## Recent Advance in Function Spaces and their Applications in Fractional Differential Equations

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Mingyue Zuo, Xinan Hao, "Existence Results for Impulsive Fractional -Difference Equation with Antiperiodic Boundary Conditions", *Journal of Function Spaces*, vol. 2018, Article ID 3798342, 9 pages, 2018. https://doi.org/10.1155/2018/3798342

# Existence Results for Impulsive Fractional -Difference Equation with Antiperiodic Boundary Conditions

**Academic Editor:**Maria Alessandra Ragusa

#### Abstract

In this paper, we investigate the impulsive fractional -difference equation with antiperiodic conditions. The existence and uniqueness results of solutions are established via the theorem of nonlinear alternative of Leray-Schauder type and the Banach contraction mapping principle. Two examples are given to illustrate our results.

#### 1. Introduction

In this paper, we are concerned with the existence and uniqueness of solutions for the following impulsive fractional -difference equation with antiperiodic boundary conditionswhere , , , , , is -derivative, , and denote the Caputo -derivative of orders and , respectively. , , is the set of all real numbers, and . and are linear operators defined by where , , . , where and represent the right and left limits of at has a similar meaning.

Fractional -difference calculus plays a very important role in modern applied mathematics due to their deep physical background and has been studied extensively [1â€“4]. Impulsive differential equations are important in both theory and applications. Considerable effort has been devoted to differential equations with or without impulse, for example, [5â€“21]. In recent years, impulsive fractional difference and differential equations with antiperiodic conditions have received much attention; see [22â€“27] and the references therein. Zhang and Wang [24] have applied cone contraction fixed point theorem to establish the existence of solutions to nonlinear fractional differential equation with impulses and antiperiodic boundary conditions where is the Caputo fractional derivative, , . By using Banach fixed point theorem, Schaefer fixed point theorem, and nonlinear alternative of Leray-Schauder type theorem, some existence results of solutions for problem (3) are obtained in [25]. Ahmad et al. [28] studied existence of solutions for the following antiperiodic boundary value problem (BVP for short) of impulsive fractional -difference equation where denotes the Caputo -fractional derivative of order on , , , , , . and denote the Riemann-Liouville -integral of orders and , respectively.

In this paper we are concerned with the existence and uniqueness of solutions for impulsive fractional -difference equation antiperiodic BVP. By applying the theorem of nonlinear alternative of Leray-Schauder type and Banach contraction mapping principle, we show the existence and uniqueness of solutions for the BVP (1). Some ideas of this paper are from [29, 30].

#### 2. Preliminaries and Lemmas

For , letWe define the -analogue of the power function with is and, for , The -derivative of is defined byand -derivative of higher order by The -integral of is defined by

Lemma 1 (see [31]). *(1) If is -integral on the interval , then .**(2) If and are -integral on the interval , for all , then .*

*Definition 2 (see [2]). *Let and be a function defined on . The fractional -integral of the Riemann-Liouville type is defined by and

*Definition 3 (see [3]). *The fractional -derivative of the Caputo type of order is defined by where is the smallest integer greater than or equal to . If , , then .

Lemma 4 (see [2, 3]). *Let and be a function defined on . The following formulas hold:**(1) ;**(2) .*

Lemma 5 (see [3]). *Let and . Then If , then *

Lemma 6 (see [3]). *For , and , In particular, when and , using -integration by part,*

Lemma 7 (see [32] (nonlinear alternative of Leray-Schauder type)). *Let be a Banach space, be a bounded open subset of with , and be a completely continuous operator. Then, either there exists such that for or there exists a fixed point .**Let ia a map from into such that is continuous at , left continuous at and its right limit at exists for ; then is a Banach space with the norm .*

Lemma 8. *For , the solution of impulsive BVP, *

is given by

*Proof. *In view of Definitions 2 and 3 and Lemma 5, for , , we have and It follows from Definition 3, Lemma 5, and (18) that Applying in (20), we obtain Note boundary conditions and ; we get Applying (21) and (22), we have Thanks to , it is derived that and By (22), we get Combining (25) and (26), we have Therefore, for , , and, for ,

#### 3. Main Results

Define an operator by

Theorem 9. *Assume that** There exist nonnegative functions such that for , , .** There exist positive numbers and such that **where , , , .*

Then BVP (1) has a unique solution.

*Proof. *For and , we have and then , and hence is a contraction operator. It follows from Banach contraction mapping principle that BVP (1) has a unique solution.

Theorem 10. *Assume the following:** There exist continuous and nondecreasing function and such that ** There exist continuous and nondecreasing functions such that ** There exists constant such that where .*

Then BVP (1) has at least one solution.

*Proof. *The continuity of implies that operator is continuous. Let be bounded; then there exist positive constants , , and such that , , and for all , , . Thus, we have Consequently, operator is uniformly bounded on .

On the other hand, for , we have