Unique solution for a new system of fractional differential equations

R E S E A R C H

Open Access

Unique solution for a new system of

fractional differential equations

Chengbo Zhai

1*

_{and Xiaolin Zhu}

1

*_{Correspondence:} [email protected]

1_{School of Mathematical Sciences,}

Shanxi University, Taiyuan, P.R. China

Abstract

In this article, we discuss a new system of fractional differential equations ⎧

⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Ds1

0+u(t) +f(t,u(t),v(t)) =z1(t), 0 <t< 1,

Ds2

0+v(t) +g(t,u(t),v(t)) =z2(t), 0 <t< 1,

u(0) =u(1) =u(0) =u(1) = 0, Dβ1

0+u(0) = 0, D

β1

0+u(1) =b1D

β1 0+u(

η

1),

v(0) =v(1) =v(0) =v(1) = 0, Dβ2

0+v(0) = 0, D

β2

0+v(1) =b2D

β2 0+v(

η

2), wheresi=

α

i+

β

i,

α

i∈(1, 2],

β

i∈(3, 4],zi: [0, 1]→[0, +∞) is continuous,Dα₀+i andD

βi 0+ are the standard Riemann–Liouville derivatives,

η

i∈(0, 1),bi∈(0,

η

i1–αi),i= 1, 2, and f,g∈C([0, 1]×R2,R). We establish the existence and uniqueness of solutions for the problem by a recent fixed point theorem of increasing

Ψ

-(h,e)-concave operators defined on ordered sets. Furthermore, the results obtained are well proven by means of a specific example.

MSC: 26A33; 34B18; 34B15

Keywords: Existence and uniqueness; Fractional differential system;

Ψ

-(h,e)-concave operator

1 Introduction

In this article, we investigate the following system of fractional differential equations:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Ds1

0+u(t) +f(t,u(t),v(t)) =z1(t), 0 <t< 1, Ds2

0+v(t) +g(t,u(t),v(t)) =z2(t), 0 <t< 1, u(0) =u(1) =u(0) =u(1) = 0, Dβ1

0+u(0) = 0, D

β1

0+u(1) =b1Dβ₀+1u(η1), v(0) =v(1) =v(0) =v(1) = 0, Dβ2

0+v(0) = 0, Dβ₀+2v(1) =b2Dβ0+2v(η2),

(1.1)

wheresi=αi+βi,αi∈(1, 2],βi∈(3, 4],zi: [0, 1]→[0, +∞) is continuous,Dαi0+andDβi₀+are

the standard Riemann–Liouville derivatives,ηi∈(0, 1),bi∈(0,ηi1–αi),i= 1, 2, andf,g∈

C([0, 1]×R2, R). A pair of functions (u,v)∈C([0, 1])×C([0, 1]) is called a solution of system (1.1) if it satisfies (1.1). We seek a new method which is a recent fixed point theorem forΨ-(h,e)-concave operators to discuss system (1.1).

In the last few decades, fractional problems have attracted wide attention by scholars because of their wide applications and important positions in biology, physics, chemistry,

and engineering, see [1–28] and the references therein. In [16], Xu and Dong considered the following fractional equation:

⎧ ⎨ ⎩

–Dα₀+(ϕp(D

β

0+u(t))) =f(t,u(t)), 0 <t< 1,

u(0) =u(1) =u(0) =u(1) = 0, Dβ₀+u(0) = 0, Dβ₀+u(1) =bDβ₀+u(η),

(1.2)

where α ∈ (1, 2], β ∈(3, 4], Dα

0+ and Dβ₀+ are the Riemann–Liouville derivatives, b ∈

(0,η1–p–1α_{), f} ∈_{C([0, 1]}×_{[0, +}∞_{), [0, +}∞_{)). By using Schauder’s fixed point theorem and}

the upper and lower solutions method, the existence and uniqueness of solutions were established. On the other hand, fractional system have been investigated extensively, see [29–38]. In [33], the authors considered the existence of positive solutions for the follow-ing fractional system with parameters:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

