Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 21, pp. 1–21.

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

DOI: 10.58997/ejde.2024.21

EXISTENCE OF PERIODIC SOLUTIONS AND STABILITY FOR

A NONLINEAR SYSTEM OF NEUTRAL DIFFERENTIAL

EQUATIONS

YANG LI, GUILING CHEN

Abstract. In this article, we study the existence and uniqueness of periodic

solutions, and stability of the zero solution to the nonlinear neutral system

d

dt
x(t) = A(t)h

(
x(t− τ1(t))

)
+

d

dt
Q
(
t, x(t− τ2(t))

)
+G

(
t, x(t), x(t− τ2(t))

)
.

We use integrating factors to transform the neutral differential equation into
an equivalent integral equation. Then we construct appropriate mappings and

employ Krasnoselskii’s fixed point theorem to show the existence of a periodic
solution. We also use the contraction mapping principle to show the existence

of a unique periodic solution and the asymptotic stability of the zero solution.

Our results generalize the corresponding results in the existing literature. An
example is given to illustrate our results.

1. Introduction

Functional differential equations are widely applied in fields, such as neural net-
works, population dynamics, control theory, and many other fields. Recently, the
theory about these equations has been an object of active research, mostly because
by understanding the properties of solutions we can see the trend of events in real
world problems.

Investigators have given special attention to the study of equations in which the
delay occurs in the derivative of the state variable as well as in the independent
variables. These equations are called neutral differential equations, and describe
actual problems more accurately than other differential equations. In particular,
qualitative analysis, such as periodicity and stability of solutions of neutral differ-
ential equations, has been studied extensively by many authors. For a long time,
the direct Lyapunov method or Lyapunov function was the main tool for determin-
ing stability in many differential equations without solving the equations explicitly.
However, there are a lot of problems when using this method: Lyapunov’s direct
method requires pointwise conditions while many practical problems do not meet
these conditions; a suitable Lyapunov function is not easy to construct; and there
are problems with ascertaining limit sets when the equation becomes unbounded

2020 Mathematics Subject Classification. 34K13, 34K20, 34K40.
Key words and phrases. Neutral equation; periodic solution; existence; uniqueness;

stability; Krasnoselskii’s fixed point theorem; contraction mapping principle.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted December 12, 2023. Published March 4, 2024.

1



2 Y. LI, G. CHEN EJDE-2024/21

or the derivative is not finite. Fortunately, Burton and other authors have applied
fixed point theory to investigate the stability of systems and obtained some appli-
cable techniques. Moreover, the advantage of fixed point theory is that it can prove
the existence, uniqueness, boundedness, and stability of the equation at the same
time. For recent works on periodicity and stability of neutral equations, we refer
the reader to the references in this article.

In 2007, Islam and Raffoul [9] studied the existence of periodic solutions of the
system

d

dt
x(t) = A(t)x(t) +

d

dt
Q
(
t, x(t− g(t))

)
+G

(
t, x(t), x(t− g(t))

)
. (1.1)

Using Krasnoselskii’s fixed point theorem, they obtained the existence of a unique
periodic solution of (1.1) . Existence and uniqueness of solutions has been also
investigated using the contraction mapping principle; see [9] for linear systems,
and [14] for nonlinear systems. Mesmouli, Ardjouni, and Djoudi [14] extended the
results of [9] by studying a nonlinear system with two variable delays

d

dt
x(t) = A(t)x(t− τ(t)) +

d

dt
Q(t, x(t− g(t))) +G(t, x(t), x(t− g(t))).

Motivated by the works mentioned above, we study the following system where
we replace the linear term A(t)x(t−τ(t)) by the nonlinear term A(t)h

(
x(t−τ1(t))

)
,

and has two variable delays:

d

dt
x(t) = A(t)h

(
x(t− τ1(t))

)
+
d

dt
Q
(
t, x(t− τ2(t)

)
+G

(
t, x(t), x(t− τ2(t))

)
, (1.2)

where A(·) is a nonsingular n× n matrix with continuous real-valued functions as
entries. The functions h : R → Rn, Q : R× Rn → Rn, and G : R× Rn × Rn → Rn
are continuous in their respective arguments, and τ2 is continuously differentiable.
In our analysis we use the fundamental matrix solution of x′(t) = A(t)x(t) coupled
with Floquet theory to transform (1.2) into an integral system. The integral system
obtained is the sum of two mappings, one completely continuous, and the other a
contraction.

The organization of this article is as follows. In section 2 we present some
definitions and transform (1.2) into an integral system. In section 3 we study the
existence and uniqueness of a periodic solutions. In section 4 study the stability
of the zero solution. And in section 5 we present an example that illustrates our
results.

2. Preliminaries

For T > 0, let PT be the set of all n-vector valued functions x(t), which are
continuous and periodic in t of period T . Then (PT , ∥ · ∥) is a Banach space with
the supremum norm

∥x(·)∥ = sup
t∈R

|x(t)| = sup
t∈[0,T ]

|x(t)|,

where | · | denotes the norm for x ∈ Rn. Also, if A is an n× n real matrix, then we
define the norm |A| = max1≤i≤n

∑n
j=1 |aij |.

Definition 2.1. Let matrix A(·) be periodic of period T . The linear system

y′(t) = A(t)y(t), (2.1)



EJDE-2024/21 NONLINEAR SYSTEM OF NEUTRAL DIFFERENTIAL EQUATIONS 3

is said to be noncritical with respect to T , if it has no periodic solution of period
T except for the trivial solution y = 0.

In this article, we assume that

A(t+T ) = A(t), τ1(t+T ) = τ1(t) ≥ τ∗1 > 0, τ2(t+T ) = τ2(t) ≥ τ∗2 > 0, (2.2)

with τ1 is continuously differentiable and τ∗1 , τ
∗
2 are constants. For t ∈ R, x, y ∈ Rn,

the functions Q(t, x) and G(t, x, y) are periodic in t of period T . That is

Q(t+ T, x) = Q(t, x), G(t+ T, x, y) = G(t, x, y). (2.3)

The functions Q,G, h are also globally Lipschitz continuous. That is, for x, y, z, w ∈
Rn, there are positive constants k1, k2, k3, k4 such that

|Q(t, x)−Q(t, y)| ≤ k1∥x− y∥, (2.4)

|G(t, x, y)−G(t, z, w)| ≤ k2∥x− z∥+ k3∥y − w∥, (2.5)

|h(x)− h(y)| ≤ k4∥x− y∥. (2.6)

Throughout this article we assume system (2.1) is noncritical. LetK(t) represent
the fundamental matrix of (2.1) withK(0) = I, where I is the n×n identity matrix.
Then:

(a) detK(t) ̸= 0.
(b) There exists a constant matrix B such thatK(t+T ) = K(t)eTB , by Floquet

theory.
(c) System (2.1) is noncritical if and only if det(I −K(T )) ̸= 0.

The following lemma is fundamental for our results.

Lemma 2.2. Suppose (2.2) and (2.3) hold, then x is a solution of (1.2) if and
only if

x(t)

= Q(t, x(t− τ2(t)))−
∫ t

t−τ1(t)
A(s)h(x(s))ds

+K(t)U(T )

∫ t+T

t

K−1(s)A(s)
[
Q(s, x(s− τ2(s)))− (x(s)− h(x(s)))

−
∫ s

s−τ1(s)
A(u)h(x(u))du

]
ds+K(t)U(T )

∫ t+T

t

K−1(s)[F (s)h(x(s− τ1(s)))

+G(s, x(s), x(s− τ2(s)))]ds,

(2.7)
where

U(T ) = (K−1(T )− I)−1,

F (t) = A(t)− (1− τ ′1(t))A(t− τ1(t)),
(2.8)

where K(·) is the fundamental matrix solution of (2.1).

