Journal of Function Spaces
Volume 2018 (2018), Article ID 7643123, 11 pages
https://doi.org/10.1155/2018/7643123

Research Article

Antiperiodic Boundary Value Problems for Impulsive Fractional Functional Differential Equations via Conformable Derivative

Jingfeng Wang and Chuanzhi Bai

Department of Mathematics, Huaiyin Normal University, Huaian, Jiangsu 223300, China

Correspondence should be addressed to Chuanzhi Bai; czbai8@sohu.com

Received 22 October 2018; Accepted 12 December 2018; Published 31 December 2018

Academic Editor: Ismat Beg

Copyright © 2018 Jingfeng Wang and Chuanzhi Bai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 norm

and 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 as

where .

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 by

provided 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:

where

and

with .

Proof. For convenience, let

If is a solution of BVP (11), then for , we have by Lemma 6 that

Thus, multiplying by both sides of the first equation of (11), we obtain

The conformable fractional integral of order from to () of (17) yields

For , multiplying by both sides of the first equation of (11), we get

Applying the conformable fractional integral of order to both sides of (19) for , we obtain

By using Property 2, , and

(20) implies that

In the same way, for , we derive

where .
Note that , and we impose in (23). From antiperiodic boundary value condition , we deduce that

Substituting (24) into (23), we have

By using Property 2 (ii), we get

which 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 , , , , . If

then (11) has a unique solution.

Proof. For each , we define an operator by

where are given by Lemma 8; then .
It is easy to check that

and

For , , and by (29)-(30), we obtain

Moreover, we get

Thus, , we obtain

By (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 satisfies

where constants , , . Assume in addition that

Then for all .

Proof. Suppose, to the contrary, that there must be such that . Let with , , such that . It is easy to show that

Suppose for . It is easy to know that ; then . By Theorem 7, we have

Summing up the above inequalities, we obtain

which implies that

Thus

which contradicts (35). The proof is completed.

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

And

For , 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 satisfies

for , ,
(H2) All functions satisfy

where ,
(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 that

on , where is a solution of BVP (3) such that on .

Proof. We construct two sequences , which are the solutions of

and

It 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) that

and

and

By 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 have

and by (H2)

Thus, from (H3) and Lemma 10, we obtain , which yields . Still by mathematical induction, we have , .
Following (i) and (ii) above, we have

Obviously, 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 have

and

By 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 problem

where

Thus

for all , , , .

Let , ; then condition (H1) holds. Since , condition (H2) also holds with . We easily know that , , , , , , , , , and for . Moreover, we have

and

which implies that hypothesis () is satisfied.

Setting

Obviously,

and

We see that

Moreover, we have

and

which implies that

Hence, 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).

References

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.

I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

V. Lakshmikantham, S. Leela, and V. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009.

L. Debnath and D. D. Bhatta, “Solutions to few linear fractional inhomogeneous partial differential equations in fluid mechanics,” Fractional Calculus and Applied Analysis, vol. 7, no. 1, pp. 21–36, 2004.

W. Chen, H. Sun, X. Zhang, and D. Korošak, “Anomalous diffusion modeling by fractal and fractional derivatives,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1754–1758, 2010.

M. Di Paola, F. P. Pinnola, and M. Zingales, “Fractional differential equations and related exact mechanical models,” Computers & Mathematics with Applications, vol. 66, no. 5, pp. 608–620, 2013.

V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.

A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.

S. T. Zavalishchin and A. N. Sesekin, Dynamic Impulse Systems: Theory and Applications, vol. 394, Kluwer Academic, Dodrecht, The Netherlands, 1997.

G. Wang, B. Ahmad, and L. Zhang, “Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 3, pp. 792–804, 2011.

B. Ahmad and J. J. Nieto, “Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory,” Topological Methods in Nonlinear Analysis, vol. 35, no. 2, pp. 295–304, 2010.

X. Li, F. Chen, and X. Li, “Generalized anti-periodic boundary value problems of impulsive fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 1, pp. 28–41, 2013.

B. Ahmad and G. Wang, “Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order,” Comput. Math. Appl, vol. 59, pp. 1341–1349, 2010.

R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, “A new definition of fractional derivative,” Journal of Computational and Applied Mathematics, vol. 264, pp. 65–70, 2014.

D. R. Anderson and D. J. Ulness, “Newly defined conformable derivatives,” Advances in Dynamical Systems and Applications (ADSA), vol. 10, no. 2, pp. 109–137, 2015.

T. Abdeljawad, “On conformable fractional calculus,” Journal of Computational and Applied Mathematics, vol. 279, pp. 57–66, 2015.

H. Batarfi, J. Losada, J. J. Nieto, and W. Shammakh, “Three-Point Boundary Value Problems for Conformable Fractional Differential Equations,” Journal of Function Spaces, vol. 2015, Article ID 706383, 6 pages, 2015.

B. Abdalla, “Oscillation of differential equations in the frame of nonlocal fractional derivatives generated by conformable derivatives,” Advances in Difference Equations, Paper No. 107, 15 pages, 2018.

A. Akbulut and M. Kaplan, “Auxiliary equation method for time-fractional differential equations with conformable derivative,” Computers & Mathematics with Applications, vol. 75, no. 3, pp. 876–882, 2018.

D. Zhao, X. Pan, and M. Luo, “A new framework for multivariate general conformable fractional calculus and potential applications,” Physica A: Statistical Mechanics and its Applications, vol. 510, pp. 271–280, 2018.

W. Zhong and L. Wang, “Basic theory of initial value problems of conformable fractional differential equations,” Advances in Difference Equations, Paper No. 321, 14 pages, 2018.

S. Asawasamrit, S. K. Ntouyas, P. Thiramanus, and J. Tariboon, “Periodic boundary value problems for impulsive conformable fractional integro-differential equations,” Boundary Value Problems, Paper No. 122, 18 pages, 2016.

W. Ding, Y. Xing, and M. Han, “Anti-periodic boundary value problems for first order impulsive functional differential equations,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 45–53, 2007.