–Dα1 0+(ϕp1(D

β1

0+u(t))) =λf(t,u(t),v(t)), 0 <t< 1,

–Dα2 0+(ϕp2(D

β2

0+v(t))) =μg(t,u(t),v(t))), 0 <t< 1, u(0) =u(1) =u(0) =u(1) = 0, Dβ1

0+u(0) = 0, D

β1

0+u(1) =b1Dβ0+1u(η1), v(0) =v(1) =v(0) =v(1) = 0, Dβ2

0+v(0) = 0, Dβ₀+2v(1) =b2Dβ0+2v(η2),

(1.3)

whereαi∈(1, 2],βi∈(3, 4],D0αi+andDβi₀+are the Riemann–Liouville derivatives,ηi∈(0, 1),

bi∈(0,ηi

1–αi

pi–1_{),i= 1, 2.f,g}∈_{C([0, 1]}×_R2_{, R). The authors applied Guo–Krasnosel’skii’s}

fixed point theorem to get various existence results for positive solutions in terms of dif-ferent values ofλandμ.

In the existing literature, most of the scholars have studied the existence of solutions, but there is little discussion about the uniqueness. Moreover, the usual methods used are Guo–Krasnosel’skii’s fixed point theorem, the upper and lower solutions method, mono-tone iterative method, and fixed index theory. In 2017, the authors [29] used a new method to study the existence and uniqueness of solutions for the following fractional system:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dα_{u(t) +f(t,v(t)) =a, 0 <t< 1,}

Dβ_{v(t) +g(t,u(t)) =b, 0 <t< 1,}

u(0) = 0, u(1) =₀1φ(t)u(t)dt,

v(0) = 0, v(1) =₀1ψ(t)v(t)dt,

(1.4)

where 1 <α,β≤2,f,g∈C([0, 1]×(–∞, +∞), (–∞, +∞)),φ,ψ ∈L1_{[0, 1],a,bare}

con-stants andDdenotes the usual Riemann–Liouville derivatives. The authors gave the exis-tence and uniqueness of solutions for the coupled system dependent on constantsaand

bby using a fixed point theorem of increasingΨ-(h,e)-concave operators. We found that the method can resolve some new differential systems and can obtain some good unique results.

The paper is organized as follows. In Sect.2, we propose not only some definitions and lemmas to be used to prove our main results, but also some useful properties of Green functions. In Sect.3, we discuss the existence and uniqueness of solutions to sys-tem (1.1). Finally, in Sect.4, a concrete example is given as the application of our main results.

2 Preliminaries and lemmas

We present here some definitions and related properties of Riemann–Liouville fractional derivatives and integrals. Some auxiliary results which will be used to prove our main results are also given.

Definition 2.1([39,40]) The Riemann–Liouville fractional integral of orderα> 0 of a

continuous functionf : (0, +∞)→(–∞, +∞) is given by

I₀α+f(t) =

1 Γ(α)

t

0

(t–s)α–1f(s)ds,

provided the right-hand side is pointwise defined on (0, +∞).

Definition 2.2([39,40]) The Riemann–Liouville fractional derivative of orderα> 0 of a

continuous functionf : (0, +∞)→(–∞, +∞) is given by

Dα₀+f(t) =

1 Γ(n–α)

d dt

n t

0

(t–s)n–α–1f(s)ds,

wheren= [α] + 1, [α] denotes the integer part of the numberα, provided that the right-hand side is pointwise defined on (0, +∞).

From Lemmas 2.3, 2.5 in [33], we can easily get the following conclusions.

Lemma 2.1 Let s1=α1+β1,α1∈(1, 2],β1∈(3, 4],η1∈(0, 1),b1∈(0,η11–α1). If y1 ∈

C[0, 1],then the following fractional boundary value problem

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Ds1

0+u(t) +y1(t) = 0, 0 <t< 1, u(0) =u(1) =u(0) =u(1) = 0,

Dβ1

0+u(0) = 0, D

β1

0+u(1) =b1Dβ₀1+u(η1),

(2.1)

has a unique solution

u(t) =

1

0

G1(t,s) 1

0

where

G1(t,s) =

1 Γ(β1)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tβ1–2_{(1 –s)}β1–2_{[(s–t) + (β}