Proof. Let x(t) ∈ PT be a solution of (1.2). We rewrite (1.2) as

d

dt
x(t) = A(t− τ1(t))h(x(t− τ1(t)))(1− τ ′1(t))

−A(t− τ1(t))h(x(t− τ1(t)))(1− τ ′1(t)) +A(t)h(x(t))−A(t)h(x(t))

+A(t)h(x(t− τ1(t))) +
d

dt
Q(t, x(t− τ2(t))) +G(t, x(t), x(t− τ2(t)))



4 Y. LI, G. CHEN EJDE-2024/21

=
d

dt
Q(t, x(t− τ2(t)))−

d

dt

∫ t

t−τ1(t)
A(s)h(x(s))ds+A(t)h(x(t))

+ h(x(t− τ1(t)))[A(t)−A(t− τ1(t))(1− τ ′1(t))]

+G(t, x(t), x(t− τ2(t))).

Putting F (t) = A(t)− (1− τ ′1(t))A(t− τ1(t)) we have

d

dt

[
x(t) +

∫ t

t−τ1(t)
A(s)h(x(s))ds−Q(t, x(t− τ2(t)))

]
= A(t)

[
x(t) +

∫ t

t−τ1(t)
A(s)h(x(s))ds−Q(t, x(t− τ2(t)))

]
+ F (t)h(x(t− τ1(t)))−A(t)[x(t)− h(x(t))] +A(t)Q(t, x(t− τ2(t)))

−A(t)

∫ t

t−τ1(t)
A(s)h(x(s))ds+G(t, x(t), x(t− τ2(t))).

Since K(t)K−1(t) = I, it follows that

0 =
d

dt

[
K(t)K−1(t)

]
= A(t)K(t)K−1(t) +K(t)

d

dt
K−1(t)

= A(t) +K(t)
d

dt
K−1(t).

This implies

d

dt
K−1(t) = −K−1(t)A(t).

If x(t) is a solution of (1.2) with x(0) = x0, then

d

dt

[
K−1(t)

(
x(t) +

∫ t

t−τ1(t)
A(s)h(x(s))ds−Q(t, x(t− τ2(t)))

)]
=

d

dt
K−1(t)

[
x(t) +

∫ t

t−τ1(t)
A(s)h(x(s))ds−Q(t, x(t− τ2(t)))

]
+K−1(t)

d

dt

[
x(t) +

∫ t

t−τ1(t)
A(s)h(x(s))ds−Q(t, x(t− τ2(t)))

]
= K−1(t)A(t)

[
Q(t, x(t− τ2(t)))− (x(t)− h(x(t)))−

∫ t

t−τ1(t)
A(s)h(x(s))ds

]
+K−1(t)[F (t)h(x(t− τ1(t))) +G(t, x(t), x(t− τ2(t)))].



EJDE-2024/21 NONLINEAR SYSTEM OF NEUTRAL DIFFERENTIAL EQUATIONS 5

Integrating of the above equation from 0 to t yields

x(t) = Q(t, x(t− τ2(t)))−
∫ t

t−τ1(t)
A(s)h(x(s))ds+K(t)

(
x(0)

+

∫ 0

−τ1(0)
A(s)h(x(s))ds−Q(0,−τ2(0))

)
+K(t)

∫ t

0

K−1(s)A(s)
[
Q(s, x(s− τ2(s)))− (x(s)− h(x(s)))

−
∫ s

s−τ1(s)
A(u)h(x(u))du

]
ds

+K(t)

∫ t

0

K−1(s)[F (s)h(x(s− τ1(s))) +G(s, x(s), x(s− τ2(s)))]ds.

(2.9)

Since x(T ) = x0 = x(0) and

(I −K(T ))
−1

=
(
K(T )

(
K−1(T )− I

))−1
=

(
K−1(T )− I

)−1
K−1(T ),

using (2.9) we obtain

x(0) +

∫ 0

−τ1(0)
A(s)h(x(s))ds−Q(0,−τ2(0))

=
(
K−1(T )− I

)−1
∫ T

0

K−1(s)A(s)
[
Q(s, x(s− τ2(s)))− (x(s)− h(x(s)))

−
∫ s

s−τ1(s)
A(u)h(x(u))du

]
ds

+
(
K−1(T )− I

)−1
∫ T

0

K−1(s)
[
F (s)h(x(s− τ1(s)))

+G(s, x(s), x(s− τ2(s)))
]
ds.

(2.10)

Substituting (2.10) into (2.9) yields

x(t)

= Q(t, x(t− τ2(t)))−
∫ t

t−τ1(t)
A(s)h(x(s))ds

+K(t)
(
K−1(T )− I

)−1
∫ T

0

K−1(s)A(s)
[
Q(s, x(s− τ2(s)))− (x(s)− h(x(s)))

−
∫ s

s−τ1(s)
A(u)h(x(u))du

]
ds

+K(t)
(
K−1(T )− I

)−1
∫ T

0

K−1(s)
[
F (s)h(x(s− τ1(s)))

+G(s, x(s), x(s− τ2(s)))
]
ds+K(t)

∫ t

0

K−1(s)A(s)
[
Q(s, x(s− τ2(s)))

− (x(s)− h(x(s)))−
∫ s

s−τ1(s)
A(u)h(x(u))du

]
ds

+K(t)

∫ t

0

K−1(s)
[
F (s)h(x(s− τ1(s))) +G(s, x(s), x(s− τ2(s)))

]
ds.



6 Y. LI, G. CHEN EJDE-2024/21

Then

x(t) = Q(t, x(t− τ2(t)))−
∫ t

t−τ1(t)
A(s)h(x(s))ds

+K(t)
(
K−1(T )− I

)−1
{∫ T

0

K−1(s)A(s)
[
Q(s, x(s− τ2(s)))

− (x(s)− h(x(s)))−
∫ s

s−τ1(s)
A(u)h(x(u))du

]
ds

+

∫ T

0

K−1(s) [F (s)h(x(s− τ1(s))) +G(s, x(s), x(s− τ2(s)))] ds

+
(
K−1(T )− I

) ∫ t

0

K−1(s)A(s)
[
Q(s, x(s− τ2(s)))− (x(s)− h(x(s)))

−
∫ s

s−τ1(s)
A(u)h(x(u))du

]
ds

+
(
K−1(T )− I

) ∫ t

0

K−1(s)
[
F (s)h(x(s− τ1(s)))

+G(s, x(s), x(s− τ2(s)))
]
ds
}

= Q(t, x(t− τ2(t)))−
∫ t

t−τ1(t)
A(s)h(x(s))ds

+K(t)
(
K−1(T )− I

)−1
{∫ T

t

K−1(s)A(s)
[
Q(s, x(s− τ2(s)))

− (x(s)− h(x(s)))−
∫ s

s−τ1(s)
A(u)h(x(u))du

]
ds

+

∫ T

t

K−1(s) [F (s)h(x(s− τ1(s))) +G(s, x(s), x(s− τ2(s)))] ds

+

∫ t

0

K−1(T )K−1(s)A(s)
[
Q(s, x(s− τ2(s)))− (x(s)− h(x(s)))

−
∫ s

s−τ1(s)
A(u)h(x(u))du

]
ds

+

∫ t

0

K−1(T )K−1(s) [F (s)h(x(s− τ1(s))) +G(s, x(s), x(s− τ2(s)))] ds
}
.

