Existence and Ulam Type Stability for Impulsive Fractional Differential Systems with Pure Delay

Abstract

Through literature retrieval and classification, it can be found that for the fractional delay impulse differential system, the existence and uniqueness of the solution and UHR stability of the fractional delay impulse differential system are rarely studied by using the polynomial function of the fractional delay impulse matrix. In this paper, we firstly introduce a new concept of impulsive delayed Mittag–Leffler type solution vector function, which helps us to construct a representation of an exact solution for the linear impulsive fractional differential delay equations (IFDDEs). Secondly, by using Banach’s and Schauder’s fixed point theorems, we derive some sufficient conditions to guarantee the existence and uniqueness of solutions of nonlinear IFDDEs. Finally, we obtain the Ulam–Hyers stability (UHs) and Ulam–Hyers–Rassias stability (UHRs) for a class of nonlinear IFDDEs.

Full text

Citation: Chen, C.; Li, M. Existence
and Ulam Type Stability for
Impulsive Fractional Differential
Systems with Pure Delay. Fractal
Fract. 2022,6, 742. https://doi.org/
10.3390/fractalfract6120742
Academic Editors: Oana Brandibur
and Eva Kaslik
Received: 10 November 2022
Accepted: 13 December 2022
Published: 15 December 2022
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4.0/).
fractal and fractional
Article
Existence and Ulam Type Stability for Impulsive Fractional
Differential Systems with Pure Delay
Chaowen Chen and Mengmeng Li *
Department of Mathematics, Guizhou University, Guiyang 550025, China
*Correspondence: mmli@gzu.edu.cn
Abstract:
Through literature retrieval and classification, it can be found that for the fractional delay
impulse differential system, the existence and uniqueness of the solution and UHR stability of the
fractional delay impulse differential system are rarely studied by using the polynomial function of the
fractional delay impulse matrix. In this paper, we firstly introduce a new concept of impulsive delayed
Mittag–Leffler type solution vector function, which helps us to construct a representation of an exact
solution for the linear impulsive fractional differential delay equations (IFDDEs). Secondly, by using
Banach’s and Schauder’s fixed point theorems, we derive some sufficient conditions to guarantee
the existence and uniqueness of solutions of nonlinear IFDDEs. Finally, we obtain the Ulam–Hyers
stability (UHs) and Ulam–Hyers–Rassias stability (UHRs) for a class of nonlinear IFDDEs.
Keywords:
fractional differential equations; impulsive delayed Mittag–Leffler type vector function;
existence of solution; Ulam–Hyers stability
MSC: 34A08; 34A37; 34D20
1. Introduction
Fractional calculus and fractional differential equations (FDEs) have been widely ap-
plied in mechanics, physics, biological and the other fields of science and engineering
[1–6]
.
In recent decades, there has been an explosion in searching for the existence, uniqueness,
stability and controllability of impulsive differential equations (IDEs) as researchers in
epidemic, optimal control, mechanical and engineering studies are pouring into the field of
research; we refer the reader to [7–12].
Impulsive fractional differential equations (IFDEs) have attracted great interest due to
their potential applications in modeling dynamical systems involving genetic phenomena
and mutations. Among the numerous research results, it is worth noting that the authors
in [
13
] introduced a formula for solutions of the Cauchy problem of IFDEs and gave a
counter example to prove that the previous results were incorrect. For more research results
on the recent advances of existence, uniqueness, exponential stability, uniform stability and
continuous dependence of IFDEs, one can see the research papers [13–20].
Meanwhile, impulsive fractional differential delay equations and fractional differential
delay equations (FDDEs) are widely used to characterize the situation of their states
depending on the previous time interval subject to abrupt changes. In [
21
], the authors
obtained finite-time stability of solution for FDDEs by using the delayed single parameter
Mittag–Leffler type matrix function. In [
22
], the authors introduced a concept of a delayed
two parameter Mittag–Leffler type matrix function and gave an explicit formula of a
solution for FDDEs. In [
23
], an explicit solution of the conformable FDDEs was given and
the UH and UHR stability were discussed.
In [
24
], some sufficient conditions for the finite-time stability of IFDDEs were obtained
by using the generalized Bellman–Gronwall’s inequality, which extended some known
results. In [
25
], the authors proposed a class of linear fractional difference equations with
Fractal Fract. 2022,6, 742. https://doi.org/10.3390/fractalfract6120742 https://www.mdpi.com/journal/fractalfract

Fractal Fract. 2022,6, 742 2 of 16
discrete-time delay and impulse effects. In [
26
], the authors studied the controllability
of an impulsive fractional differential equation with infinite state-dependent delay in an
arbitrary Banach space.
Motivated by [
21
–
23
], we first study the analytic representation of a solution of lin-
ear IFDDEs:





(CDα
0+z)(x) = Az(x−h) + g(x),x6=xi,h>0, x∈J,
z(x+
i) = z(x−
i) + Ci,x=xi,i=1, 2, . . . , r(T, 0),
z(x) = v(x),−h≤x≤0,
(1)
where
CDα
0+z(·)(
0
<α<
1
)
is the Caputo derivative,
A∈Rn×n
,
g∈C(J
,
Rn)
,
J:= [
0,
T]
,
T=k∗h
,
k∗∈N:={
0, 1, 2,
. . .}
,
r(T
, 0
)
denotes the finite number of impulsive points
which belong to
(
0,
T)
and
v∈C1([−h
, 0
]
,
Rn)
. The symbols
z(x+
i) = lim
e→0+z(xi+e)
and
z(x−
i) = lim
e→0−z(xi+e)represent the right and left limits of z(x)at x=xi, respectively.
In a conference held at Wisconsin University in 1940, Ulam [
27
] first raised the question
of the stability of functional equations. The first answer to the question of Ulam [
28
] was
given by Hyers in 1941 in the case of Banach spaces. In recent years, many researchers
have been interested in the UHs and UHR of IFDEs and FDDEs. In [
29
], the authors
introduced the concept of piecewise continuous solutions for impulsive Cauchy problems
and discussed UHs for IFDEs. In [
30
], the authors gave existence and uniqueness of
solutions as well as UH results for IFDEs. In [
31
], the authors established the existence,
uniqueness, UHs and UHRs of solutions for FDEs. In [
32
], the authors introduced four
Ulam type stability concepts for non-instantaneous IFDEs with state dependent delay and
obtained sufficient conditions for Ulam type stability. However, there are few studies on
UHs and UHRs of IFDDE, which is complex.
Therefore, we attempt to investigate the existence, uniqueness, UHs and UHRs of the
nonlinear IFDDEs in this paper:





