Monotone Iterative Technique for Conformable Fractional Differential Equations with Deviating Arguments

Chen, Huiling; Meng, Shuman; Cui, Yujun

doi:https://doi.org/10.1155/2020/5827127

Discrete Dynamics in Nature and Society

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Fractional Difference and Differential Operators and their Applications in Nonlinear Systems

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Research Article | Open Access

Volume 2020 | Article ID 5827127 | https://doi.org/10.1155/2020/5827127

Monotone Iterative Technique for Conformable Fractional Differential Equations with Deviating Arguments

Huiling Chen,¹Shuman Meng,¹and Yujun Cui²

Academic Editor: Qasem M. Al-Mdallal

Received21 Feb 2020

Revised05 Apr 2020

Accepted22 Sept 2020

Published15 Oct 2020

Abstract

This paper is concerned with the existence of extremal solutions for periodic boundary value problems for conformable fractional differential equations with deviating arguments. We first build two comparison principles for the corresponding linear equation with deviating arguments. With the help of new comparison principles, some sufficient conditions for the existence of extremal solutions are established by combining the method of lower and upper solutions and the monotone iterative technique. As an application, an example is presented to enrich the main results of this article.

1. Introduction

In recent years, people have been paying attention to the progress of the fractional differential equations. In fact, it is the generalization of the ordinary differential equations to a noninteger order. Significantly, fractional differential equations appear more frequently in different fields of science and engineering, such as viscoelasticity, circuit, and neuron modeling [1–3]. Gradually, fractional differential equations are increasingly regarded as effective assistants. We have observed that many papers are exploring the existence of solutions of boundary value problems for fractional differential equations by using nonlinear functional analysis methods such as fixed point theorems, fixed point index on cone, variational methods and critical point theory, the theory of Mawhin coincidence degree, and the upper and lower solution method; see the monographs of Kilbas et al. [1], Podlubny [2], Diethem [3], the papers [4–26], and the references therein. Among them, the monotone iterative technique is an ingenious and effective method that offers theoretical, as well constructive existence results for nonlinear problems via linear iterates [9–15, 17, 23, 26]. It yields monotone sequences that converge to the extremal solutions in a sector generated by the upper and lower solutions. For example, the authors of [22] adopted the method of monotone iteration combined with the method of upper and lower solutions to consider the following system of nonlinear fractional differential equations:where , , , and . In addition, [15, 24] used these methods to study the initial value problems for nonlinear fractional differential equations with no deviating arguments. On the basis of [22], Jian et al. [13] successfully investigated the following nonlinear fractional order differential systems with deviating arguments:where . They introduce two well-defined monotone sequences that converge to the solution of the system and, then, establish the existence and uniqueness of the solution of the system. Finally, a numerical iterative scheme is introduced to obtain an accurate approximate solution for the systems.

Motivated by the abovementioned papers, in this paper, we devote ourselves to the existence of solutions to the following boundary value problems with deviation arguments:where , , and , and is the conformable fractional derivative of order . The conformable fractional calculus which was introduced in the work of Khalil et al. [27], then developed by Abdeljawad [28], have been receiving a lot of attention due to the wide application in physics and engineering [29, 30]. The reader is referred to [14, 16, 17, 27–33] and references therein for some recent advances in conformable fractional calculus and its applications.

In this paper, by establishing two comparison results and using the monotone iterative technique combined with the method of upper and lower solutions, some sufficient conditions are presented for the existence of extremal solutions for periodic boundary value problem (3).

2. Preliminaries

Definition 1 (See [27]). Let and , and the conformable fractional derivative of order is defined byfor , and the conformable fractional derivative at 0 is defined as . If is differentiable, then .

Definition 2 (See [27]). Let . The conformable fractional integral of a function of order is denoted as

Lemma 1 (See [32]). Let . Assume that and with . Then, we have

Lemma 2 (See [27]). Let , , and the functions , be -differentiable on . Then,(a) for all constant functions (b)(c)(d)(e) when

Lemma 3 (See [34]). Let linear operator, be the spectral radius of , and . Then,(1)(2)if , then exists and , where stands for the identity operatorIt is given that . Let ; then, is a Banach space with the norm .

Let us introduce the following values and functions which will be used in the rest paper.

For the forthcoming analysis, we first consider the following two boundary value problems for a linear differential fractional equations:

Lemma 4. Let , , and . Then, problem (8) has the unique solution:where and

Proof. Multiply both sides of the first equation of (8) by , namely,By using Lemma 2 (d), equation (12) is equivalent toIn view of Lemma 1 and Definition 2, we getsoThe boundary condition leads toClearly,Substituting (17) into (15), it follows that linear problem (8) has the following integral representation of the solution:This completes the proof.
For all , Green’s function admits the following properties:Namely,In addition, for given in Lemma 4, we can getWe define the operator on byIt is easy to see that is a positive linear continuous operator.

