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

Authors: Jungang Chen; Xi Qin
Journal: Journal of Mathematics
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

Differential equations with integral boundary conditions have been applied in many fields such as thermoelasticity, blood flow phenomena, and groundwater systems. For specific details, readers interested in this topic can see papers [1–6] and the references therein. In addition, the advantages of fractional derivatives make fractional differential equations a hot topic. At present, it exhibits great vitality and splendor in a number of applications of interest such as biophysics, hemodynamics, complex media circuit analysis and simulation, control optimization theory, and earthquake prediction models. For more details, refer to books in [7–12]. For the latest developments and trends, refer to [13–24]. Fractional differential system, as an important branch of differential system, is attracting more and more scholars research interest, which comes from its good practical application background (see [25–32]).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:(1)Dαφε=Xε,φε,ψε,ε∈0,ℬ,Dαψε=Yε,ψε,φε,ε∈0,ℬ,ε1−αφε|ε=0=x0,ε1−αψε|ε=0=y0,where 0<ℬ<∞,X,Y∈C0,ℬ×ℝ×ℝ,ℝ,x0,y0∈ℝ and x0≤y0.In [32], Ahmad and Nieto studied a three point-coupled nonlinear Riemann–Liouville fractional differential system given by(2)Dαφε=Xε,ψε,Dδψε,ε∈0,1,Dβψε=Yε,φε,Dσφε,ε∈0,1,φ0=0,φ1=γφη,ψ0=0,ψ1=γψη,where 1<α,β<2,δ,σ,γ>0,0<η<1,α−σ≥1,β−δ≥1,γηα−1<1,γηβ−1<1,X,Y∈C0,1×ℝ×ℝ,ℝ,x0,y0∈ℝ. 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 order0<α≤1:(3)Dαφε=ℱε,φε,ψε,φθε,Dαψε=Gε,ψε,φε,ψθε,where ε∈0,L0<L<∞, J=0,L, ℱ,G∈CJ×ℝ×ℝ×ℝ,ℝ,θ∈CJ,J. Notice that our system contains the unknown functions φε,ψε and deviating arguments φθε,ψθε.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. The advantage of the technique needs no special emphasis [33–40]. 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:(4)ε1−αφε|ε=0=ε1−α∫0τWsψsds+x0,ε1−αψε|ε=0=ε1−α∫0τWsφsds+y0,where0<τ<L,Ws∈CJ,−∞,0,x0,y0∈ℝ and x0≤y0.In the present study, nonlocal type of integral boundary condition with limits of integration involving the parameters0<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:(5)ε1−αφε|ε=0=ε1−α∫ντVsφsds+x0,ε1−αψε|ε=0=ε1−α∫ντVsψsds+y0,where 0<ν<τ<L,Vs∈CJ,0,+∞,x0,y0∈ℝ and x0≤y0.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. Ifν⟶0,τ⟶L, the condition is degenerated to a classic integral boundary condition (see [42] for details).


## 2. Comparison Theorem: The Unique Solution of Linear System

Let(6)C1−αJ=φ∈C0,L;ε1−αφ∈CJ,with the norm(7)φC1−α=ε1−αφC.Next, we provide a comparison result from Wang’s paper [43]. Notice that the comparison result is valid for Iε which is a nonnegative bounded integrable function.Lemma 1.
Letξ∈C1−αJbe locally Hölder continuous, andξsatisfies(8)Dαξε≥−Iεξε,ε1−αξε|ε=0≥0,whereIεis a nonnegative bounded integrable function and satisfiessupε∈JIεLαΓ1−α<1. Then,ξε≥0,for allε∈0,L.Now, we are in a position to prove the following new comparison result for fractional differential system.Lemma 2 (comparison theorem).
Letφ,ψ∈C1−αJbe locally Hölder continuous and satisfy(9)Dαφε≥−Iεφε+Jεψε,ε∈0,L,Dαψε≥−Iεψε+Jεφε,ε∈0,L,ε1−αφε|ε=0≥0,ε1−αψε|ε=0≥0,whereIε,Jεare nonnegative bounded integrable functions andIε≥Jε≥0,for allε∈J.If(10)supε∈JI+JεLαΓ1−α<1,then φε≥0,ψε≥0,for allε∈0,L.Proof.
Putξε=φε+ψε,for allε∈0,L. Then, by (9), we have(11)Dαξε≥−I−Jεξε,ε∈0,L,ε1−αξε|ε=0≥0.
Thus, by (11) and Lemma 1, we have that(12)ξε≥0,forε∈0,L,i.e.φε+ψε≥0,forε∈0,L.
Next, we show thatφε≥0,ψε≥0,forε∈0,L.
In fact, by (9) and (12), we have that(13)Dαφε≥−I+Jεφε,ε∈0,L,ε1−αφε|ε=0≥0.
By (13) and Lemma 1, we have that φε≥0,forε∈0,L. Similarly, we can show that vε≥0,forε∈0,L.Finally, we consider the linear system:(14)Dαφε=σ1ε−Iεφε−Jεψε,ε∈0,L,Dαψε=σ2ε−Iεψε−Jεφε,ε∈0,L,ε1−αφε|ε=0=r1,ε1−αψε|ε=0=r2,where Iε,Jε are nonnegative bounded integrable functions σ1,σ2∈CJ,ℝ,r1,r2∈ℝ.Lemma 3.
If10holds, then the problem14has a unique system of solutions inC1−α0,L×C1−α0,L.Proof.
Let(15)ξ1ε=φε+ψε2,ξ2ε=φε−ψε2,ε∈0,L,where ξ1 and ξ2 solve the problems(16)Dαξ1ε=σ1+σ2ε−I+Jεξ1ε,ε∈0,L,ε1−αξ1ε|ε=0=r1+r2,and(17)Dαξ2ε=σ1−σ2ε−I−Jεξ2ε,ε∈0,L,ε1−αξ2ε|ε=0=r1−r2.
It is obvious that the problems (16) and (17) have the unique solution ξ∗,ξ∗∗∈C1−α0,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 C1−α0,L×C1−α0,L.