(CDα
0+z)(x) = Az(x−h) + g(x,z(x)),x6=xi,x∈J,
z(x+
i) = z(x−
i) + Ci,x=xi,i=1, 2, . . . , r(T, 0),
z(x) = v(x),−h≤x≤0.
(2)
Compared with [21–23], the novelties of this paper are as follows:
In this paper, the explicit solution of the Caputo fractional time delay impulse differen-
tial equation is given. A different from the system studied in [
23
], the fractional derivative
is Caputo type and adds the impulsive condition to the system in this paper. In view of this
difference, the impulsive delayed Mittag–Leffler type vector function newly constructed is
important to solving the problem.
Although the ideas and methods adopted in the study of existence and UHs of solu-
tions are similar to [
21
–
23
], the considered system is different, here, we not only give the
representation of solutions via the new constructed impulsive delayed Mittag–Leffler type
vector function but also study the existence and uniqueness of solutions, UHs and UHRs
of (2).
The structure of this paper is as follows. Firstly, we seek for the fundamental solution
vector for the linear homogeneous IFDDEs and give its exact solution. Secondly, we derive
the exact representation of solution of
(1)
by using the delayed Mittag–Leffler type matrix
functions and impulsive delayed Mittag–Leffler type vector function. Furthermore, we
prove the existence and uniqueness of solutions of
(2)
. Finally, we establish the conditions
for the existence of the UHs and UHRs for the nonlinear IFDDEs.
2. Preliminaries
Set
PC(J
,
Rn):={z:J→Rn:z∈C((xi
,
xi+1]
,
Rn)
,
i=
1, 2,
. . .
,
r(T
, 0
)
; there
exist
z(x+
i)
and
z(x−
i)
,
z(x−
i) = z(xi)}
with
kzkPC :=sup
x∈J
kz(x)k
;
C(J
,
Rn)
is the space

Fractal Fract. 2022,6, 742 3 of 16
of all the continuous functions from
J
to
Rn
with
kzkC=max
x∈Jkz(x)k
and
C1(J
,
Rn) =
{z∈C(J
,
Rn):z0∈C(J
,
Rn)}
. Let
z∈Rn
and
A∈Rn×n
; we introduce vector norm
kzk=n
∑
i=1
|zi|
and matrix norm
kAk=max
1≤j≤n
n
∑
i=1
|aij |
. Denote
kvkC=max
x∈[−h,0]kv(x)k
and
kv0kC=max
x∈[−h,0]kv0(x)k.
Definition 1
(see [
2
])
.
Let
α∈(
0, 1
)
and
g:[
0,
+∞)→Rn
. The Caputo fractional derivative of
g can be written as
(CDα
0+g)(x) = 1
Γ(1−α)Zx
0(x−t)−αg0(t)dt,x>0.
Definition 2
(see [
2
])
.
Let
α∈(
0, 1
)
and
g:[
0,
+∞)→Rn
. The Riemann–Liouville fractional
integral of g can be written as
(Iα
0+g)(x) = 1
Γ(α)Zx
0(x−t)α−1g(t)dt,x>0.
Definition 3
(see [
21
])
.
Delayed one-parameter Mittag–Leffler type matrix function
EA·α
h:R→
Rn×nis defined by
EAxα
h=















Θ,−∞<x<−h,
E,−h≤x≤0,
E+Axα
Γ(α+1)+A2(x−h)2α
Γ(2α+1)+. . . +Aj(x−(j−1)h)jα
Γ(jα+1),
(j−1)h<x≤jh,j∈N,
(3)
where Θis a zero matrix and E is an identity matrix.
Definition 4
(see [
21
])
.
The delayed two-parameter Mittag–Leffler type matrix function
EA·α
h,β:
R→Rn×nis defined by
EAxα
h,β=





























Θ,−∞<x<−h,
E(h+x)α−1
Γ(α),−h≤x≤0,
E(h+x)α−1
Γ(α)+Ax2α−1
Γ(α+β)+A2(x−h)3α−1
Γ(2α+β)
+. . . +Aj(x−(j−1)h)(j+1)α−1
Γ(jα+β),
(j−1)h<x≤jh,j∈N.
(4)
Definition 5.
Let
x∈((j−
1
)h
,
jh]
,
j=
1, 2,
. . .
,
k∗
; the impulsive delayed Mittag–Leffler type
vector function Zh,α(·):R→Rnis defined by
Zh,α(x) = ∑
0<xi<x
EB(x−xi−h)α
hCi. (5)
Remark 1.
Let
x∈(jh
,
(j+
1
)h]
,
xi=jh
,
j=
1, 2,
. . .
,
k∗−
1; the impulsive delayed Mittag–
Leffler type vector function Zh,α(·):R→Rnis defined by
Zh,α(x) =
j
∑
k=1
EA(x−(k+1)h)α
hCk.
Lemma 1
(see [
21
])
.
Let
x∈((j−
1
)h
,
jh]
;
j∈N
, one can obtain
(CDα
0+EA·α
h)(x) = AEA(x−h)α
h
.

