# Extremal Solutions for Caputo Conformable Differential Equations withp-Laplacian Operator and Integral Boundary Condition

Authors: Zhongqi Peng; Yuan Li; Qi Zhang; Yimin Xue
Journal: Complexity
 (2021)

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## Abstract

The Caputo conformable derivative is a new Caputo-type fractional differential operator generated by conformable derivatives. In this paper, using Banach fixed point theorem, we obtain the uniqueness of the solution of nonlinear and linear Cauchy problem with the conformable derivatives in the Caputo setting, respectively. We also establish two comparison principles and prove the extremal solutions for nonlinear fractionalp-Laplacian differential system with Caputo conformable derivatives by utilizing the monotone iterative technique. An example is given to verify the validity of the results.


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## Body

## 1. Introduction

In recent years, fractional calculus has been widely developed in pure mathematics and applied mathematics [1–7]. The characteristic of fractional calculus is that there are many different fractional derivatives or integrals, like Riemann–Liouville (RL), Caputo, Hadamard, Caputo–Hadamard types, and so on [1, 2, 8, 9]. So, the scholars have the opportunity to choose the most appropriate operators to describe complex problems in the real world. We recall some definitions from the traditional fractional calculus [1, 2].The left RL fractional integral of orderβ>0 is given by(1)IaRβft=1Γβ∫abt−sβ−1fsds.The left RL fractional derivative of orderβ>0 is defined as(2)Daβft=ddtnIan−βft=d/dtnΓn−β∫att−sn−β−1fsds.The left Caputo fractional derivative of orderβ>0 is given by(3)DaCβft=Ian−βfnt=1Γn−β∫att−sn−β−1fnsds.However, some basic properties such as product rule and chain rule are not valid for the RL and Caputo-type fractional derivatives. In 2014, Khalil et al. [10] defined a new fractional differential operator named the conformable derivative which satisfies the product rule and some other properties. In 2015, Abdeljawad [11] defined the left conformable integral Iaα and derivative Taα as(4)Iaαft=1Γα∫abs−aα−1fsds,Taαft=limɛ⟶0ft+ɛt−a1−α−ftɛ,where α∈0,1,t>0,f:a,+∞⟶ℝ. If f is differentiable, then Taαft=t−a1−αf′t.In 2017, Jarad et al. [12] established the conformable calculus in both RL and Caputo setting based on the work of Abdeljawad.The left RL conformable integral of orderβ∈ℂ,Reβ>0, is given by(5)Iaβαft=1Γβ∫att−aα−s−aααβ−1fsdss−a1−α.The fractional integral (5) coincides with the traditional RL fractional integral (1) if α=1.LetIαa,b=f:a,b⟶ℝ;ft=Iaαψt+fafor someψ∈Lαa, where Lαa=ψ:a,b⟶ℝ;Iaαψtexists for anyt∈a,b. If n∈ℕ+, f∈Cα,ana,b=f:a,b⟶ℝ;Tan−1αf∈Iαa,b, the left conformable derivative of order β∈ℂ,Reβ>0, in the Caputo setting is defined by(6)DaCβαft=Ian−βαTanαft=1Γn−β∫att−aα−s−aααn−β−1Tanαfss−a1−αds,where n=β+1, Tanα=TaαTaα…Taα︸ntimes and Taαft=t−a1−αf′t. The fractional derivative (6) coincides with the traditional Caputo fractional derivative (3) if α=1. Readers can see [13, 14] for more details.It is well known that the monotone iterative technique coupled with the method of upper and lower solutions is an effective mechanism to obtain extremal solutions for nonlinear problems [15]. By using this method, scholars have studied the periodic boundary value problems (BVPs) [16–24], anti-periodic BVPs [25–27], and integral BVPs [28, 29] of integer-order differential equations. Later, this method was widely used to study the initial value problems, periodic BVPs or integral BVPs of RL and Caputo fractional differential equations [30–35].Mathematical modeling of the real world in physics and mechanical and dynamical systems often involves thep-Laplacian operator. In order to study the turbulent flow in a porous medium, Lejbenson [36] introduced the model of differential equation with the p-Laplacian operator. Many results about the fractional differential equations with the p-Laplacian operator were also studied [37–39]. However, the Caputo conformable fractional differential equations with the p-Laplacian operator have not been considered.In [37], Liu et al. studied the following problem:(7)D0+βϕpDC0+αxt=ht,xt,DC0+αxt,t∈0,1,DC0+αx0=x′0=0,x1=r1xη,DC0+αx1=r2DC0+αxξ,where 1<α,β≤2,r1,r2≥0, h∈C0,1×0,+∞×−∞,0,0,+∞, and ϕps is the p-Laplacian operator. The extremal solutions were obtained under the assumption that(8)ht,w1,z1≤ht,w2,z2,for0≤w1<w2,z1>z2≥0.Inspired by the above work, we study the nonlinear fractionalp-Laplacian differential system involving the Caputo conformable derivatives as follows:(9)DaCβαϕpDaCγαxt=ht,xt,DaCγαxt,t∈a,b,TakαϕpDaCγαxa=bk,xa=∫abws,xsds+ρ,where n−1<β≤n,n=β+1,0<γ,α≤1, ρ≥0, h∈Ca,b×ℝ2,ℝ, w∈Ca,b×ℝ,ℝ, bkk=0,1,…,n−1 are real numbers, ϕps=sp−2sp>1 is the p-Laplacian operator, ϕp−1=ϕq,1/p+1/q=1, and DaCδα is Caputo conformable derivative with order δ=β,γ.To obtain the extremal solutions of problem (9), we need consider the nonlinear Cauchy problem(10)DaCβαzt=gt,zt,Takαza=bk,t∈a,b,and the linear Cauchy problem(11)DaCβαzt=σt−λzt,Takαza=bk,t∈a,b.The main contributions of this paper are as follows:(i)
We obtain the unique solution to problem (10) and construct the approximate solutions to problem (11) in terms of Mittag-Leffler function. The corresponding results of problem (10) and problem (11) can be seen as a generalization of Theorem 3.25 and Theorem 4.3 in [1], respectively.(ii)
Based on two comparison principles, we obtain the extremal solutions to problem (9) by using the monotone iterative technique. Different from [37], the restrictive condition of function h is no longer needed in this paper.The remainder of this paper is organized as follows. In Section2, we give some preliminaries and define some special spaces. In Section 3, we show the uniqueness of the solution for linear and nonlinear Cauchy problems. In Section 4, two comparison principles are established. In Section 5, the extremal solutions for problem (9) are obtained. In Section 6, a numerical example is given.


