Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 07, pp. 1–18.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2024.07

EXISTENCE AND STABILIZATION FOR IMPULSIVE

DIFFERENTIAL EQUATIONS OF SECOND ORDER WITH

MULTIPLE DELAYS

SANDRA PINELAS, OSMAN TUNÇ, ERDAL KORKMAZ, CEMIL TUNÇ

Abstract. Existence and stability of solutions are important parts in the
qualitative study of delay differential equations. The stabilizing by impos-

ing proper impulse controls are used in many areas of natural sciences and

engineering. This article provides sufficient conditions for the existence and
exponential stabilization of solutions to delay impulsive differential equations

of second-order with multiple delays. The main tools in this article are the

Schaefer fixed point theorem, fixed impulse effects, and Lyapunov-Krasovskii
functionals. The outcomes extend earlier results in the literature.

1. Introduction

Impulsive differential equations have become a very important part in the study
of differential equations with and without delays. This includes ordinary and par-
tial differential equations, functional equations, integro-differential differential, etc.
Various mathematical models of impulsive differential equations appear in real
world applications, and they are very effective in modeling these problems. For
applied impulsive mathematical models, see the book by Stamova and Stamov [34].
For recent advances in stability and control of impulsive delay systems, see the book
by Li and Song [25]. For optimal impulsive control in cancer therapy medicine, see
the book by Belfo and Lemos [4]. For periodic solutions and applications, see the
article by Li et al. [20]. For applications of switched and impulsive systems, see the
book by Li et al. [26]. For results on stability and nonlinear dynamics of high-order
delayed cellular neural systems, see Huang et al. [13, 14] and Zhao et al. [51].

In the previous decades, significant progress has been made in the qualitative
theory of impulsive differential equations. This progress includes areas such as net-
works, neutral-type functional differential equations, global asymptotic stability of
periodic patch-constructed Nicholson’s blowflies systems with time varying delays,
etc. See the various works and interesting results mention in our references and the
references therein.

An important fact about stability is that the effect of impulses can cause stable
systems to become unstable, and inversely, unstable systems to become stable. The

2020 Mathematics Subject Classification. 34A37, 34K45.
Key words and phrases. Impulsive delay differential equations, exponentially stabilization,

existence of solutions; multiple delays.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted August 23, 2023. Published January 23, 2024.

1



2 S. PINELAS, O. TUNÇ, E. KORKMAZ, C. TUNÇ EJDE-2024/07

problem of stabilizing ordinary differential equations (ODEs) by imposing impulse
controls can be used in several fields such as biotechnology, chemical technology,
economics, medicine, physics, and population dynamics. From the mathematical
point of view, also partial differential equations (PDEs) with delay can be stabilized
by adding impulsive effects. For this, we mention the recent papers by Columbu et
al. [7] and Li et al. [17, 18, 19].

Now we want to mention some papers that serve as source for the stabilization
problems we study in this article.

In 2006, Gimenes and Federson [9] considered the impulsive delay differential
equations

x′′ +

N∑
i=1

ai(t)xi(t− τi) + f(x, x′) = 0, t ≥ t0, t 6= tk (1.1)

and

x′′ +

N∑
i=1

∫ t

t−τi
bi(t− u)x(u)du+ f(x, x′) = 0, t ≥ t0, t 6= tk, k ∈ N, (1.2)

with suitable initial data and impulsive controls at t = tk; see equation (2.1) below.
Gimenes and Federson [9] derived sufficient conditions for the existence of solu-

tions on a closed time interval. The authors also show that the non-impulsive forms
of (1.1) and (1.2) can be stabilized via impulse controls.

Later, in 2007, Gimenes et al. [10] dealt with the equations

x′′ + f(t, x, x′) + g(t, x, x(t− τ)) = 0, t ≥ t0, t 6= tk, (1.3)

x′′ +

N∑
i=1

ai(t)x(t− τi) + f(t, x, x′) = 0, t ≥ t0, t 6= tk (1.4)

and

x′′ +

N∑
i=1

∫ t

t−τi
bi(t− u)x(u)du+ f(t, x, x′) = 0,

t ≥ t0, t 6= tk, k = 1, 2, . . . ,

(1.5)

with suitable initial data and the impulsive controls at t = tk; see equation (2.1)
below.

Gimenes et al. [10] proved that non-impulsive forms of (1.3), (1.4), and (1.5) can
be stabilized via impulsive controls. The basic tools utilized in the proofs of their
results are Lyapunov functions, stability theory, and control by impulses.

Recently, Tunç et al. [43] dealt with the equations

x′′ + c(t)f(x, x′) + b(t)g(x) +

N∑
i=1

ai(t)hi(x(t− τi)) + p(x, x′) = 0,

t ≥ t0, t 6= tk

(1.6)

and

x′′ + c(t)f(x, x′) + b(t)g(x) +

N∑
i=1

∫ t

t−τi
di(t− u)h(x(u))du+ q(x, x′) = 0,

t ≥ t0, t 6= tk, k ∈ N,
(1.7)

with suitable the initial data and the impulsive controls at t = tk. The authors
obtained results on the existence of solutions to (1.6) and (1.7).



EJDE-2024/07 EXISTENCE AND STABILIZATION 3

In recent years, various qualitative results about impulsive differential equations
have also been obtained. See, Graef and Tunç [12], Pinelas and Tunç [38], Tunç
[39], Tunç and Tunç [36, 37, 41, 38], Tunç et al. [40, 42, 44], and their references.
In the above publications the main tools are the Lyapunov-Razumikhin technique,
fixed point theorems, direct method of Lyapunov, and impulsive perturbations. In
particular, see Benchohra et al. [5], Smart [32], Tunç and Tunç [36], Xie [45].