Fractal Fract. 2022,6, 742 4 of 16
Lemma 2
(see [
21
])
.
Let
x∈[(j−
1
)h
,
jh]
;
j∈N
; one can obtain

EAxα
h
≤Eα(kAkxα)
, where
Eα(·)is defined by Eα(z) =
∞
∑
k=0
zk
Γ(kα+1).
Lemma 3 (see [22]).Let x ∈((j−1)h,jh]; j ∈N;0≤t<x; we have

EA(x−h−t)α
h,α
≤
j
∑
m=1
kAkm−1
Γ(mα)(x−(m−1)h)mα−1.
Lemma 4. Let x ∈((j−1)h,jh], j ∈N, xi∈
j−1
S
r=2
(x−(j+1−r)h,x−(j−r)h]; we have
Zx
xi+(j−r)h(x−t)−α(t−xi−(j−r)h)(j−r)α−1dt
= (x−xi−(j−r)h)(j−r−1)αB[(j−r)α, 1 −α],
where B[l,m] = R1
0sl−1(1−s)m−1ds.
Proof. By calculation, one can obtain
Rx
xi+(j−r)h(x−t)−α(t−xi−(j−r)h)(j−r)α−1dt
= (x−xi−(j−r)h)(j−r−1)αR1
0ξ(j−r−1)α−1(1−ξ)−αdξ
= (x−xi−(j−r)h)(j−r−1)αB[(j−r)α, 1 −α].
Lemma 5. Let x ∈((j−1)h,jh], j ∈Nand xi∈(0, x); we obtain

EA(x−xi−h)α
h
≤Eα(kAk(x−xi)α).
Proof. Let x∈((j−1)h,jh]and xi∈(0, x−(j−1)h]; one can obtain

EA(x−xi−h)α
h
≤
j−1
∑
m=0


Am(x−xi−mh)mα
Γ(mα+1)



≤
j−1
∑
m=0
kAkm(x−xi)mα
Γ(mα+1)
≤Eα(kAk(x−xi)α).
Let
x∈((j−
1
)h
,
jh]
and
xi∈(x−(j+
1
−r)h
,
x−(j−r)h]
,
r=
2, 3
. . . j−
1; one
can obtain 
EA(x−xi−h)α
h
≤
j−r
∑
m=0


Am(x−xi−mh)mα
Γ(mα+1)



≤
j−r
∑
m=0
kAkm(x−xi)mα
Γ(mα+1)
≤Eα(kAk(x−xi)α).
Let x∈((j−1)h,jh]and xi∈(x−h,x); one can obtain

EA(x−xi−h)α
h
=0<Eα(kAk(x−xi)α).

Fractal Fract. 2022,6, 742 5 of 16
3. The General Solution of Homogeneous System
In this part, we discuss the exact solution of





(CDα
0+z)(x) = Az(x−h),x6=xi,x∈J,
z(x+
i) = z(x−
i) + Ci,x=xi,i=1, 2, . . . r(T, 0),
z(x) = v(x),−h≤x≤0,
(6)
by using (3)–(5).
Lemma 6.
Let
x∈((j−
1
)h
,
jh]
;
j∈N
and
xi∈(
0,
x)
is arbitrarily fixed impulsive points; we
have
CDα
0+(EA(·−xi−h)α
hCi)(x) = AEA(x−xi−2h)α
hCi.
Proof. Let xi∈(x−h,x]; we have
EA(x−xi−h)α
hCi=ECi. (7)
By Definition 1and (7), one can obtain
CDα
0+(EA(·−xi−h)α
hCi)(x) = 1
Γ(1−α)Rx
0(ECi)0(x−t)−αdt
=AEA(x−xi−2h)α
hCi.
For any xi∈(x−(j+1−r)h,x−(j−r)h]and r=2, 3, . . . , j−1, we have
EA(x−xi−h)α
hCi=
j−r
∑
m=0
Am(x−xi−mh)mα
Γ(mα+1)Ci. (8)
By Definition 1and (8), one can obtain
(CDα
0+EA(·−xi−h)α
hCi)(x)
=1
Γ(1−α)Rx
0(x−t)−α(EA(t−xi−h)α
h)0Cidt
=1
Γ(1−α)Rxi+2h
xi+h(x−t)−αAα(t−xi−h)α−1
Γ(α+1)Cidt+
. . . +Rx
xi+(j−r)h(x−t)−αj−r
∑
m=0
Ammα(t−xi−mh)mα−1
Γ(mα+1)Cidt
=1
Γ(1−α)Rx
xi+h(x−t)−αAα(t−xi−h)α−1
Γ(α+1)Cidt
+Rx
xi+2h(x−t)−αA22α(t−xi−2h)2α−1
Γ(2α+1)Cidt
. . . +Rx
xi+(j−r)h(x−t)−αAj−r(j−r)α(t−xi−(j−r)h)(j−r)α−1
Γ((j−r)α+1)Cidt
=AECi+A2(x−xi−2h)α
Γ(α+1)Ci+. . . +Aj−r(x−xi−(j−r)h)(j−r−1)α
Γ((j−r−1)α+1)Ci
=A
j−r−1
∑
m=0
Bm(x−h−xi−mh)mα
Γ(mα+1)Ci
=AEA(x−xi−2h)α
hCi.
For any fixed xi∈(0, x−(j−1)h], one can obtain
EA(x−xi−h)α
hCi=
j−1
∑
m=0
Am(x−xi−mh)mα
Γ(mα+1)Ci. (9)

