Applied Mathematics, 2020, 11, 1041-1051

https://www.scirp.org/journal/am
ISSN Online: 2152-7393
ISSN Print: 2152-7385

On the Nonlinear Neutral Conformable

Fractional Integral-Differential Equation

Rui Li, Wei Jiang, Jiale Sheng, Sen Wang

School of Mathematical Sciences, Anhui University, Hefei, China

How to cite this paper: Li, R., Jiang, W., Abstract

Sheng, J.L. and Wang, S. (2020) On the
Nonlinear Neutral Conformable Fraction- In this paper, we investigate the nonlinear neutral fractional integral-differential
al Integral-Differential Equation. Applied equation involving conformable fractional derivative and integral. First of all,
Mathematics, 11, 1041-1051. we give the form of the solution by lemma. Furthermore, existence results for
https://doi.org/10.4236/am.2020.1110069
the solution and sufficient conditions for uniqueness solution are given by the
Received: September 29, 2020 Leray-Schauder nonlinear alternative and Banach contraction mapping prin-
Accepted: October 25, 2020 ciple. Finally, an example is provided to show the application of results.
Published: October 28, 2020
Keywords
Copyright © 2020 by author(s) and
Scientific Research Publishing Inc. Conformable Fractional Derivative, Delay, Existence and Uniqueness,
This work is licensed under the Creative Functional Differential Equation
Commons Attribution International
License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access 1. Introduction
The theory of fractional calculus has played a major role in control theory, fluid
dynamics, biological systems, economics and other fields [1] [2] [3]. It serves as
a valuable tool for the description of memory and hereditary properties of vari-
ous materials and processes. In recent years, plenty of interesting results have
been observed for the Riemann-Liouville and Caputo type fractional derivatives.
The definition of the conformable fractional derivative and integral was in-
troduced in 2014 by Khalil et al. [4]. Compared with Riemann-Liouville and
Caputo type fractional derivatives, the conformable fractional derivative satis-
fies the Leibniz rule and chain rule, and can be converted to classical derivative
[5]. This is of great help to study fractional differential equations. In the past
few years, the conformable fractional derivative has been used in the field of frac-
tional newtonian mechanics, heat equation, biology and so on, and the results are
abundant. The conformable fractional optimal control problems with time-delay
were studied, which proved that the embedding method, embedding the admiss-
ible set into a subset of measures, can be successfully applied to nonlinear prob-

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R. Li et al.

lems [6]. Alharbi et al. utilized the homotopy perturbation method to solve a
model of Ambartsumian equation with the conformable derivative and gave the
approximate solution of equation [7]. The conformable fractional derivative was
utilized to solve the time-fractional Burgers type equations approximately [8].
Over these years, there has been a significant development in fractional func-
tional differential equations. Among them, Li, Liu and Jiang gave sufficient con-
ditions of the existence of positive solutions for a class of nonlinear fractional
differential equations with caputo derivative [9]. Guo et al. studied fractional func-
tional differential equation with impulsive, then they obtained existence, unique-
ness, and data-dependent results of solutions to the equation [10]. The existence
of positive periodic solutions was given by Zhang and Jiang, for n-dimensional
impulsive periodic functional differential equations [11]. In addition, the frac-
tional stochastic functional system driven by Rosenblatt process was investigated
by Shen et al., and they obtained controllability and stability results [12].
Time-delay is part of the theoretical fields investigated by many authors, includ-
ing unbounded time-delay, bounded time-delay, state-dependent time-delay and
others. In 2009, the existence and uniqueness of solutions of the Caputo
fractional neutral differential equations with unbounded delays were dis-
cussed [13]. In 2011, Li and Zhang considered the Caputo fractional neutral
integral-differential equations with unbounded delay, which used the fixed
point theorem to study the existence of mild solutions of equations [14]. The
caputo fractional neutral integral-differential equations with unbounded delay
were discussed in 2013 [15]. The paper used Monch’s fixed point theorem via
measures of non-compactness to study the existence of solutions of equations.
Based on the above research background and relevant discussions, we found
that few people used conformable derivative to study fractional differential equ-
ations with time-delay. In 2019, Mohamed I. Abbas gave the existence of solu-
tions and uniqueness of solution for fractional neutral integro-differential equa-
tions by the Hadamard fractional derivative of order α ∈ ( 0,1) and the Rie-
mann-Liouville integral [16]. In this paper, we will discuss the nonlinear neutral
fractional integral-differential equation in the frame of the conformable deriva-
tive of order α ∈ (1, 2 ) and the comformable integral. Then we make the
condition 3 weaker to improve feasibility. Considering the following equation:
 α p

