The Existence and Uniqueness of Solution to Sequential Fractional Differential Equation with Afﬁne Periodic Boundary Value Conditions

: The solution to a sequential fractional differential equation with afﬁne periodic boundary value conditions is investigated in this paper. The existence theorem of solution is established by means of the Leray–Schauder ﬁxed point theorem and Krasnoselskii ﬁxed point theorem. What is more, the uniqueness theorem of solution is demonstrated via Banach contraction mapping principle. In order to illustrate the main results, two examples are listed.


Introduction
Recently, the investigation of fractional differential system has attracted extensive interest of researchers. Due to the past effects of the phenomenon under consideration, fractional differential system can build more accurate and precise models than integer differential systems; therefore, it is widely used in many domains, for instance, physics, biology, chemistry, astronomy, economics, control theory, and ecology. For relevant research on this results, we refer the interested readers to see [1][2][3][4].
Boundary value problem of fractional differential equation constitutes a very important and interesting class of problems, which arise in underground water flow, heat conduction, electromagnetic waves, membranes in nuclear reactors, etc. More and more scholars pay attention to this subject and achieve many excellent results. For instance, see [5][6][7][8][9][10][11] and the references therein. There are many types of boundary value problem, including the integral boundary value problem, multipoint boundary value problem, and periodic boundary value problem. In 2013, Li et al. [12] first proposed the affine-periodic system, which describes some physical phenomenon that is periodic in time and symmetric in space. Since then, scholars have done a lot of work on the affine periodic boundary value problem. In [13], Xu et al. have proved the existence of the affine-periodic solution to a Newton affine-periodic system by the lower and upper solutions method. For more details about affine periodic boundary value problem, we refer readers to see [14][15][16][17].
The research of sequential fractional differential equation has aroused widespread interests among scholars, since Miller and Ross first proposed the notion of sequential fractional derivative in [18] (p. 209). Many scholars have studied different types of fractional derivative, for instance, Riemann-Liouville fractional derivative, Caputo fractional derivative, and Hadamard fractional derivative. For Riemann-Liouville sequential fractional derivative, Bai studied the existence of solutions to a nonlinear impulsive fractional differential equation supplemented with periodic boundary value condition in [19]. For Caputo sequential fractional derivative, Ahmad et al. [20] applied the fixed point theorem to research the existence of solution to a fractional differential equation with integral boundary conditions. In [21], Ahmad et al. considered a nonlinear fractional differential equation involving Hadamard sequential fractional derivative, under multi-point boundary conditions, they established the theory of existence and uniqueness of solution. For more research results on sequential fractional derivative, readers can be referred to the papers [22][23][24][25][26].
To our best knowledge, the boundary value problem of sequential fractional differential equation has been studied by many authors. However, there have not been any research results on the affine periodic boundary value problem of sequential fractional differential equation. According to the above analysis, we investigate the sequential fractional differential equation with affine periodic boundary value conditions: where C D p expresses the Caputo fractional derivative, the order p ∈ {α, β} with 0 < α < 1 < β < 2, and β = α + 1. λ, a ∈ R, with a = 1, a = e −λT , and g(t, z) : [0, T ] × C([0, T ]; R) → R is a continuous function. The contribution of this paper is the investigation of the issue of solution to the sequential fractional differential equation with affine periodic boundary value conditions. Firstly, we use two different methods to prove the existence theorem of the solution. On the basis of improving the condition, we prove the uniqueness of the solution to the equation. Most of the previous studies on affine periodic system are of integer-order derivative, this paper provides an idea for the study of fractional order affine periodic system. The structure of this paper is as follows. Some definitions and lemmas are introduced in Section 2. The main results and the processes of proofs are presented in Section 3. In Section 4, we list two examples to illustrate our results. |z|. In the following, we will introduce a number of basic definitions and lemmas, which will be used thereafter. For more results, we refer the interested readers to see [27][28][29][30][31]. Definition 1. The Riemann-Liouville fractional integral of order p > 0 for a function w is defined as

Preliminaries
where Γ(·) is the Gamma function.