Lemma 5. .

Proof. By direct computation, one hasThen, for any , we havewhich implies that . On the other hand, take , then , , andThis yields . Therefore, . This completes the proof.
We recall that . Then, . For , it follows from (20) thatThe abovementioned two inequalities show thatBased on the above analysis, we have the following result on (9).

Lemma 6. Let , , , , and . Then, problem (9) has a unique solution.

Proof. From Lemma 4, it follows that is a solution of (9) if and only ifNow, we introduce an operator as follows:It is easy to see that is a positive linear operator with . Thus, (28) reduces toNote from Lemma 5 that . Thus, it follows from Lemma 3 that exists andTherefore, the unique solution of (9) is given byThe proof is complete.
Now, we present two comparison results.

Lemma 7. Let , , , and . Assume that satisfies andThen, for all .

Proof. Take , . Then,Applying Lemma 6, (32) holds, and (32) can be expressed bySince , it implies that . Thus, from (27), we obtainWith the help of positivity of operator , the definition of operator , and (23), we haveConsequently, we conclude thatOn the other hand, by (21), we infer thatHence, holds for all that follow from and (35). This completes the proof.

Lemma 8. Let , , , , , and . Assume that satisfies and (33). Then, for all .

Proof. Take again , . Then,Applying Lemma 6, (32) holds, and (32) can be expressed byTaking notice of the fact that , by (20), we haveand for ,and for ,These lead us toBy (20)–(23) and the positivity of operator , we haveand for ,and for ,These, together with the fact that , ensure thatThus, by (41), (45), and (49), we have that for all , and the lemma is proved.

3. Main Results

Now, we are in the position to prove the existence of extremal solutions of (3) by using the monotone iterative method of lower and upper solutions. To this end, we define the lower and upper solutions of (3).

Definition 3. A function satisfying is called a lower solution of problem (3) if it satisfiesAnalogously, a function satisfying is called an upper solution of (3) if the inequalitieshold.

Theorem 1. Assume that the following conditions hold: : the functions and are lower and upper solutions of problem (3), respectively, such that on and there exist , such that for all , , : the inequality holds or the inequalities , , , holdThen, (3) has minimal and maximal solution in the sector , which can be obtained by monotone iterative sequences starting from and , where .

Proof. For , let us defineBy Lemma 6, for any , we know that linear problems (53) and (54) have a unique solution , , respectively, which implies that the sequences , are well defined. Furthermore, , can be expressed aswhere is a bounded operator defined byBy the integral expression of operator , it is easy to see that is completely continuous. Hence, is completely continuous.
Firstly, let us prove thatTo do this, let . By the definition of the lower solution, we getThis shows, by Lemma 7 or Lemma 8, that on , and hence, . Similarly, we can deduce that .
Now, let ; by and , we obtainThen, from Lemma 7 or Lemma 8, we get , which yields .
Secondly, we need to show that and are the lower and upper solutions of problem (3), respectively. In fact, it follows from and thatwhich show that is a lower solution of problem (3). Similarly, we can conclude that is an upper solution of problem (3).
Repeating the foregoing arguments, we can prove that the sequences , are lower and upper solutions of problem (3), respectively, and satisfy the following inequality:Obviously, the sequences , are uniformly bounded in and by (55) and the complete continuity of operator , and it follows that , are relatively compact. This, together with the monotonicity of the sequences , , guarantees that the sequences , converge uniformly to , respectively, and that are solutions of (3).
Finally, we prove the minimal and maximal property of and on . We assume that is any solution of (3) and there exists a positive integer such that for .
Let , thenundoubtedly, , namely, . By a similar method, we can show that . Thus, , . It is easy to find that when . That is , are minimal and maximal solutions of (1) in the sector . The proof is completed.
Then, by applying Lemma 7 or Lemma 8, we get , that is on . Similarly, we can show that on . Notice that on . So, hold for every from mathematical induction. Hence, by taking , we have on . The proof is complete.

Example 1. We consider the following BVP:Obviously, , , , andLet ; then,This shows that are lower and upper solutions of (63). On the other hand, it is easy to verify that holds for and . Furthermore, we haveHence, all conditions of Theorem 1 hold. Therefore, equation (63) has the extremal solution in .

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11571207 and 51774197), the Shandong Natural Science Foundation (ZR2018MA011), SDUST graduate innovation project (SDKDYC190238), and the Tai’shan Scholar Engineering Construction Fund of Shandong Province of China.

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Copyright © 2020 Huiling Chen et al. 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.

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