## 2. Preliminaries

In this section, we introduce some definitions and lemmas to be used in the sequel.LetCnJ,ℝ be a Banach space of all n-order continuously differentiable functions on J=a,b. For n−1<β≤n,0<γ,α≤1, and 0≤τ<1 such that τ≤β−k, we define the spaces CγJ,Cα,τJ,CTnJ,CT,τnJ, and CT,τβ,nJ as follows:(12)CγJ=f:ft∈CJ,DaCγαfJ∈CJ,under the norm fγ=fC+DaCγαfC, where fC=maxt∈Jft and DaCγαfC=maxt∈JDaCγαft.(13)Cα,τJ=ft∈Ca,b:t−aαατft∈CJ,Cα,0J=CJ,under the norm(14)fCα,τ=t−aαατfC=maxt∈Jt−aαατft,CTnJ=ft∈CJ:Tanαft∈CJ,under the norm(15)fCTn=∑k=0nTanαfC=∑k=0nmaxt∈JTanαft,CT,τnJ=ft∈CJ:Tan−1αft∈CJ,Tanαft∈Cα,τJ,CT,τβ,nJ=ft∈CTn−1J:DaCβαft∈Cα,τJ.For convenience, we present the following assumptions:(H1) For t∈a,b, z1,z2∈CTn−1J, assume that function g satisfies(16)gt,z1t−gt,z2t≤Mz1t−z2t,M≥0.(H2) Assume that x0t≤y0t,t∈J, where x0t,y0t∈CγJ are lower and upper solutions of (9), respectively.(H3) Assume that a constant λ≤0 such that(17)ht,yt,DaCγαyt−ht,xt,DaCγαxt≥−λϕpDaCγαyt−ϕpDaCγαxt,wherex0t≤xt≤yt≤y0t, t∈J.(H4) Assume that a constant 0≤η<1/b−a such that(18)wt,yt−wt,xt≥ηyt−xt,where x0t≤xt≤yt≤y0t, t∈J.Definition 1.
The functionx0t∈CγJ satisfying ϕpDaCγαx0t∈CTnJ is a lower solution of problem (9) if it satisfies(19)DaCβαϕpDaCγαx0t≤ht,x0t,DaCγαx0t,t∈J,TakαϕpDaCγαx0a≤bk,xa≤∫abws,x0sds+ρ.The functiony0t∈CγJ satisfying ϕpDaCγαy0t∈CTnJ is an upper solution of problem (9) if the above inequalities are reversed.Lemma 1 (see [12]).
Forα>0, the space Cα,anJ consists of those and only those functions which are represented in the form(20)ft=1n−1!∫att−aα−s−aααn−1ψsdss−a1−α+∑k=0n−1Takαfak!x−aαkαk,where ψt=Tanαft and ψt∈Lαa.Lemma 2 (see [12]).
Letf∈Cα,anJ, β∈ℂ. Then,(21)IaβαDaCβαft=ft−∑k=0n−1Takαfat−aαkαkk!.Remark 1.
Lemma1 still holds if we replace the space Cα,anJ with CT,τnJ. In such case, ψt=Tanαft and ψt∈Cα,τJ. In particular, ψt∈CJ when τ=0. Lemma 2 is also valid for f∈CTnJ.Lemma 3 (see [12]).
LetReβ>0, Reγ>0,0<m<Reβ,m∈ℕ. Then,(a)
IaβαIaγαft=Iaβ+γαft.(b)
Iaβαt−aαγ−1=1/αβΓγ/Γβ+γt−aαβ+γ−1.(c)
TamαIaβαft=Iaβ−mαft.Lemma 4 (see [1]) (Banach fixed point theorem).
LetU,d be a nonempty complete metric space, let 0≤ρ<1, and let A:U⟶U be the map such that, for every x,y∈U, the relation(22)dAx,Ay≤ρdx,y,holds. Then, the operator A has a unique fixed point x∗∈U. Moreover, if Akk∈ℕ is the sequence of operators defined by(23)A1=A,Ak=AAk−1k∈ℕ\1,then for any x0∈U, the sequence Akx0k=1∞ converges to the above fixed point x∗.


