CUBO A Mathematical Journal
Vol.16, No¯ 02, (111–119). June 2014

On the uniform asymptotic stability to certain first order
neutral differential equations

Cemil Tunç

Department of Mathematics,

Faculty of Science,

Yüzüncü Yıl University, 65080, Van, Turkey

cemtunc@yahoo.com

ABSTRACT

In this paper, the uniform asymptotic stability of the zero solution of a kind of neutral

differential equations is discussed. Based on the Lyapunov functional approach, a new

stability criterion is derived, which is delay dependent on two positive constants. The

result to be obtained here extends and generalizes the existing ones in the literature.

RESUMEN

En este art́ıculo se discute la estabilidad asintótica uniforme de la solución cero de

un tipo de ecuaciones diferenciales neutrales. Basados en la técnica de funcional de

Lyapunov, se deriva un nuevo criterio de estabilidad, el cual es dependiente de atrasos

basados en dos constantes positivas. El resultado que se obtiene extiende y generaliza

los encontrados en la literatura.

Keywords and Phrases: Neutral differential equation; first order, uniform asymptotic stability;

Lyapunov functional.

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



112 Cemil Tunç CUBO
16, 2 (2014)

1 Introduction

In 2009, Nam and Phat [4] considered the first order neutral differential equation

d

dt
[x(t) + px(t − τ)] = −ax(t) + b tanh x(t − σ), t ≥ 0, (1.1)

where a, τ and σ are positive constants, σ ≥ τ, b and p are real numbers with |p| < 1. Using

a Lyapunov functional, the authors established some sufficient conditions for the zero solution of

Eq. (1.1) to be uniformly asymptotically stable. By this work, Nam and Phat [4] established an

improved criterion for the uniform asymptotic stability of the zero solution of Eq. (1.1).

In this paper, instead of Eq. (1.1), we consider the first order neutral differential equation of

the form

d

dt
[x(t) + px(t − τ)] = −f(x(t))x(t) − g(x(t − τ))x(t) + b tanh x(t − σ), t ≥ 0, (1.2)

where f and g are continuous functions on ℜ = (−∞, ∞), τ and σ are positive constants with

σ ≥ τ, b and p are real numbers, |p| < 1. For each solution of Eq. (1.2), we assume the initial

condition

x(t) = φ(t), t ∈ [−σ, 0], φ ∈ C([−σ, 0], ℜ).

A primary purpose of this paper is to study the uniform asymptotic stability of the zero

solution of Eq. (1.2). Motivated by Nam and Phat [4], we obtain some sufficient conditions which

guarantee the uniform asymptotic stability of the zero solution of Eq. (1.2). It is clear that the

equation discussed by Nam and Phat [4], Eq. (1.1), is a special case of our equation, Eq. (1.2).

That is, our equation, Eq. (1.2), includes and extends the equation discussed in [4]. By this paper,

we generalize and extend the result obtained by Nam and Phat [4].

Finally, in paticular, one can refer to the papers of Agarwal and Grace [1], Park [5], Sun and

Wang [6], Tunç [7], Tunç and Sirma [9] and the references listed in these sources to see some recent

contributions focused on the topic of this paper.

2 Description of problem

Before introducing our main result, we give some basic information relative to the topic.

Lemma 2.1. ([4]). Assume that S ∈ ℜn×n is a symmetric positive definite matrix. Then for

every Q ∈ ℜn×n,

2 < Qy, x >≤< QS−1QT x, x > + < Sy, y >, ∀x, y ∈ ℜn.



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On the uniform asymptotic stability to certain first order neutral . . . 113

Lemma 2.2. ([2]). For any symmetric positive definite matrix M ∈ ℜn×n, scalar σ ≥ 0 and

vector function w : [0, σ] → ℜn such that the integrations concerned are well defined, then

⎛

⎝

σ∫

0

w(s)ds

⎞

⎠

T

M

⎛

⎝

σ∫

0

w(s)ds

⎞

⎠ ≤ σ

σ∫

0

wT (s)Mw(s)ds.

Definition ([3]). Suppose Ω ⊆ ℜ × C is open, f : Ω → ℜn, D : Ω → ℜn are given continuous

functions with D atomic at zero (see Hale and Lunel [[3], pp.53]. The relation

d

dt
D(t, xt) = f(t, xt)

is called the neutral functional differential equation.

Theorem 2.1. ([3]). Suppose D is a stable operator, f : ℜ × C → ℜn, f : ℜ × (bounded sets of

C) into bounded sets of ℜn and suppose u(s), v(s), and w(s) are continuous, nonnegative, and

non-decreasing with u(s), v(s) > 0 for s ̸= 0, and u(0) = v(0) = 0. If there is a continuous function

V : ℜ × C → ℜn such that

u(|Dφ|) ≤ V(t, φ) ≤ v(|φ|)

and

V̇(t, φ) ≤ −w(|Dφ|),

then the solution x = 0 of d
dt

D(t, xt) = f(t, xt) are uniformly stable. If u(s) → ∞ as s → ∞,

then the solutions of d
dt

D(t, xt) = f(t, xt) are uniformly bounded. If w(s) > 0 for s > 0, then the

solution x = 0 of d
dt

D(t, xt) = f(t, xt) is uniformly asymptotically stable. The same conclusion

holds if the upper bound of V̇(t, φ) is given by −w(|φ(0)| .