By letting s = v − T and U(T ) = (K−1(T )− I)−1, the above expression yields

x(t)

= Q(t, x(t− τ2(t)))−
∫ t

t−τ1(t)
A(s)h(x(s))ds

+K(t)U(T )

∫ T

t

K−1(s)A(s)
[
Q(s, x(s− τ2(s)))− (x(s)− h(x(s)))

−
∫ s

s−τ1(s)
A(u)h(x(u))du

]
ds



EJDE-2024/21 NONLINEAR SYSTEM OF NEUTRAL DIFFERENTIAL EQUATIONS 7

+K(t)U(T )

∫ T

t

K−1(s) [F (s)h(x(s− τ1(s))) +G(s, x(s), x(s− τ2(s)))] ds

+K(t)U(T )

∫ t+T

T

K−1(T )K−1(v − T )A(v − T )
[
Q(v − T, x(v − T − τ2(v − T )))

− (x(v − T )− h(x(v − T )))−
∫ v−T

v−T−τ1(v−T )

A(u)h(x(u))du
]
dv

+K(t)U(T )

∫ t+T

T

K−1(T )K−1(v − T )
[
F (v − T )h(x(v − T − τ1(v − T )))

+G(v − T, x(v − T ), x(v − T − τ2(v − T )))
]
ds. (2.11)

By assumption (b) we have K(t− T ) = K(t)e−TB and K(T ) = eTB . Hence

K−1(T )K−1(v − T ) = K−1(v).

Consequently, since (2.2) and (2.3) hold, equation (2.11) becomes

x(t)

= Q(t, x(t− τ2(t)))−
∫ t

t−τ1(t)
A(s)h(x(s))ds

+K(t)U(T )

∫ T

t

K−1(s)A(s)
[
Q(s, x(s− τ2(s)))− (x(s)− h(x(s)))

−
∫ s

s−τ1(s)
A(u)h(x(u))du

]
ds

+K(t)U(T )

∫ T

t

K−1(s) [F (s)h(x(s− τ1(s))) +G(s, x(s), x(s− τ2(s)))] ds

+K(t)U(T )

∫ t+T

T

K−1(s)A(s)
[
Q(s, x(s− τ2(s)))− (x(s)− h(x(s)))

−
∫ s

s−τ1(s)
A(u)h(x(u))du

]
ds

+K(t)U(T )

∫ t+T

T

K−1(s) [F (s)h(x(s− τ1(s))) +G(s, x(s), x(s− τ2(s)))] ds.

(2.12)
By combining the two integrals of (2.12), we can easily obtain (2.7). By finding the
derivative of (2.7), we can obtain (1.2). The converse implication is easily obtained
and the proof is complete. □

We end this section by stating some theorems which will be useful for obtaining
our main results. Next we state Krasnoselskii’s fixed point theorem whose proof
can be found in [19].

Theorem 2.3. Let M be a closed convex non-empty subset of a Banach space
(X, ∥ · ∥). Suppose that A and B map M into X such that the following conditions
hold

(i) Ax+By ∈M for all x, y ∈M ;
(ii) A is continuous, and AM is contained in a compact set;
(iii) B is a contraction.



8 Y. LI, G. CHEN EJDE-2024/21

Then there is a z ∈M , with z = Az +Bz.

Theorem 2.4 (Contraction mapping principle). Let (X, ρ) a complete metric space
and let P : X → X. If there exists a constant α < 1, such that for x, y ∈ X we
have

ρ(Px, Py) ≤ αρ(x, y),

then there exists a unique point z ∈ X with Pz = z.

3. Existence and uniqueness of periodic solutions

In this section, we discuss the existence and uniqueness of periodic solution
of system (1.2) by using Krasnoselskii’s fixed point theorem and the contraction
mapping principle. To apply Theorem 2.3, we need to define a Banach space B, a
closed bounded convex subsetM of B and construct two mappings; one completely
continuous and the other a contraction. So we let (B, ∥ · ∥) = (PT , ∥ · ∥) and

M = {φ ∈ PT : ∥φ∥ ≤ L}, (3.1)

where L is a positive constant.
Define the mapping P : PT → PT by

(Pφ)(t)

= Q(t, φ(t− τ2(t)))−
∫ t

t−τ1(t)
A(s)h(φ(s))ds

+K(t)U(T )

∫ t+T

t

K−1(s)A(s)
[
Q(s, φ(s− τ2(s)))− (φ(s)− h(φ(s)))

−
∫ s

s−τ1(s)
A(u)h(φ(u))du

]
ds

+K(t)U(T )

∫ t+T

t

K−1(s)[F (s)h(φ(s− τ1(s)))

+G(s, φ(s), φ(s− τ2(s)))]ds.

(3.2)

Therefore, we express the above equation as

(Pφ)(t) = (Rφ)(t) + (Bφ)(t), (3.3)

where R,B : PT → PT are given by

(Rφ)(t) = Q(t, φ(t− τ2(t)))−
∫ t

t−τ1(t)
A(s)h(φ(s))ds

+K(t)U(T )

∫ t+T

t

K−1(s)A(s)
[
Q(s, φ(s− τ2(s)))

− (φ(s)− h(φ(s)))−
∫ s

s−τ1(s)
A(u)h(φ(u))du

]
ds,

(3.4)

and

(Bφ)(t) = K(t)U(T )

∫ t+T

t

K−1(s)[F (s)h(φ(s− τ1(s)))

+G(s, φ(s), φ(s− τ2(s)))]ds.

(3.5)

Lemma 3.1. Let R be defined by (3.4), and assume that (2.2)-(2.6) hold. Then R
is continuous and RM is contained in a compact set.



EJDE-2024/21 NONLINEAR SYSTEM OF NEUTRAL DIFFERENTIAL EQUATIONS 9

Proof. Firstly, by (2.4)-(2.6), we obtain

|Q(t, x)| ≤ |Q(t, x)−Q(t, 0) +Q(t, 0)| ≤ k1∥x∥+ |Q(t, 0)| , (3.6)

|G(t, x, y)| ≤ |G(t, x, y)−G(t, 0, 0) +G(t, 0, 0)|
≤ k2∥x∥+ k3∥y∥+ |G(t, 0, 0)| ,

(3.7)

|h(x)| ≤ k4∥x∥+ |h(0)| . (3.8)

Let R defined by (3.4). For φ ∈M , we have

(Rφ)(t) ≤ |Q(t, φ(t− τ2(t)))|+
∫ t

t−τ1(t)
|A||h(φ(s))|ds

+

∫ t+T

t

∣∣[K(s)U−1(T )K−1(t)
]−1∣∣ |A|[|Q(s, φ(s− τ2(s)))|

+ |φ(s)|+ |h(φ(s))|+
∫ s

s−τ1(s)
|A||h(φ(u))|du

]
ds

≤ k1L+ β + α|A|(k4L+ η) + cT |A|[k1L+ β + L+ k4L+ η

+ α|A|(k4L+ η)] = E,

(3.9)

where E is a constant, and

α = sup
t∈[0,T ]

|τ1(t)|, β = sup
t∈[0,T ]

|Q(t, 0)|, γ = sup
t∈[0,T ]

|G(t, 0, 0)|,

η = |h(0)|, c = sup
t∈[0,T ]

(
sup

s∈[t,t+T ]

∣∣[K(s)U−1(T )K−1(t)]−1
∣∣).

