Existence Results for Generalized Bagley-Torvik Type Fractional Differential Inclusions with Nonlocal Initial Conditions

Abstract

In this article, we prove the existence of solutions for the generalized Bagley-Torvik type fractional order differential inclusions with nonlocal conditions. It allows applying the noncompactness measure of Hausdorff, fractional calculus theory, and the nonlinear alternative for Kakutani maps fixed point theorem to obtain the existence results under the assumptions that the nonlocal item is compact continuous and Lipschitz continuous and multifunction is compact and Lipschitz, respectively. Our results extend the existence theorems for the classical Bagley-Torvik inclusion and some related models.

1. Introduction

In this article, we will consider the following generalized Bagley-Torvik type fractional differential inclusions: where and are Caputo fractional derivatives with and , is a constant, and is a multifunction.

By introducing nonlocal conditions into the initial-value problems, Byszewski and Lakshmikantham [1] provided a more accurate model for the nonlocal initial valued problem since more information was incorporated in the experiment. As a result, the negative impact of single initial value can be significantly reduced. Concerning the initial-value problems, the most recent developments can be referred to in [2, 3]. On the other hand, the fractional calculus and fractional differential equations have many real applications in biology, physics, and natural sciences and a number of results on this topic have emerged in the last decade [4–6]. In [7], EI-Sayed and Ibrahim initiated the research on fractional multivalued differential inclusions. After that, many authors were devoted to study the existence of solutions for fractional differential inclusions [8–11]. Very recently, Wang et al. [12, 13] studied the controllability and topological structure of the solution set for fractional impulsive differential inclusions.

For the problem of fractional differential inclusions, multiterm fractional differential equations are a hot research direction owning to their wide use in practice and technique sciences, for example, physics, mechanics, and chemistry. An important result on multiterm fractional calculus is formulated by Bagley and Torvik in [14]. Here the authors deduced and tested a relation , where is a function describing the motion of thin plates in Newtonian fluids, , the mass of thin rigid plate, , where is area of the plate immersed in Newtonian fluid, is viscosity, is the fluid density, , the stiffness of the spring, and is an external force. Later, the above equation was called Bagley-Torvik equation [15]. Based on this model, the nonlinear multiterm fractional differential equations were rediscovered and popularized by Kaufmann and Yao in [16]. As far as the author knows, there are few papers on the existence of the generalized Bagley-Torvik type fractional differential inclusion (1) except for Ibrahim, Dong, and Fan [17]. They studied the following equation: where and are the Caputo fractional derivatives, , .

In this article, we shall be concerned with the existence of the generalized Bagley-Torvik type fractional differential inclusions (1) by employing the noncompactness measure of Hausdorff, fractional calculus, and the nonlinear alternative for Kakutani maps fixed point theorem, when is compact continuous and Lipschitz continuous and is compact and Lipschitz, respectively. Our theory improves the results in [6, 8–10, 16, 17] and extends (generalizes) the corresponding results of Bagley-Torvik equation.

The structure of this article is as follows: some preliminary knowledge is introduced in Section 2; some existence criteria are derived from (1) in Section 3; in the end, we use an example to illustrate an application of the main result.

2. Preliminaries

Let be a metric space and a normed space; ; ; ; ; .

Let be a Banach space. . For , , the norm is defined as , and is the conjugate exponent; that is, Denote the Hausdorff measure of noncompactness (MNC) as

Proposition 1 (see [18]). The MNC enjoy the following properties:(1)monotone, if , (2)nonsingular, if , for all and (3)regular, if is relative compact(4)algebraically semiadditive, if , for (5)

We now denote the sequential MNC as follows: One can easily have that Moreover, , if is separable.

Proposition 2 (see [18]). For every bounded set , , the inequality holds, where and is the MNC defined in . Moreover, if is equicontinuous, then is continuous. Moreover,

Lemma 3 (see [19]). Suppose , ; the inequality holds; for every , . Then , and

In what follows, we will recall some necessary definitions and lemmas of the fractional order differential and integra theory, which can be found in the literature [15].

Definition 4. Suppose , . If , then is called order Riemann-Liouville fractional integral of .

Definition 5. Suppose , . we define as the order Caputo fractional derivative of , where .

Lemma 6. Suppose and . Consider the following differential equation: Then there exists some constants , such that where .

Lemma 7. Suppose , and ; then there exist , satisfying

For simplicity, we denote and by and , respectively. And let , .

Lemma 8. Let , . Assume that . Then the solution of the equation, has the following form:

Proof. In light of , from Lemma 7, there exists satisfying Moreover, By Lemma 6, it follows that So we have Since , we obtain . Substituting the value of into (19), the proof is complete.

Given , . Define the operator by the formula that is, is the solution of the above system (14).

For any belonging to , set For given , we define as a set of solutions to the generalized Bagley-Torvik type fractional differential system

Before ending this section, we define the solution of the generalized Bagley-Torvik type fractional differential inclusions (1).

Definition 9. If and where , then is a solution of the generalized Bagley-Torvik type fractional differential inclusions (1).