In this article, C([a, b],R) denotes the Banach space of continuous functions,
endowed with the usual supremum norm; k ∈ N = {1, 2, 3, . . . }; i = 1, 2, . . . , N ; x
denotes x(t); x′ denotes x′(t).

Motivated by the articles of Gimenes and Federson [9, Theorem 3.1 and 3.2],
Gimenes et al. [10], and Tunç et al. [43], we study the impulsive initial-value problem

x′′ +

N∑
i=1

ai(t)Fi(t, x, x
′, x(t− τi), x′(t− τi)) +G(t, x, x′) + b(t)H(x)

+

N∑
i=1

Gi(t, x, x(t− τi)) = 0, t ≥ t0, t 6= tk,

x(t0) = ψ(t), t0 − τN ≤ t ≤ t0, x′(t0) = y0.

(1.8)

Letting x′(t) = y(t), problem (1.8) can be rewritten as

x′ = y,

y′ = −
N∑
i=1

ai(t)Fi(t, x, y, x(t− τi), y(t− τi))−G(t, x, y)

− b(t)H(x)−
N∑
i=1

Gi(t, x, x(t− τi)) = 0, t ≥ t0,

x(t0) = ψ(t), t0 − τN ≤ t ≤ t0, y(t0) = y0,

(1.9)

where 0 < τ1 < τ2 < · · · < τN , x(t) : [t0 − τ,+∞) → R, τ = max{τi},
Fi(t, 0, 0, 0, 0) = 0, Fi ∈ C(R+ × R4,R), R+ = [0,∞), G,Gi ∈ C(R+ × R2,R),
G(t, 0, 0) = 0, Gi(t, 0, 0) = 0, H ∈ C(R,R), H(0) = 0, ai : [t0, T ]→ R are piecewise
continuous functions, b ∈ C[R+, R], {tk}∞k=0 is a monotone increasing unbounded
sequence of real numbers and φ, φ′ : [−τ, 0] → R have at most a finite number
of discontinuity points such that all of them being of the first kind, and are right
continuous at these points.

The remaining of this article is structured as follows: In Section 2, two new
results with regard to the existence and exponentially stabilization of solutions
of (1.8) are given. Section 3 deals with the contributions of the study, and the
conclusion of this article.

2. Existence of solutions and exponentially stabilization

We now construct sufficient conditions for the existence of solutions of the prob-
lem (1.8), (2.1), and for exponentially stabilization of the ODE included by (1.8).
See Theorems 2.1 and 2.2 below. Hence, for the existence result, we use the follow-
ing conditions

(A1) The functions φ(t) and φ′(t) are continuous on I1, I1 = [t0− τN , t0], except
at the most at a finite set Φ of points at which the lateral limits φ(t−),



4 S. PINELAS, O. TUNÇ, E. KORKMAZ, C. TUNÇ EJDE-2024/07

φ(t+), φ′(t−) and φ′(t+) exist and φ and φ′ are right continuous at these
points;

(A2) Let a0, b0 ∈ R, a0 > 0, b0 > 0 be such that

a0 = max
1≤i≤N

‖ai(t)‖, |b(t)| ≤ b0, ∀t ∈ R+;

(A3) Let α1, . . . , αN be positive constants such that

|Fi(t, u, v, w, z)| ≤ αi|u|, ∀t ∈ R+, ∀u, v, w, z ∈ R,
N∑
i=1

|Fi(·)− Fi(··)| ≤
N∑
i=1

αi|v1 − u1| ≤ α0N |v1 − u1|,

where

Fi(·) = Fi(t, v1, v2, v3, v4), Fi(··) = Fi(t, u1, u2, u3, u4),

α0 = max{αi}, ∀t ∈ R+, ∀u1, . . . , u4, v1, . . . , v4 ∈ R;

(A4) Let β1, . . . , βN be positive constants such that

Gi(t, 0, 0) = 0, |Gi(t, u, v)| ≤ βi|u|,
N∑
i=1

|Gi(t, un, vn)−Gi(t, u, v)| ≤
N∑
i=1

βi|un − u| ≤ β0N |un − u|,

where β0 = max{βi : ∀t ∈ R+,∀u, un, v, vn ∈ R};
(A5) β0N(T − t0)2 < 1;
(A6) Let g0, h0 ∈ R be positive constants such that

G(t, 0, 0) = 0, |G(t, u, v)| ≤ g0|u|, ∀t ∈ R+, ∀u, v ∈ R,
|G(t, v1, v2)−G(t, u1, u2)| ≤ g0|v1 − u1|, ∀t ∈ R+, u1, u2, v1, v2 ∈ R,

H(0) = 0, |H(u)| ≤ h0|u|, ∀u ∈ R;

According to (A1)–(A6), the impulses at times tk, k ∈ N, satisfy

x(tk) = Ik(x(t−k )),

x′(tk) = Jk(x′(t−k ));
(2.1)

(A7) Ik ∈ C[R,R], Ik(0) = 0, Jk ∈ C[R,R], Jk(0) = 0, k ∈ N, and there are
non-negative constants ck, dk ∈ R which allow Ik = Ik(x) and Jk = Jk(x)
to be bounded with upper bounds ck, dk, i.e.,

|Ik(x)| ≤ ck, |Jk(x)| ≤ dk, ∀k ∈ N, x ∈ R.