Fractal Fract. 2022,6, 742 6 of 16
By Definition 1and (9), one can obtain
(CDα
0+EA(·−xi−h)α
hCi)(x)
=1
Γ(1−α)Rx
xi+h(x−t)−αAα(t−xi−h)α−1
Γ(α+1)Cidt
+Rx
xi+2h(x−t)−αA22α(t−xi−2h)2α−1
Γ(2α+1)Cidt
. . . +Rx
xi+(j−1)h(x−t)−αAj−1jα(t−xi−(j−1)h)jα−1
Γ(jα+1)Cidt
=AECi+A2(x−xi−2h)α
Γ(α+1)Ci+. . . +Aj−1(x−xi−(j−1)h)(j−2)α
Γ((j−2)α+1)Ci
=A
j−2
∑
m=0
Am(x−h−xi−mh)mα
Γ(mα+1)Ci
=AEA(x−xi−2h)α
hCi.
Lemma 7.
Impulsive delayed Mittag–Leffler type vector function
Zh,α(·)
is the fundamental
solution of (6).
Proof. By Definition 1and Lemma 6, we have
CDα
0+∑
0<xi<x
EA(·−xi−h)α
hCi(x) = 1
Γ(1−α)Rx
0(x−t)−α∑
0<xi<x
(EA(t−xi−h)α
h)0Cidt
=∑
0<xi<x
CDα
0+EA(t−xi−h)α
hCi(x)
=A∑
0<xi<x
EA(x−xi−2h)α
hCi.
For any xi∈(0, x)and i=1, 2, . . . , r(x, 0), we verify that Zh,α(x+
i) = Zh,α(x−
i) + Ci.
Zh,α(x+
i) = ∑i
k=1EA(x+
i−xk−h)α
hCk
=∑i−1
k=1EA(x−
i−xk−h)α
hCk+Ci,x+
i∈(xi,xi+1],
Zh,α(x−
i) = ∑i−1
k=1EA(x−
i−xk−h)α
hCk,x−
i∈(xi−1,xi],
which implies that Zh,α(x+
i) = Zh,α(x−
i) + Ci.
Theorem 1. The solution z ∈PC([−h,T],Rn)of (6)has the following form:
z(x) = EAxα
hv(−h) + Z0
−h
EA(x−h−s)α
hv0(s)ds +∑
0<xi<x
EA(x−xi−h)α
hCi.
Proof. The argument is similar to that in ([21] Theorem 3.2).
We find the exact solution of (6) that satisfies z(x) = v(x),−h≤x≤0, in the form
z(x) = EAxα
hc+Z0
−h
EA(x−h−s)α
hy(s)ds +Zh,α(x),−h≤x≤T.
where
c
is an unknown constant vector,
y(·)∈C1([
0,
+∞)
,
Rn)
is an unknown function.
Note that
EAxα
hc+Z0
−h
EA(x−h−s)α
hy(s)ds =v(x),−h≤x≤0.
Let x=−h; we have
EA(−h)α
h=E,EA(−2h−s)α
h=Θ,−h<s≤0, EA(−2h−s)α
h=E,s=−h.

Fractal Fract. 2022,6, 742 7 of 16
Thus, we obtain c=v(−h)and
v(x) = v(−h) + Z0
−h
EA(x−h−s)α
hy(s)ds,−h≤x≤0.
For
−h≤x≤
0, when
−h≤s≤x
, one can obtain
−h≤x−h−s≤x
and when
x≤s≤0, one can obtain x−h≤x−h−s≤ −h. Let −h≤x≤0, we obtain
v(x) = v(−h) + Rx
−hEA(x−h−s)α
hy(s)ds
=v(−h) + Rx
−hy(s)ds.
Thus, one can obtain y(x) = v0(x).
Let x∈[0, T]and xi∈(0, x]; we have
z(x+
i) = EA(x+
i)α
hv(−h) + R0
−hEA(x+
i−h−s)α
hv0(s)ds +∑i
k=1EA(x+
i−xk−h)α
hCk,
=EA(xi)α
hv(−h) + R0
−hEA(xi−h−s)α
hv0(s)ds +∑i−1
k=1EA(xi−xk−h)α
hCk+Ci,
=z(xi) + Ci.
Theorem 2.
The particular solution
e
z(x)∈PC([−h
,
T]
,
Rn)
of
(1)
with
e
z(x)≡0= (
0, 0,
. . .
, 0
)>
,
−h≤x≤0can be written as
e
z(x) = Zx
0
EA(x−h−t)α
h,αg(t)dt.
Proof.
The proof is analogous to the one in ([
22
] Theorem 3.1), so we omit the details.
Combining with Theorems 1and 2, any expression of the exact solution of
(1)
is
obtained.
Theorem 3.
The exact solution
z∈PC([−h
,
T]
,
Rn)
of
(1)
with
z(x) = v(x)
;
−h≤x≤
0can
be written as
z(x) = EAxα
hv(−h) + R0
−hEA(x−h−s)α
hv0(s)ds +∑
0<xi<x
EA(x−xi−h)α
hCi
+Rx
0EA(x−h−t)α
h,αg(t)dt.
According to the Theorems 2and 3, the function
z(·)
is called a solution of
(2)
if
z(x)
satisfies the following form:
z(x) = EAxα
hv(−h) + R0
−hEA(x−h−s)α
hv0(s)ds +∑
0<xi<x
EA(x−xi−h)α
hCi
+Rx
0EA(x−h−t)α
h,αg(t,z(t))dt.
(10)
4. Existence and Uniqueness of the Solution
In this part, we establish the existence and uniqueness results of (2). Let
Φ(x) = ∑j
m=1
kAkm−1
Γ(mα)(x−(m−1)h)mα−1,x∈((j−1)h,jh],
M=R0
−hkv0(s)kds <+∞.
We assume that the following conditions:
[H1]
There exists an
L>
0 such that
kg(x
,
z)k ≤ Lkzk+N
satisfies in the case of
g∈C(J×Rn,Rn),x∈J,z∈Rn.
[H2]Let ρ=LTΦ(T)<1.

