CUBO A Mathematical Journal
Vol.20, No¯ 3, (49–63). October 2018

http: // dx. doi. org/ 10. 4067/ S0719-06462018000300049

Study of global asymptotic stability in nonlinear neutral
dynamic equations on time scales

Abdelouaheb Ardjouni
1,2

and Ahcene Djoudi
2

1Department of Mathematics and Informatics,

University of Souk Ahras,

P.O. Box 1553, Souk Ahras, 41000, Algeria.
2Applied Mathematics Lab, Faculty of Sciences, Department of Mathematics,

University of Annaba,

P.O. Box 12, Annaba 23000, Algeria.

abd ardjouni@yahoo.fr, adjoudi@yahoo.com

ABSTRACT

This paper is mainly concerned the global asymptotic stability of the zero solution of

a class of nonlinear neutral dynamic equations in C1rd. By converting the nonlinear

neutral dynamic equation into an equivalent integral equation, our main results are

obtained via the Banach contraction mapping principle. The results obtained here

extend the work of Yazgan, Tunc and Atan [17].

RESUMEN

Este art́ıculo está mayormente interesado en la estabilidad global asintótica de la

solución cero de una clase de ecuaciones nolineales neutrales dinámicas en C1rd. Trans-

formando la ecuación nolineal neutral dinámica en una ecuación integral equivalente,

nuestros resultados principales son obtenidos a través del principio de la aplicación

contractiva de Banach. Los resultados obtenidos aqúı son una extensión del trabajo de

Yazgan, Tunc y Atan [17].

Keywords and Phrases: Fixed points, neutral dynamic equations, asymptotic stability, time

scales.

http://dx.doi.org/10.4067/S0719-06462018000300049


50 Abdelouaheb Ardjouni and Ahcene Djoudi CUBO
20, 3 (2018)

2010 AMS Mathematics Subject Classification: 34K20, 34K30, 34K40.



CUBO
20, 3 (2018)

Study of global asymptotic stability in nonlinear neutral dynamic . . . 51

The concept of time scales analysis is a fairly new idea. In 1988, it was introduced by the

German mathematician Stefan Hilger in his Ph.D. thesis [13]. It combines the traditional areas of

continuous and discrete analysis into one theory. After the publication of two textbooks in this

area by Bohner and Peterson [7] and [8], more and more researchers were getting involved in this

fast-growing field of mathematics. The study of dynamic equations brings together the traditional

research areas of differential and difference equations. It allows one to handle these two research

areas at the same time, hence shedding light on the reasons for their seeming discrepancies. In

fact, many new results for the continuous and discrete cases have been obtained by studying the

more general time scales case (see [1, 3, 4, 6, 14] and the references therein).

There is no doubt that the Lyapunov method have been used successfully to investigate sta-

bility properties of wide variety of ordinary, functional and partial equations. Nevertheless, the

application of this method to problem of stability in differential equations with delay has encoun-

tered serious difficulties if the delay is unbounded or if the equation has unbounded term. It has

been noticed that some of theses difficulties vanish by using the fixed point technic. Other advan-

tages of fixed point theory over Lyapunov’s method is that the conditions of the former are average

while those of the latter are pointwise (see [2, 5, 9, 10, 11, 12, 15, 17] and references therein).

In paper, we consider the following neutral nonlinear dynamic equations with variable delays

given by

x△ (t) = −a (t) xσ (t) + b (t) g (x (t)) + c (t) f
(
x△̃ (t − τ1 (t))

)
+ q (t, x (t) , x (t − τ2 (t))) , (0.1)

with the initial condition

x (t) = ϕ (t) , t ∈ [dt0, t0] ∩ T,

where

dt0 = inf
t∈[t0,∞)∩T

{t − τ1 (t) , t − τ2 (t)} ,

for each t0 ∈ [0, ∞) ∩ T and T is an unbounded above and below time scale and such that t0 ∈ T.