Let D ⊂ R, which is an open set such that I = [t0 − τN , T ] ⊂ D, and x(t; t0, ψ, y0)
represent the solutions of (1.8), (2.1) through a point (t0, ϕ, y0).

(A8) We assume that

|ai(t)| ≤ a0, |b(t)| ≤ b0,
|Fi(t, x1, y1, x2, y2)| ≤ βi|fi(x2)|

|fi(x2)| ≤ σi|x2|,
|G(t, x1, y1)| ≤ α0|y1|, |H(x1)| ≤ h0|x1|,

|Gi(t, x1, x2)| ≤ gi|x2|, ∀x1, y1, x2, y2 ∈ R, ∀t ∈ R+



EJDE-2024/07 EXISTENCE AND STABILIZATION 5

where

a0 ∈ R, a0 > 0, b0 ∈ R, b0 > 0, α0 ∈ R, α0 > 0,

h0 ∈ R, h0 > 0, σi ∈ R, σi > 0, βi ∈ R, βi > 0.

The first existence result of this article reads as follows.

Theorem 2.1. If (A1)–(A5), (A7) are satisfied, then (1.8) with (2.1) admits a
solution on I.

Proof. Let the operator Q0 : C1
Ψ(I,R)→ C1

Ψ(I,R) be defined by

Q0(x) =



ψ(t), if t ∈ I1,
ψ(t0) + (t− t0)y0 −

∫ t
t0

(t− s)
[∑N

i=1 ai(s)Fi(s, x(s), x′(s),

x(s− τi), x′(s− τi))
]

ds

−
∫ t
t0

(t− s)
[
G(s, x(s), x′(s)) + b(s)H(x(s))

+
∑N
i=1Gi(s, x(s), x(s− τi))

]
ds, if t ∈ I2,

where I1 = [t0 − τN , t0], I2 = [t0, T ], and Ψ is the (finite) set, which includes the
discontinuity points of ψ.

We show that the operator Q0 has a fixed point. Note that ψ is known a fixed
point of the restriction of the operator Q0 to I1, Q0

∣∣
I1

. Hence, we will verify that

Q0

∣∣
I2

admits a fixed point.

In the rest of the paper, let

Fi(s, xn(s), . . . , x′n(s− τi)) = Fi(s, xn(s), x′n(s), xn(s− τi), x′n(s− τi))
and

Fi(s, x(s), . . . , x′(s− τi)) = Fi(s, x(s), x′(s), x(s− τi), x′(s− τi)) .
We consider the operator Q0 = Q0

∣∣
I2

. Then, the operator Q0 : C1(I2,R) →
C1(I2,R) satisfies

Q0(x) = ψ(t0) + (t− t0)y0 −
∫ t

t0

(t− s)[
N∑
i=1

ai(s)Fi(s, x(s), . . . , x′(s− τi))] ds

−
∫ t

t0

(t− s)
[
G(s, x(s), x′(s)) + b(s)H(x(s))

+

N∑
i=1

Gi(s, x(s), x(s− τi))
]

ds.

We will prove Theorem 2.1 at four steps using the operator Q0.

Step 1: Operator Q0 is continuous. Let {xn} be a sequence in C1(I2,R) such that
xn tends to x. Then, x′n → x′ converges uniformly in C1(I2,R). From Q0 and
(A1)–(A6), we derive

|Q0(xn)(t)−Q0(x)(t)|

≤ (T − t0)

∫ t

t0

[ N∑
i=1

|ai(s)||Fi(s, xn(s), . . . , x′n(s− τi))

− Fi(s, x(s), . . . , x′(s− τi))|
]

ds



6 S. PINELAS, O. TUNÇ, E. KORKMAZ, C. TUNÇ EJDE-2024/07

+ (T − t0)

∫ t

t0

[|G(s, xn(s), x′n(s))−G(s, x(s), x′(s))|] ds

+ (T − t0)

∫ t

t0

|b(s)|[|H(xn(s))−H(x(s))|] ds

+ (T − t0)

∫ t

t0

[ N∑
i=1

|Gi(s, xn(s), xn(s− τi))−Gi(s, x(s), x(s− τi))|
]

ds

≤ (T − t0)a0

∫ t

t0

[

N∑
i=1

αi|xn(s)− x(s)|] ds

+ (T − t0)g0

∫ t

t0

|xn(s)− x(s)|ds+ (T − t0)

∫ t

t0

|b(s)|[|H(xn(s))−H(x(s))|] ds

+ (T − t0)

∫ t

t0

[

N∑
i=1

βi|xn(s)− x(s)|] ds

≤ α0a0N(T − t0)2‖xn − x‖+ g0(T − t0)2‖xn − x‖

+ (T − t0)b0

∫ t

t0

[‖H(xn(s))−H(x(s))|] ds+ β0N(T − t0)2‖xn − x‖.

Since H is a continuous function, the last inequality above allows that

|Q0(xn)(t)−Q0(x)(t)| → 0 as n→∞.
Hence, the operator Q0 is continuous.

