Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 118, pp. 1–19.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

STABILITY FOR CONFORMABLE IMPULSIVE DIFFERENTIAL

EQUATIONS

YUANLIN DING, MICHAL FEČKAN, JINRONG WANG

Abstract. In this article, we study impulsive differential equations with con-

formable derivatives. Firstly, we derive suitable formulas for solving linear
impulsive conformable Cauchy problems. Then, we show that the linear prob-

lem has asymptotic stability, and the nonlinear problem has generalized Ulam-

Hyers-Rassias stability. Also we illustrate our results with examples.

1. Introduction

Among the new mathematical tools, we have the conformable derivative which
was introduced in [1, 14]. It has been used in Newton mechanics [9], cobweb models
[7], logistic models [2], and other branches of physics [20] and mathematics [4, 18,
22, 24, 25, 26].

Impulsive differential equations have been applied to many problems; see [5, 6,
12, 29, 30]. In particular, [3, 8, 19] consider impulsive differential equations with a
conformable derivative of the form

Da
βy(t) = g(t, y(t)), t ∈ I := [a, b]\{t1, . . . , tm}, 0 < β < 1,

∆y(tk) = Ik(y(t−k )), k = 1, 2, . . . ,m,

where Da
β is called the conformable derivative with low index a, the function g :

[a, b]× R → R is continuous, Ik : R → R is an (instantaneous) impulsive function,
a = t0 < t1 < · · · < tm < tm+1 = b, b > 0, y(t−k ) = limε→0− y(tk + ε) and

y(t+k ) = limε→0+ y(tk + ε).
Motivated by the works [13, 16, 17, 23, 27, 28, 32], we consider the conformable

linear non-instantaneous impulsive differential equation

Da
βy(t) = µy(t), t ∈ (sk, tk+1], k = 0, 1, 2, . . . ,m,

y(t+k ) = ξy(t−k ), k = 1, 2, . . . ,m,

y(t) = ξy(t−k ), t ∈ (tk, sk], k = 1, 2, . . . ,m,

y(s+
k ) = y(s−k ), k = 1, 2, . . . ,m.

(1.1)

2010 Mathematics Subject Classification. 34A37, 34A08, 34D20.
Key words and phrases. Conformable derivative; impulsive differential equation;

asymptotic stability; generalized Ulam-Hyers-Rassias stability.
c©2020 Texas State University.

Submitted October 4, 2020. Published December 8, 2020.

1



2 Y. DING, M. FEČKAN, J. WANG EJDE-2020/118

Note that y(t+k ) = ξy(t−k ) is the classical impulsive condition that affects y at the

point tk; meanwhile y(t) = ξy(t−k ) for t ∈ (tk, sk] affects y on the interval (tk, sk]
and is called non-instantaneous impulsive equation.

Next, we consider the conformable non-linear non-instantaneous impulsive dif-
ferential equation

Da
βy(t) = g(t, y(t)), t ∈ (sk, tk+1], k = 0, 1, . . . ,m,

y(t+k ) = hk(tk, y(t−k )), k = 1, . . . ,m,

y(t) = hk(t, y(t)), t ∈ (tk, sk], k = 1, . . . ,m,

y(s+
k ) = y(s−k ), k = 1, 2, . . . ,m,

(1.2)

where µ and ξ are constants, 0 < β < 1. For k = 1, 2, . . . ,m: the sk are called
junction points while the tk are called impulse points, t0 = s0 = a < t1 < s1 <
t2 · · · < sm < tm+1 = b, b > 0, g : [a, b]×R→ R is continuous, hk : [tk, sk]×R→ R
is continuous and is called a non-instantaneous impulsive function. For details on
the non-instantaneous impulsive equations, see [27, eq. (1.6)]. Equations (1.1),
(1.2) are used in the dynamics of evolution processes in pharmacotherapy: the first
equation denotes the health status of a patient; the second equation denotes the
doctor takes some actions to test medicine for the patient practicably; the third
equation denotes the testing medicine is valid for this patient and then begin to
deal with the effect of patient for some time. The final equation shows the effect of
testing medicine disappeared in the health of the patient.

The article is organized as follows. In Section 2, we present some basic defini-
tions, and derive the solutions for two kinds of non-instantaneous impulsive frac-
tional Cauchy problems. In Section 3, we define asymptotic stability and give some
conditions for (1.1) to be asymptotically stable. In Section 4, we define generalized
Ulam-Hyers-Rassias stability for (1.2), and use a fixed point theorem to study this
stability. In Section 5, we illustrates our main results by examples.

2. Preliminaries

Let PC(I,R) = {y : I → R : y ∈ C((tk, tk+1],R), k = 0, 1, . . . , y(t−k ) = y(tk)},
where C((tk, tk+1],R). This is the space of piecewise continuous functions endowed
with the norm ‖y‖ = supt∈I |y(t)|.
Definition 2.1 ([15, Definition 2.1]). The conformable derivative with lower index
a of a function y : [a, b]→ R is defined as

Da
βy(t) = lim

ε→0

y(t+ ε(t− a)1−β)− y(t)

ε
, a < t, 0 < β < 1,

Da
βy(a) = lim

t→a+
Da
βy(t).

A function y is called β-differentiable at t0 if Da
βy(t0) exists and is finite.

If y ∈ C1([a, b],R), then Da
βy(t) = (t − a)1−βy′(t). For t > a the conformable

derivative Da
βy(t) exists if and only if y is differentiable at t and Da

βy(t) = (t −
a)1−βy′(t); see [1]

Definition 2.2 (see [15, Definition 2.3]). The conformable integral with lower index
a of a function y : [a, b]→ R is defined as

Iaβy(t) =

∫ t

a

y(s)dβ(s, a) =

∫ t

a

(s− a)β−1y(s)ds, a ≤ t; 0 < β < 1.



EJDE-2020/118 CONFORMABLE IMPULSIVE DIFFERENTIAL EQUATIONS 3

When a = 0, we write dβ(s) = dβ(s, 0).

Lemma 2.3 (see [15, Definition 3.3]). Let y : I → R be a continuous function. A
solution y ∈ C(I,R) of the linear problem

Da
βy(t) = µy(t) + g(t), t ∈ I, 0 < β < 1,

y(a) = ya

has the form

y(t) = yae
µ(t−a)β/β +

∫ t

a

eµ(t−a)β/βe−µ(s−a)β/βg(s)(s− a)β−1ds.

The result in Lemma 2.3 is also valid when continuous function is replaced by
integrable functions with finitely many points of discontinuity.

Remark 2.4. Consider the multi-dimensional case

Da
βy(t) = f(y(t), t), t ≥ a

y(a) = ya,
(2.1)

where f ∈ C(Rn × [a,∞),Rn). Then we consider the associate ODE

Y ′(z) = f
(
Y (z), β

√
βz + a

)
, z ≥ 0

Y (0) = ya.
(2.2)

For a solution Y (z) of (2.2), by defining

y(t) = Y
( (t− a)β

β

)
, (2.3)

for t > a, we obtain

Da
βy(t) = (t− a)1−βy′(t)

= (t− a)1−βY ′
( (t− a)β

β

)
(t− a)β−1

= f
(
Y
( (t− a)β

β

)
, β

√
β

(t− a)β

β
+ a
)
,

= f(y(t), t),

y(a) = Y (0) = ya.

Note that
Da
βy(a) = lim

t→a+
Da
βy(t) = f(y(a), a) = f(ya, a).