Our results are obtained with no need of further assumptions on the delta-differentiable of the

neutral coefficient c and the twice delta-differentiable of τi with τ
△

i
(t) 6= 1 for t ∈ [0, ∞) ∩ T, so

that for a given initial function ϕ ∈ Φt0 a mapping P for (0.1) is constructed in such a way to map

a, carefully chosen, closed convex nonempty subset D of a Banach space X into itself on which P

is a contraction mapping possessing a fixed point. This procedure will enable us to establish and

prove by means of the contraction mapping theorem ([16], p. 2) the global asymptotic stability

in C1rd for the zero solution of (0.1) with a less restrictive conditions. In the special case T = R,

Yazgan, Tunc and Atan in [17] show that the zero solution of (0.1) is globally asymptotically stable

in C1rd by using the contraction mapping theorem. Then, the results obtained here extend the work

of Yazgan, Tunc and Atan [17].



52 Abdelouaheb Ardjouni and Ahcene Djoudi CUBO
20, 3 (2018)

1 Preliminaries

In this section, we consider some advanced topics in the theory of dynamic equations on a time

scales. Again, we remind that for a review of this topic we direct the reader to the monographs of

Bohner and Peterson [7] and [8].

A time scale T is a closed nonempty subset of R. For t ∈ T the forward jump operator

σ, and the backward jump operator ρ, respectively, are defined as σ (t) = inf {s ∈ T : s > t} and

ρ (t) = sup {t ∈ T : s < t}. These operators allow elements in the time scale to be classified as

follows. We say t is right scattered if σ (t) > t and right dense if σ (t) = t. We say t is left

scattered if ρ (t) < t and left dense if ρ (t) = t. The graininess function µ : T → [0, ∞), is

defined by µ (t) = σ (t) − t and gives the distance between an element and its successor. We set

inf ∅ = sup T and sup ∅ = inf T. If T has a left scattered maximum M, we define Tk = T� {M}.

Otherwise, we define Tk = T. If T has a right scattered minimum m, we define Tk = T� {m}.

Otherwise, we define Tk = T.

Let t ∈ Tk and let f : T → R. The delta derivative of f (t), denoted f△ (t), is defined to be

the number (when it exists), with the property that, for each ǫ > 0, there is a neighborhood U of

t such that ∣∣f (σ (t)) − f (s) − f△ (t) [σ (t) − s]
∣∣ ≤ ǫ |σ (t) − s| ,

for all s ∈ U. If T = R then f△ (t) = f′ (t) is the usual derivative. If T = Z then f△ (t) = △f (t) =

f (t + 1) − f (t) is the forward difference of f at t.

A function f is right dense continuous (rd-continuous), f ∈ Crd = Crd (T, R), if it is continuous

at every right dense point t ∈ T and its left-hand limits exist at each left dense point t ∈ T. The

function f : T → R is differentiable on Tk provided f△ (t) exists for all t ∈ Tk. f ∈ C1rd = C
1
rd (T, R)

if f△ ∈ Crd (T, R).

We are now ready to state some properties of the delta-derivative of f. Note fσ (t) = f (σ (t)).

Theorem 1.1 ([7, Theorem 1.20]). Assume f, g : T → R are differentiable at t ∈ Tk and let α be

a scalar.

(i) (f + g)
△
(t) = g△ (t) + f△ (t).

(ii) (αf)
△
(t) = αf△ (t).

(iii) The product rules

(fg)
△
(t) = f△ (t) g (t) + fσ (t) g△ (t) ,

(fg)
△
(t) = f (t) g△ (t) + f△ (t) gσ (t) .

(iv) If g (t) gσ (t) 6= 0 then

(
f

g

)△
(t) =

f△ (t) g (t) − f (t) g△ (t)

g (t) gσ (t)
.



CUBO
20, 3 (2018)

Study of global asymptotic stability in nonlinear neutral dynamic . . . 53

The next theorem is the chain rule on time scales ([7, Theorem 1.93]).

Theorem 1.2 (Chain Rule). Assume ν : T → R is strictly increasing and T̃ := ν (T) is a time

scale. Let ω : T̃ → R. If ν△ (t) and ω△̃ (ν (t)) exist for t ∈ Tk, then (ω ◦ ν)
△

=
(
ω△̃ ◦ ν

)
ν△.

In the sequel we will need to differentiate and integrate functions of the form f (t − τ (t)) =

f (ν (t)) where, ν (t) := t − τ (t). Our next theorem is the substitution rule ([7, Theorem 1.98]).