Fractal Fract. 2022,6, 742 8 of 16
Theorem 4.
If
[H1]
and
[H2]
are satisfied, then
(2)
has at least one solution
z∈PC([−h
,
T]
,
Rn)
.
Proof. According to Theorem 3and (10), the operator Λon Brcan be written as
(Λz)(x) = EAxα
hv(−h) + R0
−hEA(x−h−s)α
hv0(s)ds +∑
0<xi<x
EA(x−xi−h)α
hCi
+Rx
0EA(x−h−t)α
h,αg(t,z(t))dt.
(11)
where
Br:={z∈PC([−h
,
T]
,
Rn)
,
kzkPC ≤r
and
r>κ
1−ρ}
and
κ=Eα(kAkTα)(kv(−h)k
+M) + ∑
0<xi<x
Eα(kAk(T−xi)α)kCik+Φ(T)NT.
Firstly, we show that Λ(Br)⊂Br. For any z∈Br, we have
k(Λz)(x)k ≤ kEAxα
hkkv(−h)k+R0
−hkEA(x−h−s)α
hkkv0(s)kds +∑
0<xi<x
kEA(x−xi−h)α
hCik
+Rx
0kEA(x−h−t)α
h,αkkg(t,z(t))kdt
≤Eα(kAkxα)kv(−h)k+Eα(kAkxα)R0
−hkv0(s)kds
+∑
0<xi<x
Eα(kAk(x−xi)α)kCik+Rx
0kEA(x−h−t)α
h,αk(LkzkPC +N)dt
≤Eα(kAkTα)(kv(−h)k+M) + Φ(T)NT +LTΦ(T)r
+∑
0<xi<x
Eα(kAk(T−xi)α)kCik
≤κ+ρr<r,
which implies that Λ(Br)⊂Br.
Secondly, we check that continuity of
Λ
. Let
{zn(·)}∞
n=1
be a Cauchy sequence such
that
zn(·)→z(·)(n→∞)
in
Br
,
gn(·) = g(·
,
zn(·))
and
g(·) = g(·
,
z(·))
. For any
x∈J
, we
have
k(Λzn)(x)−(Λz)(x)k ≤ Rx
0kEA(x−h−t)α
h,αkkg(t,zn(t)) −g(t,z(t))kdt
≤TΦ(T)kgn−gkPC,
this yields that k(Λzn)−(Λz)kPC ≤TΦ(T)kgn−gkPC .
Finally, we show that
Λ
is equicontinuous. For any
z∈Br
and 0
<x≤x+4x≤T
,
we obtain
k(Λz)(x+4x)−(Λz)(x)k ≤ k(EA(x+4x)α
h−EAxα
h)kkv(−h)k
+R0
−hkEA(x+4x−h−s)α
h−EA(x−h−s)α
hkkv0(s)kds
+∑
0<xi<x
kEA(x+4x−xi−h)α
h−EA(x−xi−h)α
hkkCik
+Rx
0kEA(x+4x−h−s)α
h,α−EA(x−h−s)α
h,αkkg(s,z(s))kds
+Rx+4x
xkEA(x+4x−h−s)α
h,αkkg(s,z(s))kds
≤I1+I2+I3+I4+I5,
where
I1=k(EA(x+4x)α
h−EAxα
h)kkv(−h)k,
I2=R0
−hkEA(x+4x−h−s)α
h−EA(x−h−s)α
hkkv0(s)kds
≤ kv0kCR0
−hkEA(x+4x−h−s)α
h−EA(x−h−s)α
hkds,
I3=∑
0<xi<x
kEA(x+4x−xi−h)α
h−EA(x−xi−h)α
hkkCik,
I4=Rx
0kEA(x+4x−h−s)α
h,α−EA(x−h−s)α
h,αkkg(s,z(s))kds
≤(LkzkPC +N)Rx
0kEA(x+4x−h−s)α
h,α−EA(x−h−s)α
h,αkds,
I5=Rx+4x
xkEB(x+4x−h−s)α
h,αkkg(s,z(s))kds.

Fractal Fract. 2022,6, 742 9 of 16
Let −h<x≤x+4x<Tas 4x→0; we have
EA(x+4x)α
h→EAxα
h,
EA(x+4x−h−s)α
h→EA(x−h−s)α
h,
∑
0<xi<x
EA(x+4x−xi−h)α
h→∑
0<xi<x
EA(x−xi−h)α
h,
EA(x+4x−h−t)α
h,α→EA(x−h−t)α
h,α,
This yields that I1→0, I2→0, I3→0, I4→0 as 4x→0. For I5, one can obtain
I5≤(LkzkPC +N)Rx+4x
xkEA(x+4x−h−s)α
h,αkds
=kEA(x+4x−h−ξ)α
h,αk(LkzkPC +N)4x→0as 4x→0.
Therefore, one can obtain k(Λz)(x+4x)−(Λz)(x)k → 0 as 4x→0.
[H3]
There exists an
e
L>
0 that
kg(x
,
z)−g(x
,
e
z)k ≤ e
Lkz−e
zk
satisfies in the case of
z,e
z∈Rn.
[H4]Let e
ρ=e
LTΦ(T)<1.
Theorem 5. If [H3]and [H4]are satisfied, then (2)has a unique solution z ∈PC(J,Rn).
Proof.
It is easy to prove that
Λ:Be
r→Be
r
defined in
(11)
is uniformly bounded by
using the Theorem 4. Now, we check that
Λ
is a Banach operator. For any
z
,
e
z∈
Be
r
, where
Be
r:={z∈PC([−h
,
T]
,
Rn)
,
kzkPC ≤e
rwith e
r>e
κ
1−e
ρ}
,
e
κ= (kv(−h)k+
M)Eα(kAkTα) + ∑
0<xi<x
Eα(kAk(T−xi)α)kCik+Φ(T)Tke
gk
and
ke
gk=supx∈Jkg(x
,
0)k
.
For any x∈[−h,T], one can obtain
k(Λz)(x)−(Λe
z)(x)k ≤ Rx
0kEA(x−h−t)α
h,αkkg(t,z(t)) −g(t,e
z(t))kdt
≤e
LRx
0kEA(x−h−t)α
h,αkkz(t)−e
z(t)kdt
≤e
Lkz−e
zkPC Rx
0kEA(x−h−t)α
h,αkdt
≤e
LTΦ(T)kz−e
zkPC,
which implies that kΛz−Λe
zkPC ≤e
ρkz−e
zkPC.
5. Ulam Type Stability Results of (2)
In this part, we establish the Ulam type stability results of nonlinear IFDDEs. Let
ε
,
φ>
0,
e
J:= [−h
, 0
]∪J
and
ψ∈C(J
,
R+:= (
0,
+∞))
. Consider
(2)
and the following
inequalities: 




kCDα
0+Ψ(x)−AΨ(x−h)−g(x,Ψ(x))k ≤ ε,x∈J,
kΨ(x+
i)−Ψ(x−
i)−Cik ≤ φ,i=1, 2, . . . , r(T, 0),
Ψ(x) = v(x),x∈[−h, 0],
(12)
and 




kCDα
0+Ψ(x)−AΨ(x−h)−g(x,Ψ(x))k ≤ εψ(x),x∈J
kΨ(x+
i)−Ψ(x−
i)−Cik ≤ εφ,i=1, 2, . . . , r(T, 0),
Ψ(x) = v(x),x∈[−h, 0].
(13)
Definition 6.
System
(2)
is said to be Ulam–Hyers stable if there exists
K>
0such that for every
ε>
0and for any solution
Ψ∈PC(e
J
,
Rn)
satisfying
(12)
, there exists a solution
z∈PC(e
J
,
Rn)
of
(2)such that
kΨ(x)−z(x)k ≤ K(ε+φ),x∈e
J.