So all solutions of (2.1) are determined by (2.2) and viceversa. For instance, when

f(y, t) = Ay + g(t),

for a matrix A. Then (2.1) becomes

y′(z) = Ay(t) + g(t), t ≥ a
y(t) = ya,

(2.4)

and (2.2) becomes

Y ′(z) = AY (z) + g( β
√
βz + a), z ≥ 0

Y (0) = ya,



4 Y. DING, M. FEČKAN, J. WANG EJDE-2020/118

with solution

Y (z) = eAzya +

∫ z

0

eA(z−u)g( β
√
βu+ a)du.

Thus by (2.3), a solution of (2.4) is

y(t) = eA
(t−a)β

β ya +

∫ (t−a)β

β

0

eA(
(t−a)β

β −u)g( β
√
βu+ a)du

(
u =

(s− a)β

β

)
= eA

(t−a)β

β ya +

∫ t

a

eA(
(t−a)β

β − (s−a)β

β )g
(
β

√
β

(s− a)β

β
+ a
)

(s− a)β−1ds,

= e
1
βA(t−a)βya +

∫ t

a

e
1
βA((t−a)β−(s−a)β)g(s)(s− a)β−1ds.

This is a generalization of Lemma 2.3 to higher dimensions.

Next, we establish two standard frameworks and derive appropriate formulas for
solving the impulsive Cauchy problem (1.1), and the problem

Da
βy(t) = g(t), t ∈ (sk, tk+1], k = 0, 1, . . . ,m, 0 < β < 1,

y(t) = hk(t), t ∈ (tk, sk], k = 1, . . . ,m,

y(a) = ya.

(2.5)

Lemma 2.5. Let y(t, s, ys) be the solution of (1.1) with initial value y(s) = ys.
Then

y(t) := y(t, s, ys) = W (t, s)ys, 0 ≤ s ≤ t,
where

W (t, s) = ξn(a,t)−n(a,s) exp
(µ
β

[(
((t− a)β − (sn(a,t) − a)β)+ −

(
(s− a)β

− (sn(a,s) − a)β
)+)

+

n(a,t)−1∑
k=n(a,s)

((tk+1 − a)β − (sk − a)β)
])
,

where n(a, t) denotes the number of the impulse points that belong to (a, t) and

z+ := max{0, z}, z ∈ R. Note that when n(a, t) = n(a, s), we have
∑n(a,t)−1
k=n(a,s) = 0.

In particular,

y(t) = ξn(a,t)e
µ
β

[
((t−a)β−(sn(a,t)−a)β)++

∑n(a,t)−1

k=n(a,s)
((tk+1−a)β−(sk−a)β)

]
ya.

Proof. Depending on the number of pulse and junction points between times t and
s, we have the following 8 cases.

Case 1: There are no pulse or junction points between t and s, i.e. n(a, t) = n(a, s).
(i) Let t, s ∈ (sk, tk+1] for k = 0, 1, 2, . . . , n(a, t). When t ∈ (a, t1], we have

y(t) = yae
µ(t−a)β/β .

When t ∈ (t1, s1], we have

y(t) = ξy(t−1 ) = ξeµ(t1−a)β/βya.

When t ∈ (s1,2 ], according to

y(s1) = ξy(t−1 ) = ξeµ(t1−a)β/βya = eµ(s1−a)β/βya1
,



EJDE-2020/118 CONFORMABLE IMPULSIVE DIFFERENTIAL EQUATIONS 5

we have

y(t) = eµ(t−a)β/βya1
= eµ(t−a)β/β ξe

µ(t1−a)β/βya

eµ(s1−a)β/β
,

y(s) = e
µ
β (s−a)βya1 = eµ(t−a)β/β ξe

µ(t1−a)β/βya

eµ(s1−a)β/β
,

so

W (t, s) = eµ
(

(t−a)β−(s−a)β
)
/β .

(ii) Let us set t, s ∈ (tk, sk], k = 1, 2, . . . , n(a, t). From

y(t) = ξy(t−k ), k = 1, 2, . . . , n(a, t),

we obtain y(t) = y(s), so W (t, s) = 1, a constant.

Case 2: There is only one junction point between t and s, i.e. n(a, t) = n(a, s).
For every s ∈ (tn(a,s), sn(a,s)] and t ∈ (sn(a,t), tn(a,t)+1), we have

y(t) = eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βy(s+

n(a,t))

= eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βy(s−n(a,t))

= eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βy(s),

so

W (t, s) = e
µ
β

(
(t−a)β−((sn(a,t)−a)β

)
.

Case 3: There is only one pulse point between time t and s, i.e. n(a, t) = n(a, s)+1.
Let us select every s ∈ (sn(a,s), tn(a,s)+1] and t ∈ (tn(a,t), sn(a,t)]. When y(t) =

ξy(t−n(a,t)), we have

y(t) = ξy(t−n(a,t)) = ξeµ
(

(tn(a,t)−a)β−(s−a)β
)
/βy(s),

so

W (t, s) = ξeµ
(

(tn(a,t)−a)β−(s−a)β
)
/β .

Case 4: There are one pulse and one junction points between t and s, i.e. n(a, t) =
n(a, s) + 1.

(i) By selecting every s ∈ (sn(a,s), tn(a,s)+1] and t ∈ (sn(a,t), tn(a,t)+1], we have

y(t) = eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βy(s+

n(a,t))

= eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βy(s−n(a,t))

= eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βξy(t−n(a,t))

= eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βξeµ

(
(tn(a,t)−a)β−(s−a)β

)
/βy(s),

so

W (t, s) = eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βξeµ

(
(tn(a,t)−a)β−(s−a)β

)
/β .

(ii) For every s ∈ (tn(a,s), sn(a,s)] and t ∈ (tn(a,t), sn(a,t)], we have

y(t) = ξy(t−n(a,t))



6 Y. DING, M. FEČKAN, J. WANG EJDE-2020/118

= ξeµ
(

(tn(a,t)−a)β−(sn(a,s)−a)β
)
/βy(s+

n(a,s))

= ξeµ
(

(tn(a,t)−a)β−(sn(a,s)−a)β
)
/βy(s−n(a,s))

= ξeµ
(

(tn(a,t)−a)β−(sn(a,s)−a)β
)
/βy(s),

so

W (t, s) = ξeµ
(

(tn(a,t)−a)β−(sn(a,s)−a)β
)
/β .

Case 5: There are two pulse and one junction points between t and s, i.e. n(a, t) =
n(a, s) + 2. For every t ∈ (tn(a,t), sn(a,t)] and s ∈ (sn(a,s), tn(a,s)+1], we have

y(t) = ξy(t−n(a,t))

= ξeµ
(

(tn(a,t)−a)β−(sn(a,t)−1−a)β
)
/βy(s+

n(a,s)+1)

= ξeµ
(

(tn(a,t)−a)β−(sn(a,t)−1−a)β
)
/βy(s−n(a,s)+1)

= ξeµ
(

(tn(a,t)−a)β−(sn(a,t)−1−a)β
)
/βξy(t−n(a,s)+1)

= ξeµ
(

(tn(a,t)−a)β−(sn(a,t)−1−a)β
)
/βξe

µ
β

(
(tn(a,s)+1−a)β−(s−a)β

)
y(s)

= ξ2 exp
(µ
β
β
(
(tn(a,t) − a)β − (sn(a,t)−1 − a)β

)
+
(
(tn(a,s)+1 − a)β − (sn(a,s) − a)β

)
−
(
(s− a)β − (sn(a,s) − a)β

))
y(s),

so

W (t, s) = ξ2 exp
(µ
β

(
(tn(a,t) − a)β − (sn(a,t)−1 − a)β

)
+
(
(tn(a,s)+1 − a)β − (sn(a,s) − a)β

)
−
(
(s− a)β − (sn(a,s) − a)β

))
.

