# Existence and Uniqueness of Positive Solutions for a New System of Fractional Differential Equations

Authors: Hongyu Li; Yang Chen
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

By virtue of a recent existing fixed point theorem of increasingφ−h,e-concave operator by Zhai and Wang, we consider the existence and uniqueness of positive solutions for a new system of Caputo-type fractional differential equations with Riemann–Stieltjes integral boundary conditions.


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## Body

## 1. Introduction

In this paper, we consider the following nonlinear Caputo-type fractional system:(1)Dc0+θ1xt+f1t,xt,yt=a1t,t∈0,1,Dc0+θ2yt+f2t,xt,yt=a2t,t∈0,1,x0=x′′0=0,x1=∫01xtdA1t,y0=y′′0=0,y1=∫01ytdA2t,where θ1∈2,3, θ2∈2,3; Dc0+θ1 and Dc0+θ2 are the Caputo fractional derivative; f1 and f2:0,1×−∞,+∞×−∞,+∞⟶−∞,+∞ are continuous; a1,a2:0,1⟶0,+∞ are continuous; and A1 and A2 are bounded variation functions with positive measures with B1=∫01tdA1t<1, B2=∫01tdA2t<1.In recent decades, fractional-order calculus has been widely used in engineering, biology, physics, and so on. Based on it, many scholars have been interested in the study of the existence of nontrivial solutions for various fractional boundary value problems. For some recent works, we can refer to [1–30] and the references therein. For example, in [10], by virtue of Guo–Krasnosel’skii fixed point theorem, Ma and Cui studied the following fractional boundary value problem:(2)Dc0+θpy+μfy,py=0,y∈0,1,p0=p′′0=0,p1=∫01pydAy,where f∈C0,1×0,+∞,0,+∞, Dc0+θ is the Caputo fractional derivative, θ∈2,3, and μ>0 is a parameter. In the case that the parameter μ satisfied some conditions, the existence of positive solutions for the boundary value problem (2) was proved. Meanwhile, many scholars considered some various fractional systems, such as [16–30] and the reference therein. For instance, in [16], the authors obtained the existence of two positive solutions for a nonlinear Caputo-type fractional system by virtue of fixed point index theory. In [17], by using monotone iterative approach, the authors investigated the iterative positive solutions of a system of fractional Riemann–Liouville-type equations with four-point boundary conditions. In [19], by using Banach’s contraction principle, the authors studied the uniqueness of solution for a system of Hadamard-type fractional differential equations with integral boundary conditions. In [21], by using Guo–Krasnosel’skii fixed point theorem, the authors studied the existence of positive solutions for an infinite system of fractional Caputo-type differential equations.In [22], Zhai and Wang introduced a new concept of φ−h,e-concave operator and obtained fixed point theorems of increasing φ−h,e-concave operator. In recent years, by using the fixed point theorems of φ−h,e-concave operator, Zhai and Jiang investigated the uniqueness of positive solutions for a Riemann–Liouville-type fractional system with integral boundary conditions in [23]; Zhai and Wang considered the uniqueness of positive solutions for a system of Hadamard fractional differential equations with integral equations in [24]; Zhai and Zhu considered the uniqueness of positive solutions for a system of Riemann–Liouville fractional differential equations in [25].Inspired by [10, 22–25], we introduce a new system of nonlinear Caputo-type fractional differential equations (1). There are few papers about the application of φ−h,e-concave operator in nonlinear Caputo-type fractional boundary value problems. So, in this paper, we use the recent fixed point theorems of φ−h,e-concave operator by Zhai and Wang to study system (1). The result of the existence and uniqueness of positive solutions for system (1) is obtained.