Theorem 1.3 (Substitution). Assume ν : T → R is strictly increasing and T̃ := ν (T) is a time

scale. If f : T → R is rd-continuous function and ν is differentiable with rd-continuous derivative,

then for a, b ∈ T,
∫b

a

f (t) ν△ (t) △t =

∫ν(b)

ν(a)

(
f ◦ ν−1

)
(s) △̃s.

A function p : T → R is said to be regressive provided 1 + µ (t) p (t) 6= 0 for all t ∈ Tk. The

set of all regressive rd-continuous function f : T → R is denoted by R. The set of all positively

regressive functions R+, is given by R+ = {f ∈ R : 1 + µ (t) f (t) > 0 for all t ∈ T}.

Let p ∈ R and µ (t) 6= 0 for all t ∈ T. The exponential function on T is defined by

ep (t, s) = exp

(∫t

s

1

µ (z)
log (1 + µ (z) p (z)) ∆z

)
.

It is well known that if p ∈ R+, then ep (t, s) > 0 for all t ∈ T. Also, the exponential function

y (t) = ep (t, s) is the solution to the initial value problem y
△ = p (t) y, y (s) = 1. Other properties

of the exponential function are given by the following lemma.

Lemma 1.4 ([7, Theorem 2.36]). Let p, q ∈ R. Then

(i) e0 (t, s) = 1 and ep (t, t) = 1,

(ii) ep (σ (t) , s) = (1 + µ (t) p (t)) ep (t, s),

(iii) 1
ep(t,s)

= e⊖p (t, s), where ⊖p (t) = −
p(t)

1+µ(t)p(t)
,

(iv) ep (t, s) =
1

ep(s,t)
= e⊖p (s, t),

(v) ep (t, s) ep (s, r) = ep (t, r),

(vi) e
△
p (., s) = pep (., s) and

(
1

ep(.,s)

)△
= −

p(t)

eσp(.,s)
.

Lemma 1.5 ([1]). If p ∈ R+, then

0 < ep (t, s) ≤ exp

(∫t

s

p (u) △u

)
, ∀t ∈ T.

2 Global asymptotic stability

In this section, we shall study the global asymptotic stability in C1rd of the zero solution to (0.1).

We introduce the following hypotheses.



54 Abdelouaheb Ardjouni and Ahcene Djoudi CUBO
20, 3 (2018)

(H1) a, b, c ∈ Crd ([0, ∞) ∩ T, R), g, f ∈ C (R, R), q ∈ Crd ([0, ∞) ∩ T × R×R, R), τi ∈

Crd([0, ∞) ∩ T, (0, ∞) ∩ T) and (id − τi) ([0, ∞) ∩ T) is closed with t − τi (t) → ∞ as t → ∞,

i = 1, 2.

(H2) For t ∈ [0, ∞) ∩ T, g (0) = f (0) = q (t, 0, 0) = 0, and there exist Lg, Lf > 0, L1, L2 ∈

Crd ([0, ∞) ∩ T, (0, ∞)) such that

|g (x1) − g (x2)| ≤ Lg |x1 − x2| ,

|f (x1) − f (x2)| ≤ Lf |x1 − x2| ,

|q (t, x1, y1) − q (t, x2, y2)| ≤ L1 (t) |x1 − x2| + L2 (t) |y1 − y2| ,

for any xi, yi ∈ R, i = 1, 2.

(H3) a ∈ R
+ is bounded on [0, ∞) ∩ T and lim

t→∞
inf

∫t
0

1
µ(s)

log (1 + µ (s) a (s)) ∆s > −∞.

(H4) There exists α ∈ (0, 1) such that for t ∈ [0, ∞) ∩ T,

∫t

0

e⊖a (t, u) [Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] ∆u ≤ α,

and

|a (t)|

∫σ(t)

0

e⊖a (σ (t) , u) [Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] ∆u

+ Lg |b (t)| + Lf |c (t)| + L1 (t) + L2 (t) ≤ α.

For each t0 ∈ [0, ∞) ∩ T denote C
1
rd (t0) = C

1
rd ([dt0, t0] ∩ T, R) with the norm defined by

|x|
t0

= max
t∈[dt0 ,t0]∩T

{
|x (t)| ,

∣∣x△ (t)
∣∣}

for x ∈ C1rd (t0). In addition, denote

Φt0 = {ϕ ∈ C
1
rd (t0) : ϕ

△
(t0) = −a (t0) ϕ

σ
(t0) + b (t0) g (ϕ (t0))

+ c (t0) f
(
ϕ△̃ (t0 − τ1 (t0))

)
+ q (t0, ϕ (t0) , ϕ (t0 − τ2 (t0)))}.