Next, we show that R is continuous in the supremum norm. Let φ1, φ2 ∈M , for
ϵ > 0, choose η = ϵ/∆, where

∆ = k1 + α|A|k4 + cT |A|k1 + cT |A|+ cT |A|k4 + cT |A|(α|A|k4) + cT |F |k4.

For ∥φ1 − φ2∥ < η, we obtain

|(Rφ1)(t)− (Rφ2)(t)|

≤ |Q(t, φ1(t− τ2(t)))−Q(t, φ2(t− τ2(t)))|+
∫ t

t−τ1(t)
|A| |h(φ1(s))− h(φ2(s))| ds

+

∫ t+T

t

∣∣[K(s)U−1(T )K−1(t)]−1
∣∣ |A| |Q(s, φ1(s− τ2(s)))−Q(s, φ2(s− τ2(s)))| ds

+

∫ t+T

t

∣∣[K(s)U−1(T )K−1(t)]−1
∣∣ |A||φ1(s)− φ2(s)|ds

+

∫ t+T

t

∣∣[K(s)U−1(T )K−1(t)]−1
∣∣ |A||h(φ1(s))− h(φ2(s))|ds

+

∫ t+T

t

∣∣[K(s)U−1(T )K−1(t)]−1
∣∣ |A|∫ s

s−τ1(s)
|A| |h(φ1(u))− h(φ2(u))| du ds

+

∫ t+T

t

∣∣[K(s)U−1(T )K−1(t)]−1
∣∣ |F | |h(φ1(s− τ1(s)))− h(φ2(s− τ1(s)))| ds

≤ k1∥φ1 − φ2∥+ α|A|k4∥φ1 − φ2∥+ cT |A|k1∥φ1 − φ2∥+ cT |A|∥φ1 − φ2∥
+ cT |A|k4∥φ1 − φ2∥+ cT |A| (α|A|k4∥φ1 − φ2∥) + cT |F |k4∥φ1 − φ2∥

=
[
k1 + α|A|k4 + cT |A|k1 + cT |A|+ cT |A|k4 + cT |A|(α|A|k4)



10 Y. LI, G. CHEN EJDE-2024/21

+ cT |F |k4
]
∥φ1 − φ2∥ < ϵ,

which shows the continuity of R.
Finally, we show that RM is contained in a compact set. Let φ ∈ M , then by

(3.9), we see that Rφ is uniformly bounded. Then, let φ ∈ M , without loss of
generality, we can pick ω < t such that t− ω < T . Then we have

|(Rφ)(t)− (Rφ)(ω)|

≤ |Q(t, φ(t− τ2(t)))−Q(ω, φ(ω − τ2(ω)))|+
∣∣∣ ∫ t

t−τ1(t)
A(s)h(φ(s))ds

−
∫ ω

ω−τ1(ω)
A(s)h(φ(s))ds

∣∣∣
+
∣∣∣ ∫ t+T

t

[K(s)U−1(T )K−1(t)]−1A(s)Q(s, φ(s− τ2(s)))ds

−
∫ ω+T

ω

[K(s)U−1(T )K−1(ω)]−1A(s)Q(s, φ(s− τ2(s)))ds
∣∣∣

+
∣∣∣ ∫ t+T

t

[K(s)U−1(T )K−1(t)]−1A(s)φ(s)ds

−
∫ ω+T

ω

[K(s)U−1(T )K−1(ω)]−1A(s)φ(s)ds
∣∣∣

+
∣∣∣ ∫ t+T

t

[K(s)U−1(T )K−1(t)]−1A(s)h(φ(s))ds

−
∫ ω+T

ω

[K(s)U−1(T )K−1(ω)]−1A(s)h(φ(s))ds
∣∣∣

+
∣∣∣ ∫ t+T

t

[K(s)U−1(T )K−1(t)]−1A(s)

∫ s

s−τ1(s)
A(u)h(φ(u)) du ds

−
∫ ω+T

ω

[K(s)U−1(T )K−1(ω)]−1A(s)

∫ s

s−τ1(s)
A(u)h(φ(u)) du ds

∣∣∣.
Since (3.6)-(3.8) hold, we have∣∣∣ ∫ t

t−τ1(t)
A(s)h(φ(s))ds−

∫ ω

ω−τ1(ω)
A(s)h(φ(s))ds

∣∣∣
≤ (k4L+ η)

(∫ t

ω

|A|ds+
∫ t−τ1(t)

ω−τ1(ω)
|A|ds

)
,

and ∣∣∣ ∫ t+T

t

[K(s)U−1(T )K−1(t)]−1A(s)Q(s, φ(s− τ2(s)))ds

−
∫ ω+T

ω

[K(s)U−1(T )K−1(ω)]−1A(s)Q(s, φ(s− τ2(s)))ds
∣∣

=
∣∣∣ ∫ ω

t

[K(s)U−1(T )K−1(t)]−1A(s)Q(s, φ(s− τ2(s)))ds

+

∫ t+T

ω+T

[K(s)U−1(T )K−1(t)]−1A(s)Q(s, φ(s− τ2(s)))ds



EJDE-2024/21 NONLINEAR SYSTEM OF NEUTRAL DIFFERENTIAL EQUATIONS 11

+

∫ ω+T

ω

[K(s)U−1(T )K−1(t)]−1 − [K(s)U−1(T )K−1(ω)]−1

×A(s)Q(s, φ(s− τ2(s)))ds
∣∣∣

≤ |K(T )− I|
∣∣∣ ∫ t

ω

[K(s)U−1(T )K−1(t)]−1A(s)Q(s, φ(s− τ2(s)))ds
∣∣∣

+
∣∣∣ ∫ ω+T

ω

(
[K(s)U−1(T )K−1(t)]−1 − [K(s)U−1(T )K−1(ω)]−1

)
×A(s)Q(s, φ(s− τ2(s)))ds

∣∣∣
≤ c|K(T )− I|(k1L+ β)

∫ t

ω

|A|ds+ (k1L+ β)

×
∫ T

0

|K(t)−K(ω)|
∣∣U(T )K(T )K−1(s)

∣∣ |A|ds,
and∣∣∣ ∫ t+T

t

[K(s)U−1(T )K−1(t)]−1A(s)φ(s)ds

−
∫ ω+T

ω

[K(s)U−1(T )K−1(ω)]−1A(s)φ(s)ds
∣∣∣

=
∣∣∣ ∫ ω

t

[K(s)U−1(T )K−1(t)]−1A(s)φ(s)ds

+

∫ t+T

ω+T

[K(s)U−1(T )K−1(t)]−1A(s)φ(s)ds

+

∫ ω+T

ω

∣∣[K(s)U−1(T )K−1(t)]−1 − [K(s)U−1(T )K−1(ω)]−1
∣∣A(s)φ(s)ds∣∣∣

≤ |K(T )− I|
∣∣∣ ∫ t

ω

[K(s)U−1(T )K−1(t)]−1A(s)φ(s)ds
∣∣∣

+
∣∣∣ ∫ ω+T

ω

(
[K(s)U−1(T )K−1(t)]−1 − [K(s)U−1(T )K−1(ω)]−1

)
A(s)φ(s)ds

∣∣∣
≤ c |K(T )− I|L

∫ t

ω

|A|ds+ L

∫ T

0

|K(t)−K(ω)|
∣∣U(T )K(T )K−1(s)

