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

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 [1–4]. Impulsive differential equations are important in both theory and applications. Considerable effort has been devoted to differential equations with or without impulse, for example, [5–21]. In recent years, impulsive fractional difference and differential equations with antiperiodic conditions have received much attention; see [22–27] 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 which tends to zero as ; then is equicontinuous on . Hence by -type Arzela-Ascoli Theorem ([33]), operator is completely continuous.
Let be such that for some ; then and hence Let ; then operator is completely continuous. By , one has for any and . By Lemma 7, BVP (1) has at least one solution.

4. Examples

Example 1. Consider the BVP LetBy direct computation, . For any and , we have Let , , , , and ; then Then, - hold. It follows from Theorem 9 that BVP (42) has a unique solution.

Example 2. Consider the BVP Let then Let , ; then . Choose and ; we have Let ; then condition holds. Therefore, by Theorem 10, BVP (46) has at least one solution.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

The authors contributed equally to this paper. All authors read and approved the final manuscript.

Acknowledgments

Supported financially by the National Natural Science Foundation of China (11501318, 11871302), the China Postdoctoral Science Foundation (2017M612230), the Natural Science Foundation of Shandong Province of China (ZR2017MA036), and the International Cooperation Program of Key Professors by Qufu Normal University.