T  w ( t ) − ∑ I ui ( t , wt )  =
βi
l ( t , wt ) , t ∈ [ 0, ρ ] ,
  i =1  (1.1)
=
 w ( t ) ψ ( t ) , t ∈ [ −υ , 0] ,
where T α denotes the conformable fractional derivative of order α , 1 < α < 2 ,
I βi denotes the conformable fractional integral with order βi , βi ∈ ( 0,1) ,
i = 1, 2,3, , p , p ∈ N + . ρ ,υ > 0 are constants. And for any t ∈ [ 0, ρ ] , we de-
note by wt the element of C ([ −υ , 0] , R ) and is defined by wt (= θ ) w (t + θ ) ,
θ ∈ [ −υ , 0] . Here wt (⋅) represents the history of the state from time t − υ up
to the present time t. l , ui : [ 0, ρ ] × C ([ −υ , 0] , R ) → R are continuous functions
that satisfy some hypotheses given later, ψ ∈ C ([ −υ , 0] , R ) .

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 R. Li et al.

The rest of this paper is organized as follows: In Section 2, we introduce the

concepts and basic properties of conformable fractional integral and derivative.
In Section 3, we give existence results for the solution and sufficient conditions
for uniqueness solution by Leray-Schauder nonlinear alternative and Banach
contraction mapping principle. In Section 4, the numerical simulation is showed
to illustrate the results.
Notations: C ([ 0, ρ ] , R ) denotes all continuous functions that mapped from
[ ρ ] to R and R denotes all real numbers. R + denotes all positive real num-
0,
bers.

2. Preliminaries
In this section, we present some necessary definitions and lemmas to establish
our main results.
Definition 2.1. ([5]) For a function w : [ a, +∞ ) → R , the conformable frac-
tional integral of order α ( n < α ≤ n + 1, n ∈ N ) of the function w is defined as
follows:
1 t
I aα w ( t ) = I an +1 ( t − a ) w ( t ) =
 n ! ∫a (
t − x) ( x − a) w ( x ) dx.
α − n −1 n α − n −1


If a = 0 , I aα w ( t ) can be written as I α w ( t ) .
Definition 2.2. ([5]) For a function w : [ a, +∞ ) → R , the conformable frac-
tional derivative of order α ( n < α ≤ n + 1, n ∈ N ) of the function w is defined as
follows:

Ta w ( t ) = lim
α
α −1
(
w[ ] t + ε ( t − a )
[α ]−α
− w[ ] ( t )
α −1
) ,
ε →0 ε
where [α ] denotes the smallest integer greater than or equal to α . If a = 0 ,
Taα w ( t ) can be written as T α w ( t ) .
Lemma 2.3 ([4]) If function l is α times differential at a point t > 0 for
n < α ≤ n + 1 , n ∈ N , then
1) T α l ( t ) = 0 , for all constant functions l ( t ) = λ ;
2) l is n + 1 times differential simultaneously, then T α ( l )( t ) = t [α ]−α l [α ] ( t ) .
Lemma 2.4. ([5])
1) For a function w : [ a, +∞ ) → R , if wn ( t ) is continuous, then for any t > a ,
we have
) w ( t ) , α ∈ ( n, n + 1] ;
Taα I aα w ( t=

2) For a function w : [ a, +∞ ) → R , if w is n + 1 times differentiable, then for

any t > a , we have
w( ) ( a )( t − a )
i i
n
I aα Taα w ( t ) = w ( t ) − ∑ , α ∈ ( n, n + 1].
i =0 i!
Lemma 2.5. ([4]) If α ∈ ( 0,1] , functions w1 , w2 are α times differentia-
ble at a point t > 0 , then
1) T α ( a1 w1 + a2 w2=
) a1T α ( w1 ) + a2T α ( w2 ) ;
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R. Li et al.

