Some Existence Results for High Order Fractional Impulsive Differential Equation on Infinite Interval

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                  </jats:inline-formula> By applying Schauder fixed points and Altman fixed points, we obtain some new results on the existence of solutions. The nonlinear term of the equation contains fractional integral operator <jats:inline-formula>
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                  </jats:inline-formula> and lower order derivative operator <jats:inline-formula>
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                  </jats:inline-formula>. An example is presented to illustrate our results.</jats:p>

Since the last century, the dynamics of populations subject to abrupt changes was described by impulsive differential system. And other phenomena, for instance, harvesting, diseases, and so on, also have been described by using impulsive differential systems. Impulsive differential equations of fractional order play an important role in fractional differential equations theory and applications. Recently, impulsive fractional differential equations have been studied extensively. For example, Wang et al. studied the existence and multiplicity of solutions for impulsive fractional boundary value problem with p-Laplacian in [4], and Liu considered fractional impulsive differential equations using bifurcation techniques in [5]. For more articles related to impulsive fractional differential equations, refer to [6][7][8][9][10][11][12].
In [10], Liu and Ahmad studied the following problems: x(t), k � 1, 2, . . . ,. By using Schauder's fixed-point theorem, the authors studied the existence of solution. And the authors also considered the uniqueness of solution under some appropriate conditions.
In [9], Zhao and Ge considered the following boundary value problem: where α is a real number with 1 < α ≤ 2, D α 0+ is the standard Riemann-Liouville fractional derivative, Wang and Ge proved that the problem they studied has at least three positive solutions.
Motivated by the aforementioned work, we studied existence of solution of problem (1) by Schauder's fixedpoint theorem and Altman's fixed-point theorem. e main features of this paper are as follows. Firstly, the nonlinear term not only involved fractional order derivative but also contained fractional integral. Compared with [9,10,13], our nonlinear terms are more general. Many articles contain derivatives for nonlinear terms, but few articles contain both derivatives and integrals. Secondly, we studied the problem on the infinite interval. To the best of our knowledge, there are few articles involving the impulsive fractional order differential equations on the infinite interval. If the nonlinear term contained fractional integral and t ∈ [0, ∞), it will bring new obstacles to solve the problem. For this purpose, we overcome obstacles by constructing a special cone.
irdly, our problem is higher order impulsive fractional equation. Compared with [9], we allowed α ∈ (n − 1, n], where n > 2. It is obvious that our problem is more general.
is paper is organized as follows. In Section 2, we introduce some definitions and lemmas. In Section 3, we give our main results by fixed-point theorem. In Section 4, one example is presented to illustrate the main results.

Preliminaries and Lemmas
Let C(J, R) be the Banach space of continuous functions from J to R. Let us to introduce the Banach spaces with the norm 2 Mathematical Problems in Engineering with the norm Definition 1. e Riemann-Liouville fractional integral of order α > 0 of a function f: (0, ∞) ⟶ R is given by where the right side is pointwise defined on (0, ∞).

Definition 2.
e Riemann-Liouville fractional derivative of order α > 0 of a function f: (0, ∞) ⟶ R is given by (9) where n is the smallest integer greater than or equal to α and the right side is pointwise defined on (0, ∞). In particular,
en, u is also the solution of problem (13).

Main Results
In this section, we will prove the existence of solution of (1) by using Schauder fixed-point theorem and Altman theorem. According to Lemma 3, we obtain the following lemma first. (1) if and only if u ∈ PC 1 (J, R) is a solution of the impulsive fractional integral equation Define an operator T: PC 1 (J, R) ⟶ PC 1 (J, R) as follows: en, problem (1) has a solution if and only if the operator T has a fixed point.
en, problem (1) has at least one solution u(t) in PC 1 (J, R).
Proof. We will use five steps to prove our conclusion. Firstly, we will show T: Let u n , u ∈ PC 1 (J, R) be such that u n ⟶ u(n ⟶ ∞). en, ‖u n ‖ PC 1 < ∞ and ‖u‖ PC 1 < ∞. By (36) and the Lebesgue dominated convergence theorem, we get Hence, according to (37)-(39) and Lebesgue dominated convergence theorem, we can easily get For any u(t) ∈ B r , by (41) and condition (H1), we have (43) So, for u ∈ B r , it is easy to know that ‖Tu‖ PC 1 < ∞. Hence, TB r is uniformly bounded.
Our second result is based on Altman fixed-point theorem.

Proof. Let us choose
and define U � u ∈ PC 1 |‖u‖ PC 1 ≤ R . According to eorem 3, we know T: U ⟶ U is a completely continuous operator. For any u ∈zU, by (H3), we have us, from (49) and (50), we have TU ⊂ U and ‖Tu‖ PC 1 ≤ ‖u‖ PC 1 , ∀u ∈zU. So, by eorem 2, we know that problem (1) has at least one solution. Proof. Let us take and define U � u ∈ PC 1 |‖u‖ PC 1 < R . For any u ∈zU, we have Mathematical Problems in Engineering us, from (52) and (53), we have TU ⊂ U and ‖Tu‖ PC 1 ≤ ‖u‖ PC 1 , ∀u ∈zU. So, by eorem 2, we know that problem (1) has at least one solution.

Example
In this section, we give an example to illustrate of our main result. Example 1. Consider the following impulsive boundary value problem of fractional order: By computing, we know that (1 + t) 1/2 t 3/2 Γ(5/2) 1 20e t ≈ 64.5850 < ∞.
(56) us, the conditions of eorem 3 are satisfied, and hence problem (54) has at least one solution.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.