## 2. Preliminaries

In this section, we briefly introduce Caputo’s fractional derivative and the fixed point theorem ofφ−h,e-concave operator. For details, we can refer to the literature [1, 22]. And we give some lemmas about the relevant Green’s functions.Definition 1.
(see [1]). For a function x∈Cn0,+∞, we define Caputo’s fractional derivative of order θ>0 as follows:(3)Dc0+θxt=1Γn−θ∫0tt−sn−θ−1xnsds,n−1<θ<n,where n is the smallest integer greater than or equal to θ.
By [10], the following lemmas are listed.Lemma 1 (see [10]).
Letu∈C0,1 and θ1,θ2∈2,3. Then, the linear Caputo fractional differential equation(4)Dc0+θipt+ut=0,t∈0,1,p0=p″0=0,p1=∫01ptdAithas a unique solution(5)pt=∫01Kit,susds,where(6)Kit,s=1Γθit1−Bi1−sθi−1−∫s1t−sθi−1dAit−t−sθi−1,0≤s≤t≤1,t1−Bi1−sθi−1−∫s1t−sθi−1dAit,0≤t≤s≤1,and Bi=∫01tdAit<1,i=1,2.Lemma 2.
The above Green’s functionKit,si=1,2 has the following properties:(i)
Kit,s≥0 and Kit,s is continuous on 0,1×0,1.(ii)
t1−sθi−1/Γθi1−Bi∫01t−tθi−1dAit≤Kit,s≤t1−sθi−1/Γθi1−Bi,t,s∈0,1.Proof.
By (6), we know that Kit,s≥0 and Kit,s is continuous on 0,1×0,1.
By (6), we easily know that(7)Kit,s≤t1−sθi−1Γθi1−Bi.
From (6), when 0≤s≤t≤1, we have(8)t1−Bi1−sθi−1−∫s1t−sθi−1dAit−t−sθi−1≥t1−Bi1−sθi−1−∫s1t−tsθi−1dAit−t−tsθi−1=t1−Bi+Bi1−Bi1−sθi−1−t1−Bi∫s1t−tsθi−1dAit−tθi−11−sθi−1=1−sθi−1t−tθi−1+t1−Bi1−sθi−1∫01tdAit−∫s1t−tsθi−1dAit≥1−sθi−1t−tθi−1+t1−Bi1−sθi−1∫01t−tθi−1dAit≥t1−Bi1−sθi−1∫01t−tθi−1dAit.
Whent≤s, we have(9)t1−Bi1−sθi−1−∫s1t−sθi−1dAit≥t1−Bi1−sθi−1−∫s1t−tsθi−1dAit≥t1−Bi1−sθi−11−∫s1tθi−1dAit≥t1−Bi1−sθi−11−∫01tθi−1dAit≥t1−Bi1−sθi−1∫01t−tθi−1dAit.
By (8) and (9), we have(10)Kit,s≥t1−sθi−1Γθi1−Bi∫01t−tθi−1dAit.
LetE be a real Banach space and P⊂E be a cone. A partial order on E is induced by P. For any x,y∈E, the notation x∼y denotes that there are λ>0 and μ>0 such that λx≤y≤μx. Take h>θ (i.e., h≥θ and h≠θ); let Ph=x∈Ex∼h; then obviously, Ph⊂P. Choose e∈P with θ≤e≤h, and let Ph,e=x∈Ex+e∈Ph.Definition 2.
(see [22]). Let T:Ph,e⟶E be an operator. If for any x∈Ph,e and η∈0,1, there exists φη>η such that(11)Tηx+η−1e≥φηTx+φη−1e.
Then,T is called a φ−h,e-concave operator.Lemma 3 (see [22]).
LetP be a normal cone. Suppose that T is an increasing φ−h,e-concave operator and Th∈Ph,e; then, T has a unique fixed point u∈Ph,e. For any u0∈Ph,e, the sequence un=Tun−1,n=1,2,…, and then un−u⟶0 as n⟶∞.


