#### Abstract

The existence of solutions for a coupled system of time-fractional differential equations including continuous functions and the Caputo-Fabrizio fractional derivative is examined. After that we investigated a coupled system of time-fractional differential inclusions including compact- and convex-valued -Caratheodory multifunctions and the Caputo-Fabrizio fractional derivative.

#### 1. Introduction

The fractional calculus is nowadays an excellent mathematical tool which opens the gates for finding hidden aspects of the dynamics of the complex processes which appear naturally in many branches of science and engineering [16]. The methods and techniques of this type of calculus are continuously generalized and improved especially during the last few decades. We recall that the existence and multiplicity of positive solutions corresponding to singular fractional boundary value problems were discussed in [7]. Also, the existence results for several nonlinear fractional differential equations were reported in [8]. Besides, the existence of positive solutions corresponding to a coupled system of multiterm singular fractional integrodifferential boundary value problems was shown in [9]. Inventing new derivatives and applying them to study the dynamics of complex systems are an important priority for researchers. As a result, very recently, a new fractional derivative without singular kernel has been provided [10, 11]. By using the main results presented in these two new works, we present the next definition.

Definition 1 (see [10]). The order Caputo-Fabrizio time-fractional differential derivative of the function is written aswhere represents a normalization function, , and .

Note that whenever is a constant function and the kernel has no singularity at [10, 11]. Also, Losada and Nieto defined the new time-fractional integral based on the new definition of Caputo-Fabrizio fractional derivative [11]. By using this idea, we provide the notion of Caputo-Fabrizio time-fractional integral.

Definition 2. The order time-fractional integral of a function has the form [11]where represents a normalization function and .

Losada and Nieto showed that for all [11]. By substituting in (1), we obtain the definition of the time-fractional Caputo-Fabrizio derivative of order for a function as follows:They proved that solution of is given bywhere and [11].

The next step is to consider being a metric space. Let us denote by and the class of all subsets and the class of all nonempty subsets of , respectively. Hence , , , , and are the class of all closed subsets, the class of all bounded subsets, the class of all convex subsets, the class of all compact subsets, and the class of all compact and convex subsets of , respectively. We claim that is a fixed point of the multifunction whenever [12]. A multifunction is called measurable whenever the function is measurable for all [12]. The Pompeiu-Hausdorff metric is defined by such that [13]. is a metric space and depicts a generalized metric space. Here denotes the set of closed and bounded subsets of and represents the set of closed subsets of [12, 13]. We recall that is said to be convex-valued (compact-valued) whenever is convex (compact) set for each [12]. We mention that a multifunction is a contraction whenever there exists a constant such that for all [12]. In 1970, Covitz and Nadler proved that each closed-valued contractive multifunction on a complete metric space has a fixed point [14].

Below we examine the existence of solutions for two coupled systems of nonlinear time-fractional differential equations and inclusions within Caputo-Fabrizio time-fractional derivative. First, we discuss the coupled system, namely,such thatwhere , , , and the mappings are continuous functions. In addition, we discuss the existence of solutions for the coupled system of nonlinear time-fractional differential inclusionssuch thatwhere are some multivalued maps.

We say that is a Caratheodory multifunction whenever is measurable for all and is upper semicontinuous (u.s.c) for almost all and [12]. A Caratheodory multifunction is said to be an -Caratheodory whenever for each there exists such that for all and for almost all [12]. The set of selections of at is defined by for all for . The sets are nonempty for all whenever [12, 15]. The graph of the multifunction is defined by the set (see [12, 13]). We say that the graph of is a closed subset of whenever for all sequences in and in with , , and for all we have [12]. Below we introduce the following results which will be required in our proofs.

Theorem 3 (see [12]). Suppose that is a Banach space, is a completely continuous operator, and the set is bounded. Then, has a fixed point.

Lemma 4 (see [12, Proposition ]). If is upper semicontinuous, then is a closed subset of . If is completely continuous with a closed graph, then it is upper semicontinuous.

Lemma 5 (see [12]). Let be a separable Banach space and an -Caratheodory function. Then the operator defined by is a closed graph operator, where is a linear continuous mapping from into .