Fractal Fract. 2022,6, 742 10 of 16
Remark 2.
If
Ψ∈PC(e
J
,
Rn)
is a solution of inequality
(12)
, then there exist
Di∈Rn
and
Z ∈ C(J,Rn)such that
(a)
kZ (x)k ≤ ε,kDik ≤ φ,x∈J,i=1, 2, . . . , r(T, 0).
(b)
(CDα
0+Ψ)(x) = AΨ(x−h) + g(x,Ψ(x)) + Z(x),x∈J.
(c)
Ψ(x+
i) = Ψ(x−
i) + Ci+Di,i=1, 2, . . . , r(T, 0).
(d)
Ψ(x) = v(x),x∈[−h, 0].
Definition 7.
System
(2)
is said to be Ulam–Hyers–Rassias stable with respect to
ψ(·)
and
φ
if
there exists
e
K>
0such that for every
ε>
0and for every solution
Ψ∈PC(e
J
,
Rn)
of inequality
(13)
,
there exists a solution z ∈PC(e
J,Rn)of (2)such that
kΨ(x)−z(x)k ≤ εe
K(ψ(x) + φ),x∈e
J.
Remark 3.
The function
Ψ∈PC(e
J
,
Rn)
is said to be a solution of inequality
(13)
if there exist
Ei∈Rnand Z ∈ PC(J,Rn)such that
(a)
kZ (x)k ≤ εψ(x),kEik ≤ εφ,x∈J,i=1, 2, . . . , r(T, 0).
(b)
(CDα
0+Ψ)(x) = AΨ(x−h) + g(x,Ψ(x)) + Z(x),x∈J.
(c)
Ψ(x+
i) = Ψ(x−
i) + Ci+Ei,i=1, 2, . . . , r(T, 0).
(d)
Ψ(x) = v(x),x∈[−h, 0].
Lemma 8.
If
Ψ∈PC(e
J
,
Rn)
is a solution of inequality
(12)
, then
Ψ
satisfies the following integral
inequality



Ψ(x)−EAxα
hv(−h)−R0
−hEA(x−h−s)α
hv0(s)ds −∑
0<xi<x
EA(x−xi−h)α
hCi
−Rx
0EA(x−h−t)α
h,αg(t,Ψ(t))dt



≤φ∑
0<xi<x
Eα(kAk(T−xi)α) + TΦ(T)ε.
Proof. By Remark 2, one can obtain





(CDα
0+Ψ)(x) = AΨ(x−h) + g(x,Ψ(x)) + Z(x),x∈J,
Ψ(x+
i) = Ψ(x−
i) + Ci+Di,i=1, 2, . . . , r(T, 0),
Ψ(x) = v(x),x∈[−h, 0],
and
Ψ(x) = EAxα
hv(−h) + R0
−hEA(x−h−s)α
hv0(s)ds +∑
0<xi<x
EA(x−xi−h)α
h(Ci+Di)
+Rx
0EA(x−h−t)α
h,α(g(t,Ψ(t)) + Z(t))dt.
Let x∈J; we obtain



Ψ(x)−EAxα
hv(−h)−R0
−hEA(x−h−s)α
hv0(s)ds −∑
0<xi<x
EA(x−xi−h)α
hCi
−Rx
0EA(x−h−t)α
h,αg(t,Ψ(t))dt



≤∑
0<xi<x
EA(x−xi−h)α
hDi
+Rx
0kEA(x−h−t)α
h,αZ(t)kdt
≤∑
0<xi<x
Eα(kAk(x−xi)α)kDik+Φ(T)Rx
0εdt
≤φ∑
0<xi<x
Eα(kAk(T−xi)α) + εΦ(T)T.

Fractal Fract. 2022,6, 742 11 of 16
Lemma 9.
If
Ψ∈PC(e
J
,
Rn)
is a solution of inequality
(13)
, then
Ψ
satisfies the following integral
inequality



Ψ(x)−EAxα
hv(−h)−R0
−hEA(x−h−s)α
hv0(s)ds−∑
0<xi<x
EA(x−xi−h)α
hCi
−Rx
0EA(x−h−t)α
h,αg(t,Ψ(t))dt



≤εφ ∑
0<xi<x
Eα(kAk(T−xi)α) + εΦ(T)Rx
0ψ(t)dt.
Proof. By Remark 3, one can obtain





(CDα
0+Ψ)(x) = AΨ(x−h) + g(x,Ψ(x)) + Z(x),x∈J,
Ψ(x+
i) = Ψ(x−
i) + Ci+Ei,i=1, 2, . . . , r(T, 0),
Ψ(x) = v(x),x∈[−h, 0],
and
Ψ(x) = EAxα
hv(−h) + R0
−hEA(x−h−s)α
hv0(s)ds +∑
0<xi<x
EA(x−xi−h)α
h(Ci+Ei)
+Rx
0EA(x−h−t)α
h,α(g(t,Ψ(t)) + Z(t))dt.
Let x∈J; we obtain



Ψ(x)−EAxα
hv(−h)−R0
−hEA(x−h−s)α
hv0(s)ds−∑
0<xi<x
EA(x−xi−h)α
hCi
−Rx
0EA(x−h−t)α
h,αg(t,Ψ(t))dt



≤∑
0<xi<x
kEA(x−xi−h)α
hEik+Rx
0kEA(x−h−t)α
h,αZ(t)kdt
≤∑
0<xi<x
Eα(kAk(x−xi)α)kEik+Φ(x)Rx
0kZ (t)kdt
≤εφ ∑
0<xi<x
Eα(kAk(T−xi)α) + εΦ(T)Rx
0ψ(t)dt.
Theorem 6. Suppose that [H3]and [H4]are satisfied. Then (2)is UH on e
J.
Proof. Let z∈PC(e
J,Rn); we have
z(x) = EAxα
hv(−h) + R0
−hEA(x−h−s)α
hv0(s)ds +∑
0<xi<x
EA(x−xi−h)α
hCi
+Rx
0EA(x−h−t)α
h,αg(t,z(t))dt.
Let x∈[−h, 0]; we have
kΨ(x)−z(x)k=kv(x)−v(x)k=0<K(ε+φ).
According to Lemma 8, for any x∈J, one can obtain
kΨ(x)−z(x)k
≤