## 3. The Unique Solution to the Nonlinear and Linear Cauchy Problems

In this section, we first consider the unique solution of nonlinear Cauchy problem (10) and linear Cauchy problem (11), where the function g:a,b×ℝ⟶ℝ such that g∈Cα,τJ.Letzt∈CTn−1J, and by Lemma 2 and the initial value condition Takαzt=bk, problem (10) can be reduced to the Volterra-type integral equation(24)zt=∑j=0n−1bjt−aαjαjj!+1Γβ∫atKβ−1t,sgs,zsdss−a1−α,where Kt,s=t−aα−s−aα/α,a≤s≤t≤b. Denoting z0=∑j=0n−1bjt−aαj/αjj!, equation (24) can be rewritten as zt=Azt, where(25)Azt=z0t+1Γβ∫atKβ−1t,sgs,zsdss−a1−α.Theorem 1.
If (H1) holds, there exists a unique solution zt∈CT,τβ,n−1J for problem (10).Proof.
First, we chooset1a<t1<b such that(26)∑k=0n−1MΓβ−k+1t1−aααβ−k<1,and prove that (10) has a unique solution zt∈CTn−1a,t1.
Applying the operatorTakα to both sides of (25), by Lemma 3 (c), we can get(27)TakαAzt=Takαz0t+1Γβ−k∫atKβ−k−1t,sgs,zsdss−a1−α,where Takαz0t=∑j=kn−1bjt−aj−k/αj−kj−k!. It is obvious that Takαz0t is continuous on a,t1. Furthermore, for 0≤τ<β−k and g∈Cα,τJ, we get by (14) that(28)Iaβ−kαg=1Γβ−k∫atKβ−k−1t,sgs,zsdss−a1−α≤1Γβ−k∫att−aα−s−aααβ−k−1gs,zsdss−a1−α≤gCα,τa,t1Γβ−k∫att−aα−s−aααβ−k−1s−aαα−τdss−a1−α≤Γ1−τgCα,τa,t1Γβ−k+1−τt1−aααβ−k−τ,that is,(29)Iaβ−kαgCa,t1≤Γ1−τgCα,τa,t1Γβ−k+1−τt1−aααβ−k−τ.
Inequality (29) implies that the operator Iaβ−kα is bounded from Cα,τa,t1 to Ca,t1. In particular, if τ=0, then g∈CJ and Iaβ−kα is bounded from Ca,t1 to Ca,t1 such that(30)Iaβ−kαgCa,t1≤gCa,t1Γβ−k+1t1−aααβ−k.
It follows from (29) and (30) that TakαAzt is continuous on a,t1, that is, Azt∈CTn−1a,t1. By (14), (15), and (H1), we have(31)∑k=0n−1TakαAz1t−TakαAz2t=∑k=0n−11Γβ−k∫atKβ−k−1t,sgs,z1s−gs,z2sdss−a1−α≤∑k=0n−11Γβ−k∫atKβ−k−1t,sgs,z1s−gs,z2sdss−a1−α≤∑k=0n−1MΓβ−k∫att−aα−s−aααβ−k−1z1s−z2sdss−a1−α≤∑k=0n−1Mz1−z2Ca,t1Γβ−k+1t1−aααβ−k≤∑k=0n−1Mz1−z2CTn−1a,t1Γβ−k+1t1−aααβ−k,that is,(32)Az1−Az2CTn−1a,t1=∑k=0n−1TakαAz1t−TakαAz2tCa,t1≤∑k=0n−1Mz1−z2CTn−1a,t1Γβ−k+1t1−aααβ−k.
From Lemma4 and (26), we get that there exists a unique solution z1∗∈CTn−1a,t1 to problem (10). Moreover, z1∗t satisfies(33)limi⟶∞Aizg−z1∗CTn−1a,t1=0,where zgt is any function in CTn−1a,t1 and Aizgt=AAi−1zgt. Let zit=Aizgt; then,(34)limi⟶∞zit−z1∗CTn−1a,t1=0.
Next, chooset2,t3,…,tR such that a=t0<t1<t2<⋯<tR=b. Using the same arguments as above, we get that problem (10) has a unique solution zr∗∈CTn−1tr−1,trr=1,2,…,R. Therefore, (10) has a unique solution z∗=zr∗∈CTn−1J.