Step 2: Operator Q0 maps bounded sets into bounded sets. Let ω ≥ 0, ω ∈ R,
and υ ≥ 0. We define

Bω = {y ∈ C1(I2,R) : ‖y‖ ≤ ω}, x ∈ Bω,
which implies that ‖Q0(x)‖ ≤ υ. Since t ∈ I2, by the definition of Q0 and (A1)–
(A6), we derive that

|Q0(x)(t)|

≤ ‖ψ(t0)‖+ (t− t0)|y0|+
∫ t

t0

(t− s)
N∑
i=1

|ai(s)| |Fi(s, x(s), . . . , x′(s− τi))|ds

+

∫ t

t0

(t− s)|G(s, x(s), x′(s))|ds+

∫ t

t0

(t− s)b(s)|H(x(s))|ds

+

∫ t

t0

(t− s)
N∑
i=1

|Gi(s, x(s), x(s− τi))|ds

≤ ‖ψ(t0)‖+ (T − t0)|y0|+ (T − t0)a0

∫ t

t0

N∑
i=1

|Fi(s, x(s), . . . , x′(s− τi))|ds

+ (T − t0)

∫ t

t0

|G(s, x(s), x′(s))|ds+ (T − t0)b0

∫ t

t0

|H(x(s))|ds

+ (T − t0)

∫ t

t0

N∑
i=1

|Gi(s, x(s), x(s− τi))|ds

≤ ‖ψ(t0)‖+ (T − t0)2(α0a0N + g0 + h0b0) + (T − t0)|y0|+ β0N(T − t0)2‖x‖



EJDE-2024/07 EXISTENCE AND STABILIZATION 7

≤ ‖ψ(t0)‖+ (T − t0)2(α0a0N + g0 + h0b0) + (T − t0)|y0|+ ωβ0N(T − t0)2.

Let

υ = ‖ψ(t0)‖+ (T − t0)
2
(α0a0N + g0 + h0b0 + ωβ0N) + (T − t0)|y0|.

From the above inequality it follows that ‖Q0(x)‖ ≤ υ, which completes Step 2.

Step 3: Operator Q0 converts bounded sets to the equicontinuous sets, which are
included in the space C1(I2,R).

Let θ1, θ2 ∈ I2 with θ1 < θ2 and consider x ∈ Bω. Then, from the definition of
Q0 and (A1)–(A6), we have

|Q0(x)(θ2)−Q0(x)(θ1)|

≤ |y0(θ2 − θ1)|+
∣∣∣ ∫ θ2

t0

(θ2 − s)
N∑
i=1

ai(s)Fi(s, x(s), . . . , x′(s− τi)) ds

−
∫ θ1

t0

(θ1 − s)
N∑
i=1

ai(s)Fi(s, x(s), . . . , x′(s− τi)) ds
∣∣∣

+
∣∣∣ ∫ θ2

t0

(θ2 − s)G(s, x(s), x′(s)) ds−
∫ θ1

t0

(θ1 − s)G(s, x(s), x′(s)) ds
∣∣∣

+
∣∣∣ ∫ θ2

t0

(θ2 − s)b(s)H(x(s)) ds−
∫ θ1

t0

(θ1 − s)b(s)H(x(s)) ds
∣∣∣

+
∣∣∣ ∫ θ2

t0

(θ2 − s)
N∑
i=1

Gi(s, x(s), x(s− τi)) ds

−
∫ θ1

t0

(θ1 − s)
N∑
i=1

Gi(s, x(s), x(s− τi)) ds
∣∣∣

≤ |y0|(θ2 − θ1) + (θ2 − θ1)

∫ θ2

t0

N∑
i=1

|ai(s)| |Fi(s, x(s), . . . , x′(s− τi))|ds

+ (θ2 − θ1)

∫ θ2

t0

|G(s, x(s), x′(s))|ds+ (θ2 − θ1)

∫ θ2

t0

|b(s)| |H(x(s))|ds

+ (θ2 − θ1)

∫ θ2

t0

N∑
i=1

|Gi(s, x(s), x(s− τi))|ds,

which converges to 0 as θ2 → θ1. As for the next step, the equicontinuity for the
cases θ1 < θ2 ≤ 0 and θ1 ≤ 0 ≤ θ2 can be confirmed similarly.

The outcomes of the steps above allow that Q0(Bω) be bounded and equicontin-
uous for all υ > 0. Hence, using the Ascoli-Arzela theorem, it follows that Q0(Bω)
is relatively compact and therefore the operator Q0 is compact. This completes the
proof of Step 3.

Step 4: We prove that the set

Λ(Q0) = {x ∈ C1(I2,R) : x = λQ0(x), 0 < λ < 1}

is bounded. Let x ∈ Λ(Q0). Then

x = λQ0(x), 0 < λ < 1.



8 S. PINELAS, O. TUNÇ, E. KORKMAZ, C. TUNÇ EJDE-2024/07

According to this equality and Q0, for all t ∈ I2, we have

x(t) = λQ0x(t) = λ{ψ(t0) + y0(t− t0)}

− λ
∫ t

t0

(t− s)
N∑
i=1

ai(s)Fi(s, x(s), . . . , x′(s− τi)) ds

− λ
∫ t

t0

(t− s)[G(s, x(s), x′(s)) + b(s)H(x(s))] ds

− λ
∫ t

t0

(t− s)[
N∑
i=1

Gi(s, x(s), x(s− τi))] ds.

Using the above equality, 0 < λ < 1, and the outcomes of Step 2, we obtain

|x(t)| ≤ (T − t0)
2
(α0a0N + g0 + h0b0)

+ ‖ψ(t0)‖+ (T − t0)|y0|+ β0N(T − t0)
2‖x‖.

Then

‖x‖ ≤ ‖ψ(t0)‖+ (T − t0)
2
(α0a0N + g0 + h0b0) + (T − t0)|y0|

1− β0N(T − t0)
2 .