Theorem 6 (see [12]). Let be a Banach space, a closed convex subset of , an open subset of , and . Let us suppose that depicts an upper semicontinuous compact map, such that denotes the family of nonempty, compact convex subsets of . Then either admits a fixed point in or there exist and such that .

#### 2. Main Results

First, we investigate the coupled system equipped with the boundary value conditions and , where are continuous mappings, , , and and are the Caputo-Fabrizio time-fractional derivatives. Now, consider the Banach space endowed with the sup-norm . Thus, the product space is also a Banach space via the product norm . First, we prove the next key lemma.

Lemma 7. Suppose that and . The function is a solution for the time-fractional integral equationif and only if is a unique solution of the time-fractional differential equation

Proof. A solution of initial value problem (14) is denoted by . As a result and . By integrating both sides we get and so . This shows that represents the solution of time-fractional integral equation (13). If and are two distinct solutions for initial value problem (14), then and . By the property of the Caputo-Fabrizio time-fractional derivative in [11], we get . Hence, is a unique solution of initial value problem (14). Now, suppose that is a solution of time-fractional integral equation (13). Then, we conclude that +. By using (4), one can see that this function represents a solution for initial value problem (14). Note that .

Now, we consider (1)-(2). For each , define the operators byand put

Theorem 8. Suppose that are the continuous mappings in system (6)-(7) and there exist positive constants and fulfilling and for all and . Then, system (6)-(7) possesses at least one solution.

Proof. Let the operators defined by (16). We define the operator by for all . Note that is continuous because the mappings and are continuous. We prove that the operator maps bounded sets into the bounded subsets of . Let be a bounded subset of , , and . Then, we haveand so . Also, we haveand so . Thus, . This shows that the operator maps bounded sets into the bounded sets of . Now, we show that the operator is equicontinuous. Let with . Then, we haveThis implies that whenever . By utilizing the Arzela-Ascoli theorem, is completely continuous. Similarly, we have Again, by utilizing the Arzela-Ascoli theorem we observe that is completely continuous. Therefore, we get whenever tends to . Thus, is completely continuous. In the next step we prove that is bounded. Let be an arbitrary element of . Choose fulfilling . Hence, and for all . Since we get and so . Similarly, we prove that . Thus, and so is a bounded set. Now, by using Theorem 3, we get that has a fixed point which is a solution for the coupled system of the time-fractional differential equations.

Next we study the existence of solution for the coupled system of time-fractional differential inclusions with the initial value conditions and , where are some multivalued maps.

Definition 9. One says that is a solution for the system of the time-fractional differential inclusions whenever it satisfies the initial value conditions and there exists such that for almost all and and also for all and .

Theorem 10. Let be -Caratheodory multifunctions. Suppose that there exist a nondecreasing bounded continuous map and a continuous function such that for all , for . Then, coupled system of time-fractional differential inclusions (8)-(9) has at least one solution.

Proof. Define the operator by , whereBy Lemma 7, it is clear that each fixed point of the operator is a solution for system of time-fractional differential inclusions (8). First, we prove that the multifunction is convex-valued. Let , . Choose such that for almost all and . Let be given. Then, we have for . Since the operator has convex values, is a convex set and for . This implies that the operator has convex values. Now, we prove that maps bounded sets of into bounded sets. Let , be a bounded subset of , , and . Then, there exists such that and + for almost all . If , then we obtain where the constant is defined by (17). This implies that . Similarly, we get , where the constant is defined by (17). Thus, . Now, we prove that maps bounded sets into equicontinuous subsets of . Let and with . Then, we haveBy using a similar method, we obtain Hence, as . By using the Arzela-Ascoli theorem we get that is completely continuous. Here, we prove that is upper semicontinuous. By using Lemma 4, is upper semicontinuous whenever it has a closed graph. Since is completely continuous, we must show that has a closed graph.
Let be a sequence in with and with . We show that . For each , we can choose such that and for all . It is sufficient to show that there exists such that and for all . Now, consider the linear operators defined by and . Note that