Monotone Iterative Technique for Conformable Fractional Differential Equations with Deviating Arguments

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.


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][2][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 , 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: (t, v(t), w(t)), t ∈ (0, T], D α w(t) � g(t, w(t), v(t)), t ∈ (0, T], where 0 < T < ∞, f, g ∈ C([0, T] × R × R, R), x 0 , y 0 ∈ R, and x 0 ≤ y 0 . 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 θ ∈ C([0, 1], [0, 1]). ey 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: , and D δ ϕ is the conformable fractional derivative of order δ. e 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]. e reader is referred to [14,16,17,[27][28][29][30][31][32][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).
For the forthcoming analysis, we first consider the following two boundary value problems for a linear differential fractional equations: Lemma 4. Let K > 0, a ∈ R, and h ∈ E. en, problem (8) has the unique solution: where Proof. Multiply both sides of the first equation of (8) by e K 1 t δ , namely, By using Lemma 2 (d), equation (12) is equivalent to In view of Lemma 1 and Definition 2, we get so e boundary condition ϕ(0) � ϕ(T) + a leads to Clearly, Substituting (17) into (15), it follows that linear problem (8) has the following integral representation of the solution: is completes the proof. For all 0 < δ ≤ 1, Green's function G admits the following properties: Namely, In addition, for Ψ given in Lemma 4, we can get We define the operator A on E by It is easy to see that A: E ⟶ E is a positive linear continuous operator. □ Lemma 5. ‖A‖ � (1/K).

Proof. By direct computation, one has
en, for any h ∈ E, we have Discrete Dynamics in Nature and Society which implies that ‖A‖ ≤ (1/K). On the other hand, take h 0 (t) ≡ 1, then h 0 ∈ E, ‖h 0 ‖ � 1, and e abovementioned two inequalities show that Based on the above analysis, we have the following result on (9).
and h ∈ E. en, problem (9) has a unique solution.
Proof. From Lemma 4, it follows that ϕ ∈ E is a solution of (9) if and only if
us, it follows from Lemma 3 that (I + AB) − 1 exists and erefore, the unique solution of (9) is given by e proof is complete. Now, we present two comparison results.
Applying Lemma 6, (32) holds, and (32) can be expressed by With the help of positivity of operator AB, the definition of operator B, and (23), we have Consequently, we conclude that On the other hand, by (21), we infer that Hence, ϕ(t) ≤ 0 holds for all t ∈ [0, T] that follow from a ≤ 0 and (35). is completes the proof.

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).
Analogously, a function w 0 ∈ E satisfying D δ w 0 ∈ E is called an upper solution of (3) if the inequalities hold.
Theorem 1. Assume that the following conditions hold: : the functions u 0 and w 0 are lower and upper solutions of problem (3), respectively, such that Proof. For k � 1, 2, . . ., let us define By Lemma 6, for any k � 1, 2, . . ., we know that linear problems (53) and (54) have a unique solution u k (t), w k (t), respectively, which implies that the sequences u k (t) , w k (t) are well defined. Furthermore, u k (t), w k (t) can be expressed as where F: E ⟶ E is a bounded operator defined by By the integral expression of operator A, it is easy to see that A is completely continuous. Hence, (I + AB) − 1 AF is completely continuous.
Firstly, let us prove that To do this, let v(t) � u 0 (t) − u 1 (t). By the definition of the lower solution, we get is shows, by Lemma 7 or Lemma 8, that v(t) ≤ 0 on [0, T], and hence, u 0 ≤ u 1 . Similarly, we can deduce that w 1 ≤ w 0 . 6 Discrete Dynamics in Nature and Society Now, let v(t) � u 1 (t) − w 1 (t); by (H 2 ) and (H 3 ), we obtain (59) en, from Lemma 7 or Lemma 8, we get v(t) ≤ 0, which yields u 1 ≤ w 1 .
Secondly, we need to show that u 1 and w 1 are the lower and upper solutions of problem (3), respectively. In fact, it follows from (H 2 ) and (H 3 ) that which show that v 1 is a lower solution of problem (3). Similarly, we can conclude that w 1 is an upper solution of problem (3).
Repeating the foregoing arguments, we can prove that the sequences u k (t) , w k (t) are lower and upper solutions of problem (3), respectively, and satisfy the following inequality: Obviously, the sequences u k (t) , w k (t) are uniformly bounded in E and by (55) and the complete continuity of operator (I + AB) − 1 AF, and it follows that u k (t) , w k (t) are relatively compact.
is, together with the monotonicity of the sequences u k (t) , w k (t) , guarantees that the sequences u k (t) , w k (t) converge uniformly to u, w, respectively, and that u, w ∈ [v 0 , w 0 ] are solutions of (3).

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.