This result verifies that Λ(Q0) is bounded.
From steps four steps above, and the Schaefer fixed point theorem, the operator

Q0 has a fixed point, call it y1(t). In the light of this result,

x1(t) =

{
ψ(t), t0 ≤ t ≤ t0 − τN
y1(t), t0 ≤ t ≤ T

is a solution of (1.8), for all t ∈ I.
Let I1 = [t1 − τN , T ]. Consider the impulsive problem

x′′ +

N∑
i=1

ai(t)Fi(t, x, x
′, x(t− τi), x′(t− τi)) +G(t, x, x′)

+ b(t)H(x) +

N∑
i=1

Gi(t, x, x(t− τi)) = 0, t1 < t ≤ T,

x = x1(t), t1 − τN ≤ t < t1,

x(t1) = I1(x(t−1 )), x′(t1) = J1(x′(t−1 ))

(2.2)

and the operator Q1 : C1
K1

(I1,R)→ C1
K1

(I1,R) defined by

Q1(x)(t) =



x1(t), if t1 − τN ≤ t < t1,

I1(x(t−1 )) + J1(x′(t−1 ))(t− t1)

−
∫ t
t1

(t− s)
∑N
i=1 ai(s)Fi(s, x(s), . . . , x′(s− τi)) ds

−
∫ t
t1

(t− s)[G(s, x(s), x′(s)) + b(s)H(x(s))] ds

−
∫ t
t0

(t− s)
∑N
i=1Gi(s, x(s), x(s− τi)) ds, if t1 ≤ t ≤ T,

where K1 = K1(t1) ⊂ {t1} ∪Ψ including t1.
We note that x1(t) is a fixed point of the restriction of Q1 to [t1 − τN , t1),

N1

∣∣
[t1−τN ,t1)

. Then, we will show that Q1

∣∣
[t1,T ]

has a fixed point. Let I3 = [t1, T ].



EJDE-2024/07 EXISTENCE AND STABILIZATION 9

We rename Q1

∣∣
I3

and represent it by the operator Q1,

Q1 : C1(I3,R)→ C1(I3,R)

such that

Q1(x)(t) = x(t1) + x′(t1)(t− t1)

−
∫ t

t0

(t− s)
N∑
i=1

ai(s)

N∑
i=1

ai(s)Fi(s, x(s), . . . , x′(s− τi)) ds

−
∫ t

t0

(t− s)[G(s, x(s), x′(s)) + b(s)H(x(s))] ds

−
∫ t

t0

(t− s)
N∑
i=1

Gi(s, x(s), x(s− τi)) ds.

Following Steps 1–4 above, and using (A7), we easily obtain that Q1 has a fixed
point, call it y2(t). Hence, we have

x2(t) =

{
x1(t), t ∈ [t0 − τN , t1),

y2(t), t ∈ [t1, T ]
=


ψ(t), t ∈ I1,
y1(t), t ∈ [t0, t1),

y2(t), t ∈ I3

is a solution of the impulsive problem (2.2) on I = [t0 − τN , T ] ⊂ D.
Let Υk = [tk − τN , T ]. Repeating a similar way as the above, for t = tk, k ∈ N,

we now take into account the impulsive problem

x′′ +

N∑
i=1

ai(t)Fi(t, x, x
′, x(t− τi), x′(t− τi)) +G(t, x, x′)

+ b(t)H(x) +

N∑
i=1

Gi(t, x, x(t− τi)) = 0, tk < t ≤ T,

x = xk(t), tk − τN ≤ t < tk,

x(tk) = Ik(x(t−k )), x′(tk) = Jk(x′(t−k ))

(2.3)

and the operator

Qk : C1
Kk

(Υk,R)→ C1
Kk

(Υk,R),

defined by

Qk(x)(t) =



xk(t), if tk − τN ≤ t < tk,

Ik(x(t−k )) + Jk(x′(t−k ))(t− tk)

−
∫ t
t0

(t− s)
∑N
i=1 ai(s)Fi(s, x(s), . . . , x′(s− τi)) ds

−
∫ t
tk

(t− s)[G(s, x(s), x′(s)) + b(s)H(x(s))] ds

−
∫ t
tk

(t− s)
∑N
i=1Gi(s, x(s), x(s− τi)) ds, if tk ≤ t ≤ T,

where Kk ⊂ {t1, t2, . . . , tk} ∪ Ψ. For i = 1, 2, . . . , k − 1, we note that tk ∈ Kk.
However, whether ti ∈ Kk, for example, depends upon to the magnitude of the
delay τN . We write that Qk = Qk

∣∣
[tk,T ]

and have that

Qk : C1([tk, T ],R)→ C1([tk, T ],R),



10 S. PINELAS, O. TUNÇ, E. KORKMAZ, C. TUNÇ EJDE-2024/07

is defined by

Qk(x)(t) = x(tk) + x′(tk)(t− tk)−
∫ t

t0

(t− s)
N∑
i=1

ai(s)Fi(s, x(s), . . . , x′(s− τi)) ds

−
∫ t

t0

(t− s)[G(s, x(s), x′(s)) + b(s)H(x(s))] ds

−
∫ t

t0

(t− s)
N∑
i=1

Gi(s, x(s), x(s− τi)) ds.