For each t0 ∈ [0, ∞)∩T, we always assume that the initial function for (0.1) is of the type ϕ ∈ Φt0.

For convenience of stating our main result, we shall give the following definitions.

Definition 2.1. For each (t0, ϕ) ∈ [0, ∞) ∩ T×Φt0, x is said to be a solution of (0.1) through

(t0, ϕ) if x ∈ C
1
rd ([dt0, ∞) ∩ T) satisfies (0.1) on [t0, ∞)∩T and x (t) = ϕ (t) for t ∈ [dt0, t0]∩T.

We denote such a solution by x (t) = x (t, t0, ϕ).

Definition 2.2. (i) The zero solution of (0.1) is said to be stable in C1rd if, for any t0 ∈ [0, ∞)∩T,

ε > 0 there is a δ = δ (ε, t0) such that ϕ ∈ Φt0 and |ϕ|t0 < δ implies

max
s∈[dt0 ,t]∩T

{
|x (s)| ,

∣∣x△ (s)
∣∣} < ε,



CUBO
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Study of global asymptotic stability in nonlinear neutral dynamic . . . 55

for t ∈ [t0, ∞) ∩ T.

(ii) The zero solution of (0.1) is said to be globally asymptotically stable in C1rd if it is stable

in C1rd, and for any t0 ∈ [0, ∞) ∩ T, ϕ ∈ Φt0 implies

lim
t→∞

x (t, t0, ϕ) = lim
t→∞

x△ (t, t0, ϕ) = 0.

In view of the definition of solution of (0.1), it is clear that the conditions imposed on the

initial functions are very natural. From the above assumptions, it is easy to see that for each

(t0, ϕ) ∈ [0, ∞) ∩ T×Φt0, there exists a unique solution x (t) = x (t, t0, ϕ) of (0.1) defined on

[dt0, ∞) ∩ T. By (H2), (0.1) has the zero solution.

Theorem 2.3. Assume that (H1) − (H4) hold. Then the zero solution of (0.1) is globally asymp-

totically stable in C1rd if and only if

∫t

0

1

µ (s)
log (1 + µ (s) a (s)) ∆s → ∞ as t → ∞. (2.1)

Proof. (i) Suppose that (2.1) holds. For any t0 ∈ [0, ∞) ∩ T, let

X =
{
x ∈ C1rd ([dt0, ∞) ∩ T) : lim

t→∞
x (t) = lim

t→∞
x△ (t) = 0

}
,

with the norm defined by

‖x‖
t0

= max
t∈[dt0 ,∞)∩T

{
|x (t)| ,

∣∣x△ (t)
∣∣} ,

for x ∈ X. Since X is a closed vectorial subspace of C1rd ([dt0, ∞) ∩ T) and C
1
rd

(
[dt0, ∞) ∩ T, ‖.‖t0

)

is a Banach space, then
(
X, ‖.‖

t0

)
is also a Banach space. For any ϕ ∈ Φt0, let

D = {x ∈ X : x (t) = ϕ (t) for t ∈ [dt0, t0] ∩ T} .

It is easy to see that D is a nonempty, closed subset of X.

Multiply both sides of (0.1) by ea (t, t0) and then integrate from t0 to t to obtain

∫t

t0

[x (u) ea (u, t0)]
∆
∆u

=

∫t

t0

ea (u, t0)
[
b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)

+q (u, x (u) , x (u − τ2 (u)))] ∆u.

As a consequence, we arrive at

x (t) ea (t, t0) − x (t0)

=

∫t

t0

ea (u, t0)
[
b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)

+q (u, x (u) , x (u − τ2 (u)))] ∆u.