Finally, we show that the unique solutionz∗t belongs to CT,τβ,n−1J. By (15) and (H1), we have(35)DaCβαzi−DaCβαz∗Cα,τJ=gt,zit−gt,z∗tCα,τJ≤Mt1−aαατzi−z∗CJ≤Mt1−aαατzi−z∗CTn−1J.
Taking the limit asi⟶∞, we obtain(36)limi⟶∞DaCβαzi−DaCβαz∗Cα,τJ=0,which implies that z∗t∈CT,τβ,n−1J. This ends the proof.Remark 2.
Whenα=1, the RL conformable integral and Caputo conformable derivative coincide with the traditional RL fractional integral and Caputo derivative, respectively. Hence, the results of Theorem 3.25 in [1] can be seen as the special case of Theorem 1.Corollary 1.
If (H1) holds and the function g:J×ℝ⟶ℝ satisfies g∈CJ, then Cauchy problem (10) has a unique solution zt belonging to CTn−1J.Proof.
Corollary1 can be proven by replacing τ with 0 and using the same argument in Theorem 1.Theorem 2.
Ifσt∈CJ and λ is a constant, then problem (11) has a unique solution zt∈CTn−1J which is given by(37)zt=∑j=0n−1bjt−ajααjEβ,j+1−λt−aαβαβ+∫atKβ−1t,sEβ,β−λKβt,sσsdss−a1−α,where Ep,qρ=∑k=0∞ρk/Γpk+q is the Mittag-Leffler function.Proof.
Clearlygt,zt=σt−λzt satisfies (H1). By Corollary 1, there exists a unique solution zt∈CTn−1J to problem (11).
Next, we prove that this unique solution is given by (18). By Lemma 2 and the initial value condition Takαzt=bk, problem (11) can be reduced to the equation(38)zt=∑j=0n−1bjt−aαjαjj!+−λIaβαzt+Iaβασt.
We apply the successive approximations method to solve equation (38) by taking z0t=∑j=0n−1bjt−aαj/αjj! and(39)zmt=z0t+−λIaβαzm−1t+Iaβασt,m=1,2,….
By Lemma3 (b), for m=1, we have(40)z1t=z0t+−λIaβαz0t+Iaβασt=z0t+∑j=0n−1−λbjαjj!Iaβαt−aαj+Iaβασt=z0t+∑j=0n−1−λbjt−aαβ+jαβ+jΓβ+j+1+Iaβασt.
By (a) and (b) of Lemma3, for m=2,(41)z2t=z0t+−λIaβαz1t+Iaβασt=z0t+−λIaβαz0t+−λIaβαz0t+Iaβασt+Iaβασt=z0t+−λIaβαz0t+−λ2Ia2βαz0t+−λIa2βασt+Iaβασt=z0t+∑j=0n−1−λbjt−aαβ+jαβ+jΓβ+j+1+∑j=0n−1−λ2bjt−aα2β+jα2β+jΓ2β+j+1+−λIa2βασt+Iaβασt.
Continuing this process, we have(42)zmt=∑j=0n−1bjt−ajααj∑r=0m−λrt−arαβαrβΓrβ+j+1+∑r=1m−λr−1Iarβασt=∑j=0n−1bjt−ajααj∑r=0m−λrt−arαβαrβΓrβ+j+1+∫atKβ−1t,s∑r=0m−λrKrβt,sΓr+1βσsdss−a1−α.
Taking the limit asm⟶∞ and according to the definition of Mittag-Leffler function, we get formula (37). This ends the proof.Remark 3.
Ifα=1, we can get the results of Theorem 4.3 in [1].