1– 2)(1 –t)s],

0≤t≤s≤1,

tβ1–2_{(1 –s)}β1–2_{[(s–t) + (β}

1– 2)(1 –t)s] + (t–s)β1–1,

0≤s≤t≤1,

H1(t,s) =h1(t,s) +

b1tα1–1

1 –b1ηα11–1

h1(η1,s),

h1(t,s) =

1 Γ(α1)

⎧ ⎨ ⎩

[t(1 –s)]α1–1_{, 0≤t≤s≤1,}

[t(1 –s)]α1–1_{– (t–s)}α1–1_{, 0≤s≤t≤1.}

(2.2)

Lemma 2.2 Let s2=α2+β2,α2∈(1, 2],β2∈(3, 4],η2∈(0, 1),b2∈(0,η21–α2). If y2 ∈

C[0, 1],then the following fractional boundary value problem

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Ds2

0+v(t) +y2(t) = 0, 0 <t< 1, v(0) =v(1) =v(0) =v(1) = 0,

Dβ2

0+v(0) = 0, Dβ₀+2v(1) =b2Dβ0+2v(η2),

(2.3)

has a unique solution

v(t) =

1

0

G2(t,s) 1

0

H2(s,τ)y2(τ)dτds,

where

G2(t,s) =

1 Γ(β2)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tβ2–2_{(1 –s)}β2–2_{[(s–t) + (β}

2– 2)(1 –t)s],

0≤t≤s≤1,

tβ2–2_{(1 –s)}β2–2_{[(s–t) + (β}

2– 2)(1 –t)s] + (t–s)β2–1,

0≤s≤t≤1,

H2(t,s) =h2(t,s) +

b2tα2–1

1 –b2ηα22–1

h2(η2,s),

h2(t,s) =

1 Γ(α2)

⎧ ⎨ ⎩

[t(1 –s)]α2–1_{, 0≤t≤s≤1,}

[t(1 –s)]α2–1_{– (t–s)}α2–1_{, 0≤s≤t≤1.}

(2.4)

From Lemmas 2.4, 2.6 in [33], we can easily get the following lemma.

Lemma 2.3 The functions Gi(t,s), i= 1, 2,defined by(2.2)and(2.4)have the following

properties:

(i) Gi(t,s)is continuous on[0, 1]×[0, 1]andGi(t,s) > 0for all(t,s)∈(0, 1)×(0, 1);

(ii) (βi– 2)ki(t)ri(s)≤Γ(βi)Gi(t,s)≤Miri(s),(t,s)∈[0, 1]×[0, 1];

where

ki(t) =tβi–2(1 –t)2, ri(s) =s2(1 –s)βi–2,

Mi=max

βi– 1, (βi– 2)2

.

Let (X, · ) be a real Banach space with a partial order induced by a coneP⊂X. For any

x,y∈X, the notationx∼ydenotes that there areλ> 0 andμ> 0 such thatλx≤y≤μx. Given h>θ (i.e.,h≥θ andh=θ), we consider a setPh={x∈X|x∼h}. It is clear that

Ph⊂P. Lete∈Pwithθ≤e≤h, we definePh,e={x∈X|x+e∈Ph}.

Now, we present the definition ofΨ-(h,e)-concave operator and a fixed point theorem which can be easily used to study some systems of differential equations.

Definition 2.3([4]) Suppose thatT:Ph,e→Xis a given operator which satisfies: for any

x∈Ph,eandξ∈(0, 1), there existsΨ(ξ) >ξ such that

Tξx+ (ξ– 1)e≥Ψ(ξ)Tx+Ψ(ξ) – 1e.

ThenT is called aΨ-(h,e)-concave operator.

Lemma 2.4 ([4]) Assume that T is an increasing Ψ-(h,e)-concave operator satisfying

Th∈Ph,eand P is normal.Then T has a unique fixed point x∗ in Ph,e.For any w0∈Ph,e,

constructing the sequence wn=Twn–1,n= 1, 2, . . . ,thenwn–x∗ →0as n→ ∞.