∣∣ |A|ds,
and∣∣∣ ∫ t+T

t

[K(s)U−1(T )K−1(t)]−1A(s)h(φ(s))ds

−
∫ ω+T

ω

[K(s)U−1(T )K−1(ω)]−1A(s)h(φ(s))ds
∣∣∣

=
∣∣∣ ∫ ω

t

[K(s)U−1(T )K−1(t)]−1A(s)h(φ(s))ds

+

∫ t+T

ω+T

[K(s)U−1(T )K−1(t)]−1A(s)h(φ(s))ds

+

∫ ω+T

ω

∣∣[K(s)U−1(T )K−1(t)]−1 − [K(s)U−1(T )K−1(ω)]−1
∣∣A(s)h(φ(s))ds∣∣∣



12 Y. LI, G. CHEN EJDE-2024/21

≤ |K(T )− I|
∣∣∣ ∫ t

ω

[K(s)U−1(T )K−1(t)]−1A(s)h(φ(s))ds
∣∣∣

+
∣∣∣ ∫ ω+T

ω

(
[K(s)U−1(T )K−1(t)]−1 − [K(s)U−1(T )K−1(ω)]−1

)
A(s)h(φ(s))ds

∣∣∣
≤ c

∣∣∣K(T )− I
∣∣∣(k4L+ η)

∫ t

ω

|A|ds+ (k4L+ η)

×
∫ T

0

|K(t)−K(ω)||U(T )K(T )K−1(s)||A|ds,

and ∣∣∣ ∫ t+T

t

[K(s)U−1(T )K−1(t)]−1A(s)

∫ s

s−τ1(s)
A(u)h(φ(u)) du ds

−
∫ ω+T

ω

[K(s)U−1(T )K−1(ω)]−1A(s)

∫ s

s−τ1(s)
A(u)h(φ(u)) du ds

∣∣∣
=

∣∣∣ ∫ ω

t

[K(s)U−1(T )K−1(t)]−1A(s)

∫ s

s−τ1(s)
A(u)h(φ(u)) du ds

+

∫ t+T

ω+T

[K(s)U−1(T )K−1(t)]−1A(s)

∫ s

s−τ1(s)
A(u)h(φ(u)) du ds

+

∫ ω+T

ω

[K(s)U−1(T )K−1(t)]−1 − [K(s)U−1(T )K−1(ω)]−1

×A(s)

∫ s

s−τ1(s)
A(u)h(φ(u)) du ds

∣∣∣
≤ |K(T )− I|

∣∣∣ ∫ t

ω

[K(s)U−1(T )K−1(t)]−1A(s)

∫ s

s−τ1(s)
A(u)h(φ(u)) du ds

∣∣∣
+
∣∣∣ ∫ ω+T

ω

(
[K(s)U−1(T )K−1(t)]−1 − [K(s)U−1(T )K−1(ω)]−1

)
×A(s)

∫ s

s−τ1(s)
A(u)h(φ(u)) du ds

∣∣∣
≤ |K(T )− I|α|A|(k4L+ η)

∫ t

ω

|A|ds+ α|A|(k4L+ η)

×
∫ T

0

|K(t)−K(ω)| |U(T )K(T )K−1(s)| |A|ds,

which implies

|(Rφ)(t)− (Rφ)(ω)|

≤ |Q(t, φ(t− τ2(t)))−Q(ω, φ(ω − τ2(ω)))|+ (k4L+ η)
(∫ t

ω

|A|ds+
∫ t−τ1(t)

ω−τ1(ω)
|A|ds

)
+ c|K(T )− I|(k1L+ β)

∫ t

ω

|A|ds+ (k1L+ β)

×
∫ T

0

|K(t)−K(ω)|
∣∣U(T )K(T )K−1(s)

∣∣ |A|ds



EJDE-2024/21 NONLINEAR SYSTEM OF NEUTRAL DIFFERENTIAL EQUATIONS 13

+ c |K(T )− I|L
∫ t

ω

|A|ds+ L

∫ T

0

|K(t)−K(ω)|
∣∣U(T )K(T )K−1(s)

∣∣ |A|ds
+ c |K(T )− I| (k4L+ η)

∫ t

ω

|A|ds+ (k4L+ η)

×
∫ T

0

|K(t)−K(ω)|
∣∣U(T )K(T )K−1(s)

∣∣ |A|ds
+ |K(T )− I|α|A|(k4L+ η)

∫ t

ω

|A|ds+ α|A|(k4L+ η)

×
∫ T

0

|K(t)−K(ω)|
∣∣U(T )K(T )K−1(s)

∣∣ |A|ds.
Then by the dominated convergence theorem |(Rφ)(t)− (Rφ)(ω)| → 0 as t−ω → 0
independently of φ ∈ M . Thus (Rφ) is equicontinuous. Hence by Ascoli-Arzela’s
theorem, RM is contained in a compact set. □

Lemma 3.2. Suppose (2.2)-(2.6) hold and

cT (k4|F |+ k2 + k3) < 1. (3.10)

If B is defined by (3.5), and F by (2.8), then B is a contraction.

Proof. Let B defined by (3.5), and φ,ψ ∈M . By (2.4)-(2.6), we have

|(Bφ)(t)− (Bψ)(t)|

=
∣∣∣ ∫ t+T

t

[K(s)U−1(T )K−1(t)]−1
[
F (s) (h(φ(s− τ1(s)))− h(ψ(s− τ1(s))))

+ (G(s, φ(s), φ(s− τ2(s)))−G(s, ψ(s), ψ(s− τ2(s))))
]
ds
∣∣∣

≤ cT (k4|F |+ k2 + k3) ∥φ− ψ∥.
The proof is complete. □

Theorem 3.3. Let the hypothesis of Lemmas 3.1 and 3.2 hold, M defined by (3.1).
If there exist a constant L > 0 such that

k1L+ β + α|A|(k4L+ η) + cT |A|[k1L+ β + L+ (k4L+ η) + α|A|(k4L+ η)]

+ cT [|F |(k4L+ η) + (k2 + k3)L+ γ] ≤ L,

then (1.2) has a T -periodic solution in M .

Proof. By Lemma 3.1, R is continuous and RM is contained in a compact set, by
Lemma 3.2, B is a contraction. Next, we will show that if φ,ψ ∈ M , we have
Rφ+Bψ ∈M . Let φ,ψ ∈M , we have

∥Rφ+Bψ∥ ≤ k1L+ β + α|A|(k4L+ η) + cT |A|[k1L+ β + L+ (k4L+ η)

+ α|A|(k4L+ η)] + cT [|F |(k4L+ η) + (k2 + k3)L+ γ] ≤ L.

