Existence and Uniqueness Theorems for a Fractional Differential Equation with Impulsive Effect under Band-Like Integral Boundary Conditions

Researches on fractional differential equations have witnessed an unprecedented boom in recent years on account of the far-reaching application in various subjects, such as physics, biology, nuclear dynamics, chemistry, etc., for more details, see [1–3] and the references therein. Considering the impulse effect in the continuous differential equation can quantify the impact of the instantaneous mutation of the model and provide a theoretical basis for the practical application. erefore, impulsive differential equation problems also attract great attention from scholars. For the theories of impulsive differential equations, the readers can refer to [4–7]. In addition, there have been some excellent results concerning the existence, uniqueness, and multiplicity of solutions or positive solutions to some nonlinear fractional differential equations with various nonlocal boundary conditions. As for some recent bibliographies, we refer readers to see [8–11] and the reference therein. Yang and Zhang in [12] studied the following impulsive fractional differential equation where 푐 훼 0+ is the Caputo fractional derivative, 훼 ∈ R, 1 < 훼 ≤ 2. 푓 : [0, 1] × R → R is a continuous function, 퐼푘, − 퐼푘 are continuous functions, 푔(푥) = max (( 儨儨儨儨푥(휉 ) 儨儨儨儨)/(휆 + 儨儨儨儨푥(휉 ) 儨儨儨儨)), h(푥) = min (( 儨儨儨儨푥(휉 ) 儨儨儨儨)/(휅 + 儨儨儨儨푥(휉 ) 儨儨儨儨)). By transforming the boundary value problem into an equivalent integral equation and employing some fixed point theorems, existence result is obtained. e research results of fractional differential equations with integral boundary conditions are also quite rich, and the research on those questions remains as a hotpot among many scholars in recent years. We refer readers to see [13–16] and the reference therein. In [13], Song and Bai considered the following boundary value problem of fractional differential equation with Riemann–Stieltjes integral boundary condition


Introduction
Researches on fractional differential equations have witnessed an unprecedented boom in recent years on account of the far-reaching application in various subjects, such as physics, biology, nuclear dynamics, chemistry, etc., for more details, see [1][2][3] and the references therein. Considering the impulse effect in the continuous differential equation can quantify the impact of the instantaneous mutation of the model and provide a theoretical basis for the practical application. erefore, impulsive differential equation problems also attract great attention from scholars. For the theories of impulsive differential equations, the readers can refer to [4][5][6][7]. In addition, there have been some excellent results concerning the existence, uniqueness, and multiplicity of solutions or positive solutions to some nonlinear fractional differential equations with various nonlocal boundary conditions. As for some recent bibliographies, we refer readers to see [8][9][10][11] and the reference therein.
Yang and Zhang in [12] studied the following impulsive fractional differential equation where 푐 훼 0+ is the Caputo fractional derivative, 훼 ∈ ℝ, 1 < 훼 ≤ 2. e research results of fractional differential equations with integral boundary conditions are also quite rich, and the research on those questions remains as a hotpot among many scholars in recent years. We refer readers to see [13][14][15][16] and the reference therein.
In [13], Song and Bai considered the following boundary value problem of fractional differential equation with Riemann-Stieltjes integral boundary condition is a function of bounded variation, ∫ 1 0 푢(푠)푑퐴(푠) denotes the Riemann-Stieltjes integral of with respect to . By the use of fixed point theorem and the properties of mixed monotone operator theory, the existence and uniqueness of positive solutions for the problem are acquired. (1) Moreover, Zhao and Liang in [14] added impulsive effect to fractional equations with integral boundary conditions and discussed the existence of solutions where is the Riemann-iouville fractional derivative of By applying the contraction mapping principle and the fixed point theorem, some sufficient criteria for the existence of solutions are obtained.
Inspired by the works above, we will study the impulsive fractional differential equation with band-like integral boundary conditions represent the right and the le limits of at 푡 = 휉 푘 , Δ푥| 푡=휉 = 푥 휉 + 푘 − 푥 휉 − 푘 . 퐼 푥 휉 ∈ 퐶(ℝ, ℝ). By using the Leray-Schauder alternative theorem and the Banach contraction mapping principle, the existence and uniqueness theorems of solutions to problem (4) can be established.
We emphasize that the discontinuous points caused by impulse are just the upper and lower limits of the band-like integral values in the boundary conditions of (4). In other words, the value of the unknown function at the endpoint of the interval [0,1] is related to the linear combination of the integral values of the unknown function between the discontinuous points.
Another thing worth mentioning is that despite the complicated boundary conditions and the interference of the impulse, we use a piecewise function to represent the operator in a concise form based on the form of the Green's function and accurately estimate the upper bound of its absolute value, which is fully prepared for the establishment of the main theorem.

Preliminaries
In this section, we present some definitions, lemmas, and some prerequisite results that will be used to prove our results.

Advances in Mathematical Physics
Hence, we can get erefore, the operator is equicontinuous, and the operator is completely continuous. ☐
e first result is based on the Letay-Schauder alternative theorem.