Existence of Solutions for a Periodic Boundary Value Problem with Impulse and Fractional Derivative Dependence

where Dqt and D c t represent the common Caputo derivatives of orders q and c, and 1< q< 2, 0< c< 1, and J � [0, 1], 0 � t0 < t1 < t2 < · · · < tm < tm+1 � 1, J′ � J\ t1, t2, . . . , tm 􏼈 􏼉. Here, f: J × R × R⟶ R and Ik, Jk: R × R ⟶ R are continuous functions. Now, Δu(tk) � u(t+k ) − u (t−k ), where u(t + k ) and u(t − k ) denote the right limit and the left limit of u(t) at the impulsive point tk. Also, ΔDct u(tk) � D c t u(t + k ) − D c t u(t − k ), where D c t u(t + k ) and Dct u(t−k ) denote the right limit and the left limit of D c t u(t) at the impulsive point tk. If u(t−k ) and D c t u(t − k ) exist, we let u(tk) � u(t−k ) and D c t u(tk) � D c t u (t−k ), wherek � 1, 2, . . . , m. Also, a and b are two real constants with b> a> 0. -e theory of fractional differential equation has received a lot of attention because of its wide application in mathematical models (see [1–27] and the references therein). Fractional-order impulsive differential equations are a natural generalization of the case of nonimpulses and are used to describe sudden changes in their states, such as in optimal control, population dynamics, biological systems, financial systems, and mechanical systems with impact. We refer the reader to [28–36] and the references therein. In particular, Bai et al. [37] investigated a mixed boundary value problem of nonlinear impulsive fractional differential equation:


Introduction
is paper considers the existence of solutions of the following fractional-order impulsive periodic boundary value problem: c D q t u(t) � f t, u(t), c D c t u(t) , t ∈ J ′ , Δu t k � I k u t k , c D c t u t k , Δ c D c t u t k � J k u t k , cc D c t u t k , k � 1, 2, . . . , m, where c D q t and c D c t represent the common Caputo derivatives of orders q and c, and 1 < q < 2, 0 < c < 1, and J � [0, 1], 0 � t 0 < t 1 < t 2 < · · · < t m < t m+1 � 1, J ′ � J\ t 1 , t 2 , . . . , t m . Here, f: J × R × R ⟶ R and I k , J k : R × R ⟶ R are continuous functions. Now, Δu(t k ) � u(t + k ) − u (t − k ), where u(t + k ) and u(t − k ) denote the right limit and the left limit of u(t) at the impulsive point t k . Also, where c D c t u(t + k ) and c D c t u(t − k ) denote the right limit and the left limit of c D c t u(t) at the impulsive point t k . If u(t − k ) and c D c t u(t − k ) exist, we let u(t k ) � u(t − k ) and c D e theory of fractional differential equation has received a lot of attention because of its wide application in mathematical models (see  and the references therein). Fractional-order impulsive differential equations are a natural generalization of the case of nonimpulses and are used to describe sudden changes in their states, such as in optimal control, population dynamics, biological systems, financial systems, and mechanical systems with impact. We refer the reader to [28][29][30][31][32][33][34][35][36] and the references therein. In particular, Bai et al. [37] investigated a mixed boundary value problem of nonlinear impulsive fractional differential equation: and some sufficient conditions on the existence and uniqueness of solutions for problem (2) are obtained under Lipschitz conditions. In [38], Zhang and Xu studied the following impulsive periodic boundary value problem with the Caputo fractional derivative: using Green's function in [36], and via the symmetry property of Green's function and topological degree theory, the authors obtained the existence of positive solutions for (3) when the growth of f is superlinear and sublinear. Inspired by the above research studies, in this paper, we consider fractional-order impulsive differential equations with generalized periodic boundary value conditions (1), where the nonlinear term, impulse terms, and periodic boundary conditions all depend on unknown functions and the lower-order fractional derivative of unknown functions.
is is obviously more general and more widely applied, but it is also more complex and difficult to solve. Compared with (1), the nonlinear term, pulse term, and periodic boundary conditions of (3) are all independent of fractional derivatives, so it is a special form of (1). In this paper, we first give an equivalent integral form of solutions for problem (1) using some new Green's functions. Next, we present some sufficient conditions for the existence of solutions for problem (1), where the nonlinear and impulse terms satisfy some nonlinear and linear growth conditions, which are different from the conditions in [36][37][38]. Finally, we present three examples to illustrate our main results.

Preliminaries and Lemmas
In this section, we only present some necessary definitions and lemmas about fractional calculus.
e Riemann-Liouville fractional integral of order α > 0 for a function f: (0, ∞) ⟶ R is defined as where Γ(·) is the Euler gamma function.
Definition 2 (see [39,40]). e Caputo fractional derivative of order α > 0 for a continuous and n-order differentiable function f: where Γ(·) is the Euler gamma function and n � [α] + 1, where [α] is the smallest integer greater than or equal to α.
en, K 1 (t, s) + K 2 (t, t i )and K 3 (t, t i ) and H 1 (t, s)and H 2 (t, t i ) defined as in (9) and (11) are continuous, and the following inequalities hold: Proof. Directly observe that Let

Lemma 4. If the function f(t, u, c D c t u(t)) is continuous, then u ∈ E is a solution of (1) if and only if u ∈ E is a solution of the following integral equation:
Proof. Assume that u satisfies (1). From Lemma 2, we see that u satisfies integral equation (26).
Conversely, assume that u satisfies integral equation (26). Applying Definition 2, by a direct fractional derivative computation, it follows that the solution given by (26) and (2) satisfies (1).
Define an operator T: E ⟶ E as It is easy to prove that the function u is a solution of (1) if and only if u is a fixed point of the operator T.
For convenience, we list some hypotheses:  (27) is completely continuous.
Proof. We divide the proof into three steps. Set Ω r � u ∈ E, ‖u‖ ≤ r { } for some r > 0. e steps are as follows: (i) Step 1. T is continuous from the continuity of the functions which and Lemma 4 imply that Step 3. T is equicontinuous. For any t 1 , t 2 ∈ J k , k � 0, 1, . . . , m, fixed s ∈ J and for any ϵ > 0, there exists a constant δ > 0 such that for |t 1 − t 2 | < δ, we have en, us, which implies that T(Ω r ) is equicontinuous on any subinterval J k , k � 0, 1, . . . , m.
From the Arzela-Ascoli theorem, we deduce that T: E ⟶ E is completely continuous.
Lemma 6 (Schauder fixed-point theorem, see [41,42]). Let X be a real Banach space, C ⊂ X be a nonempty closed bounded and convex subset, and F: C ⟶ C be compact. en, T has at least one fixed point in C.

Existence of the Solutions
For convenience, we give the following symbols: Now, we present our main theorems.
In view of Lemma 7, there exists a u ∈ Ω R 2 such that Φu + Ψu � u, so (1) has at least one solution in E.