Clearly, all the hypotheses of Krasnoselskii’s fixed point theorem are satisfied. Thus
there exists a fixed point z ∈M such that z = Az +Bz. By Lemma 2.2, this fixed
point is a solution of (1.2). Hence (1.2) has a T -periodic solution in M . □

Theorem 3.4. Suppose (2.4)-(2.6) hold. If

k1 + α|A|k4 + cT |A|k1 + cT |A|+ cT |A|k4 + cT |A|(α|A|k4)
+ cT |F |k4 + cT (k2 + k3) < 1,

(3.11)



14 Y. LI, G. CHEN EJDE-2024/21

then (1.2) has a unique T -periodic solution in PT .

Proof. Let the mapping P given by (3.2). For φ1, φ2 ∈ PT , we have

|(Pφ1)(t)− (Pφ2)(t)|

≤ |Q(t, φ1(t− τ2(t)))−Q(t, φ2(t− τ2(t)))|+
∫ t

t−τ1(t)
|A| |h(φ1(s))− h(φ2(s))| ds

+

∫ t+T

t

∣∣[K(s)U−1(T )K−1(t)]−1
∣∣ |A| |Q(s, φ1(s− τ2(s)))−Q(s, φ2(s− τ2(s)))| ds

+

∫ t+T

t

∣∣[K(s)U−1(T )K−1(t)]−1
∣∣ |A||φ1(s)− φ2(s)|ds

+

∫ t+T

t

∣∣[K(s)U−1(T )K−1(t)]−1
∣∣ |A||h(φ1(s))− h(φ2(s))|ds

+

∫ t+T

t

∣∣[K(s)U−1(T )K−1(t)]−1
∣∣ |A|∫ s

s−τ1(s)
|A| |h(φ1(u))− h(φ2(u))| du ds

+

∫ t+T

t

∣∣[K(s)U−1(T )K−1(t)]−1
∣∣ |F | |h(φ1(s− τ1(s)))− h(φ2(s− τ1(s)))| ds

+

∫ t+T

t

∣∣[K(s)U−1(T )K−1(t)]−1
∣∣ ∣∣G(s, φ1(s), φ1(s− τ2(s)))

−G(s, φ2(s), φ2(s− τ2(s)))
∣∣ds

≤ k1∥φ1 − φ2∥+ α|A|k4∥φ1 − φ2∥+ cT |A|k1∥φ1 − φ2∥+ cT |A|∥φ1 − φ2∥
+ cT |A|k4∥φ1 − φ2∥+ cT |A| (α|A|k4∥φ1 − φ2∥) + cT |F |k4∥φ1 − φ2∥
+ cT (k2 + k3)∥φ1 − φ2∥

= [k1 + α|A|k4 + cT |A|k1 + cT |A|+ cT |A|k4 + cT |A|(α|A|k4)
+ cT |F |k4 + cT (k2 + k3)]∥φ1 − φ2∥.

Since (3.11) holds, with the contraction mapping principle we complete the proof.
□

Note that, when h(x) ≡ x, Theorem 3.3 and 3.4 reduce to [14, Theorems 3.1 and
3.2].

4. Asymptotic stability of the zero solution

In this section, we study the asymptotic stability of the zero solution of the
nonlinear system

d

dt
x(t) = A(t)h(x(t− τ1(t))) +

d

dt
Q(t, x(t− τ2(t))) +G(t, x(t), x(t− τ2(t))), (4.1)

with the initial function

x(t) = ψ(t), t ∈ [m(t0), t0], (4.2)

where ψ ∈ C([m(t0), t0],Rn). A, h,Q,G, τ1, τ2 are defined as in the previous section,
and for t ≥ t0,

mj(t0) = inf{t− rj(t), t ≥ t0}, m(t0) = inf{mj(t0), j = 1, 2}.
We assume that

Q(t, 0) = G(t, 0, 0) = h(0) = 0. (4.3)



EJDE-2024/21 NONLINEAR SYSTEM OF NEUTRAL DIFFERENTIAL EQUATIONS 15

Now we obtain sufficient conditions for the asymptotic stability of the zero solu-
tion for system (4.1) by using contraction mapping principle. We define the space

Sψ =
{
φ ∈ C(R,Rn) : φ(t) = ψ(t) if m(t0) ≤ t ≤ t0,

φ(t) → 0 as t→ ∞, φ is bounded
}
.

Then, (Sψ, ∥ · ∥) is a complete metric space with the supremum norm ∥ · ∥.

Definition 4.1. For each initial value (t0, ψ) ∈ (0,∞)× Sψ, a function x is called
a solution of (4.1) associated with (t0, ψ) if x ∈ C([m(t0),∞),Rn) satisfies (4.1)
for almost all t ≥ t0 and x = ψ for t ≤ t0. Such a solution is denoted by x(t) =
x(t, t0, ψ).

Definition 4.2. If Φ(t) is a fundamental matrix solution for system (2.1), then
Φ(t, r) := Φ(t)Φ−1(r) is the state transition matrix. Also, the state transition
matrix satisfies the Chapman-Kolmogorov identities

Φ(r, r) = I, Φ(t, s) Φ(s, r) = Φ(t, r),

Φ−1(t, s) = Φ(s, t),
∂Φ(t, s)

∂s
= −Φ(t, s)A(s).

In our analysis we use the fundamental matrix solution of (2.1) to transform
(4.1) into an integral equation. Then we employ the contraction mapping principle
to show the asymptotic stability of the zero solution of (4.1).

Lemma 4.3. x is a solution of the (4.1) if and only if

x(t) = Q(t, x(t− τ2(t)))−
∫ t

t−τ1(t)
A(s)h(x(s))ds

+Φ(t, t0)
[
x(t0) +

∫ t0

t0−τ1(t0)
A(s)h(x(s))ds−Q(t0, x(t0 − τ2(t0)))

]
+

∫ t

t0

Φ(t, s)
{
F (s)h(x(s))−A(s)[x(s)− h(x(s))] +A(s)Q(s, x(s− τ2(s)))

−A(s)

∫ t

s−τ1(s)
A(u)h(x(u))du+G(s, x(s), x(s− τ2(s)))

}
ds,

(4.4)
where

F (t) = A(t)− (1− τ ′1(t))A(t− τ1(t)).

Proof. Let x be a solution of (4.1) and Φ(t) is a fundamental matrix solution of
(2.1). We rewrite (4.1) as

d

dt

[
x(t) +

∫ t

t−τ1(t)
A(s)h(x(s))ds−Q(t, x(t− τ2(t)))

]
= A(t)

[
x(t) +

∫ t

t−τ1(t)
A(s)h(x(s))ds−Q(t, x(t− τ2(t)))

]
+ F (t)h(x(t− τ1(t)))−A(t)[x(t)− h(x(t))] +A(t)Q(t, x(t− τ2(t)))

−A(t)

∫ t

t−τ1(t)
A(s)h(x(s))ds+G(t, x(t), x(t− τ2(t))),

where F (t) = A(t)− (1− τ ′1(t))A(t− τ1(t)).



16 Y. LI, G. CHEN EJDE-2024/21

Defining a new function

z(t) = Φ−1(t)
[
x(t) +

∫ t

t−τ1(t)
A(s)h(x(s))ds−Q(t, x(t− τ2(t)))

]
,

we have

d

dt
z(t) =

d

dt
Φ−1(t)

[
x(t) +

∫ t

t−τ1(t)
A(s)h(x(s))ds−Q(t, x(t− τ2(t)))

]
+Φ−1(t)

d

dt

[
x(t) +

∫ t

t−τ1(t)
A(s)h(x(s))ds−Q(t, x(t− τ2(t)))

]
.