( )
2) T α t p = pt p −α ;
3) T=α
( w1w2 ) w1T α ( w2 ) + w2T α ( w1 ) ;
 w  w T α ( w1 ) − w1T α ( w2 )
4) T α  1  = 2 .
( w2 )
2
 w2 
Lemma 2.6. ([17]) (The nonlinear alternative of Leray-Schauder type) Let C
be a Banach space, C1 be a closed, convex subset of C, and P be an open subset
of C1 , 0 ∈ P . If A : P → C1 is a continuous, compact map. Then
1) A has a fixed point in P, or
2) There is a w ∈ ∂P (the boundary of P in C1 ) and λ ∈ ( 0,1) , such that
w = λ A ( w) .
Lemma 2.7. Let l ( t ) be a continuous function, then the fractional differen-
tial equation
) l ( t ) , t ∈ [0, ρ ] , 1 < α < 2,
T  w ( t ) − η ( t =
α

 (2.1)
 w ( 0 ) ψ=
= 0 , w (0) ψ 0 ,
′ ′

is equivalent to the integral-differential equation

w ( t ) =ψ 0 − η ( 0 ) + ψ 0′ − η ′ ( 0 )  t + η ( t ) + ∫0 ( t − s ) sα − 2 l ( s ) ds,
t
(2.2)

where ψ 0 = w ( 0 ) , ψ 0′ = w′ ( 0 ) .
Proof. Consider Equation (2.1), for any t ∈ [ 0, ρ ] ,
l (t ).
T α  w ( t ) − η ( t )  =

Transforming α times conformable fractional integral on both sides of the

equation and using Lemma 2.4, since 1 < α < 2 , we have

w ( t ) =ψ 0 − η ( 0 ) + ψ 0′ − η ′ ( 0 )  t + η ( t ) + ∫0 ( t − s ) sα − 2 l ( s ) ds,
t

where ψ 0 = w ( 0 ) , ψ 0′ = w′ ( 0 ) . The proof is completed. □

3. Main Results
In this section, we give several results about the fractional integral-differential
Equation (1.1).
{ }
If X= w | w ∈ C ([ −υ , ρ ] ,  ) is Banach Space, the norm is defined as
= {
w sup w ( t ) , t ∈ [ −υ , ρ ] .
p
}
Let η ( t ) = ∑ I βi ui ( t , wt ) . Giving the definition of the operator A : X → X ,
i =1

 p

ψ 0 + ψ 0′ − η ′ ( 0 )  t + ∑ I ui ( t , wt ) + ∫0 ( t − s ) s l ( s, ws ) ds, t ∈ [ 0, ρ ] ,
βj t α −2
Aw ( t ) =  i =1 (3.1)
ψ ( t ) , t ∈ [ −υ , 0] .

where ψ 0 = w (0) , ψ 0′ = w′ ( 0 ) , w0 = w ( 0 + θ ) = ψ (θ ) , θ ∈ [ −υ , 0] ,
p
=η (0) I β ui ( t , wt )
∑= i
0.
i =1 t =0

It should be noticed that Equation (1.1) has solutions if and only if the opera-
tor A has fixed points. So as to achieve the desired goals, we impose the follow-

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 R. Li et al.

ing assumptions for the Equation (1.1).

(H1) There exist functions γ ( t ) , δ i ( t ) : [ 0, ρ ] → R + and continuous non
decreasing functions ζ ( t ) , φi ( t ) : [ 0, ∞ ] → [ 0, ∞ ] , such that, for any ( t , wt ) ∈
[0, ρ ] ×C ([ −υ , 0] , R ) ,
l ( t , wt ) ≤ γ ( t ) ζ ( wt ) ,
ui ( t , wt ) ≤ δ i ( t ) φi ( wt ) .

(H2) Functions l , ui : [ 0, ρ ] × C ([ −υ , 0] , R ) → R are

continuous. There exist
positive functions λi , µ with bounds λi , µ , i = 1, 2, , p , p ∈ N + , respec-
tively such that
ui ( t , x ) − ui ( t , y ) ≤ λi ( t ) x − y ,

l (t, x ) − l (t, y ) ≤ µ (t ) x − y .