Case 6: There are one pulse and two junction points between t and s, i.e. n(a, t) =
n(a, s) + 1. For every s ∈ (tn(a,s), sn(a,s)] and t ∈ (sn(a,t), tn(a,t)+1), we have

y(t) = eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βy(s+

n(a,t))

= eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βy(s−n(a,t))

= eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βξy(t−n(a,t))

= eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βξeµ

(
(tn(a,t)−a)β−(sn(a,s)−a)β

)
/βy(s+

(a,t))

= eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βξeµ

(
(tn(a,t)−a)β−(sn(a,s)−a)β

)
/βy(s),

so

W (t, s) = eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βξeµ

(
(tn(a,t)−a)β−(sn(a,t)−1−a)β

)
/β .

Case 7: There are two pulse and two junction points between t and s, i.e. n(a, t) =
n(a, s) + 2.



EJDE-2020/118 CONFORMABLE IMPULSIVE DIFFERENTIAL EQUATIONS 7

(i) For every s ∈ (sn(a,s), tn(a,s)+1] and t ∈ (sn(a,t), tn(a,t)+1), we have

y(t) = eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βy(s+

n(a,t))

= eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βy(s−n(a,t))

= eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βξy(t−n(a,t))

= eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βξeµ

(
(tn(a,t)−a)β−(sn(a,t)−1−a)β

)
/βy(s+

n(a,t)−1)

= eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βξeµ

(
(tn(a,t)−a)β−(sn(a,t)−1−a)β

)
/βy(s−n(a,s)+1)

= eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βξeµ

(
(tn(a,t)−a)β−(sn(a,t)−1−a)β

)
/βξy(t−n(a,s)+1)

= eµ
(

(t−a)β−(sn(a,t)−a)β
)
/βξeµ

(
(tn(a,t)−a)β−(sn(a,t)−1−a)β

)
/β

× ξe
µ
β

(
(tn(a,t)−1−a)β−(s−a)β

)
y(s)

= ξ2 exp
(µ
β

(
(t− a)β − (sn(a,t) − a)β

)
+
(
(tn(a,t) − a)β − (sn(a,t)−1 − a)β

)
+
(
(tn(a,t)−1 − a)β − (s− a)β

))
y(s),

so

W (t, s) = ξ2 exp
(µ
β

(
(t− a)β − (sn(a,t) − a)β

)
+
(
(tn(a,t) − a)β − (sn(a,t)−1 − a)β

)
+
(
(tn(a,t)−1 − a)β − (s− a)β

))
.

(ii) For every s ∈ (tn(a,s), sn(a,s)] and t ∈ (tn(a,t), sn(a,t)], we have

y(t) = ξy(t−n(a,t))

= ξe
µ
β

(
(tn(a,t)−a)β−(sn(a,s)+1−a)β

)
y(s+

n(a,s)+1)

= ξe
µ
β

(
(tn(a,t)−a)β−(sn(a,s)+1−a)β

)
y(s−n(a,s)+1)

= ξe
µ
β

(
(tn(a,t)−a)β−(sn(a,s)+1−a)β

)
ξy(t−n(a,s)+1)

= ξe
µ
β

(
(tn(a,t)−a)β−(sn(a,s)+1−a)β

)
ξe

µ
β

(
(tn(a,s)+1−a)β−(sn(a,s)−a)β

)
y(s)

= ξ2e
µ
β

(
(tn(a,t)−a)β−(sn(a,s)+1−a)β

)
+
(

(tn(a,s)+1−a)β−(sn(a,s)−a)β
)
y(s),

so

W (t, s) = ξ2e
µ
β

(
(tn(a,t)−a)β−(sn(a,s)+1−a)β

)
+
(

(tn(a,s)+1−a)β−(sn(a,s)−a)β
)
.

Case 8: There are several pulse and several junction points between t and s.
(i) For every s ∈ (tn(a,s), sn(a,s)] and t ∈ (sn(a,t), tn(a,t)+1) we have

W (t, s) = ξn(a,t)−n(a,s)e
µ
β

[(
(t−a)β−(sn(a,t)−a)β

)
+
∑n(a,t)−1

k=n(a,s)

(
(tk+1−a)β−(sk−a)β

)]
.

(ii) For every s ∈ (sn(a,s), tn(a,s)+1] and t ∈ (tn(a,t), sn(a,t)], we have

W (t, s) = ξn(a,t)−n(a,s)e
µ
β

[∑n(a,t)−1

k=n(a,s)

(
(tk+1−a)β−(sk−a)β

)
−
(

(s−a)β−(sn(a,s)−a)β
)]
.



8 Y. DING, M. FEČKAN, J. WANG EJDE-2020/118

(iii) For every s ∈ (sn(a,s), tn(a,s)+1] and t ∈ (sn(a,t), tn(a,t)+1], we have

W (t, s) = ξn(a,t)−n(a,s) exp
(µ
β

[(
((t− a)β − (sn(a,t) − a)β)

− ((s− a)β − (sn(a,s) − a)β)
)

+

n(a,t)−1∑
k=n(a,s)

((tk+1 − a)β − (sk − a)β)
])
.

(iv) For every s ∈ (tn(a,s), sn(a,s)] and t ∈ (tn(a,t), sn(a,t)], we have

W (t, s) = ξn(a,t)−n(a,s)e
µ
β

∑n(a,t)−1

k=n(a,s)

(
(tk+1−a)β−(sk−a)β

)
.

Summarizing the 8 cases above, we can write

W (t, s) = ξn(a,t)−n(a,s) exp
(µ
β

[(
((t− a)β − (sn(a,t) − a)β)+

− ((s− a)β − (sn(a,s) − a)β)+
)

+

n(a,t)−1∑
k=n(a,s)

((tk+1 − a)β − (sk − a)β)
])
.

(2.6)

In particular when s = a,

W (t, a) = ξn(a,t)e
µ
β [
(

(t−a)β−(sn(a,t)−a)β
)+

+
∑n(a,t)−1

k=n(a,s)
((tk+1−a)β−(sk−a)β)]

.

The proof is complete. �

Lemma 2.6. A function y ∈ PC(I,R), is a solution of the fractional integral
equations

y(t) =

∫ t

a

(s− a)β−1g(s)ds+ ya, t ∈ (a, t1];

y(t) =

∫ t

sk

(s− a)β−1g(s)ds+ hk(sk), t ∈ (sk, tk+1], k = 1, . . . ,m;

y(t) = hk(t), t ∈ (tk, sk], k = 1, . . . ,m,

if and only if y is a solution of (2.5).

Proof. Assume y is the solution of (2.5). When t ∈ [a, t1], we have

Da
βy(t) = g(t), t ∈ (a, t1] with y(a) = ya. (2.7)

By Definition 2.2 and integrating (2.7), we obtain

y(t) =

∫ t

a

(s− a)β−1g(s)ds+ c.

Obviously, y(a) = ya so c = ya. Therefore

y(t) =

∫ t

a

(s− a)β−1g(s)ds+ ya, t ∈ [a, t1].

