Afﬁne-Periodic Boundary Value Problem for a Fractional Differential Inclusion

: In the article, afﬁne-periodic boundary value problem involving fractional derivative is considered. Existence of solutions to a Caputo-type fractional differential inclusion is researched by some ﬁxed-point theorems and set-valued analysis theory. Speciﬁcally, we consider two cases in which the multifunction has convex values and nonconvex values, respectively


Introduction
In recent years, the study of fractional calculus has aroused wide enthusiasm from scholars. Fractional calculus is widely used in physics, biology, control theory, celestial mechanics, economics and many other fields. The mathematical model established by fractional calculus can describe the phenomena in natural life more accurately and in more detail, so as to solve some problems that the integer order calculus mathematical model cannot solve. For the latest research on fractional calculus, we refer the readers to see [1][2][3][4][5].
Differential equation is a deterministic model used to describe systems in physics, engineering, mechanics, economics, etc. However, in real life and scientific practice, many phenomena cannot be described using the deterministic model, such as when describing some dynamic systems or uncertain objects. Differential inclusion is a dynamic system based on a certain but incomplete understanding of a system's process, which is used to reveal the laws of uncertain dynamical systems and discontinuous dynamical systems. Differential inclusion theory is an important application in a lot of fields, for example, in automatic control systems, economic dynamic systems, adaptive control theory, etc. As a branch of the general theory of differential equation, differential inclusion theory is developing rapidly. For the research results of differential inclusion theory, readers can refer to [6][7][8][9][10].
The affine-periodic was firstly proposed by Professor Li in 2013 [11], which describes a physical phenomenon with symmetry in space and periodicity in time. The affine period is widely used in astrophysics. For some interesting results on the affine period, please refer to [12][13][14][15] and the references therein. In [12], using the lower and upper solutions method and topological degree theory, Xu et al. claimed that a Newton affine-periodic system admits an affine-periodic; In [13], Liu et al. showed that every first-order dissipative-(T, a)affine-periodic system also has a dissipative-(T, a)-affine-periodic solution in [0, ∞) using topological degree theory and the lower and upper solutions method; In [14], Xu et al. firstly gave some extremum principles for higher-order affine-periodic systems. Then, using these extremum principles, the authors studied the existence of affine-periodic solutions for n(n ∈ N)-order ordinary differential equations. A class of nonlinear fractional dynamical systems with affine-periodic boundary conditions were considered by Xu et al. in [15]. Using the homotopy invariance of the Brouwer degree, the authors gained the existence of solutions to the fractional dynamical systems, while using Gronwall-Bellman inequality, the uniqueness of the solution was also obtained. However, fractional-order differential systems do not have affine-periodic solutions, but can only study the solutions with affine-periodic boundary value conditions. In [16], Gao et al. investigated the well-posedness of the affine-periodic boundary value solution to a sequential fractional differential equation. The existence results were obtained via Leray-Schauder and Krasnoselskii fixed-point theorems, while the uniqueness result was gained via the Banach contraction mapping principle.
Inspired by [16], in this article, we study the case when the nonlinear function is a multifunction. Precisely speaking, the (K, λ)-affine-periodic boundary value problem of a fractional differential inclusion is expressed by where C D α denotes the Caputo fractional derivative with the order 0 < α < 1. b ∈ R and λ are constants that satisfy The contribution of this article is to research the existence theorem of solutions to the (K, λ)-affine-periodic boundary value problem of fractional differential inclusion. For the case of multifunction G with convex and nonconvex values, we obtain the existence theorem of solutions by applying the Leray-Schauder alternative theorem and Covitz-Nadler fixedpoint theorem, respectively.
The rest of this article is organized as follows. In Section 2, the definitions and useful lemmas are stated. The main results are presented in Section 3, and some examples are listed in Section 4. In the last Section 5, the conclusion is given.

Preliminaries
In this section, we put forward the definitions of fractional calculus and some basic theory of set-valued analysis. If the readers are interested, more details can be found in [17][18][19][20]. Definition 1. The Riemann-Liouville fractional integral of order γ > 0 for a function u is defined as where Γ(·) is the Gamma function.

Definition 2.
The Caputo fractional derivative of order γ > 0 for a function u can be written as for t > 0.
Throughout this paper, let C([0, K]; R) be a Banach space that is composed of all continuous functions x : [0, K] → R with the usual norm

|x|.
Let AC([0, K]; R) be the space of absolutely continuous functions and L 1 ([0, K]; R) be a Banach space of measurable functions x : [0, K] → R that are Lebesgue integrable and normed by Furthermore, we introduce the notations:

Proposition 1.
If the multifunction F is completely continuous with nonempty compact values, then F is u.s.c. if, and only if, F has a closed graph.
The set of selections of F is defined by where χ T denotes the characteristic function of T. Let (Θ, d) be a metric space; for U, V ⊂ Θ, the Hausdorff metric is gained by The multifunction F has a fixed point if there is u ∈ Θ such that u ∈ F (u). Next, we propose the following lemmas, which are crucial to our research.

Main Results
In this section, we will use fixed-point theorems to prove the existence results of the problem (1); for convenience, we let where where g ∈ S G(·,z(·)) . What follows is to transform problem (1) into a fixed-point problem z ∈ Φ(z) and prove the existence of the fixed point. The proof is divided into four steps: Step 1. Φ is convex.
As the second result, we consider that the multifunction G is not necessarily convexvalued. Thanks to the Bressan-Colombo selection theorem and the Leray-Schauder alternative theorem, we gain the existence result of the problem (1).
Then, the solution set of the (K, λ)-affine-periodic boundary value problem (1) is nonempty.
Let us define the operator Φ to transform problem (8) into a fixed-point problem, Similar to the previous analysis, it is easy to know that Φ is convex, completely continuous and has a closed graph. The proof process is analogous to Theorem 1, so we do not repeat it here.
As the last result of this article, we change the convex value condition of the multifunction G into the nonconvex value condition.
Proof. As G(·, z) is measurable, S G(·,z(·)) is nonempty for each z ∈ C([0, K]; R), which implies that G has a measurable selection. We divided the proof process into two steps: Step 1. Φ is closed. Let {z n } n≥0 be a sequence where z n is convergent to z * in C([0, K]; R), then z * ∈ C([0, K]; R). For any t ∈ [0, K], there exists x n ∈ S G(·,z n (·)) such that, In view of G as compact, we may pass to a subsequence to understand that x n → x * ∈ L 1 ([0, K]; R). It is easy to obtain that x * ∈ S G(·,z * (·)) and for each t ∈ [0, K]. That is, z * ∈ Φ(z), which means that Φ is closed.