Then, using the four steps above, and using (A7), we conclude that Qk has a fixed
point, call it yk(t). Then

xk(t) =

{
xk−1(t), t0 − τN , tk−1),

yk(t), t ∈ [tk−1, T ]
=



ψ(t), t ∈ I1,
y1(t), t ∈ [t0, t1),

y2(t), t ∈ [t1, t2),

. . .

yk(t), t ∈ [tk−1, T ]

is a solution of (2.3) on I1.
Next, in the same way as above, it follows that

x(t) =

{
xm(t), t0 − τN , tm),

ym+1(t), t ∈ [tm, T ]
=



ψ(t), t ∈ I1,
y1(t), t ∈ [t0, t1),

y2(t), t ∈ [t1, t2),

. . .

yk(t), t ∈ [tk−1, tk),

. . .

ym+1(t), t ∈ [tm, T ]

is a solution of (1.8) satisfying (2.1) on I, where m = max{k ∈ N : tk ≤ T}. The
proof of Theorem 2.1 is complete. �

The second result of this article reads as follows.

Theorem 2.2. Let (A1), (A7), (A8) be satisfied and

N∑
i=1

τi(gi + a0σi) < exp
[
−

N∑
i=1

(τiK2)
]

(2.4)

with

K2 = max
{ N∑
i=1

(gi + a0σi) + b0h0, 1 + α0

}
.

Then the zero solution of (1.8) is exponentially stabilized by impulses.

Proof. Assume that (2.4) is satisfied. Then by (2.4) there exist ᾱ ∈ R, ᾱ > 0, and
¯̀≥

∑N
i=1 (τi) such that

N∑
i=1

τi(gi + a0σi) ≤ exp
[
− ᾱ

(
¯̀+

N∑
i=1

(τi)
)]

exp[−K2
¯̀]. (2.5)



EJDE-2024/07 EXISTENCE AND STABILIZATION 11

Let ᾱ > 0 and ¯̀≥
∑N
i=1 (τi) be as in (2.5). Consider the sequence {tk}k∈N such

that t0 < t1 < . . . < tk < . . ., with lim
k→∞

tk =∞ and

N∑
i=1

(τi) ≤ tk − tk−1 ≤ ¯̀.

Let

Ik(u) = d̄ku, Jk(u) = d̄ku,

d̄k = p̄k −
N∑
i=1

τi(gi + a0σi),

p̄k = exp
[
− ᾱ(tk+1 − tk)− ᾱ

N∑
i=1

τi(gi + a0σi)
]

exp[−K2(tk+1 − tk)].

Then, d̄k ≥ 0, d̄k ∈ R, since p̄k ≥
∑N
i=1 τi(gi + a0σi) by (2.5).

For a given ε > 0, let

δ =
ε

1 +
∑N
i=1 τi(gi + a0σi)

exp[−(ᾱ+K2)(t1 − t0)].

We now prove that, for every solution x(t) = x(t; t0, ψ, y0) of (1.8) satisfying (2.1),
if

‖ψ(t0)‖+ |y(t0)| ≤ δ,
then

|x(t)|+ |y(t)| ≤ ε exp[−α(t− t0)], t0 ≤ t ≤ T.
Let zt = (xt, yt) be a solution of (1.8). We define a new Lyapunov-Krasovskii
functional

W0(zt) = W0(t)

= |x|+ |y|+
N∑
i=1

gi

∫ t

t−τi
|x(s)|ds

+

N∑
i=1

γi

∫ t

t−τi
|ai(s+ τi)| |fi(x(s))|ds,

(2.6)

where γi ∈ R, γi > 0 are arbitrary constants.
Firstly, we show that (2.6) fulfill the following steps.

Step 5: From (2.6), it follows that W0(t) ≥ |x|+ |y|.
Step 6: According to (A8), from (2.6), we obtain

W0(t) ≤ |x|+ |y|+
N∑
i=1

(τigi)‖xt‖+

N∑
i=1

γi

∫ t

t−τi
(a0σi)|x(s)|ds

≤ |x|+ |y|+
N∑
i=1

(τigi)‖xt‖+

N∑
i=1

(a0γiτiσi)‖xt‖

≤ (1 +

N∑
i=1

τi(gi + a0γiσi))‖xt‖+ |y|,

where ‖xt‖ = supt−τN≤s≤t |x(s)|.



12 S. PINELAS, O. TUNÇ, E. KORKMAZ, C. TUNÇ EJDE-2024/07

Step 7: Let t0 < t < t1. Calculating the right upper derivative of W0(t) along
solutions of (1.8) and using (A1), (A7), and (A8), we have

d

dt
W0(t)

= x′ sgnx+ y′ sgn y +

N∑
i=1

(gi)|x| −
N∑
i=1

(gi)|x(t− τi)|

+

N∑
i=1

γi|ai(t+ τi)| |fi(x)| −
N∑
i=1

γi|ai(t)| |fi(x(t− τi))|

= y sgnx− sgn y
[ N∑
i=1

ai(t)Fi(t, x, y, x(t− τi), y(t− τi)) +G(t, x, y)
]

− sgn y
[
b(t)H(x) +

N∑
i=1

Gi(t, x, x(t− τi))
]

+

N∑
i=1

(gi)|x| −
N∑
i=1

(gi)|x(t− τi)|

+

N∑
i=1

γi|ai(t+ τi)| |fi(x)| −
N∑
i=1

γi|ai(t)| |fi(x(t− τi))|

≤ |y|+
N∑
i=1

|ai(t)| |Fi(t, x, y, x(t− τi), y(t− τi))|+ |G(t, x, y)|

+ |b(t)| |H(x)|+
N∑
i=1

|Gi(t, x, x(t− τi))|

+

N∑
i=1

(gi)|x| −
N∑
i=1

(gi)|x(t− τi)|

+

N∑
i=1

γi|ai(t+ τi)| |fi(x)| −
N∑
i=1

γi|ai(t)| |fi(x(t− τi))|

≤ (1 + α0)|y|+
N∑
i=1

γi|ai(t)| |fi(x(t− τi))|

+ (b0h0)|x|+
N∑
i=1

(gi)|x(t− τi)|+
N∑
i=1

(gi)|x| −
N∑
i=1

(gi)|x(t− τi)|

+

N∑
i=1

γi|ai(t+ τi)| |fi(x)| −
N∑
i=1

γi|ai(t)| |fi(x(t− τi))|.