From Definition 4.2, it follows that d
dtΦ

−1(t) = −Φ−1(t)A(t). Then

d

dt
z(t) =

d

dt
Φ−1(t)

[
x(t) +

∫ t

t−τ1(t)
A(s)h(x(s))ds−Q(t, x(t− τ2(t)))

]
+Φ−1(t)

d

dt

[
x(t) +

∫ t

t−τ1(t)
A(s)h(x(s))ds−Q(t, x(t− τ2(t)))

]
= −Φ−1(t)A(t)

[
x(t) +

∫ t

t−τ1(t)
A(s)h(x(s))ds−Q(t, x(t− τ2(t)))

]
+Φ−1(t)

d

dt

[
x(t) +

∫ t

t−τ1(t)
A(s)h(x(s))ds−Q(t, x(t− τ2(t)))

]
= Φ−1(t)

{
F (t)h(x(t− τ1(t)))−A(t) [x(t)− h(x(t))] +A(t)Q(t, x(t− τ2(t)))

−A(t)

∫ t

t−τ1(t)
A(s)h(x(s))ds+G(t, x(t), x(t− τ2(t)))

}
.

Also note that

z(t0) = Φ−1(t0)
[
x(t0) +

∫ t0

t0−τ1(t0)
A(s)h(x(s))ds−Q(t0, x(t0 − τ2(t0)))

]
.

Integrating from t0 to t, we have

z(t)− z(t0)

=

∫ t

t0

Φ−1(s)
{
F (s)h(x(s− τ1(s)))−A(s) [x(s)− h(x(s))] +A(s)Q(s, x(s− τ2(s)))

−A(s)

∫ s

s−τ1(s)
A(u)h(x(u))du+G(s, x(s), x(s− τ2(s)))

}
ds.

This yields

Φ−1(t)
[
x(t) +

∫ t

t−τ1(t)
A(s)h(x(s))ds−Q(t, x(t− τ2(t)))

]
= Φ−1(t0)

[
x(t0) +

∫ t0

t0−τ1(t0)
A(s)h(x(s))ds−Q(t0, x(t0 − τ2(t0)))

]
+

∫ t

t0

Φ−1(s)
{
F (s)h(x(s− τ1(s)))−A(s) [x(s)− h(x(s))]

+A(s)Q(s, x(s− τ2(s)))



EJDE-2024/21 NONLINEAR SYSTEM OF NEUTRAL DIFFERENTIAL EQUATIONS 17

−A(s)

∫ s

s−τ1(s)
A(u)h(x(u))du+G(s, x(s), x(s− τ2(s)))

}
ds,

which yields

x(t) = Q(t, x(t− τ2(t)))−
∫ t

t−τ1(t)
A(s)h(x(s))ds

+Φ(t, t0)
[
x(t0) +

∫ t0

t0−τ1(t0)
A(s)h(x(s))ds−Q(t0, x(t0 − τ2(t0)))

]
+

∫ t

t0

Φ(t, s)
{
F (s)h(x(s− τ1(s)))−A(s) [x(s)− h(x(s))]

+A(s)Q(s, x(s− τ2(s)))

−A(s)

∫ s

s−τ1(s)
A(u)h(x(u))du+G(s, x(s), x(s− τ2(s)))

}
ds.

The converse implication is easily obtained. the proof is complete. □

To obtain a sufficient condition of the asymptotic stability of the zero solution
of (4.1), we assume that

Φ(t) → 0, t→ ∞, (4.5)

t− τ1(t) → ∞, t→ ∞, (4.6)

t− τ2(t) → ∞, t→ ∞, (4.7)

and that there is β > 0, such that

k1 + k4

∫ t

t−τ1(t)
|A|ds+

∫ t

t0

|Φ(t, s)|
[
k4|F |+ (1 + k1 + k4)|A|

+ k4|A|
∫ s

s−τ1(s)
|A|du+ k2 + k3

]
ds ≤ β < 1, for t ≥ t0.

(4.8)

Theorem 4.4. Let us assume that (2.4)-(2.6) and (4.5)-(4.8) hold. Then every
solution x(t, t0, ψ) of (4.1), with small continuous initial function ψ, is bounded
and asymptotically stable.

Proof. We define the mapping P based on Lemma 4.3:

(Pφ)(t) = Q(t, φ(t− τ2(t)))−
∫ t

t−τ1(t)
A(s)h(φ(s))ds

+Φ(t, t0)
[
φ(t0) +

∫ t0

t0−τ1(t0)
A(s)h(φ(s))ds−Q(t0, φ(t0 − τ2(t0)))

]
+

∫ t

t0

Φ(t, s)
{
F (s)h(φ(s)− τ1(s))−A(s)[φ(s)− h(φ(s))]

+A(s)Q(s, φ(s− τ2(s)))

−A(s)

∫ t

s−τ1(s)
A(u)h(φ(u))du+G(s, φ(s), φ(s− τ2(s)))

}
ds.

Since Q,G, h are continuous, it is easy to show that P is continuous. Let ψ be a
small given continuous initial function with ∥ψ∥ < δ(δ > 0). Since φ ∈ Sψ, there is



18 Y. LI, G. CHEN EJDE-2024/21

a constant L > 0 such that ∥φ∥ ≤ L. By choosing a suitable δ, we have

|(Pφ)(t)| ≤ |Q(t, φ(t− τ2(t)))|+
∫ t

t−τ1(t)
|A||h(φ(s))|ds

+ |Φ(t, t0)|
[
|φ(t0)|+

∫ t0

t0−τ1(t0)
|A||h(φ(s))|ds+ |Q(t0, φ(t0 − τ2(t0)))|

]
+

∫ t

t0

|Φ(t, s)|
{
|F ||h(φ(s)− τ2(s))|+ |A|[|φ(s)|+ |h(φ(s))|]

+ |A||Q(s, φ(s− τ2(s)))|

+ |A|
∫ s

s−τ1(s)
|A(u)||h(φ(u))|du+ |G(s, φ(s), φ(s− τ2(s)))|

}
ds

≤ k1L+ k4L

∫ t

t−τ1(t)
|A|ds+ |Φ|δ(1 + k4

∫ t0

t0−τ1(t0)
|A|ds+ k1)

+ L

∫ t

t0

|Φ(t, s)|
[
k4|F |+ (1 + k1 + k4)|A|

+ k4|A|
∫ s

s−τ1(s)
|A|du+ (k2 + k3)

]
≤ βL+ |Φ|δ(1 + k4

∫ t0

t0−τ1(t0)
|A|ds+ k1),

which implies that Pφ is bounded.
Then we show that (Pφ)(t) → 0 as t → ∞. By (4.3) and (4.5)-(4.7), we can

easily have that Q(t, φ(t− τ2(t))) → 0,
∫ t
t−τ1(t)A(s)h(φ(s))ds→ 0 and

Φ(t, t0)
[
φ(t0) +

∫ t0

t0−τ1(t0)
A(s)h(φ(s))ds−Q(t0, φ(t0 − τ2(t0)))

]
→ 0 as t→ ∞.