(H3) There exists constant M > 0 , υ1 < υ , such that

M
> 1.
p
ρ βi ρα
ψ 0 + ψ 0′ − η ′ ( 0 )  ρ + ∑ δ i φi (υ1 ) + γ ζ (υ1 )
i =1 βi α (α − 1)

(H4 )
p
ρ βi ρα
∑ λi
βi
+ µ
α (α − 1)
< 1.
i =1

We give an existence result based on the nonlinear alternative of Leray-Schauder

type applied to a completely continuous operator.
Theorem 3.1. Suppose that the assumptions (H1)-(H3) are satisfied, then the
Equation (1.1) has at least one solution.
Proof. The operator A is defined as (3.1). Define
{
D = w ∈ C ([ −υ , ρ ] , R ) : wt ≤ υ1 . }
Firstly, we prove that operator A is uniformly bounded. For any ( t , wt ) ∈
[0, ρ ] ×C ([ −υ , 0] , R ) , i = 1, 2, , p , p ∈ N + , and w ∈ D , by (H1), we have
p
Aw ( t ) = ψ 0 + ψ 0′ − η ′ ( 0 )  t + ∑ I βi ui ( s, ws ) + ∫ ( t − s ) sα − 2 l ( s, ws ) ds
t

0
i =1
p
≤ ψ 0 + ψ 0′ − η ′ ( 0 ) t + ∑ I βi ui ( s, ws ) + ∫ ( t − s ) sα − 2 l ( s, ws ) ds
t

0
i =1
p
≤ ψ 0 + ψ 0′ − η ′ ( 0 ) t + ∑ I βi δ i ( t ) φi ( wt ) + ∫0 ( t − s ) sα − 2γ ( t ) ζ ( wt ) ds
t

i =1
p
≤ ψ 0 + ψ 0′ − η ′ ( 0 ) t + ∑ δ i φi (υ1 ) ∫ s βi −1ds + γ ζ (υ1 ) ∫ ( t − s ) sα − 2 ds
t t

0 0
i =1
p
t βi tα
≤ ψ 0 + ψ 0′ − η ′ ( 0 ) t + ∑ δ i φi (υ1 ) + γ ζ (υ1 )
i =1 βi α (α − 1)
βi
p
ρ ρα
≤ ψ 0 + ψ 0′ − η ′ ( 0 ) ρ + ∑ δ i φi (υ1 ) + γ ζ (υ1 ) < M.
i =1 βi α (α − 1)

For any t ∈ [ −υ , 0] and w ∈ D , we have

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R. Li et al.

=
Aw (t ) ψ (t ) ≤ ψ .
Denote M 1 = max {M , ψ } , then
Aw ( t ) ≤ M 1 , t ∈ [ −υ , ρ ] , w ∈ D.

This implies that the operator A is uniformly bounded in D.

Besides, we need to prove that AD is an equicontinuous set. Let wn be a se-
quence such that lim w = w in D. Then, for any t ∈ [ 0, ρ ] , we have
n
n →∞

( )
p
lim Awn ( t ) =ψ 0 + ψ 0′ − η ′ ( 0 )  t + lim ∑ ∫ s βi −1ui s, wsn ds
t

n →∞ n →∞ 0
i =1

+ lim ∫ ( t − s ) sα − 2 l s, wsn ds. ( )

t

n →∞ 0

By (H2), functions l , ui are uniformly continuous, thus, we have

( )
p
lim Awn ( t ) =ψ 0 + ψ 0′ − η ′ ( 0 )  t + ∑ ∫ lim s βi −1ui s, wsn ds
t

n →∞ 0 n →∞
i =1

+ ∫ lim ( t − s ) sα − 2 l s, wsn ds ( )
t

0 n →∞
p
=ψ 0 + ψ 0′ − η ′ ( 0 )  t + ∑ ∫ s βi −1ui ( s, ws ) ds
t

0
i =1

+ ∫ (t − s ) s l ( s, ws ) ds
t α −2
0

= Aw ( t ) .
For any t ∈ [ −υ , 0] , it is obvious that
( t ) ψ=
lim Awn=
n →∞
( t ) Aw ( t ) .
Therefore, Aw ( t ) is continuous and uniformly continuous for t ∈ [ −υ , ρ ] ,
which implies that Aw ( t ) is equicontinuous for t ∈ [ −υ , ρ ] , A is continuous in
C ([ −υ , ρ ] , R ) .
Furthermore, we consider Aw ( t2 ) − Aw ( t1 ) , for any t1 , t2 ∈ [ −υ , ρ ] , t1 < t2 .
Case 1. If 0 ≤ t1 < t2 ≤ ρ , for any ( t , wt ) ∈ [ 0, ρ ] × C ([ −υ , 0] , R ) and w ∈ D ,
i = 1, 2, , p , we have
Aw ( t2 ) − Aw ( t1 )
p
= ψ 0 + ψ 0′ − η ′ ( 0 )  t2 + ∑ ∫ s βi −1ui ( s, ws ) ds
t2