Ψ(x)−EAxα
hv(−h)−R0
−hEA(x−h−s)α
hv0(s)ds −∑
0<xi<x
EA(x−xi−h)α
hCi
−Rx
0EA(x−h−t)α
h,αg(t,Ψ(t))dt


+Rx
0kEA(x−h−t)α
h,αkkg(t,Ψ(t)) −g(t,z(t))kdt
≤φ∑
0<xi<T
Eα(kAk(T−xi)α) + εTΦ(T) + e
LTΦ(T)kΨ−zkPC,

Fractal Fract. 2022,6, 742 12 of 16
which implies that
kΨ(x)−z(x)k ≤
φ∑
0<xi<T
Eα(kAk(T−xi)α)
1−e
LTΦ(T)+εTΦ(T)
1−e
LTΦ(T)
≤K(ε+φ),
where
K=1
1−e
LTΦ(T)max ∑
0<xi<T
Eα(kAk(T−xi)α),TΦ(T).
[H5]For any t∈J, there is a monotone function ψ(·)∈C(J,R+)such that
Zx
0ψ(t)dt ≤e
Mψ(x).
Theorem 7. Suppose that [H3]–[H5]are satisfied. Then (2)is UHR on e
J.
Proof. Let x∈[−h, 0]; one can obtain
kΨ(x)−z(x)k=kv(x)−v(x)k=0<εe
K(φ+ψ(x)).
According to Lemma 9, for any x∈J, one can obtain
kΨ(x)−z(x)k
≤


Ψ(x)−EAxα
hv(−h)−R0
−hEA(x−h−s)α
hv0(s)ds −∑
0<xi<x
EA(x−xi−h)α
hCi
−Rx
0EA(x−h−t)α
h,αg(t,Ψ(t))dt


+Rx
0kEA(x−h−t)α
h,αkkg(t,Ψ(t)) −g(t,z(t))kdt
≤εφ ∑
0<xi<T
Eα(kAk(T−xi)α) + εe
MΦ(T)ψ(x) + e
LTΦ(T)kΨ−zkPC,
which implies that
kΨ(x)−z(x)k ≤
εφ ∑
0<xi<T
Eα(kAk(T−xi)α)
1−e
LTΦ(T)+εe
MΦ(T)ψ(x)
1−e
LTΦ(T)
≤εe
K(φ+ψ(x)),
where
e
K=1
1−e
LTΦ(T)max ∑
0<xi<T
Eα(kAk(T−xi)α),e
MΦ(T).
6. Examples
In this part, we illustrate the obtained results with a couple of examples.
Example 1.
Let
α=
0.3,
h=
0.4,
k∗=
5,
r(T
, 0
) =
4,
T=
2,
xi=
0.4
i
and
i=
1, 2, 3, 4.
Consider 




(CDα
0+z)(x) = Az(x−0.4) + g(x),x∈[0, 2],
z(x+
i) = z(x−
i) + Ci,x=xi,i=1, 2, 3, 4,
v(x) = (2x2+1, x2+2)>,−0.4 ≤x≤0,
(14)
where
z(x) = z1(x)
z2(x),A=0.2 0.8
0.3 0.5 ,Ci=i
2
i
4,g(x) = x
x2. (15)

Fractal Fract. 2022,6, 742 13 of 16
By Theorem 1, we have
z(x) = EAx0.3
0.4 v(−0.4) + R0
−0.4 EA(x−0.4−s)0.3
0.4 v0(s)ds +∑
0<xi<x
EA(x−0.4(i+1))0.3
0.4 Ci
+Rx
0EA(x−0.4−t)0.3
0.4,0.3 g(t)dt,
where
EAx0.3
0.4 =















































E+Ax0.3
Γ(1.3),x∈[0, 0.4],
E+Ax0.3
Γ(1.3)+A2(x−0.4)0.6
Γ(1.6),x∈(0.4, 0.8],
E+Ax0.3
Γ(1.3)+A2(x−0.4)0.6
Γ(1.6)+A3(x−0.8)0.9
Γ(1.9),x∈(0.8, 1.2],
E+Ax0.3
Γ(1.3)+A2(x−0.4)0.6
Γ(1.6)+A3(x−0.8)0.9
Γ(1.9)+A4(x−1.2)1.2
Γ(2.2),x∈(1.2, 1.6],
E+Ax0.3
Γ(1.3)+A2(x−0.4)0.6
Γ(1.6)+A3(x−0.8)0.9
Γ(1.9)+A4(x−1.2)1.2
Γ(2.2)
+A5(x−1.6)1.5
Γ(2.5),x∈(1.6, 2],
and
∑
0<xi<x
EA(x−0.4(i+1))0.3
0.4 Ci=





































































0
0,x∈[0, 0.4],
1
2
1
4,x∈(0.4, 0.8],
E+A(x−0.8)0.3
Γ(1.3) 1
2
1
4+1
1
2,x∈(0.8, 1.2],
E+A(x−0.8)0.3
Γ(1.3)+A2(x−1.2)0.6
Γ(1.6) 1
2
1
4
+E+A(x−1.2)0.3
Γ(1.3) 1
1
2+3
2
3
4,x∈(1.2, 1.6],
E+A(x−0.8)0.3
Γ(1.3)+A2(x−1.2)0.6
Γ(1.6)+A3(x−1.6)0.9
Γ(1.9) 1
2
1
4
+E+A(x−1.2)0.3
Γ(1.3)+A2(x−1.6)0.6
Γ(1.6) 1
1
2
+E+A(x−1.6)0.3
Γ(1.3) 3
2
3
4+2
1,x∈(1.6, 2].
Example 2.
Let
α=
0.3,
h=
0.4,
k∗=
3,
T=
1.2,
r(T
, 0
) =
3,
x1=
0.1,
x2=
0.6 and
x3=1.1. Consider





(CDα
0+z)(x) = Az(x−0.4) + g(x,z(x)),x∈[0, 1.2],
z(x+
i) = z(x−
i) + Ci,x=xi,i=1, 2, 3,
v(x) = (2x2+1, x2+2)>,−0.4 ≤x≤0,
(16)
where A and Ciare defined in (15)and g(x,z(x)) = x
12 z1(x),x
12 z2(x)>.

