Abstract

In this paper, by using the lower and upper solution method and the monotone iterative technique, we investigate the existence of solutions to antiperiodic boundary value problems for impulsive fractional functional equations via a recent novel concept of conformable fractional derivative. An example is given to illustrate our theoretical results.

1. Introduction

In the past decades, fractional differential equations play important roles in describing many phenomena and processes occurring in engineering and scientific disciplines; for instance, see [1–7]. Besides the research of fractional differential equations, the impulsive differential equation is found to be an effective tool to study some problems of medicine, engineering, biology, and physics [8–10]. In various fields, such as physics, engineering, and chemistry, many models come down to antiperiodic boundary value problems, so there have been many papers focused on the subject of fractional differential equations with impulsive antiperiodic boundary value conditions; one can refer to [11–14].

Recently, Khalil et al. [15] introduced the conformable fractional derivative, which was a new well-behaved definition, depending just on the basic limit definition of the derivative. For details and applications of this concept, see [16–18]. Fractional differential equations via conformable fractional derivatives have recently received considerable attention; see [19–22] and the references therein. In [23], the authors studied the following periodic boundary value problem for impulsive conformable fractional integrodifferential equations:where denotes the conformable fractional derivative of order starting from , , :They obtained the existence of solutions for (1) by using the monotone iterative method.

Motivated by the above-mentioned work and a recent paper [24], in this article, we discuss the existence of solutions to antiperiodic boundary value problems for impulsive conformable fractional functional differential equations:where , denotes the conformable fractional derivative of order starting from , , , , and , , , , , .

To our knowledge, the work on the antiperiodic boundary value problems for impulsive fractional conformable functional differential equations is not to be initiated yet. By applying the method of lower and upper solutions coupled with the monotone iterative technique, we obtain the existence of extreme solutions for problem (3).

The rest of this paper is arranged as follows. Section 2 contains some preliminary notations, definitions, and basic results about conformable fractional calculus. In Section 3, we establish comparison principle and define the upper and lower solutions, and then we obtain the existence of extreme solutions for problem (3) by means of the monotone iterative technique. Finally, in Section 4, an example is given to show the effectiveness of the results obtained.

Remark 1. If , then BVP (3) is reduced to BVP (1) in [24].

2. Preliminaries

Let , , for be subinterval of :And let with normand then is a Banach space. A function is called a solution of problem (3) if it satisfies (3).

For with , new notations are introduced in [23] as follows:where , for some and .

Let be a function given by

The impulsive integral notation is defined aswhere .

Property 2 (see [17]). Let be nonnegative real numbers. The following relations hold:(i),(ii)

Definition 3 (see [17]). The conformable fractional derivative starting from a point of a function of order is defined byprovided that the limit exists.

Definition 4 (see [17]). Let . The conformable fractional integral starting from a point of a function of order is defined as

Remark 5. If is differentiable, then . In addition, if exists on , then we say that is -differentiable on .

Lemma 6 (see [17]). Let , , and the functions be -differentiable on . Then(i)(ii)(iii) for all constant functions (iv)(v) for all functions

Theorem 7 (see [23]). Let an interval and let be given function satisfying the following:(i) is continuous on (ii) is -differentiable for some Then there exists a constant , such that .

3. Main Results

We now consider the following antiperiodic boundary value problem:where , , are constants and .

Lemma 8. is a solution of BVP (11) if and only if is a solution of the impulsive integral equation:whereandwith .

Proof. For convenience, letIf is a solution of BVP (11), then for , we have by Lemma 6 thatThus, multiplying by both sides of the first equation of (11), we obtainThe conformable fractional integral of order from to () of (17) yieldsFor , multiplying by both sides of the first equation of (11), we getApplying the conformable fractional integral of order to both sides of (19) for , we obtainBy using Property 2, , and (20) implies thatIn the same way, for , we derivewhere .
Note that , and we impose in (23). From antiperiodic boundary value condition , we deduce thatSubstituting (24) into (23), we haveBy using Property 2 (ii), we getwhich implies that (12) holds.
Conversely, assume is a solution of (12); then by direct calculus, we can easily obtain that satisfies fractional impulsive antiperiodic boundary value problem (11). The proof is completed.

Denote . Now we establish the comparsion result.

Lemma 9. Assume that , , , , . Ifthen (11) has a unique solution.

Proof. For each , we define an operator bywhere are given by Lemma 8; then .
It is easy to check thatandFor , , and by (29)-(30), we obtainMoreover, we getThus, , we obtainBy (27) and Banach fixed point theorem, has a unique fixed point which is the unique solution of problem (11). The proof is completed.

Lemma 10. Let . Suppose that satisfieswhere constants , , . Assume in addition thatThen for all .

Proof. Suppose, to the contrary, that there must be such that . Let with , , such that . It is easy to show thatSuppose for . It is easy to know that ; then . By Theorem 7, we haveSumming up the above inequalities, we obtainwhich implies thatThuswhich contradicts (35). The proof is completed.

Definition 11. The functions are said to be related lower and upper solutions for BVP (3) ifAndFor , we write if for all , and

Theorem 12. Let be coupled lower and upper solutions of BVP (3). Suppose that the following conditions hold:
(H1) The function satisfiesfor , ,
(H2) All functions satisfywhere ,
(H3) Constants , , in (H1) and (H2) satisfy (27) and (35).
Then there exists monotone sequence which converges uniformly to the minimal and maximal solutions of BVP (3), respectively, such thaton , where is a solution of BVP (3) such that on .

Proof. We construct two sequences , which are the solutions ofandIt follows from (H3) and Lemma 9 that problem (47) has a unique solution. Similarly, we also conclude that problem (48) has a unique solution too.
We prove that these sequences satisfy the following properties:(i)(ii)if , then To prove (i), put ; then we get by Definition 11 and (47) thatandandBy using (H3) and Lemma 10, we deduce that , which implies for all , i.e., . Analogously, . By mathematics induction, we can obtain that is a nondecreasing sequence and is a nonincreasing sequence.
To prove (ii), we first show that , if . Let , using (47), (48), and (H1), we haveand by (H2)Thus, from (H3) and Lemma 10, we obtain , which yields . Still by mathematical induction, we have , .
Following (i) and (ii) above, we haveObviously, and satisfy (47) and (48), respectively. Thus, there exist and on , such that and uniformly on . It is easy to see that and are solutions of BVP (3) in .
Finally, we prove and are extreme solutions of BVP (3) in . Assume that is any solution of problem (3), which satisfies , . In the following, we will prove that if for some positive integral , then there holds on . Let ; then for we haveandBy Lemma 10, we have for all , which leads to on . Similarly, we can show that on . By mathematical induction, we obtain that for each . Taking , we have for . The proof is finished.

Remark 13. Let ; then Theorem 12 is generalization of the result of Theorem 3.1 in [24].

4. An Example

Example 1. Consider the problemwhereThusfor all , , , .

Let , ; then condition (H1) holds. Since , condition (H2) also holds with . We easily know that , , , , , , , , , and for . Moreover, we haveandwhich implies that hypothesis () is satisfied.

SettingObviously,andWe see thatMoreover, we haveandwhich implies thatHence, and are coupled lower and upper solutions of (57). Thus, applying Theorem 12, we have the existence of monotone sequences that approximate the extremal solutions of (57) in the interval .

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

This work is supported by the National Natural Science Foundation of China (11271364, 10771212).