0
i =1

( t2 − s ) s l ( s, ws ) ds −ψ 0 − ψ 0′ − η ′ ( 0 )  t1
t2
+∫ α −2
0
p
− ∑ ∫ s βi −1ui ( s, ws ) ds − ∫ ( t1 − s ) sα − 2 l ( s, ws ) ds
t1 t1

0 0
i =1

p
≤ ψ 0′ − η ′ ( 0 ) ( t2 − t1 ) + ∑ ∫t s βi −1 ui ( s, ws ) ds
t2

i =1 1

+ ∫0 ( t2 − t1 ) s l ( s, ws ) ds + ∫t ( t2 − s ) sα − 2 l ( s, ws ) ds
t1 α −2 t2

1
p
≤ ψ 0′ − η ′ ( 0 ) ( t2 − t1 ) + ∑ ∫t s βi −1δ i ( t ) φi ( wt ) ds
t2

i =1 1

+ ∫0 ( t2 − t1 ) s γ ( t ) ζ ( wt ) ds + ∫t ( t2 − s ) sα − 2γ ( t ) ζ ( wt ) ds
t1 α −2 t2

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 R. Li et al.

p
≤ ψ 0′ − η ′ ( 0 ) ( t2 − t1 ) + ∑ δ i φi (υ1 ) ∫ s βi −1ds
t2

t1
i =1

+ γ ζ (υ1 ) ∫0 ( t2 − t1 ) sα − 2 ds + γ ζ (υ1 ) ∫t ( t2 − s ) sα − 2 ds
t1 t2

p
t2βi − t1βi t2α − t1α
≤ ψ 0′ − η ′ ( 0 ) ( t2 − t1 ) + ∑ δ i φi (υ1 ) + γ ζ (υ1 ) .
i =1 βi α (α − 1)

If t2 − t1 → 0 , then Aw ( t2 ) − Aw ( t1 ) → 0 .
Case 2. If −υ < t1 < 0 ≤ t2 < ρ , for any ( t , wt ) ∈ [ 0, ρ ] × C ([ −υ , 0] , R ) and
w ∈ D , i = 1, 2, , p , Aw ( t1 ) = ψ ( t1 ) holds for any −υ < t1 < 0 , we have
Aw ( t2 ) − Aw ( t1 )
p
= ψ 0 + ψ 0′ − η ′ ( 0 )  t2 + ∑ ∫0 s βi −1ui ( s, ws ) ds
t2

i =1

+ ∫0 ( t2 − s ) s l ( s, ws ) ds −ψ ( t1 )
t2 α −2

p
≤ ψ 0′ − η ′ ( 0 )  t2 + ψ 0 −ψ ( t1 ) + ∑ ∫0 s βi −1 ui ( s, ws ) ds
t2

i =1

+ ∫0 ( t2 − s ) s l ( s, ws ) ds
t2 α −2

p
≤ ψ 0′ − η ′ ( 0 )  t2 + ψ 0 −ψ ( t1 ) + ∑ δ i φi (υ1 ) ∫ s βi −1ds
t2

0
i =1

+ γ ζ (υ1 ) ∫ ( t 2 − s ) s α − 2 ds
t2

0
p
t2βi
≤ ψ 0′ − η ′ ( 0 )  t2 + ψ 0 −ψ ( t1 ) + ∑ δ i φi (υ1 )
i =1 βi
α
t2
+ δ ζ (υ1 ) .
α (α − 1)

Since ψ ( t ) is a continuous function, if t1 → 0 , then we have ψ ( t1 ) → ψ 0 . If

t2 − t1 → 0 , then Aw ( t2 ) − Aw ( t1 ) → 0 .
Case 3. If −υ ≤ t1 < t2 < 0 , for any −υ ≤ t1 < t2 < 0 and w ∈ D ,
Aw ( t2 ) − Aw ( t1 ) =ψ ( t2 ) −ψ ( t1 ) .