Let ϵ > 0 be given, then there exist a t1 > t0 such that for t > t1, |φ(t−τ1(t))| < ϵ.
By (4.5), there exist a t2 > t1 such that for t > t2 implies |Φ(t, t2)| < ϵ

βL . Thus for

t > t2, we have∫ t

t0

|Φ(t, s)|
{
|F ||h(φ(s)− τ2(s))|+ |A|[|φ(s)|+ |h(φ(s))|] + |A||Q(s, φ(s− τ2(s)))|

+ |A|
∫ s

s−τ1(s)
|A||h(φ(u))|du+ |G(s, φ(s), φ(s− τ2(s)))|

}
ds

≤ L

∫ t1

t0

|Φ(t, s)|
[
k4|F |+ (1 + k1 + k4)|A|+ k4|A|

∫ s

s−τ1(s)
|A|du+ (k2 + k3)

]
ds

+ ϵ

∫ t

t1

|Φ(t, s)|
[
k4|F |+ (1 + k1 + k4)|A|+ k4|A|

∫ s

s−τ1(s)
|A|du+ (k2 + k3)

]
ds

≤ L|Φ(t, t2)|
∫ t1

t0

|Φ(t2, s)|
[
k4|F |+ (1 + k1 + k4)|A|

+ k4|A|
∫ s

s−τ1(s)
|A|du+ (k2 + k3)

]
ds+ βϵ

≤ βL|Φ(t, t2)|+ βϵ < ϵ+ βϵ.



EJDE-2024/21 NONLINEAR SYSTEM OF NEUTRAL DIFFERENTIAL EQUATIONS 19

Hence, (Pφ)(t) → 0 as t→ ∞.
Now we show that P is a contraction under the supremum norm. Let φ1, φ2 ∈

Sψ, then

|(Pφ1)(t)− (Pφ2)(t)|

≤ |Q(t, φ1(t− τ2(t)))−Q(t, φ2(t− τ2(t)))|+
∫ t

t−τ1(t)
|A| |h(φ1(s))− h(φ2(s))| ds

+

∫ t

t0

|Φ(t, s)|
[
k4|F |+ (1 + k1 + k4)|A|

+ k4|A|
∫ s

s−τ1(s)
|A|du+ k2 + k3

]
ds∥φ1 − φ2∥

≤ β∥φ1 − φ2∥.

Since β < 1 as defined by (4.8), the contraction mapping principle implies that P
has a unique fixed point in Sψ which satisfies (4.1).

Lastly, we need to show that the zero solution of (4.1) is stable. Choose a δ such
that 2δK + βϵ < ϵ, where K = supt∈[t0,∞) Φ(t, t0), and β is defined by (4.8). For

∥ψ∥ ≤ δ, we claim that |x(t)| < ϵ. Suppose that there exists a t′ > t0 such that
|x(t′)| ≥ ϵ, and t∗ = inf {t′ : x(t′) ≥ ϵ}. By the integral representation of x(t), we
have

|x(t∗)|

≤ k1ϵ+ k4L

∫ t

t−τ1(t)
|A|ds+ |Φ(t, t0)|δ(1 + k4

∫ t0

t0−τ1(t0)
|A|hds+ k1)

+ ϵ

∫ t

t0

|Φ(t, s)|
[
k4|F |+ (1 + k1 + k4)|A|+ k4|A|

∫ s

s−τ1(s)
|A|du+ (k2 + k3)

]
≤ 2δK + βϵ < ϵ,

which contradicts the definition of t∗. Therefore, the zero solution of (4.1) is stable.
Hence, the fixed point is bounded and asymptotically stable. □

5. An example

We consider the two-dimensional system(
x′1(t)
x′2(t)

)
=

(
p(t) 1
0 q(t)

)(
h(x1(t− τ1(t)))
h(x2(t− τ1(t)))

)
+
d

dt

(
0

V (t, x1(t− τ2(t)))

)
+

(
0

W (t, x1(t− τ2(t)))

)
,

(5.1)

where p and q are positive periodic continuous functions with periodic T . The
functions h : R → R, V : R × R → R, and W : R × R × R → R are continuous in
their respective arguments, τ1(·), τ2(·) satisfy (2.2).

Functions V (t, x) and W (t, x, y) are periodic in t with period T . They are also
globally Lipschitz continuous in x and y respectively. That is

V (t+ T, x) = V (t, x), W (t+ T, x, y) =W (t, x, y), (5.2)

and there are positive constants k1, k2, k3 such that

|V (t, x)− V (t, y)| ≤ k1∥x− y∥, (5.3)



20 Y. LI, G. CHEN EJDE-2024/21

|W (t, x, y)−W (t, z, w)| ≤ k2∥x− z∥+ k3∥y − w∥, (5.4)

|h(x)− h(y)| ≤ k4∥x− y∥. (5.5)

Let q(t) = p(t) = −1, h(x) = 1
3 sin(x + π

3 ), τ1(t) = π
50 sin

2(πt), τ2(·) is a non-

negative and continuous function with period of T , V (t, x) = 1
5sin(2πt) sin(x+

π
6 ),

W (t, x, y) = 1
8 cos(2πt) sin(x) +

1
7 sin(y +

π
3 ). Consider the Banach space (C1, ∥ · ∥)

C1 = {φ ∈ C(R,R), φ(t+ 1) = φ, t ∈ R},

and the closed bounded convex subset M = {φ ∈ C1 : ∥φ∥ ≤ π}. Then for
x, y, z, w ∈M , we have

|V (t, x)− V (t, y)| ≤ 1

5
∥x− y∥, |W (t, x, y)−W (t, z, w)| ≤ 1

8
∥x− z∥+ 1

7
∥y − w∥,

|h(x)− h(y)| ≤ 1

3
∥x− y∥, α = sup

t∈[0,T ]

|τ1(t)| =
π

50
,

β = sup
t∈[0,T ]

|V (t, 0)| = 1

10
, γ = sup

t∈[0,T ]

|W (t, 0, 0)| = 1

7
,

η = |h(0)| =
√
3

6
, |F | = |A(t)− (1− τ ′1(t))A(1− τ1(t))| < 0.04, c ≤ 0.34.

Consequently,

k1L+ β + α|A|(k4L+ η) + cT |A|[k1L+ β + L+ (k4L+ η) + α|A|(k4L+ η)]

+ cT [|F |(k4L+ η) + (k2 + k3)L+ γ]

≤ π

5
+

1

10
+

π

50
(
π

3
+

√
3

6
) + 0.34

[π
5
+

1

10
+ π +

π

3
+

√
3

6
+

π

50
(
π

3
+

√
3

6
)
]

+ 0.34[0.04(
π

3
+

√
3

6
) + (

1

7
+

1

8
)π +

1

7
] ≤ π.

Then (5.1) has a 1-periodic solution in M . Moreover,

k1 + α|A|k4 + cT |A|k1 + cT |A|+ cT |A|k4 + cT |A|(α|A|k4) + cT |F |k4
+ cT (k2 + k3)

≤ 1

5
+

π

50

1

3
+ 0.34

1

5
+ 0.34 + 0.34

1

3
+ 0.34

π

50
(
π

50

1

3
) + 0.34(0.04)

1

3
+ 0.34

15

56
≤ 1.

Then (5.1) has a unique 1-periodic solution in M .

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Yang Li
School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China

Email address: yang@my.swjtu.edu.cn

Guiling Chen
School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China

Email address: guiling@swjtu.edu.cn


	1. Introduction
	2. Preliminaries
	3. Existence and uniqueness of periodic solutions
	4. Asymptotic stability of the zero solution
	5. An example
	References