Let γ1 = γ2 = · · · = γN = 1. It follows that

d

dt
W0(t) ≤

( N∑
i=1

(gi + a0σi) + b0h0

)
|x|+ (1 + α0) |y| ≤ K2(|x|+ |y|), (2.7)



EJDE-2024/07 EXISTENCE AND STABILIZATION 13

where

K2 = max
{ N∑
i=1

(gi + a0σi) + b0h0, 1 + α0

}
.

From (2.6) and (2.7), we derive that d
dtW0(t) ≤ K2W0(t). Integrating this inequal-

ity, we have

W0(t) ≤W0(t0) exp[K2(t− t0)], t0 < t < t1. (2.8)

According to Steps 5 and 6, (2.8), and the above inequalities, we obtain

|x|+ |y| ≤W0(t)

≤W0(t0) exp[K2(t− t0)]

≤W0(t0) exp[K2(t1 − t0)]

≤
(

1 +
N∑
i=1

τi(gi + a0σi)
)

[‖xt0‖+ |y(t0)|] exp[K2(t1 − t0)]

≤
(

1 +

N∑
i=1

τi(gi + a0σi)
)
δ exp[K2(t1 − t0)]

≤ ε exp[−ᾱ(t1 − t0)]

≤ ε exp[−ᾱ(t− t0)].

Hence, we conclude that

|x|+ |y| ≤ ε exp[−ᾱ(t− t0)], t ∈ (t0, t1). (2.9)

Since the right continuity of x and x′ holds on [t0, t1), from (2.9), it follows that

|x|+ |y| ≤ ε exp[−ᾱ(t− t0)], t ∈ [t0, t1).

Secondly, we will now show that (2.6) satisfies the following step:

Step 7: For t1 < t < t2 we have the results in Steps 5 and 6. We repeat similar
calculations as in the lines above. When we consider

N∑
i=1

(τi) ≤ tk − tk−1 ≤ ¯̀

and the definition of p̄k. It follows that

N∑
i=1

(τi) ≤ t1 − t0 ≤ ¯̀,

d̄1 = p̄1 −
N∑
i=1

τi(gi + a0σi).

Hence, repeating similar calculations as above, for t1 < t < t2, we obtain

W0(t) ≤W0(t+1 ) exp[K2(t2 − t1)]

≤
(
|x(t+1 )|+ |y(t+1 )|+

N∑
i=1

gi

∫ t1

t1−τi
|x(s)|ds

+

N∑
i=1

∫ t1

t1−τi
|ai(s+ τi)| |fi(x(s))|ds

)
exp[K2(t2 − t1)]



14 S. PINELAS, O. TUNÇ, E. KORKMAZ, C. TUNÇ EJDE-2024/07

=
(
|x(t1)|+ |y(t1)|+

N∑
i=1

gi

∫ t1

t1−τi
|x(s)|ds

+

N∑
i=1

∫ t1

t1−τi
|ai(s+ τi)| |fi(x(s))|ds

)
exp[K2(t2 − t1)]

=
(
|I1(x(t−1 ))|+ |J1(y(t−1 ))|+

N∑
i=1

gi

∫ t1

t1−τi
|x(s)|ds

+

N∑
i=1

∫ t1

t1−τi
|ai(s+ τi)| |fi(x(s))|ds

)
exp[K2(t2 − t1)]

≤
(
d̄1[|x(t−1 )|+ |y(t−1 )|] +

N∑
i=1

gi

∫ t1

t1−τi
|x(s)|ds

+

N∑
i=1

∫ t1

t1−τi
|ai(s+ τi)| |fi(x(s))|ds

)
exp[K2(t2 − t1)]

≤ d̄1 sup
t1−τi≤t≤t1

[|x(t)|+ |y(t)|] exp[K2(t2 − t1)]

+ sup
t1−τi≤t≤t1

|x|
N∑
i=1

(τigi) exp[K2(t2 − t1)]

+ sup
t1−τi≤t≤t1

|x|
N∑
i=1

(a0τiσi) exp[K2(t2 − t1)]

≤ (d̄1 +

N∑
i=1

τi(gi + a0σi)) sup
t1−τi≤t≤t1

[|x(t)|+ |y(t)|] exp[K2(t2 − t1)]

≤
(
d̄1 +

N∑
i=1

τi(gi + a0σi)
)
ε exp[−ᾱ(t1 − t0 +

N∑
i=1

(τi))]

× exp[K2(t2 − t1)]

= p̄1ε exp
[
− ᾱ(t1 − t0 +

N∑
i=1

(τi))
]

exp[K2(t2 − t1)]

≤ ε exp[−ᾱ(t2 − t0)]

≤ ε exp[−ᾱ(t− t0)].

From the calculations above, we have

|x|+ |y| ≤W0(t) ≤ ε exp[−ᾱ(t− t0)], t ∈ (t1, t2).

It follows that
|x|+ |y| ≤ ε exp[−ᾱ(t− t0)], t ∈ (t1, t2). (2.10)

From the right continuity of x and x′ on [t0, t1), inequality (2.10) holds for
t ∈ [t1, t2), i.e., we have

|x|+ |y| ≤ ε exp[−ᾱ(t− t0)], t ∈ [t1, t2).