## 3. Main Results

LetE¯=C0,1 with the norm x=max0≤t≤1xt. Let E=E¯×E¯ with the norm x,yE=maxx,y. Let P¯=x∈E¯:xt≥0,t∈0,1. Then, P¯ is a normal cone. Let P=P¯×P¯. It is obvious that P is a normal cone of E. We define the following partial order on space E: x1,y1≤x2,y2⟺x1t≤x2t,y1t≤y2t,t∈0,1. For the detailed knowledge about the cone, we can refer to [31].Define the following operatorsT1:E⟶E¯, T2:E⟶E¯ and T:E⟶E:(12)T1x,yt=∫01K1t,sf1s,xs,ysds−∫01K1t,sa1sds,t∈0,1,T2x,yt=∫01K2t,sf2s,xs,ysds−∫01K2t,sa2sds,t∈0,1,Tx,yt=T1x,y,T2x,yt,x,y∈E,t∈0,1,where Kit,si=1,2 is defined by (6).By [10], we easily know that fixed points of the operator T are solutions of system (1).Let(13)e1t=∫01K1t,sa1sds,e2t=∫01K2t,sa2sds,t∈0,1,h1t=C1t,h2t=C2t,t∈0,1,where(14)C1>1Γθ11−B1∫011−sθ1−1a1sds,C2>1Γθ21−B2∫011−sθ2−1a2sds.Let(15)P¯h1=x∈E¯:x∼h1,P¯h2=x∈E¯:x∼h2,P¯h1,e1=x∈E¯:x+e1∼h1,P¯h2,e2=x∈E¯:x+e2∼h2.Setht=h1t,h2t and et=e1t,e2t. By [29], we have(16)Ph=P¯h1×P¯h2,Ph,e=P¯h1,e1×P¯h2,e2.LetM1=maxt∈0,1e1t,M2=maxt∈0,1e2t.For convenience, the following conditions are given.(i)
S1f1:0,1×−M1,+∞×−M2,+∞⟶−∞,+∞ is increasing about the second and third variables; f2:0,1×−M1,+∞×−M2,+∞⟶−∞,+∞ is increasing about the second and third variables.(ii)
S2 For η∈0,1, there exists φη>η such that(17)f1t,ηu1+η−1v1,ηu2+η−1v2≥φηf1t,u1,u2,f2t,ηu1+η−1v1,ηu2+η−1v2≥φηf2t,u1,u2,where(18)t∈0,1,u1,u2∈−∞,+∞,v1∈0,M1,v2∈0,M2.(iii)
S3f1t,0,0≥0, f2t,0,0≥0 and f1t,0,0≡0, f2t,0,0≡0, ∀t∈0,1.Theorem 1.
Suppose that conditionsS1,S2, and S3 are satisfied. Then, system (1) has a unique solution x∗,y∗∈Ph,e, and for any given x0,y0∈Ph,e, we have xn+1t⟶x∗tn⟶∞, yn+1t⟶y∗tn⟶∞, where the sequences:(19)xn+1t=∫01K1t,sf1s,xns,ynsds−∫01K1t,sa1sds,yn+1t=∫01K2t,sf2s,xns,ynsds−∫01K2t,sa2sds,where n=0,1,2,….Proof.
By Lemma2, we know that Kit,s≥0,i=1,2. So,(20)e1t=∫01K1t,sa1sds≥0,t∈0,1,e2t=∫01K2t,sa2sds≥0,t∈0,1.
By Lemma2, we have(21)t1−sθi−1Γθi1−Bi∫01t−tθi−1dAit≤Kit,s≤t1−sθi−1Γθi1−Bi,t,s∈0,1,i=1,2.
From (21), for t∈0,1, we have(22)e1t=∫01K1t,sa1sds≤tΓθ11−B1∫011−sθ1−1a1sds≤C1t=h1t,e2t=∫01K2t,sa2sds≤tΓθ21−B2∫011−sθ2−1a2sds≤C2t=h2t.
Thus, we obtain that(23)0≤e1≤h1,0≤e2≤h2.