For h1,h2∈P with h1,h2=θ.Suppose h= (h1,h2),then h∈ ¯P:=P×P.Takeθ≤e1≤h1,

θ ≤e2≤h2,and writeθ¯= (θ,θ),e= (e1,e2).Thenθ¯= (θ,θ)≤(e1,e2)≤(h1,h2) =h.That is,θ¯≤e≤h.If P is normal,thenP¯= (P,P)is normal.

Lemma 2.5([31]) P¯h=Ph1×Ph2.

Lemma 2.6([31]) P¯h,e=Ph1,e1×Ph2,e2.

3 Main results

In this section, letX={u|u∈C[0, 1]}, a Banach space with the normu=sup{|u(t)|:t∈

[0, 1]}. We will consider (1.1) inX×X. For (u,v)∈X×X, let(u,v)=max{u,v}. Then (X×X,(u,v)) is a Banach space. LetP¯ ={(u,v)∈X×X|u(t)≥0,v(t)≥0,t∈[0, 1]},

P={u∈X|u(t)≥0,t∈[0, 1]}, then the coneP¯⊂X×XandP¯=P×Pis normal, and the spaceX×Xhas a partial order: (u1,v1)≤(u2,v2) if and only ifu1(t)≤u2(t),v1(t)≤v2(t), t∈[0, 1].

Lemma 3.1 Suppose that f(t,u,v),g(t,u,v)are continuous.By Lemmas2.1,2.2and the

result of[33],we can claim that(u,v)∈X×X is a solution of problem(1.1)if and only if

(u,v)∈X×X is a solution of the following equations:

⎧ ⎨ ⎩

u(t) =₀1G1(t,s) 1

0 H1(s,τ)f(τ,u(τ),v(τ))dτds– 1

0 G1(t,s) 1

0H1(s,τ)z1(τ)dτds, v(t) =₀1G2(t,s)

1

0 H2(s,τ)g(τ,u(τ),v(τ))dτds– 1

0G2(t,s) 1

For(u,v)∈X×X,we consider three operators A1,A2:X×X→X and T:X×X→X×X by

A1(u,v)(t) = 1

0

G1(t,s) 1

0

H1(s,τ)f

τ,u(τ),v(τ)dτds

–

1

0

G1(t,s) 1

0

H1(s,τ)z1(τ)dτds,

A2(u,v)(t) = 1

0

G2(t,s) 1

0

H2(s,τ)g

τ,u(τ),v(τ)dτds

–

1

0

G2(t,s) 1

0

H2(s,τ)z2(τ)dτds,

T(u,v)(t) =A1(u,v)(t),A2(u,v)(t)

.

According to the above results,we can easily get that(u,v)∈X×X is a solution of system

(1.1)if and only if(u,v)∈X×X is a fixed point of operator T.Let

e1(t) = 1

0

G1(t,s) 1

0

H1(s,τ)z1(τ)dτds,

e2(t) = 1

0

G2(t,s) 1

0

H2(s,τ)z2(τ)dτds,

h1(t) =N1tβ1–2(1 –t)2, h2(t) =N2tβ2–2(1 –t)2, t∈[0, 1],

(3.1)

where

N1≥ M1

Γ(β1) 1

0 1

0

H1(s,τ)z1(τ)dτds,

N2≥ M2

Γ(β2) 1

0 1

0

H2(s,τ)z2(τ)dτds,

with Mi,i= 1, 2,given as in Lemma2.3.

Theorem 3.1 Letαi∈(1, 2],βi∈(3, 4],zi: [0, 1]→(0, +∞)be continuous and e1,e2,h1,

h2 be given as in(3.1).Assume that f,g∈C([0, 1]×R2, R)and z1,z2∈C([0, 1], [0, +∞]). Moreover,

(H1) f : [0, 1]×[–e1∗, +∞)×[–e2∗, +∞)→(–∞, +∞) and g : [0, 1]×[–e1∗, +∞)×

[–e2∗, +∞)→(–∞, +∞)are both increasing with respect to the second and third

variables,wheree1∗=max{e1(t) :t∈[0, 1]}ande2∗=max{e2(t) :t∈[0, 1]};