From this results, we obtain

|x|+ |y| ≤ ε exp[−ᾱ(t− t0)], t ∈ [t0, t2).



EJDE-2024/07 EXISTENCE AND STABILIZATION 15

Repeating these calculations, for k ∈ N, we arrive at

|x|+ |y| ≤ ε exp[−ᾱ(t− t0)], t ∈ [t0, tk).

Then, we conclude that

|x|+ |y| ≤ ε exp[−ᾱ(t− t0)], t ≥ t0.

Then, the proof of Theorem 2.2 is complete. �

3. Conclusion

The new results in this paper, Theorems 2.1 and 2.2, are derived from concise
derivations and provide strong conditions. The impulsive problem (1.8) includes
N constant delays and has a more general nonlinear form than the ones found in
the literature. Theorems 2.1 and 2.2 extend earlier results from particular cases to
general multiple-delays equations. It is expected naturally that Theorems 2.1 and
2.2 can be extended to some stronger results.

Additionally, we would like to suggest the study of existence and stabilization
of nonlinear impulsive differential systems. Also suggest the study of impulsive
integro-differential equations of high order and fractional order including multiple
constant or variable time delays.

Acknowledgments. This research was supported by the Center for Research and
Development in Mathematics and Applications (CIDMA), through the Portuguese
Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tec-
nologia), references UIDB/04106/2020 and UIDP/0410
6/2020.

The authors would like to thank the anonymous referees and the handling Editor
for many useful comments and suggestions, leading to a substantial improvement
in the presentation of this article.

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ics/Ouvrages de Mathematiques de la SMC. Springer, Cham, 2016.

[35] Stamova, I. M., Stamov, G. T.; Functional and impulsive differential equations of fractional

order. Qualitative analysis and applications. CRC Press, Boca Raton, FL, 2017.
[36] Tunç, C.; Tunç, O.; On the stability, integrability and boundedness analyses of sys-

tems of integro-differential equations with time-delay retardation. Rev. R. Acad. Cienc.

Exactas Fis. Nat. Ser. A Mat. RACSAM, 115 (2021), no. 3, Paper No. 115, 17 pp.
https://doi.org/10.1007/s13398-021-01058-8

[37] Tunç, C.; Tunç, O.; On the Fundamental Analyses of Solutions to Nonlinear

Integro-Differential Equations of the Second Order. Mathematics, 2022; 10(22):4235.
https://doi.org/10.3390/math10224235

[38] Tunç, C.; Tunç, O.; Ulam stabilities of nonlinear iterative integro-differential equa-
tions. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 118 (2023).

https://doi.org/10.1007/s13398-023-01450-6

[39] Tunç, O.; On the fundamental analyses of solutions to nonlinear integro-differential equations
of second order. J. Nonlinear Convex Anal., 24 (2023), no. 1, 17-32.

[40] Tunç, O.; Tunç, C.; Wen C.-F., Yao, J.-C.; On the qualitative analyses solutions of new

mathematical models of integro-differential equations with infinite delay. Math. Meth. Appl.
Sci. (2023), 1-17. https://doi.org/10.1002/mma.9306

[41] Tunç, O.; Tunç, C.; Solution estimates to Caputo proportional fractional derivative delay

integro-differential equations. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM,
117 (2023), no. 1, Paper No. 12, 13 pp. https://doi.org/10.1007/s13398-022-01345-y

[42] Tunç, C.; Tunç, O.; Yao, J.-C.; On the Enhanced New Qualitative Re-

sults of Nonlinear Integro-Differential Equations. Symmetry 2023, 15, 109.
https://doi.org/10.3390/sym15010109

[43] Tunç, O.; Tunç, C; Yao, J.-C.; On the existence of results for multiple retarded differential
and integro-differential equations of second order. J. Nonlinear Convex Anal., 25 (2024),

(accepted).

[44] Tunç, C., Wang, Y.; Tunç, O.; Yao, J.-C.; New and Improved Criteria on Fundamental
Properties of Solutions of Integro-Delay Differential Equations with Constant Delay. Math-

ematics. 2021; 9(24):3317. https://doi.org/10.3390/math9243317

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[51] Zhao, X.; Liu, B.; Qian, C.; Cao, J.; Stability analysis of delay patch-
constructed Nicholson’s blowflies system, Math. Comput. Simulation (2023), (in press).
https://doi.org/10.1016/j.matcom.2023.09.012

Sandra Pinelas

Departamento de Ciências Exatas e Engenharia, Academia Militar, Av. Conde Castro
Guimarães, 2720-113 Amadora, Portugal.

Center for Research and Development in Mathematics and Applications (CIDMA), De-

partment of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
Email address: sandra.pinelas@gmail.com



18 S. PINELAS, O. TUNÇ, E. KORKMAZ, C. TUNÇ EJDE-2024/07

Osman Tunç

Department of Computer Programing, Baskale Vocational School, Van Yuzuncu Yil

University, Campus, 65080 Van-Turkey
Email address: osmantunc89@gmail.com

Erdal Korkmaz
Department of Mathematics, Faculty of Arts and Sciences, Muş Alparslan University,

49250, Muş, Turkey

Email address: korkmazerdal36@hotmail.com

Cemil Tunç

Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080,

Campus, Van, Turkey
Email address: cemtunc@yahoo.com


	1. Introduction
	2. Existence of solutions and exponentially stabilization
	3. Conclusion
	Acknowledgments

	References