56 Abdelouaheb Ardjouni and Ahcene Djoudi CUBO
20, 3 (2018)

By dividing both sides of the above equation by ea (t, t0), we obtain

x (t) = ϕ (t0) e⊖a (t, t0) +

∫t

t0

e⊖a (t, u)
[
b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)

+q (u, x (u) , x (u − τ2 (u)))] ∆u. (2.2)

Use (2.2) to define the operator P : D → Crd ([dt0, ∞) ∩ T) by (Px) (t) = ϕ (t) for t ∈ [dt0, t0] ∩ T

and

(Px) (t) = ϕ (t0) e⊖a (t, t0) +

∫t

t0

e⊖a (t, u)
[
b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)

+q (u, x (u) , x (u − τ2 (u)))] ∆u, (2.3)

for t ∈ [t0, ∞) ∩ T.

Firstly, we prove Px ∈ D for any x ∈ D. From (2.3), for t > t0,

(Px)
△
(t) = −ϕ (t0) a (t) e⊖a (σ (t) , t0)

+ e⊖a (σ (t) , t)
[
b (t) g (x (t)) + c (t) f

(
x△̃ (t − τ1 (t))

)
+ q (t, x (t) , x (t − τ2 (t)))

]

− a (t)

∫t

t0

e⊖a (σ (t) , u)
[
b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)

+q (u, x (u) , x (u − τ2 (u)))] ∆u

= −a (t) ϕ (t0) e⊖a (σ (t) , t0)

− a (t)

∫σ(t)

t0

e⊖a (σ (t) , u)
[
b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)

+q (u, x (u) , x (u − τ2 (u)))] ∆u

+ b (t) g (x (t)) + c (t) f
(
x△̃ (t − τ1 (t))

)
+ q (t, x (t) , x (t − τ2 (t)))

= −a (t) (Px)
σ
(t) + b (t) g (x (t)) + c (t) f

(
x△̃ (t − τ1 (t))

)
+ q (t, x (t) , x (t − τ2 (t))) .

(2.4)

By the definition of Φt0, (2.4) yields on a time scale

(Px)
△

+
(t0) = −a (t0) ϕ

σ
(t0) + b (t0) g (ϕ (t0)) + c (t0) f

(
ϕ△̃ (t0 − τ1 (t0))

)

+ q (t0, ϕ (t0) , ϕ (t0 − τ2 (t0)))

= ϕ
△
− (t0) .

Hence, Px ∈ C1rd ([dt0, ∞) ∩ T) for x ∈ D.

For x ∈ D, limt→∞ x (t) = limt→∞ x
△ (t) = 0. Note that limt→∞ t − τi (t) = ∞, i = 1, 2.

Therefore, for any ε > 0, there exists T > 0 such that for t ≥ T,

max
{
|x (t)| , |x (t − τ2 (t))| ,

∣∣∣x△̃ (t − τ1 (t))
∣∣∣
}
< ε. (2.5)



CUBO
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Study of global asymptotic stability in nonlinear neutral dynamic . . . 57

It follows from (2.3), (2.5) and (H2) and (H4) that for t > T and x ∈ D,

|(Px) (t)|

≤ |ϕ (t0)| e⊖a (t, t0) +

∫T

t0

e⊖a (t, u)
∣∣∣b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)

+q (u, x (u) , x (u − τ2 (u)))| ∆u

+

∫t

T

e⊖a (t, u)
∣∣∣b (u) (g (x (u)) − g (0)) + c (u)

(
f
(
x△̃ (u − τ1 (u))

)
− f (0)

)

+q (u, x (u) , x (u − τ2 (u))) − q (u, 0, 0)| ∆u

≤ e⊖a (t, t0)

[
|ϕ (t0)| +

∫T

t0

ea (u, t0)
∣∣∣b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)

+q (u, x (u) , x (u − τ2 (u)))| ∆u]

+

∫t

T

e⊖a (t, u)
[
Lg |b (u)| |x (u)| + Lf |c (u)|

∣∣∣x△̃ (u − τ1 (u))
∣∣∣

+L1 (u) |x (u)| + L2 (u) |x (u − τ2 (u))|] ∆u

≤ e⊖a (t, t0)

[
|ϕ (t0)| +

∫T

t0

ea (u, t0)
∣∣∣b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)

+q (u, x (u) , x (u − τ2 (u)))| ∆u]

+ ε

∫t

T

e⊖a (t, u) [Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] ∆u

≤ e⊖a (t, t0)

[
|ϕ (t0)| +

∫T

t0

ea (u, t0)
∣∣∣b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)

+q (u, x (u) , x (u − τ2 (u)))| ∆u] + αε.