Note that when t ∈ (t1, s1], we have y(t) = h1(t). Also when t ∈ (s1,2 ], we have

Da
βy(t) = g(t), t ∈ (s1,2 ] with y(s1) = h1(s1).

Similarly, we have

y(t) =

∫ t

s1

(s− a)β−1g(s)ds+ h1(s1), for t ∈ (s1,2 ].



EJDE-2020/118 CONFORMABLE IMPULSIVE DIFFERENTIAL EQUATIONS 9

When t ∈ (2, s2], we have y(t) = h2(t). Also when t ∈ (s2, t3], we have

Da
βy(t) = g(t), t ∈ (s2, t3] with y(s2) = h2(s2).

So, we obtain

y(t) =

∫ t

s2

(s− a)β−1g(s)ds+ h2(s2), t ∈ (s2, t3].

Summarizing,

Da
βy(t) = g(t), t ∈ (sk, tk+1] with y(sk) = hk(sk).

Then

y(t) =

∫ t

sk

(s− a)β−1g(s)ds+ hk(sk), t ∈ (tk, sk].

The remaining proofs can be done by continuing the standard steps and then verify
the conclusions. �

Lemma 2.7 (see [10]). Suppose that (Y, d) is a complete metric space, and that
W : Y → Y is a strictly contractive operator with constant L < 1. If there exists a
nonnegative integer k such that d(W k+1y,W ky) <∞ for some y ∈ Y , then:

(i) The sequence {Wny} converges to a fixed point y∗ in W ;
(ii) y∗ is the unique fixed point of W in Y ∗ = {x ∈ Y : d(W ky, x) <∞};
(iii) If x ∈ Y ∗, then d(x, y∗) ≤ 1

1−Ld(Wx, x).

3. Asymptotic stability for the linear problem

Definition 3.1. The solution y(t) of (1.1) is locally asymptotically stable if there
exists δ > 0 such that for any xa ∈ R with |ya − xa| < δ, it holds

lim
t→∞

|y(t, a, ya)− y(t, a, xa)| = 0.

If δ is arbitrary, then y(t) is globally asymptotically stable.

For the next theorem we assume that sk and tk+1 satisfy

η1 ≤
(tk+1 − a)β

β
− (sk − a)β

β
≤ η2, k = 0, 1, 2, . . . ,m (3.1)

and define

η =

{
η1, µ < 0,

η2, µ ≥ 0.

Theorem 3.2. Assume that (3.1) holds. If

Θ := µ+
1

η
ln ξ < 0, (3.2)

then (1.1) is asymptotically stable.

Proof. From (2.6) and (3.1), we have

|W (t, a)| ≤ eµ
[
(

(t−a)β

β −
(sn(a,t)−a)β

β )++
∑n(a,t)−1
k=0 (

(tk+1−a)β

β − (sk−a)β

β )
]
ξn(a,t)

≤ eµ
[
(

(t−a)β

β −
(sn(a,t)−a)β

β )++n(a,t)η
]
ξn(a,t)

≤ eµη
(
eµηξ

)n(a,t)
.



10 Y. DING, M. FEČKAN, J. WANG EJDE-2020/118

By (3.2), we have

eµηξ ≤ e
ηΘ
2 < 1,

so when t→∞, we have n(a, t)→∞, and then

|W (t, a)| ≤ eµηe
ηΘ
2 n(a,t) → 0, as t→∞.

The proof is complete. �

Theorem 3.3. Assume that λ = µ + ρ ln ξ < 0 and one of the following two
conditions holds: ξ ≥ 1, and

lim sup
t→∞

n(a, t)

( (t−a)β

β − (sn(a,t)−a)β

β )+ +
∑n(a,t)−1
k=0 ( (tk+1−a)β

β − (sk−a)β

β )
:= ρ <∞,

(3.3)
or ξ < 1 and

lim inf
t→∞

n(a, t)

( (t−a)β

β − (sn(a,t)−a)β

β )+ +
∑n(a,t)−1
k=0 ( (tk+1−a)β

β − (sk−a)β

β )
:= ρ <∞, (3.4)

then (1.1) is asymptotically stable.

Proof. By (2.6), we have

|W (t, a)| ≤ eµ
[
(

(t−a)β

β −
(sn(a,t)−a)β

β )++
∑n(a,t)−1
k=0 (

(tk+1−a)β

β − (sk−a)β

β )
]
ξn(a,t).

When ξ ≥ 1, by (3.3), we obtain

n(a, t) < ρ
[
(
(t− a)β

β
−

(sn(a,t) − a)β

β
)+ +

n(a,t)−1∑
k=0

(
(tk+1 − a)β

β
− (sk − a)β

β
)
]
,

for any t large enough. Then

|W (t, a)| ≤ e(µ+ρ ln ξ)
[
(

(t−a)β

β −
(sn(a,t)−a)β

β )++
∑n(a,t)−1
k=0 (

(tk+1−a)β

β − (sk−a)β

β )
]
.

Because µ+ ρ ln ξ < λ/2 < 0 we have

|W (t, a)| ≤ e
λ
2

[
(

(t−a)β

β −
(sn(a,t)−a)β

β )++
∑n(a,t)−1
k=0 (

(tk+1−a)β

β − (sk−a)β

β )
]
→ 0,

as t→∞. Similarly, when ξ < 1, by (3.4), we obtain

n(a, t) > ρ
[
(
(t− a)β

β
−

(sn(a,t) − a)β

β
)+ +

n(a,t)−1∑
k=0

(
(tk+1 − a)β

β
− (sk − a)β

β
)
]
,

for any t large enough. When ξ < 1, we have

|W (t, a)| ≤ e(µ+ρ ln ξ)
[
(

(t−a)β

β −
(sn(a,t)−a)β

β )++
∑n(a,t)−1
k=0 (

(tk+1−a)β

β − (sk−a)β

β )
]
,

and satisfy µ+ ρ ln ξ < λ/2 < 0, so

|W (t, a)| ≤ e
λ
2

[
(

(t−a)β

β −
(sn(a,t)−a)β

β )++
∑n(a,t)−1
k=0 (

(tk+1−a)β

β − (sk−a)β

β )
]
→ 0,

as t→∞. The proof is complete. �



EJDE-2020/118 CONFORMABLE IMPULSIVE DIFFERENTIAL EQUATIONS 11

Note that

W (t, a)

= e
µ
β

[(
(t−a)β−(sn(a,t)−a)β

)+
+
∑n(a,t)−1
k=0 ((tk+1−a)β−(sk−a)β)

]
ξn(a,t)

= e
λ
β

[(
(t−a)β−(sn(a,t)−a)β

)+
+
∑n(a,t)−1
k=0 ((tk+1−a)β−(sk−a)β)

]
× eln ξ

(
n(a,t)− ρβ

[(
(t−a)β−(sn(a,t)−a)β

)+
+
∑n(a,t)−1
k=0 ((tk+1−a)β−(sk−a)β)

])
.

(3.5)

Next we discuss the condition on λ = µ+ ρ ln ξ directly.

Theorem 3.4. Assume that

lim
t→∞

n(a, t)

( (t−a)β

β − (sn(a,t)−a)β

β )+ +
∑n(a,t)−1
k=0 ( (tk+1−a)β

β − (sk−a)β

β )
:= ρ <∞. (3.6)

Then:

(i) If λ < 0, then (1.1) is asymptotically stable.
(ii) If λ > 0, then (1.1) is unstable.