## 4. Comparison Principles

In this section, two comparison principles which will be used in the next section are established.Lemma 5.
Letη≠1/b−a and yt∈CJ. Then, the following problem:(43)DaCγαxt=yt,t∈J,xa=η∫abxsds+ρ,is equivalent to(44)xt=∫abGt,sysdss−a1−α+ρ1−ηb−a,where(45)Gt,s=1ξ1−ηb−aγb−aα−1Kγ−1t,s+ηKγb,s1−ηb−a,a≤s≤t≤b,ηKγb,s1−ηb−a,a≤t≤s≤b,ξ=b−aα−1Γγ+1.Proof.
For0<γ≤1, by Lemma 2, equation DaCγαxt=yt can be reduced to(46)xt=Iaγαytc0,where c0 is a constant. By the boundary condition, we easily get c0=η∫abxsds+ρ. Hence,(47)xt=Iaγαyt+η∫abxsds+ρ.
LetΔ=∫abxtdt, and we can deduce from (47) that(48)Δ=∫ab1Γγ∫att−aα−s−aααγ−1ysdss−a1−αdt+∫abηΔ+ρdt=∫ab1Γγ+1b−aα−1b−aα−s−aααγysdss−a1−α+ηΔ+ρb−a=∫abKγb,sysΓγ+1b−aα−1dss−a1−α+ηΔ+ρb−a.
Therefore,(49)Δ=∫abKγb,sys1−ηb−aΓγ+1b−aα−1dss−a1−α+ρb−a1−ηb−a.
Substituting (49) into (47), we have(50)xt=1Γγ∫atKγ−1t,sysdss−a1−α+ηρb−a1−ηb−a+ρ+∫abηKγb,sys1−ηb−aΓγ+1b−aα−1dss−a1−α=1ξ∫at1−ηb−aγb−aα−1Kγ−1t,s+ηKγb,s1−ηb−aysdss−a1−α+1ξ∫tbηKγb,s1−ηb−aysdss−a1−α+ρ1−ηb−a=∫abGt,sysdss−a1−α+ρ1−ηb−a.
This ends the proof.Remark 4.
Fora≤t,s≤b, 0≤η<1/b−a, the function Gt,s is continuous and nonnegative.Lemma 6.
Forxt∈CγJ, the following linear problem:(51)DaCβαϕpDaCγαt=σt−λϕpDaCγαxt,t∈J,TakαϕpDaCγαxa=bk,xa=η∫abxsds+ρ,has a unique solution.Proof.
Letzt=ϕpDaCγαxt. By Theorem 2, the problem(52)DaCβαzt=σt−λzt,Takαzt=bk,t∈Jhas a unique solution zt∈CTn−1J, that is, DaCγαxt=ϕqzt∈CTn−1J. Hence, DaCγαxt∈CJ.
By Lemma5, the following problem(53)DaCγαxt=ϕqzt,xa=η∫abxsds+ρ,t∈J,is equivalent to(54)xt=∫abGt,sϕqzsdss−a1−α+ρ1−ηb−a.
Considering (52) and (53), we obtain the conclusion that problem (51) has a unique solution which is given by (54).Based on the above work, we can get the following comparison principles.Lemma 7.
Ifλ≤0 and zt∈CTn−1J satisfy the following relation:(55)DaCβαzt≥−λzt,Takαza≥0,then for t∈J,zt≥0.Proof.
LetDaCβαzt=pt−λzt,Takαza=ak; then, pt≥0,ak≥0. From (37), we can see that zt≥0. This ends the proof.Lemma 8.
If0≤η<1/b−a and xt∈CγJ satisfy the following relation:(56)DaCγαxt≥0,xa≥η∫abxsds,then for t∈J,xt≥0.Proof.
LetDaCγαxt=qt,xa=∫abηxsds+d; then, qt≥0,d≥0. From (53) and (54), we have that(57)xt=∫abGt,sqsdss−a1−α+d1−ηb−a,which implies that xt≥0 due to Gt,s≥0. This ends the proof.