## 3. Extremal Solutions of Nonlinear System

Theorem 1.
Assume that the following holds:
H1 There exist two locally Hölder continuous functions φ0,ψ0∈C1−α0,L satisfying φ0ε≤ψ0ε such that(18)Dαφ0ε≤ℱε,φ0ε,ψ0ε,φ0θε,ε∈0,L,ε1−αφ0ε|ε=0≤ε1−α∫0τWsψ0sds+x0,Dαψ0ε≥Gε,ψ0ε,φ0ε,ψ0θε,ε∈0,L,ε1−αψ0ε|ε=0≥ε1−α∫0τWsφ0sds+y0.H2There exist nonnegative bounded integrable functionsIε,Jεwhich satisfy (10), andIε≥Jε, such that(19)ℱε,φε,ψε,φθε−ℱε,με,νε,μθε≥−Iεφ−με−Jεψ−νε,Gε,φε,ψε,φθε−Gε,με,νε,μθε≥−Iεφ−με−Jεψ−νε,Gε,φε,ψε,φθε−ℱε,με,νε,μθε≥−Iεφ−με−Jεψ−νε,where φ0ε≤μ≤φ≤ψ0ε,φ0ε≤ψ≤ν≤ψ0ε.Then, (3) and (4) have extremal systems of solutionsφ∗,ψ∗∈φ0,ψ0×φ0,ψ0. Moreover, there exist monotone iterative sequencesφn,ψn⊂φ0,ψ0such thatφn⟶φ∗,ψn⟶ψ∗n⟶∞uniformly on compact subsets of0,Land(20)φ0≤φ1≤⋯≤φn≤⋯≤φ∗≤ψ∗≤⋯≤ψn≤⋯≤ψ1≤ψ0.Proof.
For anyφn−1,ψn−1∈C1−α0,L, n≥1, considering (14) with(21)σ1nε=ℱε,φn−1ε,ψn−1ε,φn−1θε+Iεφn−1ε+Jεψn−1ε,σ2nε=Gε,ψn−1ε,φn−1ε,ψn−1θε+Iεψn−1ε+Jεφn−1ε,r1n=ε1−α∫0τWsψn−1sds+x0,r2n=ε1−α∫0τWsφn−1sds+y0,we have(22)Dαφnε=σ1nε−Iεφnε−Jεψnε,ε∈0,L,Dαψnε=σ2nε−Iεψnε−Jεφnε,ε∈0,L,ε1−αφnε|ε=0=r1n,ε1−αψnε|ε=0=r2n.
By Lemma3, we know that (22) has a unique system of solutions in C1−α0,L×C1−α0,L.
Now, we show thatφnε,ψnε satisfy(23)φn−1≤φn≤ψn≤ψn−1,n=1,2,….
Letξ=φ1−φ0,ζ=ψ0−ψ1. By condition H1 and (22), we have that(24)Dαξε≥−Iεξε+Jεζε,Dαζε≥−Iεζε+Jεξε,ε1−αξε|ε=0≥0,ε1−αζε|ε=0≥0.
Thus, by Lemma2, we have that ξε≥0,ζε≥0,forallε∈0,L.
Letw=ψ1−φ1, by condition H2 and (22), we get(25)Dαwε=Dαψ1ε−Dαφ1ε=Gε,ψ0ε,φ0ε,ψ0θε+Iεψ0ε+Jεφ0ε−Iεψ1ε−Jεφ1ε−ℱε,φ0ε,ψ0ε,φ0θε−Iεφ0ε−Jεψ0ε+Iεφ1ε+Jεψ1ε≥−Iεψ0−φ0ε−Jεφ0−ψ0ε+Iεψ0ε+Jεφ0ε−Iεψ1ε−Jεφ1ε−Iεφ0ε−Jεψ0ε+Iεφ1ε+Jεψ1ε=−I−Jεwε.Besides,(26)ε1−αwε|ε=0=ε1−α∫0τWsφ0−ψ0sds+y0−x0≥0.
By Lemma1, we can get wε≥0,forε∈0,L. Therefore, we have the relation φ0≤φ1≤ψ1≤ψ0.
Assume thatφk−1≤φk≤ψk≤ψk−1, for some k≥1. Then, using the same way as above, by Lemmas 1 and 2 again, we can obtain φk≤φk+1≤ψk+1≤ψk. By induction, it is not difficult to show that(27)φ0≤φ1≤⋯≤φn≤⋯≤ψn≤⋯≤ψ1≤ψ0.
Employing the standard arguments, we have(28)limn⟶∞φnε=φ∗ε,limn⟶∞ψnε=ψ∗εuniformly on compact subsets of 0,L, and φ∗,ψ∗∈φ0,ψ0 satisfy (3) and (4). Letting n⟶∞ in (3.15), we get that φ∗,ψ∗ is solutions of (3) and (4) in φ0,ψ0×φ0,ψ0 and (20) holds.
Finally, we prove thatφ∗,ψ∗ is extremal solutions of (3) and (4) in φ0,ψ0×φ0,ψ0. If φ,ψ∈φ0,ψ0×φ0,ψ0 is any solutions of (3) and (4), which means(29)Dαφε=ℱε,φε,ψε,φθε,ε∈0,L,Dαψε=Gε,ψε,φε,ψθε,ε∈0,L,ε1−αφε|ε=0=ε1−α∫0τWsψsds+x0,ε1−αψε|ε=0=ε1−α∫0τWsφsds+y0.
By (22), (29), H2, and Lemma 2, it is easy to prove that(30)φn≤φ,ψ≤ψn,n=1,2,….
Taking the limits in (30), we get φ∗≤φ,ψ≤ψ∗, which implies φ∗,ψ∗ is extremal solutions of (3) and (4) in φ0,ψ0×φ0,ψ0.
This completes the proof.We give the following assumption for convenience.H1′ There exist two locally Hölder continuous functions φ0,ψ0∈C1−α0,L satisfying φ0ε≤ψ0ε such that(31)Dαφ0ε≤ℱε,φ0ε,ψ0ε,φ0θε,ε∈0,L,ε1−αφ0ε|ε=0≤ε1−α∫ντVsφ0sds+x0,Dαψ0ε≥Gε,ψ0ε,φ0ε,ψ0θε,ε∈0,L,ε1−αψ0ε|ε=0≥ε1−α∫ντVsψ0sds+y0.By a proof similar to Theorem1, we have the following.Theorem 2.
Suppose that conditionsH1′andH2hold. Then, (3) and (5) have extremal system of solutionsφ∗,ψ∗∈φ0,ψ0×φ0,ψ0. Moreover, there exist monotone iterative sequencesφn,ψn⊂φ0,ψ0such thatφn⟶φ∗,ψn⟶ψ∗n⟶∞uniformly on compact subsets of0,Land(32)φ0≤φ1≤⋯≤φn≤⋯≤φ∗≤ψ∗≤⋯≤ψn≤⋯≤ψ1≤ψ0.