(H2) forξ∈(0, 1),there existsψ(ξ) >ξ such that

ft,ξx1+ (ξ– 1)y1,ξx2+ (ξ– 1)y2

≥ψ(ξ)f(t,x1,x2),

gt,ξx1+ (ξ– 1)y1,ξx2+ (ξ– 1)y2

≥ψ(ξ)g(t,x1,x2),

wheret∈[0, 1],x1,x2∈(–∞, +∞),y1∈[0,e1∗]andy2∈[0,e2∗];

Then:

(1) system(1.1)has a unique solution(u∗,v∗)inP¯h,e,where

e(t) =e1(t),e2(t)

, h(t) =h1(t),h2(t)

, t∈[0, 1];

(2) for a given point(u0,v0)∈ ¯Ph,e,construct the following sequences:

un+1(t) = 1

0

G1(t,s) 1

0

H1(s,τ)f

τ,un(τ),vn(τ)

dτds

–

1

0

G1(t,s) 1

0

H1(s,τ)z1(τ)dτds,

vn+1(t) = 1

0

G2(t,s) 1

0

H2(s,τ)g

τ,un(τ),vn(τ)

dτds

–

1

0

G2(t,s) 1

0

H2(s,τ)z2(τ)dτds,

n= 0, 1, 2, . . .,we haveun+1(t)→u∗(t),vn+1(t)→v∗(t)asn→ ∞.

Proof According to the properties ofGi(t,s), sinceHi(t,s)≥0,i= 1, 2, we can get

e1(t) = 1

0

G1(t,s) 1

0

H1(s,τ)z1(τ)dτds≥0, t∈[0, 1],

e2(t) = 1

0

G2(t,s) 1

0

H2(s,τ)z2(τ)dτds≥0, t∈[0, 1].

From Lemma2.3, one can obtain that, fort∈[0, 1],

e1(t) = 1

0

G1(t,s) 1

0

H1(s,τ)z1(τ)dτds

≤ 1 0

M1k1(t)

Γ(β1) 1

0

H1(s,τ)z1(τ)dτds

= M1 Γ(β1)

tβ1–2_{(1 –t)}2 1

0 1

0

H1(s,τ)z1(τ)dτds

≤N1tβ1–2(1 –t)2=h1(t);

e2(t) = 1

0

G2(t,s) 1

0

H2(s,τ)z1(τ)dτds

≤ 1

0

M2k2(t)

Γ(β2) 1

0

H2(s,τ)z2(τ)dτds

= M2 Γ(β2)

tβ2–2_{(1 –t)}2 1

0 1

0

H2(s,τ)z2(τ)dτds

≤N2tβ2–2(1 –t)2=h2(t).

In the following, we prove thatT :P¯h,e→X×X is a Ψ-(h,e)-concave operator. For

(u,v)∈ ¯Ph,e,ξ∈(0, 1),t∈[0, 1], one can obtain

Tξ(u,v) + (ξ– 1)e(t)

=Tξ(u,v) + (ξ– 1)(e1,e2)

(t)

=Tξu+ (ξ– 1)e1,ξv+ (ξ– 1)e2

(t)

=A1

ξu+ (ξ– 1)e1,ξv+ (ξ– 1)e2

,A2

ξu+ (ξ– 1)e1,ξv+ (ξ– 1)e2

(t).

We considerA1(ξu+ (ξ – 1)e1,ξv+ (ξ– 1)e2)(t) andA2(ξu+ (ξ – 1)e1,ξv+ (ξ– 1)e2)(t),

respectively. From (H2),

A1

ξu+ (ξ– 1)e1,ξv+ (ξ– 1)e2

(t)

=

1

0

G1(t,s) 1

0

H1(s,τ)f

τ,ξu+ (ξ– 1)e1

(τ),ξv+ (ξ– 1)e2

(τ)dτds–e1(t)

≥ 1

0

G1(t,s) 1

0

H1(s,τ)ψ(ξ)f

τ,u(τ),v(τ)dτds–e1(t)

=ψ(ξ)

1

0

G1(t,s) 1

0

H1(s,τ)f

τ,u(τ),v(τ)dτds–e1(t)

=ψ(ξ)

1

0

G1(t,s) 1

0

H1(s,τ)f

τ,u(τ),v(τ)dτds–e1(t)

+ψ(ξ) – 1e1(t)

=ψ(ξ)A1(u,v)(t) +

ψ(ξ) – 1e1(t).