In the following, we divide three parts to prove this theorem. Firstly, we prove thatT:Ph,e⟶E is a φ−h,e concave operator. By S2, for η∈0,1 and η∈0,1, we have(24)T1ηx+η−1e1,ηy+η−1e2t=∫01K1t,sf1s,ηxs+η−1e1s,ηys+η−1e2sds−∫01K1t,sa1sds≥∫01K1t,sφηf1s,xs,ysds−∫01K1t,sa1sds=φη∫01K1t,sf1s,xs,ysds−e1t=φη∫01K1t,sf1s,xs,ysds−∫01K1t,sa1sds+φη−1e1t=φηT1x,yt+φη−1e1t.
Similarly, byS2, for η∈0,1 and η∈0,1, we obtain(25)T2ηx+η−1e1,ηy+η−1e2t=∫01K2t,sf2s,ηxs+η−1e1s,ηys+η−1e2sds−∫01K2t,sa2sds≥∫01K2t,sφηf2s,xs,ysds−∫01K2t,sa2sds=φη∫01K2t,sf2s,xs,ysds−e2t=φη∫01K2t,sf2s,xs,ysds−∫01K2t,sa2sds+φη−1e2t=φηT2x,yt+φη−1e2t.
By (24) and (25), for x,y∈Ph,e, η∈0,1, and t∈0,1, we have(26)Tηx,y+η−1et=Tηx,y+η−1e1,e2t=Tηx+η−1e1,ηy+η−1e2t=T1ηx+η−1e1,ηy+η−1e2t,T2ηx+η−1e1,ηy+η−1e2t≥φηT1x,yt+φη−1e1t,φηT2x,yt+φη−1e2t=φηT1x,yt,T2x,yt+φη−1e1t,e2t=φηTx,yt+φη−1et.
Namely, forx,y∈Ph,e and η∈0,1, we obtain that(27)Tηx,y+η−1et≥φηTx,y+φη−1e.
Secondly, we show thatT:Ph,e⟶E is an increasing operator. By the definition of Ph,e, for x,y∈Ph,e, we get that x,y+e∈Ph, i.e., x+e1,y+e2∈P¯h1×P¯h2. By the definitions of P¯h1 and P¯h2, we obtain that there exist η1>0 and η2>0 such that(28)xt+e1t≥η1h1t,yt+e2t≥η2h2t,t∈0,1.
So,(29)xt≥η1h1t−e1t≥−e1t≥−M1,yt≥η2h2t−e2t≥−e2t≥−M2,t∈0,1.
FromS1, we easily know that the operators T1 and T2 are increasing, so it is obvious that T:Ph,e⟶E is increasing.
In the end, we proveTh∈Ph,e. By [22], we know that Ph,e=P¯h1,e1×P¯h2,e2. Since Tht=Th1,h2t=T1h1,h2t,T2h1,h2t, we need to prove T1h1,h2∈P¯h1,e1,T2h1,h2∈P¯h2,e2. In the following, by the definitions of P¯h1,e1 and P¯h2,e2, we prove T1h1,h2+e1∈P¯h1,T2h1,h2+e2∈P¯h2, respectively. From S1 and S3, combine (21), and we have(30)T1h1,h2t+e1t=∫01K1t,sf1s,h1s,h2sds≤tΓθ11−B1∫011−sθ1−1f1s,h1s,h2sds≤tΓθ11−B1∫011−sθ1−1f1s,C1,C2ds=h1tC1Γθ11−B1∫011−sθ1−1f1s,C1,C2ds,T1h1,h2t+e1t=∫01K1t,sf1s,h1s,h2sds≥tΓθ11−B1∫011−sθ1−1f1s,h1s,h2sds∫01t−tθ1−1dA1t≥tΓθ11−B1∫011−sθ1−1f1s,0,0ds∫01t−tθ1−1dA1t=h1tC1Γθ11−B1∫011−sθ1−1f1s,0,0ds∫01t−tθ1−1dA1t.
Let(31)λ1=1C1Γθ11−B1∫011−sθ1−1f1s,0,0ds∫01t−tθ1−1dA1t,λ2=1C1Γθ11−B1∫011−sθ1−1f1s,C1,C2ds.
Obviously,λ1>0,λ2>0. So, we have λ1h1t≤T1h1,h2t+e1t≤λ2h2t. And T1h1,h2+e1∈P¯h1. Similarly, we have T2h1,h2+e2∈P¯h2. So, Th∈Ph,e is proved.
Therefore, by Lemma3, the operator T has a unique fixed point x∗,y∗∈Ph,e. For any given x0,y0∈Ph,e, define the sequences:(32)xn+1t=∫01K1t,sf1s,xns,ynsds−∫01K1t,sa1sds,yn+1t=∫01K2t,sf2s,xns,ynsds−∫01K2t,sa2sds,where n=0,1,2,….