## 5. Extremal Solutions for Nonlinear System

The extremal solutions of problem (9) are obtained in this section.Theorem 3.
If (H2)–(H4) hold, then problem (9) has extremal solutions x∗t,y∗t in the sector x0,y0=xt∈CγJ:x0t≤xt≤y0t,t∈J. Moreover,(58)x0t≤x∗t≤y∗t≤y0t,DaCγαx0t≤DaCγαx∗t≤DaCγαy∗t≤DaCγαy0t.Proof.
Fort∈J,n=1,2,​…, define(59)DaCβαϕpDaCγαxnt=ht,xn−1t,DaCγαxn−1t−λϕpDaCγαxnt−ϕpDaCγαxn−1t,TakαϕpDaCγαxna=bk,xna=∫abws,xn−1s+ηxns−xn−1sds+ρ,(60)DaCβαϕpDaCγαynt=ht,yn−1t,DaCγαyn−1t−λϕpDaCγαynt−ϕpDaCγαyn−1t,TakαϕpDaCγαyna=bk,yna=∫abws,yn−1s+ηyns−yn−1sds+ρ.
By Lemma6, xn,yn are well defined. The proof includes three steps.Step 1.
We prove the monotone property ofxn and yn.
Letrt=ϕpDaCγαx1t−ϕpDaCγαx0t, and by (H2) and (59), we get(61)DaCβαrt≥−λrt,t∈J,Takαra≥0.
From Lemma7, we have rt≥0, i.e., ϕpDaCγαx1t≥ϕpDaCγαx0t. Moreover,(62)DaCγαx1t≥DaCγαx0tholds because of the monotone increasing property of ϕps.
Letr˜t=x1t−x0t. From (H2), (H4), (59), and (62), we have(63)DaCγαr˜t=DaCγαx1t−DaCγαx0t≥0,r˜a≥∫abηr˜sds.
From Lemma8, we have r˜t≥0, i.e., x1t≥x0t. The same argument holds that DaCγαy0t≥DaCγαy1t, y0t≥y1t. Let mt=ϕpDaCγαy1t−ϕpDaCγαx1t, and from (H3), (59), and (60), we get(64)DaCβαmt=ht,y0tDaCγαy0t−ht,x0t,DaCγαx0t−λϕpDaCγαy1t−ϕpDaCγαy0t+λϕpDaCγαx1t−ϕpDaCγαx0t≥−λmt,Takαma=0.
From Lemma7, we have mt≥0, i.e., ϕpDaCγαy1t≥ϕpDaCγαx1t. Hence,(65)DaCγαy1t≥DaCγαx1t.
Letm˜t=y1−x1. We get from (65) and (H4) that(66)DaCγαm˜t=DaCγαy1t−DaCγαx1t≥0,m˜a=∫abws,y0s+ηy1s−y0s−ws,x0s−ηx1s−x0sds≥∫abηm˜sds.
From Lemma8, we have m˜t≥0, i.e. y1≥x1. Therefore, x0≤x1≤y1≤y0 and DaCγαx0≤DaCγαx1≤DaCγαy1≤DaCγαy0.
Next we provex1t,y1t are lower and upper solutions of (9), respectively. From (H3), (H4), and (59), we have(67)DaCβαϕpDaCγαx1t=ht,x0t,DaCγαx0t−ht,x1t,DaCγαx1t+ht,x1t,DaCγαx1t−λϕpDaCγαx1t−ϕpDaCγαx0t≤−λϕpDaCγαx0t−ϕpDaCγαx1t+ht,x1t,DaCγαx1t−λϕpDaCγαx1t−ϕpDaCγαx0t=ht,x1t,DaCγαx1t,TakαϕpDaCγαx1a=bk,x1a=∫abws,x0s−ws,x1s+ws,x1s+ηx1s−x0sds+ρ≤∫abηx0s−x1s+ηx1s−x0s+ws,x1sds+ρ=∫abws,x1sds+ρ.