Similarly,

A2

ξu+ (ξ– 1)e1,ξv+ (ξ– 1)e2

(t)

=

1

0

G2(t,s) 1

0

H2(s,τ)f

τ,ξu+ (ξ– 1)e1

(τ),ξv+ (ξ– 1)e2

(τ)dτds–e2(t)

≥ 1

0

G2(t,s) 1

0

H2(s,τ)ψ(ξ)f

τ,u(τ),v(τ)dτds–e2(t)

=ψ(ξ)

1

0

G2(t,s) 1

0

H2(s,τ)f

τ,u(τ),v(τ)dτds–e2(t)

=ψ(ξ)

1

0

G2(t,s) 1

0

H2(s,τ)f

τ,u(τ),v(τ)dτds–e2(t)

+ψ(ξ) – 1e2(t)

=ψ(ξ)A2(u,v)(t) +

ψ(ξ) – 1e2(t).

So we have

Tξ(u,v) + (ξ– 1)e(t)

≥ψ(ξ)A1(u,v)(t) +

ψ(ξ) – 1e1(t),ψ(ξ)A2(u,v)(t) +

ψ(ξ) – 1e2(t)

=ψ(ξ)A1(u,v)(t),ψ(ξ)A2(u,v)(t)

+ψ(ξ) – 1e1(t),

=ψ(ξ)A1(u,v)(t),A2(u,v)(t)

+ψ(ξ) – 1e1(t),e2(t)

=ψ(ξ)T(u,v)(t) +ψ(ξ) – 1e(t).

That is,

Tξ(u,v) + (ξ– 1)e≥ψ(ξ)T(u,v) +ψ(ξ) – 1e, (u,v)∈ ¯Ph,e,ξ∈(0, 1).

Therefore,Tis aΨ-(h,e)-concave operator.

Next we show thatT:P¯h,e→X×Xis increasing. For (u,v)∈ ¯Ph,e, we have (u,v) +e∈ ¯Ph.

From Lemma2.5, (u+e1,v+e2)∈Ph1×Ph2. So there areλ1,λ2> 0 such that

u(t) +e1(t)≥λ1h1(t), v(t) +e2(t)≥λ2h2(t), t∈[0, 1].

Therefore, u(t)≥λ1h1(t) –e1(t)≥–e1(t)≥–e1∗,v(t)≥λ2h2(t) –e2(t)≥–e2(t)≥–e2∗.

This fact and (H1) imply thatT:P¯h,e→X×Xis increasing.

Now we showTh∈ ¯Ph,e, which needs to proveTh+e∈ ¯Ph. Fort∈[0, 1],

Th(t) +e(t) =T(h1,h2)(t) +e(t)

=A1(h1,h2)(t),A2(h1,h2)(t)

+e1(t),e2(t)

=A1(h1,h2)(t) +e1(t),A2(h1,h2)(t) +e2(t)

.

We considerA1(h1,h2)(t) +e1(t) andA2(h1,h2)(t) +e2(t), respectively. By Lemmas2.2,2.3

and (H1), (H3), we can get

A1(h1,h2)(t) +e1(t)

=

1

0

G1(t,s) 1

0

H1(s,τ)f

τ,h1(τ),h2(τ)

dτds

≥ 1

0

(β1– 2)k1(t)r1(s)

Γ(β1)

1

0

H1(s,τ)f

τ,N1τβ1–2(1 –τ)2,N2τβ2–2(1 –τ)2

dτds

≥(β1– 2)k1(t)

Γ(β1) 1

0 r1(s)

1

0

H1(s,τ)f(τ, 0, 0)dτds

= β1– 2

N1Γ(β1) h1(t)

1

0 r1(s)

1

0

H1(s,τ)f(τ, 0, 0)dτds

and

A1(h1,h2)(t) +e1(t)

=

1

0

G1(t,s) 1

0

H1(s,τ)f

τ,h1(τ),h2(τ)

dτds

≤ 1

0

M1k1(t)