Definition 2.
The Caputo fractional derivative of order p > 0 for a function w can be written as for t > 0.
An important proposition of the Caputo fractional derivative needs to be provided, which will play a crucial role in our later proof: Proposition 1 ( [28]). For the given definitions, we have: where p ∈ (m − 1, m), ∀m ∈ N + , a i (i = 1, 2, · · · , m) are arbitrary constants.

Definition 3 ([18]
). The sequential fractional derivative for a function w can be written as where p = (p 1 , · · · , p m ) is a multi-index.

Remark 1.
The symbol D p can denote the Grünwald-Letnikov, Riemann-Liouville, Caputo or any other kind of integro-differential operator. For more details, we refer readers to see [27] (p. 87).

Lemma 1 (Leray-Schauder fixed point theorem [29])
. Let X be a Banach space, Y ⊆ X be nonempty, bounded and convex, Z be an open subset of Y with 0 ∈ Z. Let map G : Z → Y be continuous and compact. Then, one of the following representations is true: (i) there exist z ∈ ∂Z and ε ∈ (0, 1) such that z = εG(z); (ii) G has a fixed point z ∈ Z.
Lemma 2 (Krasnoselskii fixed point theorem [30]). Let X be a Banach space, Y ⊆ X be nonempty, bounded, closed and convex. Let T 1 , T 2 be two maps and satisfy: Then, there exists w ∈ Y such that w = T 1 w + T 2 w.
Proof. By Proposition 1, we take I α on (3) and gain that using the method of constant variation, the solution to (3) can be expressed as where c 1 , c 2 are arbitrary constants. Then Following the boundary conditions for (3), we can find that Replacing the values of c 1 and c 2 into (6), the solution given by (4) is obtained.
In order to simplify the following proofs, we present the estimate of the integral inequalities as follows. For Similarly, one has For convenience, we let where µ 1 (t), µ 2 (t) are given in (4). Then, we can transform the (T , a)-affine-periodic system (1) into a fixed point problem, i.e., z = G(z). What follows is to use Lemma 1 to solve the fixed point problem. The proof is divided into four steps: |g(s, z m (s)) − g(s, z(s))|ds.
Next, we use Lemma 2 to research the existence of solution to system (1), whose nonlinear function satisfies Lipschize condition. The following hypotheses on g(t, z) are required: Then, the (T , a)-affine-periodic system (1) admits at least one solution on [0,T ] if where M is the constant given by (11).
Proof. We split the operator G : C([0, T ]; R) → C([0, T ]; R) defined by Equation (12) as G = G 1 + G 2 , where G 1 and G 2 are given by and Let us set δ = max In what follows, we use three steps to complete the proof of the theorem.
Next, we prove the uniqueness of the solution to the system (1) by Banach contraction mapping principe.
Thus, for every z ∈ Ω ρ , apply (19) to get which indicates that G(z)(t) = max That is, GΩ ρ ⊂ Ω ρ , which implies that G maps Ω ρ into itself. Secondly, we claim that the operator G : where l = 1 10 .   (23) has a unique solution.

Conclusions
In this paper, we investigate the existence and uniqueness of solution to a sequential fractional differential equation with affine periodic boundary value conditions. However, the two fractional derivatives in the text must meet the conditions: 0 < α < 1 < β < 2, and β = α + 1, which limit the application range of the differential equation. In the next study, we will research the existence of solution to a sequential fractional differential equation with the order 0 < α < 1 < β < 2, which α is independent of β. What is more, the fractional differential equation with higher order n − 1 < α < n < β < n + 1 will also be studied.
Due to the fact that the differential inclusion theory has a very wide range of applications in many fields, such as optimal control theory, dynamic system, and engineering technology. Therefore, the existence of solution to fractional differential inclusion problem deserves to be researched. We will further investigate the solution to a fractional differential inclusion with the affine periodic boundary value conditions in the future.