Since ψ ( t ) is continuous function, if t2 − t1 → 0 , then Aw ( t2 ) − Aw ( t1 ) → 0 .

From what has been discussed above, AD is equicontinuous. By Arzelá-Ascoli
theorem, AN is compact, then A is completely continuous on X.
For any ( t , wt ) ∈ [ 0, ρ ] × C ([ −υ , 0] , R ) and t ∈ [ 0, ρ ] , i = 1, 2, , p , we have
p
w ( t ) =ψ 0 + ψ 0′ − η ′ ( 0 )  t + ∑ I βi ui ( s, ws ) + ∫0 ( t − s ) sα − 2 l ( s, ws ) ds,
t

i =1

For any t ∈ [ 0, ρ ] , by assumption (H1), we have

p
w ( t ) ≤ ψ 0 + ψ 0′ − η ′ ( 0 ) t + ∑ I βi ui ( s, ws ) + ∫0 ( t − s ) sα − 2 l ( s, ws ) ds
t

i =1
p
≤ ψ 0 + ψ 0′ − η ′ ( 0 ) t + ∑ I βi δ i ( t ) φi ( wt ) + ∫0 ( t − s ) sα − 2γ ( t ) ζ ( wt ) ds
t

i =1
p
≤ ψ 0 + ψ 0′ − η ′ ( 0 ) t + ∑ δ i φi (υ1 ) ∫ s βi −1ds + γ ζ (υ1 ) ∫ ( t − s ) sα − 2 ds
t t
0 0
i =1

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R. Li et al.

p
t βi tα
≤ ψ 0 + ψ 0′ − η ′ ( 0 ) t + ∑ δ i φi (υ1 ) + γ ζ (υ1 )
i =1 βi α (α − 1)
βi
p
ρ ρα
≤ ψ 0 + ψ 0′ − η ′ ( 0 ) ρ + ∑ δ i φi (υ1 ) + γ ζ (υ1 ) .
i =1 βi α (α − 1)

Consider the assumption (H3), there exists M ≠ w ( t ) . Define set

{ }
P = w ∈ C ([ −υ , ρ ] , R ) : w ( t ) < M . We can show that A : P → D is conti-
nuous and completely continuous. Assuming there exists w ∈ ∂P and λ ∈ ( 0,1) ,
w
such that w = λ A ( w ) . Then we have = λ ≥1.
A ( w)
By Lemma 2.6, A has a fixed point in P, which implies that there exists at least
one solution to the Equation (1.1). The proof is completed. □
We will give the uniqueness result of solutions of Equation (1.1):
Theorem 3.2. Suppose that the assumptions (H2) and (H4) are satisfied, then
the Equation (1.1) has a unique solution.
Proof. The operator A is defined as (3.1). For any t ∈ [ 0, ρ ] and
w , w2 ∈ C ([ −υ , ρ ] , R ) , by (H2), we have
1

Aw1 ( t ) − Aw2 ( t )

( )
p
= ψ 0 + ψ 0′ − η ′ ( 0 )  t + ∑ I βi ui s, w1 s
i =1

( )
+ ∫0 ( t − s ) sα − 2 l s, w1s ds −ψ 0 − ψ 0′ − η ′ ( 0 )  t
t

( ) ( )
p
− ∑ I βi ui s, w2 s − ∫0 ( t − s ) sα − 2 l s, ws2 ds
t

i =1

( ) ( ) ( ) ( )
p
≤ ∑ I βi ui s, w1s − ui s, ws2 + ∫0 ( t − s ) sα − 2 l s, w1s − l s, ws2 ds
t

i =1
p
≤ ∑ I βi λi ( t ) w1 − w2 + ∫0 ( t − s ) sα − 2 µ ( t ) w1 − w2 ds
t

i =1

 p t 
≤ w1 − w2  ∑ ∫ s βi −1 λi ds + ∫ ( t − s ) sα − 2 µ ds 
t

 i =1 0 0

 p
t βi
t α 
≤ w1 − w2  ∑ λi + µ 
 i =1 βi α (α − 1) 
 p ρ βi ρα 
≤ w1 − w2  ∑ λi + µ .
 i =1 βi α (α − 1) 

For any t ∈ [ −υ , 0] ,
Aw1 ( t ) − Aw2 ( t ) = ψ ( t ) −ψ ( t ) = 0.