From (2.1), there exists T1 > T such that for t > T1,

e⊖a (t, t0)

[
|ϕ (t0)| +

∫T

t0

ea (u, t0)
∣∣∣b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)

+q (u, x (u) , x (u − τ2 (u)))| ∆u] < ε.

Hence, limt→∞ (Px) (t) = 0 for x ∈ D. In addition, it follows from (2.4) and (H2) that

∣∣∣(Px)△ (t)
∣∣∣

≤
∣∣a (t) (Px)σ (t)

∣∣ + |b (t) (g (x (t)) − g (0))| +
∣∣∣c (t)

(
f
(
x△̃ (t − τ1 (t))

)
− f (0)

)∣∣∣

+ |q (t, x (t) , x (t − τ2 (t))) − q (t, 0, 0)|

≤
∣∣a (t) (Px)σ (t)

∣∣ + Lg |b (t)| |x (t)| + Lf |c (t)|
∣∣∣x△̃ (t − τ1 (t))

∣∣∣

+ L1 (t) |x (t)| + L2 (t) |x (t − τ2 (t))| .



58 Abdelouaheb Ardjouni and Ahcene Djoudi CUBO
20, 3 (2018)

This, together with (H3) and (H4), yields limt→∞ (Px)
△
(t) = 0 for x ∈ D. Therefore, Px ∈ D for

x ∈ D, i.e. P : D → D.

Secondly, we show that P : D → D is a contraction mapping. For any x, y ∈ D, it follows from

(2.3), (H2) and (H4) that for t ∈ [t0, ∞) ∩ T,

|(Px) (t) − (Py) (t)|

≤

∫t

t0

e⊖a (t, u)
[
Lg |b (u)| |x (u) − y (u)| + Lf |c (u)|

∣∣∣x△̃ (u − τ1 (u)) − y△̃ (u − τ1 (u))
∣∣∣

+ |q (u, x (u) , x (u − τ2 (u))) − q (u, y (u) , y (u − τ2 (u)))|] ∆u

≤ ‖x − y‖
t0

∫t

t0

e⊖a (t, u) [Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] ∆u

≤ α ‖x − y‖
t0

. (2.6)

In addition, it follows from (2.4), (2.6), (H2) and (H4) that for t ∈ [t0, ∞) ∩ T,

∣∣∣(Px)△ (t) − (Py)△ (t)
∣∣∣

≤ |a (t)|
∣∣(Px)σ (t) − (Py)σ (t)

∣∣ + Lg |b (t)| |x (t) − y (t)| + Lf |c (t)|
∣∣∣x△̃ (t − τ1 (t)) − y△̃ (t − τ1 (t))

∣∣∣

+ |q (t, x (t) , x (t − τ2 (t))) − q (t, y (t) , y (t − τ2 (t)))|

≤ ‖x − y‖
t0

[
|a (t)|

∫σ(t)

t0

e⊖a (σ (t) , u) [Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] ∆u

+Lg |b (t)| + Lf |c (t)| + L1 (t) + L2 (t)]

≤ α ‖x − y‖
t0

. (2.7)

From (2.6) and (2.7), P : D → D is a contraction mapping. By the contraction mapping principle,

P has a unique fixed point x in D, which is a unique solution of (0.1) through (t0, ϕ) and satisfies

lim
t→∞

x (t) = lim
t→∞

x△ (t) = 0. (2.8)

Finally, we show that the zero solution of (0.1) is stable in C1rd. Let

K = sup
t∈[t0,∞)∩T

{e⊖a (t, t0)} and A = sup
t∈[t0,∞)∩T

{|a (t)|} .

From (2.1) and (H3), K, A ∈ (0, ∞). For any ε > 0, let δ > 0 such that

δ < ε min

{

1,
1 − α

K
,
1 − α

KA

}

.