Γ(β1) 1

0

H1(s,τ)f(τ,N1,N2)dτds

= M1

N1Γ(β1) h1(t)

1

0 1

0

According to (H1), (H3), sincer1(s) =s2(1 –s)β1–2≤1,s∈[0, 1], then

quently, according to Lemma2.5,

Th+e=A1(h1,h2) +e1,A2(h1,h2) +e2

4 An example

We consider the following fractional system:

Figure 1Image of functionf(t, 0, 0) ont∈[0, 1]

Figure 2Image of functiong(t, 0, 0) ont∈[0, 1]

We can obviously get f(t, 0, 0) ≥ 0, g(t, 0, 0)≥ 0 from Figs. 1, 2. And f(t, 0, 0) ≡ 0,

g(t, 0, 0)≡0. Moreover,

G1(t,s) =

1 Γ(10₃)

⎧ ⎨ ⎩

t43_{(1 –s)}43_{[(s–t) +}4

3(1 –t)s], 0≤t≤s≤1, t43_{(1 –s)}43_{[(s–t) +}4

3(1 –t)s] + (t–s) 7

3_{, 0}≤_s≤_t≤_1,

(4.2)

H1(t,s) =h1(t,s) + 1 2t

1 2

1 –1₂

3 2

h1

1 2,s ,

h1(t,s) =

1 Γ(3

2) ⎧ ⎨ ⎩

[t(1 –s)]12_{, 0}≤_t≤_s≤_1,

[t(1 –s)]12 – (_t–_s)12, 0≤_s≤_t≤1.

G2(t,s) =

1 Γ(10₃)

⎧ ⎨ ⎩

t43(1 –s)43[(s–t) +4

3(1 –t)s], 0≤t≤s≤1, t43_{(1 –s)}43_{[(s–t) +}4

3(1 –t)s] + (t–s) 7

3_{, 0}≤_s≤_t≤_1,

(4.3)

H2(t,s) =h2(t,s) + 1 3t

1 2

1 –1₃32 h1

1 3,s ,

h2(t,s) =

1 Γ(3₂)

⎧ ⎨ ⎩

[t(1 –s)]12_{, 0}≤_t≤_s≤_1,

Taking any point (u0,v0)∈ ¯Ph,e, we construct the following sequences:

un+1(t) = 1

0

G1(t,s) 1

0

H1(s,τ)f

τ,un(τ),vn(τ)

dτds

– 1

Γ(3₂)Γ(10₃)t

4

3_A1+B1t+C1t52 _+Dt92_,

vn+1(t) = 1

0

G2(t,s) 1

0

H2(s,τ)g

τ,un(τ),vn(τ)

dτds

– 1

Γ(3₂)Γ(10₃)t

4

3_A2+B2t+C2t52 _+Dt92_,

n= 0, 1, 2, . . . , then we haveun+1(t)→u∗(t),vn+1(t)→v∗(t) asn→ ∞.

5 Conclusion

Recently, fractional differential systems have been increasingly used to describe problems in optical and thermal systems, rheology and materials and mechanics systems, signal processing and system identification, control, robotics, and other applications. Because of their deep realistic background and important role, people are paying more and more attention. For nonlinear fractional differential systems subject to different boundary con-ditions, there are many articles studying the existence or multiplicity of solutions or pos-itive solutions. But the unique results are very rare. In this paper, we study a system of fractional differential Eqs. (1.1). By constructing two functionse andhand using fixed point theorem of increasing Ψ-(h,e)-concave operators defined on ordered setPh,e, we

establish some new existence and uniqueness criteria for system (1.1). Our result shows that the unique solution exists in a product setP¯h,e=Ph1,e1 ×Ph2,e2 and can be

approxi-mated by making an iterative sequence for any initial point inP¯h,e. Finally, an interesting

example is given to illustrate the application of our main results.

Acknowledgements

The authors wish to thank the anonymous referees for their valuable suggestions.

Funding

This paper was supported financially by the Youth Science Foundation of China (11201272) and Shanxi Province Science Foundation (2015011005).

Availability of data and materials

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 19 June 2019 Accepted: 4 September 2019

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