Clearly,x1t is a lower solution of (9). Similarly, y1t is an upper solution of (9). We obtain by applying mathematical induction that(68)x0t≤x1t≤⋅⋅⋅≤xnt≤⋅⋅⋅≤ynt≤⋅⋅⋅≤y1t≤y0t,DaCγαx0tDaCγαx1t≤⋅⋅⋅≤DaCγαxnt≤⋅⋅⋅DaCγαynt≤⋅⋅⋅≤DaCγαy1t≤DaCγαy0t.Step 2.
We conclude that the sequencesxn and yn satisfy the relations:(69)limn⟶∞xnt=x∗t,limn⟶∞DaCγαxnt=DaCγαx∗t,(70)limn⟶∞ynt=y∗t,limn⟶∞DaCγαynt=DaCγαy∗t.
LetFxnt=ht,xnt,DaCγαxnt+λϕpDaCγαxnt. We can see that the function F is continuous and nondecreasing from the assumption of h. By (37) and (54), equation (59) can be reduced to the equation(71)xnt∫abGt,sϕq∑j=0n−1bjs−ajααjEβ,j+1−λs−aαβαβ+∫asKβ−1s,θEβ,β−λKβs,θFxn−1θdθθ−a1−αdss−a1−α+∫abws,xn−1s−ηxn−1sds+ρ1−ηb−a.
Clearly,xn is uniformly bounded in CγJ. By the continuity of F,G,ϕq, and K, we can easily get that xn is equicontinuous. By the Arzelà–Ascoli theorem, we have that xn satisfies (69). In the same way, we get that yn satisfies (70). Moreover, x∗t and y∗t are solutions of (9).Step 3.
We prove thatx∗ and y∗ are extremal solutions of problem (9).
Assume that any solutionxt of problem (9) satisfies xnt≤xt≤ynt. Let ut=ϕpDaCγαxt−ϕpDaCγαxn+1t, and by (H3), we have(72)DaCβαut=ht,xt,DaCγαxt−ht,xnt,DaCγαxnt+λϕpDaCγαxn+1t−ϕpDaCγαxnt≥−λut,Takαua=0.
From Lemma7, we have ut≥0, i.e., ϕpDaCγαxt≥ϕpDaCγαxn+1t. Hence,(73)DaCγαxt≥DaCγαxn+1t.
Letu˜t=xt−xn+1t, and by (H4), (59), and (73), we have(74)DaCγαu˜t≥0,u˜a=∫abws,xs−ws,xns−ηxn+1s−xnsds≥η∫abu˜sds.
We getu˜t≥0 from Lemma 8, i.e., xt≥xm+1t. Similarly, DaCγαyn+1t≥DaCγαxt, yn+1t≥xt. Hence, xn+1t≤xt≤yn+1t holds. Therefore, x∗t≤xt≤y∗t as n⟶∞, ∀t∈J. This ends the proof.Remark 5.
In [37], the authors assume that h∈C0,1×0,+∞×−∞,0,0,+∞, ht,w1,z1≤ht,w2,z2 for 0≤w1<w2,z1>z2≥0,t∈0,1. The nonlinear term h in this paper satisfies the weaker conditions.