If x (t) = x (t, t0,ϕ) is a solution of (0.1) with |ϕ|t0 < δ, then x (t) = (Px) (t) on [t0, ∞) ∩ T. We

claim that ‖x‖
t0

< ε. Otherwise, there exists t1 > t0 such that

max
{
|x (t1)| ,

∣∣x△ (t1)
∣∣} = ε,



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Study of global asymptotic stability in nonlinear neutral dynamic . . . 59

and

max
{
|x (t)| ,

∣∣x△ (t)
∣∣} < ε,

for t ∈ [dt0, t1) ∩ T. If |x (t1)| = ε, then it follows from (2.3), (H2) and (H4) that

|x (t1)| ≤ |ϕ (t0)| e⊖a (t1, t0)

+

∫t1

t0

e⊖a (t1, u)
∣∣∣b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)
+ q (u, x (u) , x (u − τ2 (u)))

∣∣∣ ∆u

≤ Kδ + ε

∫t1

t0

e⊖a (t1, u) [Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] ∆u

≤ Kδ + αε

< ε.

This is a contradiction. If
∣∣x△ (t1)

∣∣ = ε, then it follows from (2.4) and (H2) and (H4) that
∣∣x△ (t1)

∣∣

≤ |ϕ (t0) a (t1)| e⊖a (σ (t1) , t0) + |b (t1)| |g (x (t1))|

+ |c (t1)|
∣∣∣f
(
x△̃ (t1 − τ1 (t1))

)∣∣∣ + |q (t1, x (t1) , x (t1 − τ2 (t1)))|

+ |a (t1)|

∫σ(t1)

t0

e⊖a (σ (t1) , u)
∣∣∣b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)
+ q (u, x (u) , x (u − τ2 (u)))

∣∣∣ ∆u

≤ KAδ + ε

{

|a (t1)|

∫σ(t1)

t0

e⊖a (σ (t1) , u) [Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] ∆u

+Lg |b (t1)| + Lf |c (t1)| + L1 (t1) + L2 (t1)}

≤ KAδ + αε

< ε.

This is also a contradiction. Hence, the zero solution of (0.1) is stable in C1rd. This, together with

(2.8), implies that the zero solution of (0.1) is globally asymptotically stable in C1rd.

(ii) Assume that the zero solution of (0.1) is globally asymptotically stable in C1rd. Now we

prove that (2.1) holds. Otherwise, set

l = lim
t 7→∞

inf

∫t

0

1

µ (s)
log (1 + µ (s) a (s)) ∆s, K0 = sup

t∈[0,∞)∩T

{e⊖a (t, 0)} , A0 = sup
t∈[0,∞)∩T

{|a (t)|} ,

thus it follows from (H3) that l ∈ (−∞, ∞), K0 ∈ (0, ∞), A0 ∈ [0, ∞). Hence, there exists an

increasing sequence {tn} ⊂ [0, ∞) ∩ T such that limt→∞ tn = ∞ and

lim
n7→∞

∫tn

0

1

µ (s)
log (1 + µ (s) a (s)) ∆s = l. (2.9)

Denote

In =

∫tn

0

ea (u, 0) [Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] ∆u, n = 1, 2, ....



60 Abdelouaheb Ardjouni and Ahcene Djoudi CUBO
20, 3 (2018)

From (H4), it follows that

In = ea (tn, 0)

∫tn

0

e⊖a (tn, u) [Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] ∆u ≤ αea (tn, 0) .

This, together with (2.9), implies that the sequence {In} is bounded. Further, there exists a

convergent subsequence. For brevity of notation, we may assume that {In} is convergent. Therefore,

there exists a positive integer m such that for any integer n > m,

∫tn

tm

ea (u, 0) [Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] ∆u <
1 − α

8B (e−l + 1)
, (2.10)

and

e⊖a (tn, tm) >
1

2
, e⊖a (tn, 0) < e

−l + 1, ea (tm, 0) < e
l + 1, (2.11)

where B = max
{
K0

(
el + 1

)
, K0A0

(
el + 1

)
, 1
}
.

For any δ > 0, consider the solution x (t) = x (t, tm, ϕ) of (0.1) with |ϕ|tm < δ and |ϕ (tm)| >

δ/2. It follows from (2.3), (2.4), (2.11), (H2) and (H4) that for t ∈ [tm, ∞) ∩ T,

|x (t)|

≤ |ϕ (tm)| e⊖a (t, tm)

+

∫t

tm

e⊖a (t, u)
∣∣∣b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)
+ q (u, x (u) , x (u − τ2 (u)))