3. Main Results

Here, we shall derive some existence criteria for the generalized Bagley-Torvik type differential system (1), when is compact continuous and Lipschitz continuous and is compact and Lipschitz, respectively. Our basic tools are noncompactness measure of Hausdorff, fractional calculus, and the nonlinear alternative for Kakutani maps fixed point theorem. Before proceeding, we assume that and and satisfy the following conditions:

(h1) is compact and continuous mapping, and there exist positive constants , satisfying

, admits a measurable selection

is upper semicontinuous, for almost every

For every and , , where , and is an increasing and continuous function

For almost every , there exists ; the inequality holds, where is a bounded set.

In the sequel, we introduce some important lemmas which are crucial to derive existence results.

Lemma 10 (see [20]). Under assumptions , if there exists sequence , , and , , such that and , then .

Lemma 11. Suppose that (h1) is satisfied; then, for any bounded set , we obtain

Proof. According to (5), for any , there exist and satisfying for all and Since now by (h1) and Proposition 1, it follows that By , we obtain Applying Proposition 1 and Lemma 3, one can easily achieve It follows from (25) that Since is arbitrary, we have

A nonlinear alternative for Kakutani maps, which is significant to develop our main results, is introduced as follows.

Theorem 12 (see [21]). Let be a closed convex set and be an open set with . Suppose is compact, upper semicontinuous. If there exist no and satisfying , then has at least one fixed point.

Next, we shall prove the existence result when is compact continuous.

Theorem 13. Suppose and are satisfied. If there exists such that and then the generalized Bagley-Torvik type differential inclusion (1) has at least one solution in .

Proof. From ([22]), according to conditions , is not empty, for every . Using (23), we define the following multivalued operator: as Clearly, if is a fixed point of , then is a solution of (1). Let ; is a bounded ball, defined as . Suppose is a bounded set and belongs to . In order to utilize the nonlinear alternative for Kakutani maps, we first need to show that . Let , , ; there exist and such that By Hölder inequality and , we have Taking into account the above inequality, one has Applying the above inequality, , as . Hence, and are equicontinuous.
Also because, for every , It follows from Lemma 11 that Thus, one can obtain Since and by (33), we know that .
Clearly, the multioperator has convex values. We still need to prove the multioperator is closed on . Suppose with in and with in . Moreover, assume is a sequence satisfying for any and Then the set is integrally bounded. From , one can achieve that Since in , therefore is bounded and belongs to . Invoking , we infer that So the sequence is semicompact. Applying the proposition (see [22]), is weakly compact. Therefore, there is satisfying . Moreover, one has which, together with (42) and being continuous, implies that Then, one can obtain Thus ; i.e., . Equivalently, is closed, and is closed on . Based on the above discussion, it can be concluded that is u.s.c. on (see [23]).
In the following, we are to find an open set , which satisfies the conditions of Theorem 12. Let , . Then there exist and such that From and , one can obtain Consequently, It follows from (32) that there exists which satisfies . Denote By the definition of , for every , there exists no satisfying . Since is compact and u.s.c., according to Theorem 12, has at least one fixed point in .

Next, we assume the nonlocal item is Lipschitz continuous; that is,

(h2) for every , there is satisfying .

Now, we give the existence criterion for the generalized Bagley-Torvik type fractional differential inclusions (1) when the nonlocal item satisfies (h2).

Theorem 14. Suppose conditions (h2) and are satisfied. If there exists , such that and then the generalized Bagley-Torvik type differential inclusion (1) has a solution in .

Proof. According to Theorem 13, the multioperator is upper semicontinuous. Suppose is bounded, and belongs to . We need to prove .
From (h2) and Lemma 11, one can infer that Thus, one has Together with (53), one can conclude that .
From a proof similar to Theorem 13, we can prove that the multioperator has at least one fixed point, which is a solution of the generalized Bagley-Torvik type fractional differential inclusions (1).

Remark 15. Assume , for some and ; then conditions (32) and (52) are automatically satisfied.

Subsequently, we will investigate the case where is Lipschitz-type about the Hausdorff metric. Before proceeding, let us introduce the following definition.

Suppose is metric spaces, corresponding to . Let , . Denote A function is called the Hausdorff metric on , if

Now, we make the following assumption:

there exists , satisfying and for almost every , , .

Noting that if is satisfied, and are automatically satisfied, in this case, the following theorems automatically hold.

Theorem 16. Let and satisfy conditions (h1), , , and . If there exists , such that and then the fractional differential inclusion (1) has at least one solution.

Theorem 17. Assume and satisfy conditions (h2), , , and . If there exists , such that and then the fractional differential inclusion (1) has at least one solution.

4. Example

Consider the following nonlocal problem of fractional differential inclusion.

Example 18. Discuss the following differential inclusion: where are certain constants, with , , , , , . Let and
Clearly, for all , , and then , .
Further, we suppose Obviously, Moreover, Take , Then, one can easily obtain that
It is easy to check that (h2) and hold. Further, we can find , such that and Therefore, according to Theorem 14, there exists at least one solution in for the problem (63).

5. Conclusion

This paper has studied the generalized Bagley-Torvik type fractional order differential inclusions with nonlocal conditions. By employing the noncompactness measure of Hausdorff and the nonlinear alternative for Kakutani maps fixed point theorem, the existence results have been derived when the nonlocal item is compact and Lipschitz continuous and multifunction is compact and Lipschitz. An example has been used to illustrate applications of the main result. Future research directions include the extension of the present results to other relevant cases, for example, controllability and topological structure of the solution set [12, 13].

Data Availability

No date were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Natural Science Foundation of China (11571300).