Proof. (i) Since λ < 0, there exists ζ1 such that

|eλt| ≤ e−ζ1t, t ≥ 0, (3.7)

in which ζ1 = −λ/2.
By (3.6), there exist ω1 > 0 such that for any t ≥ ω1, we have∣∣∣ n(a, t)

( (t−a)β

β − (sn(a,t)−a)β

β )+ +
∑n(a,t)−1
k=0 ( (tk+1−a)β

β − (sk−a)β

β )
− ρ
∣∣∣ ≤ ζ1

2| ln ξ|
.

Then∣∣∣eln ξ
(
n(a,t)−ρ

[(
(t−a)β

β −
(sn(a,t)−a)β

β

)+
+
∑n(a,t)−1
k=0 (

(tk+1−a)β

β − (sk−a)β

β

)])∣∣∣
≤ e| ln ξ|

∣∣n(a,t)−ρ
[(

(t−a)β

β −
(sn(a,t)−a)β

β

)+
+
∑n(a,t)−1
k=0 (

(tk+1−a)β

β − (sk−a)β

β

)])∣∣
≤ e

ζ1
2

[(
(t−a)β

β −
(sn(a,t)−a)β

β

)+
+
∑n(a,t)−1
k=0 (

(tk+1−a)β

β − (sk−a)β

β

)]
.

(3.8)

Substituting (3.7) and (3.8) into (3.5), we obtain

|W (t, a)| ≤ e−
ζ1
2

[
(

(t−a)β

β −
(sn(a,t)−a)β

β )++
∑n(a,t)−1
k=0 (

(tk+1−a)β

β − (sk−a)β

β )
]
→ 0,

as t→∞. Thus (i) is proved.
(ii) We rewrite (3.5) as

W (t, a)e− ln ξ
(
n(a,t)− ρβ

[(
(t−a)β−(sn(a,t)−a)β

)+
+
∑n(a,t)−1
k=0 ((tk+1−a)β−(sk−a)β)

])
= e

λ
β

[(
(t−a)β−(sn(a,t)−a)β

)+
+
∑n(a,t)−1
k=0 ((tk+1−a)β−(sk−a)β)

]
.

(3.9)

Since λ > 0, there exists ζ2 and y0 ∈ Rn such that

|eλtya| ≥ eζ2t, t ≥ 0, (3.10)

in which ζ2 = λ/2.



12 Y. DING, M. FEČKAN, J. WANG EJDE-2020/118

By (3.6), there exist ω2 > 0 such that for any t > ω2, we obtain∣∣∣e− ln ξ
(
n(a,t)− ρβ

[(
(t−a)β−(sn(a,t)−a)β

)+
+
∑n(a,t)−1
k=0 ((tk+1−a)β−(sk−a)β)

])∣∣∣
≤ e

ζ2
2

[(
(t−a)β

β −
(sn(a,t)−a)β

β

)+
+
∑n(a,t)−1
k=0 (

(tk+1−a)β

β − (sk−a)β

β

)]
.

(3.11)

Substituting (3.10) and (3.11) into (3.9), we obtain

eζ2
[
(

(t−a)β

β −
(sn(a,t)−a)β

β )++
∑n(a,t)−1
k=0 (

(tk+1−a)β

β − (sk−a)β

β )
]

≤
∣∣∣eλ[( (t−a)β

β −
(sn(a,t)−a)β

β )++
∑n(a,t)−1
k=0 (

(tk+1−a)β

β − (sk−a)β

β )
]
y0

∣∣∣
≤ |W (t, a)ya|

∣∣∣ exp
(
− ln ξ

(
n(a, t)− ρ

β

[(
(t− a)β − (sn(a,t) − a)β

)+
+

n(a,t)−1∑
k=0

((tk+1 − a)β − (sk − a)β)
]))∣∣∣

≤ |W (t, a)ya|e
ζ2
2

[(
(t−a)β

β −
(sn(a,t)−a)β

β

)+
+
∑n(a,t)−1
k=0 (

(tk+1−a)β

β − (sk−a)β

β

)]
,

so

|W (t, a)ya| ≥ e
ζ2
2

[
(

(t−a)β

β −
(sn(a,t)−a)β

β )++
∑n(a,t)−1
k=0 (

(tk+1−a)β

β − (sk−a)β

β )
]
→∞,

as t→∞. The proof is complete. �

4. Generalized Ulam-Hyers-Rassias stability for the nonlinear
problem

We introduce the concept of generalized Ulam-Hyers-Rassias stability through
the concept of stability in [21, 31].

Let ε > 0, ψ ≥ 0 and φ ∈ PC(I,R+) be nondecreasing, in the conditions

|Da
βx(t)− g(t, x(t))| ≤ φ(t), t ∈ (sk, tk+1], k = 0, 1, 2, . . . ,m, 0 < β < 1,

|x(t)− hk(t, x(t))| ≤ ϕ, t ∈ (tk, sk], k = 1, 2, . . . ,m.
(4.1)

Definition 4.1. Equation (1.2) has generalized Ulam-Hyers-Rassias stability if
there exists cg,β,hk,φ > 0 such that for each solution x ∈ PC(I,R) of inequality
(4.1), there exists a solution y ∈ PC(I,R) of (1.2) with

|x(t)− y(t)| ≤ cg,β,hk,φ(φ(t) + ϕ), t ∈ I.

When ε = 1, the generalized Ulam-Hyers-Rassias stability reduces to the classical
Ulam-Hyers-Rassias stability, see [27, Remark 3.5].

Remark 4.2. A function x ∈ PC(I,R) is a solution of (4.1) if and only if there
exists H ∈ PC(I,R) and a sequence Hk, k = 1, 2, . . . ,m which depends on x such
that

(i) |H(t)| ≤ φ(t) for t ∈ I, and |Hk| ≤ ϕ for k = 1, 2, . . . ,m;
(ii) Da

βx(t) = g(t, x(t)) +H(t) for t ∈ (sk, tk+1], k = 0, 1, 2, . . . ,m;

(iii) x(t) = hk(t, x(t)) +Hk, t ∈ (sk−1, tk], k = 1, 2, . . . ,m.



EJDE-2020/118 CONFORMABLE IMPULSIVE DIFFERENTIAL EQUATIONS 13

Remark 4.3. If x ∈ PC(I,R) is the solution of (4.1) then x satisfies the following
integral inequalities:

|x(t)− hk(t, x(t))| ≤ ϕ, t ∈ (tk, sk], k = 1, 2, . . . ,m,

|x(t)− x(a)−
∫ t

a

(s− a)β−1g(s, x(s)ds| ≤
∫ t

a

(s− a)β−1φ(s)ds, t ∈ (a, t1],

|x(t)−
∫ t

sk

(s− a)β−1g(s, x(s))ds− hk(sk, x(sk))| ≤
∫ t

(s− a)β−1φ(s)ds+ ϕ,

t ∈ (sk, tk+1], k = 1, 2, . . . ,m.

