The Existence and Uniqueness of Solutions for Variable-Order Fractional Differential Equations with Antiperiodic Fractional Boundary Conditions

In this paper, we discuss the existence and uniqueness of solutions for nonlinear fractional di ﬀ erential equations of variable order with fractional antiperiodic boundary conditions. The main results are obtained by using ﬁ xed point theorem.


Introduction
Fractional calculus has become one of the important tools for the development of modern society; the fractional differential equation with variable order has gained lots of interest [1][2][3][4]. Some researchers have investigated the physical background and numerical analysis of fractional differential equations of variable order [5][6][7][8]. In [9], Bushnaq et al. used Bernstein polynomials with nonorthogonal basis to establish operational matrices for variable-order integration and differentiation which convert the considered problem to some algebraic type matrix equations and obtained numerical solution to variable-order fractional differential equations by numerical simulation. In [10], Shah et al. proposed a new algorithm for numerical solutions to variable-order partial differential equations, used properties of shifted Legendre polynomials to establish some operational matrices of variable-order differentiation and integration, and got the numerical solution by numerical experiments.
In recent years, the antiperiodic boundary value problem of fractional differential equation has gradually become the focus of research, which have broad application in engineering and sciences such as physics, mechanics, chemistry, economics, and biology [11][12][13][14][15][16][17]. In [18], Ahmad and Nieto considered the following antiperiodic fractional boundary value problems: where C D p denotes the Caputo fractional derivative of order q and f is a given continuous function.
The problems related to the antiperiodic boundary value condition have been considered in [19][20][21][22][23][24][25][26], but the antiperiodic boundary value problem of fractional differential equation with variable order is almost not considered. In this paper, we investigate the existence of solutions for an antiperiodic fractional boundary value problem given by where C D p denotes the Caputo fractional derivative of order p, 0 < p < 1, C D qðtÞ denotes the Caputo fractional derivative of variable order qðtÞ, 1 < qðtÞ ≤ 2, T is a positive constant, and f : ½0, T × R ⟶ R is a given continuous function.

Preliminary Knowledge
In this section, we introduce some fundamental definitions and lemmas.
Definition 1 (see [27]). The Riemann-Liouville fractional integral of order q for a continuous function f : ½0,∞Þ ⟶ R is defined as provided the integral exists.
Definition 2 (see [27]). For ðn − 1Þ times absolutely continuous function f :½0,∞Þ ⟶ R, the Caputo derivative of fractional order q is defined as where ½q denotes the integer part of the real number q.
Definition 3 (see [3]). The Riemann-Liouville fractional integral of variable order qðtÞ for a continuous function f : ½0, ∞Þ ⟶ R is defined as provided that the right-hand side is pointwise defined.
Definition 4 (see [3]). For ðn − 1Þ times absolutely continuous function f : ½0,∞Þ ⟶ R, the Caputo fractional derivative of variable order qðtÞ is defined as Definition 5 (see [25]). Let I ⊂ R, I is called a generalized interval if it is either an interval, or fag or ∅.
A finite set θ is called a partition of I if each x in I lies in exactly one of the generalized intervals ξ in θ.
A function f : I ⟶ R is called piecewise constant with respect to partition θ of I if for any ξ ∈ θ, f is constant on ξ.
Theorem 6 (see [27]). Let E be a closed, convex, and nonempty subset of a Banach space X; let F: E ⟶ E be a continuous mapping such that FE is a relatively compact subset of X . Then, F has at least one fixed point in E.

Main Results
Let J = ½0, T. Denote CðJ, RÞ be the Banach space of all continuous functions x : J ⟶ R with the norm kxk = sup t∈J jxðtÞj and introduce the following assumption. ðH 1 Þ Let n ∈ N be an integer, θ = fJ 1 = ½0, T 1 , J 2 = ðT 1 , T 2 , ⋯, J n = ðT n−1 , T n g be a partition of the interval J, and qðtÞ: J ⟶ ð1, 2 be a piecewise constant function with respect to θ with the following forms: where 1 < q i ≤ 2 are constants, and I i is the indicator of the interval The Caputo fractional derivative of variable order qðtÞ for the function xðtÞ could be presented as a sum of Caputo fractional derivatives of constant orders q i by Definition 4, i = 1, 2, ⋯, n: Thus, according to (9), problem (2) can be written in the following form: Journal of Function Spaces Definition 9. The problem (2) has a solution, if there are functions x i , so that x i ∈ CðJ i , RÞ, satisfy (10) and . Let the function x ∈ CðJ, RÞ be such that xðtÞ ≡ xðT i−1 Þ on ½0, T i−1 , then consider (2) as the following form: Proposition 10. For any xðtÞ ∈ Ω i , f ðt, xðtÞÞ ∈ CðJ i × R, RÞ, xðtÞ is a unique solution of problem (11) if and only if x satisfy the integral equation: where G i ðt, sÞ is Green's function given by Proof. If xðtÞ ∈ Ω i is a solution of problem (11), applying t on both sides of (11), according to Lemma 8, we get according to the facts that C xðtÞ, and initial condition of problem (11), we get Þds, Thus, the solution of problem (11) is Green's function can be written as It implies that xðtÞ is the solution to the integral equation (12). In turn, if xðtÞ ∈ Ω i is the solution to the integral equation (12), according to Lemma 7, we deduce that xðtÞ is the solution of the problem (11). Hence, we complete this proof.
Proof. According to Proposition 10, problem (11) is equivalent to the following integral equation: Þds Þds: ð18Þ 3 Journal of Function Spaces , observe that B r i is a closed, bounded, and convex subset of Banach space Ω i . For any xðtÞ ∈ Ω i , we have It implies T : Ω i ⟶ Ω i is well defined. Now, we consider the continuity of operator T. Since f ðt, xðtÞÞ ∈ CðJ i × R, RÞ, given an arbitrary ε > 0, for any xð tÞ, yðtÞ ∈ Ω i ,we can findδ > 0such that jf ðt, xðtÞÞ − f ðt, yðtÞÞ Journal of Function Spaces We get the operator T is continuous. For each xðtÞ ∈ Ω i , we prove that if t 1 , t 2 ∈ J i , and 0 < t 2 − t 1 < δ, then kTxðt 2 Þ − Txðt 1 Þk < ε: By the mean value theorem, we have Therefore, kTxðt 2 Þ − Txðt 1 Þk < ε. According to the previous analysis, we know that is equicontinuous and uniformly bounded. We know by the Arzela-Ascoli theorem that T is compact on B r i , so the operator T is completely continuous. So, Theorem 11 implies that the antiperiodic boundary value problem of variable order (11) has at least a solution on J i . This completes the proof.