Fractal Fract. 2022,6, 742 14 of 16
Let x∈[0, 1.2]and z,e
z∈R2; one can obtain
kg(x,z(x)) −g(x,e
z(x))k ≤ x
12 (|z1(x)−e
z1(x)|+|z2(x)−e
z2(x)|)
≤1
10 kz−e
zkPC.
By calculation, one has
kAk=
1.3,
e
L=1
10
,
L=1
10
,
Φ(
1.2
) =
2.9819,
M=
1.92.
Hence,
[H1]
,
[H2]
,
[H3]
and
[H4]
are satisfied. By Theorems 4and 5, the solution
z∈
PC([−0.4, 1.2],R2)of (16) can be given by
z(x) = EAx0.3
0.4 v(−0.4) + R0
−0.4 EA(x−0.4−s)0.3
0.4 v0(s)ds +∑
0<xi<x
EA(x−xi−0.4)0.3
0.4 Ci
+Rx
0EA(x−0.4−t)0.3
0.4,0.3 g(t,z(t))dt,
where
∑
0<xi<x
EA(x−xi−0.4)0.3
0.4 Ci=





































































0
0, 0 <x≤0.1,
1
2
1
4, 0.1 <x≤0.5,
E+A(x−0.5)0.3
Γ(1.3) 1
2
1
4, 0.5 <x≤0.6,
E+A(x−0.5)0.3
Γ(1.3) 1
2
1
4+1
1
2, 0.6 <x≤1,
E+A(x−0.5)0.3
Γ(1.3)+A2(x−0.9)0.6
Γ(1.6) 1
2
1
4
+E+A(x−1)0.3
Γ(1.3) 1
1
2, 1 <x≤1.1,
E+A(x−0.5)0.3
Γ(1.3)+A2(x−0.9)0.6
Γ(1.6) 1
2
1
4
+E+A(x−1)0.3
Γ(1.3) 1
1
2+3
2
3
4, 1.1 <x≤1.2,
and
EA(x−0.4−t)0.3
0.4,0.3 =





















































E(x−t)−0.7
Γ(0.3),x∈[0, 0.4],t∈[0, x],
E(x−t)−0.7
Γ(0.3)+A(x−0.4 −t)−0.4
Γ(0.6),x∈[0.4, 0.8],t∈[0, x−0.4],
E(x−t)−0.7
Γ(0.3),x∈[0.4, 0.8],t∈[x−0.4, x],
E(x−t)−0.7
Γ(0.3)+A(x−0.4 −t)−0.4
Γ(0.6)+A2(x−0.8 −t)−0.1
Γ(0.9),
x∈[0.8, 1.2],t∈[0, x−0.8],
E(x−t)−0.7
Γ(0.3)+A(x−0.4 −t)−0.4
Γ(0.6),x∈[0.8, 1.2],t∈[x−0.8, x−0.4],
E(x−t)−0.7
Γ(0.3),x∈[0.8, 1.2],t∈[x−0.4, x].

Fractal Fract. 2022,6, 742 15 of 16
If
Ψ∈PC([−
0.4, 1.2
]
,
R2)
is solution of
(12)
, then there exist
Z(x)=(ε
2e−x
,
ε
2e−x)>∈
C([0, 1.2],R2)and Di= ( i
100 ,i
100 )>such that kZ (x)k ≤ εand kDik ≤ φ=0.1. Choose
K=1
1−e
LTΦ(T)max ∑
0<xi<T
Eα(kAk(T−xi)α),TΦ(T)≈96.21.
According to Theorem 6, we obtain
kΨ(x)−z(x)k ≤ K(ε+φ).
Then (2) is UH on [−0.4, 1.2].
If
Ψ∈PC([−
0.4, 1.2
]
,
R2)
is a solution of
(12)
, then there exist
Z(x)=(ε
2ex
,
ε
3ex)∈
C([
0, 1.2
]
,
R2)
and
Ei= ( i
100
,
i
100 )>
such that
kZ (x)k ≤ εex:=εψ(x)
,
kEik ≤ φ=
0.1.
Moreover, Zx
0ψ(t)dt =Zx
0etdt <ex,x∈[0, 1.2].
Choose
e
M=
1 and
e
K=1
1−e
LTΦ(T)max ∑
0<xi<T
Eα(kAk(T−xi)α)
,
e
MΦ(T)≈
96.21.
According to Theorem 7, we obtain
kΨ(x)−z(x)k ≤ εe
K(φ+ψ(x)),
then (2) is UHR on [−0.4, 1.2].
7. Conclusions
In this paper, a new concept of impulsive delayed Mittag–Leffler type vector function
was described, which helps us to construct a representation of an exact solution for Caputo
fractional time delay impulse differential systems. By using the fixed point technique,
fractional calculus, the delayed Mittag–Leffler type matrix functions and the impulsive
delayed Mittag–Leffler type vector function, the existence and Ulam type stability of the
considered systems were investigated. Moreover, we provided two examples to illustrate
the applicability of the results.
Author Contributions:
The contributions of all authors (C.C. and M.L.) are equal. All the main
results were developed together. All authors have read and agreed to the published version of the
manuscript.
Funding:
This work is partially supported by the National Natural Science Foundation of China
(12201148), Guizhou Provincial Science and Technology Projects (No.QKHJC-ZK[2022]YB069) and
Natural Science Special Project of Guizhou University (No.GZUTGHZ(2021)08).
Data Availability Statement: No applicable.
Acknowledgments: The authors thank the help from the editor too.
Conflicts of Interest: The authors declare no conflict of interest.
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