## 6. Example

We present a numerical example as follows:(75)D01/21/2ϕ2D01/21/2xt=t3/4+xt3t1/4+t−3/4+14D01/21/2xt,t∈0,1,ϕ2D01/21/2x0=0,x0=∫0117s+1xs+sds+1,where k=0,b0=0,p=2,a=0,b=1,ρ=1,β=γ=α=1/2 and(76)ht,xt,Da1/21/2xt=t3/4+xt3t1/4+t−3/4+14D01/21/2xt,wt,xt=17t+1xt+t.Takingx0t=0,y0t=3t+3, we can get(77)D01/21/2ϕ2D01/21/2x0t=0≤t3/4+xt3t1/4+t−3/4+14D01/21/2x0t,t∈0,1,ϕ2D01/21/2x00=0,x00<32,D01/21/2ϕ2D01/21/2y0t=D01/21/242πt3/4=3t1/2≥2+2πt3/4,t∈0,1,ϕ2D01/21/2y00=0,y00=3>52.Hence,x0 and y0 are lower and upper solutions of (75), respectively. Therefore, (H2) is satisfied. For x0t≤x≤y≤y0t,(78)ht,yt,D01/21/2yt−ht,xt,D01/21/2xt=yt−xt3t1/4+t−3/4+14D01/21/2yt−xt≥14D01/21/2yt−xt=14ϕ2D01/21/2yt−ϕ2D01/21/2xt,wt,yt−wt,xt≥17yt−xt.We can see thatλ=−1/4<0,η=1/7. Therefore, (H3) and (H4) hold. In light of Theorem 1, the extremal solutions of (75) can be obtained in x0,y0.


## 7. Conclusions

In this paper, we mainly use the montone iterative technique to study the Caputo conformable differential equations withp-Laplacian operator and integral boundary condition. A minimal and a maximal solution between the lower and the upper solutions are obtained. This method provides a constructive procedure for the solutions, and it is also useful for the investigation of qualitative properties of solutions. Since the Caputo conformable derivative can be reduced to the traditional Caputo derivative, some results produced from the traditional Caputo differential system can be seen as special cases of this paper. Moreover, the Caputo conformable derivative depends on two parameters naturally and the one coming from conformable operator can better describe long-memory processes. We believe that the Caputo and RL conformable fractional operators will play a key role in studying new types of fractional variational problems, Sturm–Liouville problems, optimal control problems, and modeling of complex systems.



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