(4.2)

By Remark 4.2 (i), we have

Da
βx(t) = g(t, x(t)) +H(t), t ∈ (sk, tk+1], k = 0, 1, 2, . . . ,m,

x(t) = hk(t, x(t)) +Hk, t ∈ (tk, sk], k = 1, 2, . . . ,m.
(4.3)

Obviously,

x(t) = hk(t, x(t)) +Hk, t ∈ (tk, sk], k = 1, 2, . . . ,m,

x(t) =

∫ t

a

(s− a)β−1
(
g(s, x(s)) +H(s)

)
ds+ ya, t ∈ (a, t1],

x(t) =

∫ t

sk

(s− a)β−1
(
g(s, x(s)) +H(s)

)
ds+ hk(sk, x(sk)) +Hk,

t ∈ (tk, sk], k = 1, 2, . . . ,m

is the solution of (4.3). For t ∈ (sk, tk+1], k = 1, 2, . . . ,m, we have∣∣∣x(t)−
∫ t

sk

(s− a)β−1g(s, x(s))ds− hk(sk, x(sk))
∣∣∣

≤
∣∣∣ ∫ t

sk

(s− a)β−1H(s)ds
∣∣∣+ |Hk|

≤
∫ t

sk

(s− a)β−1φ(s)ds+ ϕ.

As mentioned above, we can obtain

|x(t)− hk(t, x(t))| ≤ |Hk| ≤ ϕ, t ∈ (tk, sk], k = 1, 2, . . . ,m,

and∣∣∣x(t)− x(a)−
∫ t

a

(s− a)β−1g(s, x(s))ds
∣∣∣ ≤ ∣∣∣ ∫ t

a

(s− a)β−1H(s)ds
∣∣∣

≤
∫ t

a

(s− a)β−1φ(s)ds, t ∈ (a, t1].

For using a fixed point theorem of the alternative and for deriving our main
result, which is about contractions on a complete metric space, we consider the
following assumptions:

(H1) g ∈ C(I × R,R).
(H2) There exists a positive constant Lg such that

|g(t, v1)− g(t, v2)| ≤ Lg|v1 − v2|,
for each t ∈ I and all v1, v2 ∈ R.



14 Y. DING, M. FEČKAN, J. WANG EJDE-2020/118

(H3) hk ∈ C([tk, sk]×R,R) and there are positive constants Lhk , k = 1, 2, . . . ,m
such that

|gk(t, v1)− gk(t, v2)| ≤ Lhk |v1 − v2|,
for each t ∈ [tk, sk] and all v1, v2 ∈ R.

(H4) φ ∈ C(I,R+) is a nondecreasing function, and there exists cφ > 0 such that(∫ t

a

(
φ(s)

)1/p
ds
)p
≤ cφφ(t), p ∈ (0, 1), for each t ∈ I.

We use the concept of generalized Ulam-Hyers-Rassias to show stability of (1.2) in
the following section.

Theorem 4.4. Assume that (H1)–(H4) are satisfied and a function x ∈ PC(I,R)
that satisfies (4.1). Then there exists a unique solution x0 of (1.2) such that

x0(t) =

∫ t

a

(s− a)β−1g(s, x0(s))ds+ ya, t ∈ [a, t1],

x0(t) = hk(t, x0(t)), t ∈ (tk, sk], k = 1, 2, . . . ,m,

x0(t) =

∫ t

sk

(s− a)β−1g(s, x0(s))ds+ hk(sk, x0(sk)),

t ∈ (sk, tk+1], k = 1, 2, . . . ,m,

(4.4)

and

|x(t)− x0(t)| ≤
(
2cφ
(

1−p
β−p

)1−p
bβ−p + 1

)
(φ(t) + ϕ)

1−M
, (4.5)

for all t ∈ I provided that 0 < p < β < 1 and

M = M1 < 1, (4.6)

where

M1 = max{Lgcφ
( 1− p
β − p

)1−p
tβ−pk+1 + Lhk : k = 0, 1, 2, . . . ,m}.

Proof. Consider the space of piecewise continuous functions Y = {f : I → R : f ∈
PC(I,R)}, and the generalized metric

d(f, h) = inf
{
A1+A2 ∈ [0,+∞] : |f(t)−h(t)| ≤ (A1+A2)(φ(t)+ϕ) ∀t ∈ I

}
, (4.7)

where

A1 ∈ {A ∈ [0,+∞]
∣∣|f(t)− h(t)| ≤ Aφ(t) for all t ∈ (sk, tk+1], k = 0, 1, 2, . . . ,m},

A2 ∈ {A ∈ [0,+∞]
∣∣|f(t)− h(t)| ≤ Aϕ for all t ∈ (tk, sk], k = 1, 2, . . . ,m}.

This is a generalized metric in the sense that it can have value +∞. For the
necessity of introducing such a generalized metric and applications, we refer to [10].
One can easily show that (Y, d) is a complete generalized metric space.

We define an operator Υ : Y → Y by

(Υy)(t) =



∫ t
a
(s− a)β−1g(s, y(s))ds+ ya if t ∈ [a, t1],

hk(t, y(t)) if t ∈ (tk, sk], k = 1, 2, . . . ,m,∫ t
sk

(s− a)β−1g(s, y(s))ds+ hk(sk, y(sk))

if t ∈ (sk, tk+1], k = 1, 2, . . . ,m,

(4.8)

for all y ∈ Y and t ∈ [a, b]. Obviously, Υ is a well defined operator by (H1).



EJDE-2020/118 CONFORMABLE IMPULSIVE DIFFERENTIAL EQUATIONS 15

Next, we verify that Υ is strictly contractive. We considering the definition of
(Y, d), for any f, h ∈ Y , we find a A1, A2 ∈ [0,∞] such that

|f(t)− h(t)| ≤

{
A1φ(t), t ∈ (sk, tk+1], k = 0, 1, 2, . . . ,m,

A2ϕ, t ∈ (tk, sk], k = 1, 2, . . . ,m.
(4.9)

By the definition of Υ in (4.8), (H2), (H3) and (4.9), we obtain the following three
cases:

Case 1: For t ∈ [a, t1] we have

|(Υf)(t)− (Υh)(t)| =
∣∣∣ ∫ t

a

(s− a)β−1g(s, f(s))ds−
∫ t

a

(s− a)β−1g(s, h(s))ds
∣∣∣

≤
∫ t

a

(s− a)β−1|g(s, f(s))− g(s, h(s))|ds

≤ Lg
∫ t

a

(s− a)β−1|f(s)− h(s)|ds

≤ LgA1

∫ t

a

(s− a)β−1|φ(s)|ds

≤ LgA1

(∫ t

a

(s− a)
β−1
1−p ds

)1−p(∫ t

a

(
φ(s)

)1/p
ds
)p

≤ LgA1cφφ(t)
( 1− p
β − p

)1−p
tβ−p

≤ Lgcφ
( 1− p
β − p

)1−p
tβ−p1 A1φ(t).

Case 2: For t ∈ (tk, sk] we have

|(Υf)(t)− (Υh)(t)| = |hk(t, f(t))− hk(t, h(t))|
≤ Lhk |f(t)− h(t)| ≤ LhkA2ϕ.

Case 3: For t ∈ (sk, tk+1] we have

|(Υf)(t)− (Υh)(t)|

=
∣∣∣ ∫ t

sk

(s− a)β−1g(s, f(s))ds+ hk(sk, f(sk))

−
∫ t

sk

(s− a)β−1g(s, h(s))ds− hk(sk, h(sk))
∣∣∣

≤
∣∣∣ ∫ t

sk

(s− a)β−1g(s, f(s))ds−
∫ t

sk

(s− a)β−1g(s, h(s))ds
∣∣∣

+
∣∣hk(sk, f(sk))− hk(sk, h(sk))

∣∣
≤ Lgcφ

( 1− p
β − p

)1−p
tβ−pk+1A1φ(t) + LhkA2ϕ

≤
(
Lgcφ

( 1− p
β − p

)1−p
tβ−pk+1 + Lhk

)
(A1 +A2)(φ(t) + ϕ).



