Monotone Iterative Method for Two Types of Integral Boundary Value Problems of a Nonlinear Fractional Differential System with Deviating Arguments

This paper concerns on two types of integral boundary value problems of a nonlinear fractional diﬀerential system, i.e ., nonlocal strip integral boundary value problems and coupled integral boundary value problems. With the aid of the monotone iterative method combined with the upper and lower solutions, the existence of extremal system of solutions for the above two types of diﬀerential systems is investigated. In addition, a new comparison theorem for fractional diﬀerential system is also established, which is crucial for the proof of the main theorem of this paper. At the end, an example explaining how our studies can be used is also given.

In [25], by applying the monotone iterative method, Wang, Agarwal, and Cabada investigated the existence of extremal solutions for a nonlinear Riemann-Liouville fractional differential system: where 0 < B < ∞, X, Y ∈ C([0, B] × R × R, R), x 0 , y 0 ∈ R and x 0 ≤ y 0 . In [32], Ahmad and Nieto studied a three point-coupled nonlinear Riemann-Liouville fractional differential system given by By using the Schauder fixed-point theorem, the authors successfully obtained the existence of solution of the system. Inspired by these papers, we concern on the following nonlinear Riemann-Liouville fractional differential system of order 0 < α ≤ 1: where . Notice that our system contains the unknown functions φ(ε), ψ(ε) and deviating arguments φ(θ(ε)), ψ(θ(ε)).
In order to approximate the solution of the nonlinear Riemann-Liouville fractional differential system mentioned above, we firstly give a new comparison result for fractional differential system. Also, we develop the monotone iterative technique for the system. e advantage of the technique needs no special emphasis [33][34][35][36][37][38][39][40]. It is worth to point that, in this paper, only half pair of upper and lower solutions is assumed to the system, which is weaker than a pair of upper and lower solutions. It is believed that this is also an attempt to apply the monotone iterative method to solve nonlinear Riemann-Liouville fractional differential systems with deviating arguments and families of nonlocal coupled and strip integral boundary conditions.
To this end, we study the following two types of integral boundary conditions: (i) Nonlocal coupled integral boundary conditions of the form: where 0 < τ < L, W( s ) ∈ C( J, ( − ∞, 0 ] ), x 0 , y 0 ∈ R and x 0 ≤ y 0 . In the present study, nonlocal type of integral boundary condition with limits of integration involving the parameters 0 < τ < L has been introduced. It is worth mentioning that, in practical situations, such nonlocal integral boundary conditions may be regarded as a continuous distribution of arbitrary finite length; for instance, refer to [41]. (ii) Nonlocal strip condition of the form: where 0 < ] < τ < L, V( s ) ∈ C( J, [ 0, +∞ ) ), x 0 , y 0 ∈ R and x 0 ≤ y 0 . In fact, nonlocal strip condition is used to describe a continuous distribution of the values of the unknown function on an arbitrary finite segment of the interval. If ] ⟶ 0, τ ⟶ L, the condition is degenerated to a classic integral boundary condition (see [42] for details).

Comparison Theorem: The Unique Solution of Linear System
Let with the norm Next, we provide a comparison result from Wang's paper [43]. Notice that the comparison result is valid for I(ε) which is a nonnegative bounded integrable function.

be locally Hölder continuous, and ξ satisfies
where I(ε) is a nonnegative bounded integrable function and Now, we are in a position to prove the following new comparison result for fractional differential system. Lemma 2 (comparison theorem). Let φ, ψ ∈ C 1− α (J)be locally Hölder continuous and satisfy en, by (9), we have us, by (11) and Lemma 1, we have that In fact, by (9) and (12), we have that By (13) and Lemma 1, we have that Finally, we consider the linear system: where Lemma 3. If 10 holds, then the problem 14has a unique system of solutions in Proof. Let where ξ 1 and ξ 2 solve the problems and It is obvious that the problems (16) and (17) have the unique solution ξ * , ξ * * ∈ C 1− α ([0, L]), respectively. Since ξ * , ξ * * are unique, then by (14) and (15), we can show that the problem (14) has a unique system of solutions in

□
We give the following assumption for convenience. (H 1 ′ ) ere exist two locally Hölder continuous func- By a proof similar to eorem 1, we have the following.
On the contrary, we have us, eorem 2 is applied to the system (33), and we have the conclusion of eorem 2.

Conclusion
In this paper, by employing the method of upper and lower solutions combined with the monotone iterative technique, we studied a class of nonlinear fractional differential system involving nonlocal strip and coupled integral boundary conditions. Precisely, we considered the following nonlinear Riemann-Liouville fractional differential system: with two types of integral boundary conditions: (i) Nonlocal coupled integral boundary conditions of the form: (ii) Nonlocal strip condition of the form: We investigated the existence of extremal system of solutions for the above nonlinear fractional differential system involving nonlocal strip and coupled integral boundary conditions. A new comparison result for fractional differential system was also established, which played an important role in the proof of our main results. It is a contribution to the field of fractional differential system. As an extension of our conclusion, we present an open question, namely, how to develop the existence of extremal system of solutions for the above nonlinear fractional differential system with impulsive effect by the method of upper and lower solutions combined with the monotone iterative technique. e biggest difficulty for this is to perfectly establish new comparison result for fractional differential system with impulsive effect.

Data Availability
No data were used to support this study.

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