## 4. Application

Example 1.
We study the following fractional system with integral boundary conditions:(33)Dc0+5/2xt+2835π944x+11/516105πt5/2+32271/5t1/5+2835π1888y+11/532105πt5/2+32271/5t1/5=t,t∈0,1,Dc0+5/2yt+2835π944x+11/316105πt5/2+32271/3t1/3+2835π1888y+11/332105πt5/2+32271/3t1/3=2t,t∈0,1,x0=x″0=0,x1=12∫01xtdt,y0=y″0=0,y1=12∫01ytdt,where θ1=θ2=5/2;A1t=A2t=1/2t;B1=B2=1/4; a1t=t;a2t=2t; and(34)f1t,x,y=2835π944x+11/516105πt5/2+32271/5t1/5+2835π1888y+11/532105πt5/2+32271/5t1/5,(35)f2t,x,y=2835π944x+11/316105πt5/2+32271/3t1/3+2835π1888y+11/332105πt5/2+32271/3t1/3.
Obviously,(36)f1t,0,0=16105πt5/2+32271/5t1/5+32105πt5/2+32271/5t1/5,f2t,0,0=16105πt5/2+32271/3t1/3+32105πt5/2+32271/3t1/3,and f1t,0,0≡0,f2t,0,0≥0, f1t,0,0≥0,f2t,0,0≥0.
Green’s functions are as follows:(37)K1t−s=K2t−s=Kt−s=1Γ5/24t31−s3/2−151−s5/2−t−s3/2,0≤s≤t≤1,4t31−s3/2−151−s5/2,0≤s≤t≤1.
Then, we have(38)e1t=∫01K1t,sa1sds=4t3Γ5/2∫0t1−s3/2sds−15∫0t1−s5/2sds−1Γ5/2∫0tt−s3/2sds+4t3Γ5/2∫t11−s3/2sds−15∫t11−s5/2sds=4t3Γ5/2−25t1−t5/2−4351−t7/2+435+235t1−t7/2+43151−t9/2−4315+1Γ5/2435t7/2+4t3Γ5/225t1−t5/2+4351−t7/2−235t1−t7/2−43151−t9/2=16t105π3227+t5/2,e2t=∫01K2t,sa2sds=8t3Γ5/2∫0t1−s3/2sds−15∫0t1−s5/2sds−2Γ5/2∫0tt−s3/2sds+8t3Γ5/2∫t11−s3/2sds−15∫t11−s5/2sds=8t3Γ5/2−25t1−t5/2−4351−t7/2+435+235t1−t7/2+43151−t9/2−4315+2Γ5/2435t7/2+8t3Γ5/225t1−t5/2+4351−t7/2−235t1−t7/2−43151−t9/2=32t105π3227+t5/2,M1=maxt∈0,1e1t=maxt∈0,116t105π3227+t5/2=9442835π,M2=maxt∈0,1e2t=maxt∈0,132t105π3227+t5/2=18882835π.
Leth1t=C1t,h2t=C2t, where(39)C1>43Γ5/2∫011−s3/2sds=43Γ5/2435=16105Γ5/2,C2>83Γ5/2∫011−s3/2sds=83Γ5/2435=32105Γ5/2.
We have(40)e1t=16t105π3227+t5/2≤4t3Γ5/2∫011−s3/2sds=16t105Γ5/2≤C1t=h1t,e2t=32t105π3227+t5/2≤8t3Γ5/2∫011−s3/2sds=32t105Γ5/2≤C2t=h2t.
