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

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.


Introduction
In this article, we will consider the following generalized Bagley-Torvik type fractional differential inclusions: where   V 1 and   V 2 are Caputo fractional derivatives with 0 < ] 1 ≤ 1 and 0 < ] 2 < ] 1 ,  ∈ R 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][5][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][9][10][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   () +    3/2 () + () = (), where () is a function describing the motion of thin plates in Newtonian fluids,  = , the mass of thin rigid plate,  = 2 √ V, where  is area of the plate immersed in Newtonian fluid, V 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: 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 ℎ 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.
is called  order Riemann-Liouville fractional integral of .
Given  0 ∈ ,  ∈  1 (, ).Define the operator  by the formula that is,  is the solution of the above system (14).
For given   ∈  (), we define  ℎ()   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).
In the sequel, we introduce some important lemmas which are crucial to derive existence results.
Lemma 10 (see [20]).Under assumptions ( now by (h1) and Proposition 1, it follows that By ( 4 ), we obtain ≤  ()  (Δ ()) . ( Applying Proposition 1 and Lemma 3, one can easily achieve It follows from (25) that Since  > 0 is arbitrary, we have A nonlinear alternative for Kakutani maps, which is significant to develop our main results, is introduced as follows.
Next, we shall prove the existence result when ℎ is compact continuous.Theorem 13.Suppose (ℎ1) and ( 1 ) − ( 4 ) are satisfied.If there exists  > 0 such that and then the generalized Bagley-Torvik type differential inclusion (1) has at least one solution in (, ).
Subsequently, we will investigate the case where  is Lipschitz-type about the Hausdorff metric.Before proceeding, let us introduce the following definition.

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].