## 4. Example

Consider the following problem:(33)Dαφε=115ε3ε−φε−120ε4ψ2ε+115ε4tanπ4φε22,Dαψε=115ε3ε−ψε−120ε4φ2ε+115ε4tanπ4ψε22,ε1−αφε|ε=0=ε1−α∫1/62/313+s2φsds,ε1−αψε|ε=0=ε1−α∫1/62/313+s2ψsds,where ε∈J,α=1/2, and Dα is the standard Riemann–Liouville fractional derivative.Clearly,(34)ℱε,φε,ψε,φθε=115ε3ε−φε−120ε4ψ2ε+115ε4tanπ4φθε,Gε,ψε,φε,ψθε=115ε3ε−ψε−120ε4φ2ε+115ε4tanπ4ψθε,where θε=ε2/2.Takingφ0ε=0,ψ0ε=1, it is easy to show that condition H1′ of Theorem 2 holds.On the contrary, we have(35)ℱε,φε,ψε,φθε−ℱε,με,νε,μθε≥−215ε3φ−με,Gε,φε,ψε,φθε−Gε,με,νε,μθε≥−215ε3φ−με,Gε,φε,ψε,φθε−ℱε,με,νε,μθε≥−215ε3φ−με,where φ0ε≤μ≤φ≤ψ0ε,φ0ε≤ψ≤ν≤ψ0ε.We seeIε=2/15ε3,Jε=0, and supε∈JI+JεLαΓ1−α=2π/15<1. So, H2 holds.Thus, Theorem2 is applied to the system (33), and we have the conclusion of Theorem 2.


## 5. 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:(36)Dαφε=ℱε,φε,ψε,φθε,Dαψε=Gε,ψε,φε,ψθε,with two types of integral boundary conditions:(i)
Nonlocal coupled integral boundary conditions of the form:(37)ε1−αφε|ε=0=ε1−α∫0τWsψsds+x0,ε1−αψε|ε=0=ε1−α∫0τWsφsds+y0.(ii)
Nonlocal strip condition of the form:(38)ε1−αφε|ε=0=ε1−α∫ντVsφsds+x0,ε1−αψε|ε=0=ε1−α∫ντVsψsds+y0.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. The biggest difficulty for this is to perfectly establish new comparison result for fractional differential system with impulsive effect.



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