By (34) and (35), we have(41)f1t,x,y=2835π944x+11/516105πt5/2+32271/5t1/5+2835π1888y+11/532105πt5/2+32271/5t1/5=2835π944x+11/5e1t1/5+2835π1888y+11/5e2t1/5=2835π944xe1t+e1t1/5+2835π1888ye2t+e2t1/5,f2t,x,y=2835π944x+11/316105πt5/2+32271/3t1/3+2835π1888y+11/332105πt5/2+32271/3t1/3=2835π944x+11/3e1t1/3+2835π1888y+11/3e2t1/3=2835π944xe1t+e1t1/3+2835π1888ye2t+e2t1/3.
Takeφη=η1/3; then, φη>η,η∈0,1. For η∈0,1, u1,u2∈−∞,+∞, v1∈0,M1, and v2∈0,M2, we have(42)f1t,ηu1+η−1v1,ηu2+η−1v2=2835π944e1tηu1+η−1v1+e1t1/5+2835π1888e2tηu2+η−1v2+e2t1/5=η1/52835π944e1tu1+1−1ηv1+1ηe1t1/5+η1/52835π1888e2tu2+1−1ηv2+1ηe2t1/5=η1/52835π944e1tu1+2835π9441−1ηe1tv1+1ηe1t1/5+η1/52835π1888e2tu2+2835π18881−1ηe2tv2+1ηe2t1/5≥η1/52835π944e1tu1+1−1ηe1t+1ηe1t1/5+η1/52835π1888e2tu2+1−1ηe2t+1ηe2t1/5=η1/52835π944e1tu1+e1t1/5+η1/52835π1888e2tu2+e2t1/5=η1/5f1t,u1,u2≥φηf1t,u1,u2,f2t,ηu1+η−1v1,ηu2+η−1v2=2835π944e1tηu1+η−1v1+e1t1/3+2835π1888e2tηu2+η−1v2+e2t1/3=η1/32835π944e1tu1+1−1ηv1+1ηe1t1/3+η1/32835π1888e2tu2+1−1ηv2+1ηe2t1/3=η1/32835π944e1tu1+2835π9441−1ηe1tv1+1ηe1t1/3+η1/32835π1888e2tu2+2835π18881−1ηe2tv2+1ηe2t1/3≥η1/32835π944e1tu1+1−1ηe1t+1ηe1t1/3+η1/32835π1888e2tu2+1−1ηe2t+1ηe2t1/3=η1/32835π944e1tu1+e1t1/3+η1/32835π1888e2tu2+e2t1/3=η1/3f2t,u1,u2≥φηf2t,u1,u2.
Therefore, from Theorem1, we know that system (33) has a unique solution x∗,y∗∈Ph,e, where(43)et=16t105π3227+t5/2,32t105π3227+t5/2,t∈0,1,ht=C1t,C2t,t∈0,1.
For any givenx0,y0∈Ph,e, we obtain the following sequences:(44)xn+1t=∫01Kt,s2835π944xns+11/516105πs5/2+32271/5s1/5+2835π1888yns+11/532105πs5/2+32271/5s1/5ds−16t105π3227+t5/2,yn+1t=∫01Kt,s2835π944xns+11/316105πs5/2+32271/3s1/3+2835π1888yns+11/332105πs5/2+32271/3s1/3ds−32t105π3227+t5/2,where n=0,1,2,…, Kt,s is defined by (37), xn+1t⟶x∗n⟶∞,andyn+1t⟶y∗n⟶∞.



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