By (H4), A is a contraction mapping, then Equation (1.1) has a unique solu-

tion. The proof is completed. □

4. An Illustrative Example
This section presents an example where we apply Theorems 3.1 and 3.2 to some
particular cases.

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 R. Li et al.

Example 4.1. Consider the fractional integral-differential equation

 3 3 i
wt  1  wt 1
T 2  w ( t ) − ∑ I 4 =   +  , t ∈ [ 0, 2] ,
  i =1 ( 20i + 6t ) (1 + wt )  16 − t 2  3 (1 + wt ) 5 
  (4.1)

  1 
=w ( t ) ψ ( t ) , t ∈ − , 0 .
  2 
3 i
3
where T 2 denotes the conformable fractional derivative of order , I 4 de-
2
i
notes the conformable fractional integral of the order , i = 1, 2,3 . If
4
 1 
w ( t ) :  − , 2  → R , then for any t ∈ [ 0, 2] , we define wt (=
θ ) w (t + θ ) ,
 2 
 1 
θ ∈  − , 0  . Functions
 2 
wt
ui ( t , wt ) = ,
( 20i + 6t ) (1 + wt )
1  wt 1
l (t, =
wt )  + .
 3 (1 + wt ) 5 
16 − t 2  

The continuous function ψ ( t ) satisfies the condition that ψ= ( 0 ) ψ=

′ (0) 0 .
 1  
For any ( t , wt ) ∈ [ 0, 2] × C   − , 0  , R  , i = 1, 2,3 , we have
 2  
wt 1 1
ui ( t , wt )
= ≤ ≤ .
( 20i + 6t ) (1 + wt ) 20i + 6t 20

 1  
For any ( t , wt ) ∈ [0, 2] × C   − , 0  , R  , we have
 2  
1  wt 1 2
l ( t=
, wt )  +  ≤ .
16 − t 2  3 (1 + wt ) 5  45
 
 1  
For any ( t , wt ) ∈ [0, 2] × C   − (0) ψ =
, 0  , R  , ψ= ′ ( 0 ) η=
(0) η=
′ ( 0 ) 0 , if
 2  
M > 0.659 , we have
M
i 3
> 1, i =
1, 2,3.
3 4 2
1 2 2 2
∑ 20 +
i 45 3  3 
i =1
 − 1
4 22 
Consequently, by Theorem 3.1, the Equation (4.1) has at least one solution.
 1  
( ) ( )
For continuous functions column ui t , wt1 , ui t , wt2 : [ 0, 2] × C   − , 0  , R  ,
 2  
i = 1, 2,3 , we have

wt1 wt2
( )
ui t , wt1 − ui= (
t , wt2 ) −
( 20i + 6t ) (1 + wt1 ) ( 20i + 6t ) (1 + wt2 )

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R. Li et al.

w1 − w2
≤
( 20i + 6t ) (1 + wt1 )(1 + wt2 )
1
≤ w1 − w2 .
20i + 6t
 1  
( ) ( )
For continuous functions l t , wt1 , l t , wt2 : [ 0, 2] × C   − , 0  , R  , we have
 2  
 wt1   wt2 
(
l t ,= ) (
wt1 − l t , wt2 ) 1 
+
1
−
1  1
+ 
16 − t 2  3 1 + wt1
 (  )
5  16 − t 2 (
 3 1 + w2
 )
t
5

w1 − w2
≤
( )(
3 16 − t 2 1 + wt1 )(1 + w )
2
t

w1 − w2
≤ .
(
3 16 − t 2 )
 1  
For any ( t , wt ) ∈ [0, 2] × C   − , 0  , R  , we have
 2  
i 3
3
1 24 1 22
∑ 20i i + 36 3  3  < 0.596 < 1, i =
1, 2,3.
i =1
 − 1 
4 22 

Thus, by Theorem 3.2, the Equation (4.1) has a unique solution.

5. Conclusion
The conformable fractional derivative brings great convenience to the study of
fractional functional differential equations due to its unique properties. This pa-
per uses conformable derivative to study the fractional neutral integro-differential
equations, and obtains the results of the existence of the solution and the suffi-
cient conditions for the uniqueness of the solution.

Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this
paper.

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