16 Y. DING, M. FEČKAN, J. WANG EJDE-2020/118

In this case we have

|(Υf)(t)− (Υh)(t)| ≤M(A1 +A2)(φ(t) + ψ), t ∈ I.

Then d(Υf,Υh) ≤M(A1 +A2). Therefore,

d(Υf,Υh) ≤Md(f, h),

for any f, h ∈ Y , and because of (4.6), we verify the strictly continuous property.
Let us take f0 ∈ Y . By the piecewise continuous property of f0 and Υf0, there

exists a constant 0 < F1 <∞ such that

|(Υf0)(t)− f0(t)| =
∣∣∣ ∫ t

a

(s− a)β−1g(s, f0(s))ds+ ya − f0(t)
∣∣∣

≤ F1φ(t) ≤ F1(φ(t) + ϕ), t ∈ [a, t1].

Then there exists a constant 0 < F2 <∞ such that

|(Υf0)(t)− f0(t)| =
∣∣hk(t, f0(t))− f0(t)| ≤ F2ϕ ≤ F2(φ(t) + ϕ),

for t ∈ (tk, sk] and k = 1, 2, . . . ,m. Also there exists a constant 0 < F3 < ∞ such
that

|(Υf0)(t)− f0(t)| =
∣∣∣ ∫ t

sk

(s− a)β−1g(s, f0)ds+ hk(sk, f0(sk))− f0(t)
∣∣∣

≤ F3(φ(t) + ϕ), t ∈ (sk, tk+1], k = 1, 2, . . . ,m.

because g, hk, f0 <∞ are bounded on I and φ(·) + ϕ > 0. So (4.7) implies that

d(Υf0, f0) <∞.

Using the Banach fixed point theorem, we obtain a continuous function x0 : I → R
such that Υn(f0)→ x0 in (Y, d) as n→∞ and Υx0 → x0, and for every t ∈ I, x0

satisfies (4.4).
Next, we verify that {f ∈ Y |d(f0, f) < ∞} = Y . For any f ∈ Y , because f0,

f are bounded on I and mint∈I(φ(t) + ϕ) > 0, there is a constant 0 < Af < ∞
such that |f0(t)− f(t)| ≤ Af (φ(t) + ϕ), for any t ∈ I. So we have d(f0, f) <∞ for
any f ∈ Y ; that is, {f ∈ Y |d(f0, f) <∞} = Y . Therefore, we know that x0 is the
unique continuous function and it has the property (4.4). From (4.2) and (H4), we
have

d(x,Υx) ≤ 2cφ

( 1− p
β − p

)1−p
bβ−p + 1,

In summary, we have

d(x, x0) ≤ d(Υy, y)

1−M
≤

2cφ
(

1−p
β−p

)1−p
bβ−p + 1

1−M
,

so (4.5) holds for t ∈ I. The proof is complete. �

5. Examples

To illustrate our results we present the following examples.



EJDE-2020/118 CONFORMABLE IMPULSIVE DIFFERENTIAL EQUATIONS 17

Example 5.1. Consider the conformable linear non-instantaneous impulsive dif-
ferential equations

Da
βy(t) = υy(t), t ∈ (sk, tk+1], k = 0, 1, 2, . . . ,m,

y(t+k ) = νy(t−k ) on (tk, sk], k = 1, 2, . . . ,m,

y(t) = νy(t−k ), t ∈ (tk, sk], k = 1, 2, . . . ,m,

y(s+
k ) = y(s−k ), k = 1, 2, . . . ,m.

(5.1)

Let ta = sa = 1 and (tk+1−a)β

β − (sk−a)β

β = 1 for k = 0, 1, 2, 3, . . . ,m. Then η = 1.

Note that

n(a, t)

n(a, t) + 1
=

n(a, t)∑n(a,t)
k=0

( (tk+1−a)β

β − (sk−a)β

β

)
≤ n(a, t)( (t−a)β

β − (sn(a,t)−a)β

β

)+
+
∑n(a,t)−1
k=0

( (tk+1−a)β

β − (sk−a)β

β

)
≤ n(a, t)∑n(a,t)−1

k=0

( (tk+1−a)β

β − (sk−a)β

β

) = 1,

because n(a, t) > 1. Then

ρ = lim
t→∞

n(a, t)

( (t−a)β

β − (sn(a,t)−a)β

β )+ +
∑n(a,t)−1
k=0 ( (tk+1−a)β

β − (sk−a)β

β )
= 1.

Next, λ = υ+ ln ν. By Theorem 3.4, we know that if υ < − ln ν, (5.1) is asymptot-
ically stable. Also if υ > − ln ν, (5.1) is unstable.

Example 5.2. Consider

D0
βy(t) =

|y(t)|
10 + 4t2 + 10et

, t ∈ (0, 1] ∪ (2, 3],

y(t) =
t

6
e−y(t), t ∈ (1, 2],

and ∣∣D0
βx(t)− |x(t)|

10 + 4t2 + 10et
∣∣ ≤ et, t ∈ [0, 1] ∪ (2, 3],∣∣x(t)− t

6
e−x(t)

∣∣ ≤ 1, t ∈ (1, 2].

Let I = [0, 3], β = 1/2, p = 1/3 and 0 = t0 = s0 < 1 = t1 < 2 = s1 <2= 3. Denote

g(t, y(t)) = |y(t)|
10+4t2+10et with Lg = 1

20 , for t ∈ (0, 1]∪ (2, 3] and h1(t, y(t)) = t
6e
−y(t)

with Lh1 = 1
3 for t ∈ (1, 2]. Putting φ(t) = et, ϕ = 1 and cφ = 1, we have( ∫ t

0
(et)3ds

)1/3 ≤ et. Let M1 = { 1
2042/331/6 + 1

3} = 0.4846, so M = 0.4846 < 1. By
Theorem 4.4, there exists a unique solution x0 : [0, 3]→ R such that

x0(t) =


∫ t

0
s−1/2 |x0(s)|

10+4s2+10es ds+ y0, t ∈ [0, 1],

t
6e
−x0(t), t ∈ (1, 2],

∈t2 s−1/2 |x0(s)|
10+4s2+10es ds+ 2

6e
−x0(2), t ∈ (2, 3],

and

|x(t)− x0(t)| ≤ 2× 42/3 × 31/6 + 1

1− 0.5
(et + 1) ≈ 14.1047(et + 1),



18 Y. DING, M. FEČKAN, J. WANG EJDE-2020/118

for all t ∈ [0, 3].

Conclusion. This article gives elementary results for linear and nonlinear non-
instantaneous conformable impulsive differential equations keeping the lower limit
at a fixed point a. Representation of solutions and asymptotical stability for linear
problems are established. The generalized Ulam-Hyers-Rassias stability for nonlin-
ear problems are also derived. In a forthcoming paper, we can extend the current
results to higher dimension case based on Remark 2.4. Note there is no nonconstant
periodic solution for (2.5). We can consider (2.5) replacing a by sk, i.e., in each
impulse starting at impulsive time. Then when impulses and nonlinear terms are
periodic, existence of periodic solutions will be possible by following the idea in
[11].

