider a nonlinear implicit fractional differential equation and we use the Caputo derivative operator. We prove two theorems and an example to illustrate our results. In the first theorem, we prove the existence and uniqueness of the solution and the second theorem deals with the existence of at least one solution. The methods used are the Banach’s fixed point theorem and Schaefer’s fixed point theorem. Here, two Caputo derivative operators of different fractional orders were used in the considered equation and it would be relevant to generalize this idea by considering several Caputo operators of different fractional orders.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no known conflicts of interest or personal relationships that could have appeared to influence the work reported in this paper.