Abstract

This article studies a class of implicit fractional differential equations involving a Caputo-Fabrizio fractional derivative under Dirichlet boundary conditions (DBCs). Using classical fixed-point theory techniques due to Banch’s and Krasnoselskii, a qualitative analysis of the concerned problem for the existence of solutions is established. Furthermore, some results about the stability of the Ulam type are also studied for the proposed problem. Some pertinent examples are given to justify the results.

1. Introduction and Preliminaries

The concerned area of fractional order differential equations (FODEs) have many concentrations in real-world problems and have paid close attention to numerous researchers in the past few decades [1–5]. The mentioned area has been studied from several aspects, such as the existence and uniqueness of solutions via using the classical fixed-point theory, the numerical analysis, the optimization theory, and also the theory of stability corresponding to various fractional differential operators like Caputo, Hamdard, and Riemann-Liouville (we refer few as [6–9]). In the aforementioned operators, there exists a singular kernel. Therefore, recently some authors introduced some new types of fractional derivative operators in which they have replaced a singular kernel by a nonsingular kernel. The nonsingular kernel derivative has been proved as a good tool to model real-world problems in different fields of science and engineering [10, 11]. In fractional, it is called nonsingular exponential type or Caputo-Fabrizio fractional differential (CFFD) operator. The CFFD operator introduced two researchers, Caputo and Fabrizio for the first time in 2015 [12]. They replaced the singular kernel in the usual Caputo and Riemann-Liouville derivative by an exponential nonsingular kernel. The new operator of this type was found to be more practical than the usual Caputo and Riemann-Liouville fractional differential operators in some problems (see some detailed references such as [13–15]). Recently, many researchers have studied the existence and uniqueness of the solutions at the initial value problems for FODEs under the said operator. But the investigation has been limited to initial value problems only. On the other hand, boundary value problems have significant applications in engineering and other physical sciences during modeling numerous phenomena (we refer to see [16–19]). Furthermore, during optimization and numerical analysis of the mentioned problems, researchers need stable results from theoretical as well as practical sides. A stable result may lead us to a stable process. Therefore, the stability theory has also got proper attention during the last many decades. It is well known fact that stability analysis plays an important role. Various stability concepts such as exponential stability, Mittag-Lefler stability and Hayers-Ulam’s stability have been adopted in literature to study the stability of different systems of FODEs. The analysis of Hyers-Ulam’s stability has been recognized as a simple form of investigation. For historical background on the stability of Hyers-Ulam, we refer to see previous articles [20–23]. But recently, that type of problem has not been adequately studied for a new type of CFFD operator. Therefore, in this work, we will investigate an implicit class of FODEs involving the CFFD operator under DBCs where is used for CFFD and , is a continuous function. In this article, we investigate uniqueness and existence of solutions to the proposed problem (1) by classical fixed-point theorems due to Banach’s and Krasnoselskii. Further, we investigate some pertinent analysis about the stability theory due to Ulam, and Hyers is investigated for the mentioned problem (1). For the authenticity of the presented work, two concrete examples are also studied.

Throughout the paper, is a Banach space with norm .

Definition 1 (see [24]). For any , we defined the derivative of Caputo-Fabrizio for nonsingular kernel as where is the normalization function with satisfying.

Definition 2 (see [24]). The integral of Caputo-Fabrizio for nonsingular kernel type is given by where is used for Caputo-Fabrizio integral operator.

Definition 3 (see [25]). Let and be such that . Set . Then, and we define

Lemma 4. For defined on and , for some , we have

2. Results and Discussion

In this part, we investigate the solution of the proposed problem (1) and also study the uniqueness and existence of the solutions.

Lemma 5. The solution of is given by

Proof. Let be a solution to problem (6). Applying Caputo-Fabrizio integral on both sides and then using Lemma 4 and Definition 3, we have which implies that Using boundary conditions , we have Putting in (9), we get

For simplification, use some notations; we use and give the solution of (1) as bellow.

Corollary 6. In view of 6, the solution of the considered problem (1) is given by Further, for the existence and uniqueness of the solution of problem (1), we use some fixed point theorems. For this, we need to define an operator as by To proceed further, using Corollary (6) to convert the proposed problem (1) is to a fixed point problem as , where the operator is given by (13). Therefore, Problem (1) has a solution if and only if the operator has a fixed point, where and . We assume that
(H₁) There exist certain constant and , such that for all

Theorem 7. Under the hypothesis (H₁), the mentioned problem (1) has a unique solution if

Proof. Suppose we have where are given by and by using hypothesis (H₁), we have Repeating the above process, we get Using (18) in (16), we have Applying maximum on both sides, we have Thus, operator is a contraction; therefore, the operator has a unique fixed point. Hence, the corresponding problem (1) has a unique solution.

Our next result is to show the existence of the solution to the proposed problem (1) which is based on Krasnoselskii’s fixed-point theorem. Therefore, the given hypothesis hold.

(H₂) There exist constant with such that

Theorem 8 (see [26]). Let be a closed, convex nonempty subset of ; then, there exist operators such that (1) for all (2) is a contraction, and is compact and continuousThen, there exist at least one solution such that

Theorem 9. If the hypothesis (H₂) is satisfied, then (1) has at least one solution if

Proof. Suppose we define two operators from (13) as Let us define a set , since is continuous, so we show that the operator is contraction. For this we have using (18), and then taking the maximum on both sides, we have Hence, is contraction. Next, to prove that the operator is compact and continuous, for this , we have where , ; now, using hypothesis (H₂), we have repeating the above process, so we get Now, using (28) in (26) and then taking the maximum on both sides, we have Which implies that Therefore, is bounded. Next, let in , we have Now, using (28) in (31), we have Applying maximum on right-hand side of the above inequality, we take Obviously, from (33), we see that ; then, the right-hand side of (33) goes to zero, so as . Hence, the operator is continuous. Also, ; therefore, the operator is compact, and by the Arzela-Ascoli theorem, the operator has at least one fixed point. Therefore, the mentioned problem (1) has at least one solution.