∣∣∣ ∆u

≤ |ϕ (tm)| e⊖a (t, 0) ea (tm, 0) + ‖x‖tm

∫t

tm

e⊖a (t, u) [Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] ∆u

≤ K0
(
el + 1

)
δ + ‖x‖

tm

∫t

0

e⊖a (t, u) [Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] ∆u

≤ Bδ + α ‖x‖tm ,

and

∣∣x△ (t)
∣∣

≤ |ϕ (tm)| |a (t)| e⊖a (σ (t) , tm) + |b (t)| |g (x (t))|

+ |c (t)|
∣∣∣f
(
x△̃ (t − τ1 (t))

)∣∣∣ + |q (t, x (t) , x (t − τ2 (t)))|

+ |a (t)|

∫σ(t)

tm

e⊖a (σ (t) , u)
∣∣∣b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)
+ q (u, x (u) , x (u − τ2 (u)))

∣∣∣ ∆u

≤ K0A0
(
el + 1

)
δ

+ ‖x‖
tm

{

|a (t)|

∫σ(t)

tm

e⊖a (σ (t) , u) [Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] ∆u

+Lg |b (t)| + Lf |c (t)| + L1 (t) + L2 (t)}

≤ Bδ + α ‖x‖
tm

.



CUBO
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Study of global asymptotic stability in nonlinear neutral dynamic . . . 61

Hence, ‖x‖
tm

≤ Bδ + α ‖x‖
tm

, i.e.

‖x‖
tm

≤
B

1 − α
δ. (2.12)

It follows from (2.3),(2.10)-(2.12) and (H2) that for any n > m,

|x (tn)|

≥ |ϕ (tm)| e⊖a (tn, tm)

− e⊖a (tn, 0)

∫tn

tm

ea (u, 0)
∣∣∣b (u) g (x (u)) + c (u) f

(
x△̃ (u − τ1 (u))

)
+ q (u, x (u) , x (u − τ2 (u)))

∣∣∣ ∆u

≥ |ϕ (tm)| e⊖a (tn, tm) − ‖x‖tm e⊖a (tn, 0)

∫tn

tm

ea (u, 0) [Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] ∆u

>
1

4
δ −

B

1 − α
δ
(
e−l + 1

) 1 − α
8B (e−l + 1)

=
1

8
δ.

This contradicts the fact that limn→∞ tn = ∞ and the zero solution of (0.1) is globally asymptot-

ically stable in C1rd. The proof is complete.

Example 2.4. Let T = R. Consider the following neutral differential equation

x′ (t) = −a (t) x (t) + b (t) g (x (t)) + c (t) f (x′ (t − τ1 (t))) + q (t, x (t) , x (t − τ2 (t))) , (2.13)

where

τ1 (t) = t/2 + 1, τ2 (t) = t/3 + 2, a (t) =
1

t + 1
, b (t) =

1

15 (t + 1)
,

c (t) =
1

12 (t + 1)
, g (x) = 1 − cos (x) , f (x) = sin (x) ,

q (t, x, y) =
1

16 (t + 1)
sin (x + y) .

Obviously a, b, c ∈ C ([0, ∞) , R), g, f ∈ C (R, R), q ∈ C ([0, ∞) × R × R, R), τ1, τ2 ∈ C ([0, ∞) , (0, ∞))

with t − τi (t) → ∞ as t → ∞, i = 1, 2. A simple calculation shows that

g (0) = f (0) = q (t, 0, 0) = 0,

∫∞

0

a (s) ds = ∞,

L1 (t) = L2 (t) =
1

16 (t + 1)
, Lg = 1, Lf = 1,

∫t

0

e−
∫
t
u
a(s)ds

[Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] du ≤
11

40
,

and

a (t)

∫t

0

e−
∫
t
u
a(s)ds [Lg |b (u)| + Lf |c (u)| + L1 (u) + L2 (u)] du

+Lg |b (t)| + Lf |c (t)| + L1 (t) + L2 (t) ≤
22

40
.

It is easy to see that all the conditions of Theorem 2.3 hold for α = 22
40

< 1. Thus, Theorem 2.3

implies that the zero solution of (2.13) is globally asymptotically stable in C1.



62 Abdelouaheb Ardjouni and Ahcene Djoudi CUBO
20, 3 (2018)

Acknowledgments. The authors would like to thank the anonymous referee for his/her valuable

remarks.



CUBO
20, 3 (2018)

Study of global asymptotic stability in nonlinear neutral dynamic . . . 63

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	Preliminaries
	Global asymptotic stability