Acknowledgments. This work was partially supported by the National Natural
Science Foundation of China (grant 11661016), Training Object of High Level and
Innovative Talents of Guizhou Province ((2016)4006), Major Research Project of
Innovative Group in Guizhou Education Department ([2018]012), Guizhou Data
Driven Modeling Learning and Optimization Innovation Team ([2020]5016), by the
Slovak Research and Development Agency under the contract No. APVV-18-0308,
and by the Slovak Grant Agency VEGA Nos. 1/0358/20 and 2/0127/20.

References

[1] T. Abdeljawad; On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015),

57-66.

[2] T. Abdeljawad, Q. M. Al-Mdallal, F. Jarad; Fractional logistic models in the frame of frac-
tional operators generated by conformable derivatives, Chaos Solitons Frac., 119 (2019), 94-

101.
[3] B. Ahmad, M. Alghanmi, A. Alsaedi, R. P. Agarwal; On an impulsive hybrid system of

conformable fractional differential equations with boundary conditions, International Journal

of Systems Science, 51 (2020), 958-970.
[4] K. Alper; Explicit exact solutions to some one-dimensional conformable time fractional equa-

tions, Waves in Random and Complex Media, 29 (2019), 124-137.

[5] J. Andres; Coexistence of periodic solutions with various periods of impulsive differential
equations and inclusions on tori via Poincare operators, Topology and its applications, 255

(2019), 126-140.
[6] V. O. Bivziuk, V. I. Slyn’ko; Sufficient conditions for the stability of linear periodic impulsive

differential equations, 210 (2019), 1511-1530.

[7] M. Bohner, V. F. Hatipoǧlu; Dynamic cobweb models with conformable fractional derivatives,

Nonlinear Analysis: Hybrid Systems, 3 2(2019), 157-167.
[8] G. E. Chatzarakis, K. Logaarasi, T. Raja, V. Sadhasivam; Interval oscillation criteria for

impulsive conformable partial differential equations, Applicable Analysis and Discrete Math-
ematics, 13 (2019), 325-345.

[9] W. Chung; Fractional Newton mechanics with conformable fractional derivative, J. Comput.

Appl. Math., 290 (2015), 150-158.
[10] J. B. Diaz, B. Margolis; A fixed point theorem of the alternative, for contractions on a

generalized complete metric space, Bull. Am. Math. Soc., 74 (1968), 305-309.

[11] M. Fečkan, J. Wang; Periodic impulsive fractional differential equations, Adv. Nonlinear
Anal., 8 (2019), 482-496.

[12] J. Henderson, A. Ouahab, S. Youcefi; Existence results for phi-Laplacian impulsive differen-

tial equations with periodic conditions, Aims Mathmatics, 4 (2019), 1640-1633.
[13] E. Hernández, D. O’Regan; On a new class of abstract impulsive differential equations, Proc.

Am. Math. Soc., 141 (2013), 1641-1649.

[14] R. Khalil, M. Al. Horani, A. Yousef, M. Sababheh; A new definition of fractional derivative,
J. Comput. Appl. Math., 264 (2014), 65-70.



EJDE-2020/118 CONFORMABLE IMPULSIVE DIFFERENTIAL EQUATIONS 19

[15] M. Li, J. Wang, D. O’Regan; Existence and Ulam’s stability for conformable fractional differ-

ential equations with constant coefficients, Bull. Malay. Math. Sci. Soc., 40 (2019), 1791-1812.

[16] S. Liu, J. Wang, Y. Zhou; Optimal control of noninstantaneous impulsive differential equa-
tions, Journal of the Franklin Institute, 354 (2017), 7668-7698.

[17] A. Meraj, D. N. Pandey; Approximate controllability of non-autonomous Sobolev type integro-

differential equations having nonlocal and non-instantaneous impulsive conditions, Indian
Journal of Pure and Applied Mathematics, 51 (2020), 501-518.

[18] O. Ozan, K. Ali; Exact solutions of fractional partial differential equation systems with con-

formable derivative, Filomat, 33 (2019), 1313-1322.
[19] W. Qiu, J. Wang, D. O’Regan; Existence and Ulam stability of solutions for conformable im-

pulsive differential equations, Bull. Iran. Math. Soc., (2020), https://doi.org/10.1007/s41980-

019-00347-8.
[20] J. Rosales-Garćıa, J. A. Andrade-Lucio, O. Shulika; Conformable derivative applied to ex-

perimental Newton’s law of cooling, Revista Mexicana de F́ısica, 66 (2020), 224-227.
[21] I. A. Rus; Ulam stability of ordinary differential equations, Stud. Univ. Babeş Bolyai Math.,

54 (2009), 125-133.

[22] M. Z. Sarıkaya, F. Usta; On comparison theorems for conformable fractional differential
equations, International Journal of Analysis and Applications, 12 (2016), 207-214.

[23] B. Sundaravadivoo; Controllability analysis of nonlinear fractional order differential systems

with state delay and non-instantaneous impulsive effects, Discrete and Continuous Dynamical
Systems-S, 13 (2020), 2561-2573.

[24] K. Tadeusz; Analysis of positive linear continuous-time systems using the conformable de-

rivative, 28 (2018), 335-340.
[25] F. Usta; A conformable calculus of radial basis functions and its applications, An Interna-

tional Journal of Optimization and Control: Theories & Applications (IJOCTA), 8 (2018),

176-182.
[26] F. Usta, M. Z. Sarıkaya; The analytical solution of Van der Pol and Lienard differential equa-

tions within conformable fractional operator by retarded integral inequalities, Demonstratio
Mathematica, 52 (2019), 204-212.

[27] J. Wang, M. Fečkan; A general class of impulsive evolution equations, Topol Methods Non-

linear Anal., 46 (2015), 915-934.
[28] J. Wang, M. Fečkan, Y. Tian; Stability analysis for a general class of non-instantaneous

impulsive differential equations, Mediterranean Journal of Mathematics, 14 (2017), Art. 46.

[29] J. Wang, M. Fečkan, Y. Zhou; A survey on impulsive fractional differential equations, Fract.
Calc. Appl. Anal., 199 (2016), 806-831.

[30] J. Wang, M. Fečkan, Y. Zhou; Fractional order differential switched systems with coupled

nonlocal initial and impulsive conditions, Bull. Sci. Math., 141 (2017), 727-746.
[31] J. Wang, M. Fečkan, Y. Zhou; Ulam’s type stability of impulsive ordinary differential equa-

tions, J. Math. Anal. Appl., 395 (2012), 258-264.

[32] D. Yang, J. Wang, D. O’Regan; On the orbital Hausdorff dependence of differential equations
with non-instantaneous impulses, Comptes Rendus Mathematique, 356 (2018), 150-171.

Yuanlin Ding
Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China

Email address: yldingmath@126.com

Michal Fečkan

Department of Mathematical Analysis and Numerical Mathematics, Faculty of Math-
ematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina,

842 48 Bratislava, Slovakia.
Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava,

Slovakia
Email address: Michal.Feckan@fmph.uniba.sk

Jinrong Wang (corresponding author)

Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China.
School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165,

China

Email address: wjr9668@126.com


	1. Introduction
	2. Preliminaries
	3. Asymptotic stability for the linear problem
	4. Generalized Ulam-Hyers-Rassias stability for the nonlinear problem
	5. Examples
	Conclusion
	Acknowledgments

	References