For a real number α, Eq. (1.2) can be rewritten in the form

d

dt
[x(t) + px(t − τ) + α

t∫

t−τ

x(s)ds + b

t−τ∫

t−σ

tanh x(s)ds]

= {α − [f(x(t)) + g(x(t − τ))]}x(t) − αx(t − τ) + b tanh x(t − τ), t ≥ 0. (2.1)

Define the operators ([4]):

D1(xt) = x(t) + px(t − τ) + α

t∫

t−τ

x(s)ds + b

t−τ∫

t−σ

tanh x(s)ds

and

D2(xt) = x(t) + px(t − τ).

Our main result is the following theorem.



114 Cemil Tunç CUBO
16, 2 (2014)

Theorem 2.2. In addition to the all the basic assumptions imposed to the functions f and g that

appear in Eq. (1.2), we assume that there exist a constant α with 0 < |α| < 1 and positive constants

β, γ, η, θ, a1 and a2 such that the following conditions hold:

f(x) ≥ a1, g(x) ≥ a2, a = a1 + a2,

Ω =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

Ω11 Ω12 b Ω14 Ω15

∗ −β − 2pα pb −α −bα

∗ ∗ θ(σ − τ)2 − η b b2

∗ ∗ ∗ −γ 0

∗ ∗ ∗ ∗ −θ

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

< 0, (2.2)

where

Ω11 = 2(α − a) + β + γτ
2 + η,

Ω12 = p(α − f(x) − g(x(t − τ))) − α,

Ω14 = α − f(x(t)) − g(x(t − τ)),

Ω15 = b{α − f(x(t)) − g(x(t − τ))},

Then, the zero solution of Eq. (1.2) is uniformly asymptotically stable.

Remark. Here the symbol * represents the elements below the main diagonal of the matrix Ω in

(4).

Proof. Since Ω < 0, there is a positive constant δ such that the following inequality holds:

Ω1 =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

Ω11 Ω12 b Ω14 Ω15

∗ −β − 2pα pb −α −bα

∗ ∗ θ(σ − τ)2 − η + δ b b2

∗ ∗ ∗ −γ 0

∗ ∗ ∗ ∗ −θ

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

< 0.

We use the Lyapunov functional ([4]) defined by

V = V1 + V2 + V3 + V4 + V5 + V6 + V7,



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On the uniform asymptotic stability to certain first order neutral . . . 115

where

V1 = D
T
1 (xt)D1(xt),

V2 = β

t∫

t−τ

x2(s)ds,

V3 = γτ

t∫

t−τ

(τ − t + s)(αx(s))2ds,

V4 = θ(σ − τ)

t−τ∫

t−σ

(s − t + σ) tanh2 x(s)ds,

V5 = η

t∫

t−τ

tanh2 x(s)ds,

V6 = δ

t−τ∫

t−σ

tanh2 x(s)ds, (δ is a positive constant),

V7 = εD
2
2(xt),

ε is a positive constant that will be chosen later.

Let V0 = V1 +V2 +V3 +V4 +V5 +V6. The time derivative of the functional V0 along solutions

of Eq. (2.1) is given by

d

dt
V0 = 2

⎡

⎣x(t) + px(t − τ) + α

t∫

t−τ

x(s)ds + b

t−τ∫

t−σ

tanh x(s)ds

⎤

⎦

T

×[{α − [f(x(t)) + g(x(t − τ))]}x(t) − αx(t − τ) + b tanh x(t − τ)]

+βx2(t) − βx2(t − τ) + γτ2α2x2(t) − γτ

t∫

t−τ

(αx(s))2ds

+θ(σ − τ)2 tanh2 x(t − τ) − θ(σ − τ)

t−τ∫

t−σ

tanh2 x(s)ds

+η tanh2 x(t) − η tanh2 x(t − τ) + δ tanh2 x(t − τ) − δ tanh2 x(t − σ).



116 Cemil Tunç CUBO
16, 2 (2014)

Using Lemma 2.2 and applying the estimate tanh2 x(t) ≤ x2(t), we obtain

d

dt
V0 ≤ 2

⎡

⎣x(t) + px(t − τ) + α

t∫

t−τ

x(s)ds + b

t−τ∫

t−σ

tanh x(s)ds

⎤

⎦

T

×[{α − [f(x(t)) + g(x(t − τ))]}x(t) − αx(t − τ) + b tanh x(t − τ)]

+βx2(t) − βx2(t − τ) + γτ2α2x2(t) − γ

⎛

⎝

t∫

t−τ

αx(s)ds

⎞

⎠

2

+θ(σ − τ)2 tanh2 x(t − τ) − θ

⎛

⎝

t−τ∫

t−σ

tanh x(s)ds

⎞

⎠

2

+ηx2(t) − η tanh2 x(t − τ) + δ tanh2 x(t − τ) − δ tanh2 x(t − σ).

Choosing

α

t∫

t−τ

x(s)ds = u(t),

t−τ∫

t−σ

tanh x(s)ds = v(t)

and using the assumptions f(x) ≥ a1 , g(x) ≥ a2 and the estimate 0 < |α| < 1, it follows that

d

dt
V0 ≤ [2{α − (a1 + a2)} + β + γτ

2 + η]x2(t)

+[−2α + 2p(α − f(x(t)) − g(x(t − τ)))]x(t)x(t − τ)

+2bx(t) tanh x(t − τ) + 2{α − f(x(t)) − g(x(t − τ))}x(t)u(t)

+2b{α − f(x(t)) − g(x(t − τ))}x(t)v(t)

−(2pα + β2)x2(t − τ) + 2pbx(t − τ) tanh x(t − τ) − 2αx(t − τ)u(t)

−2αbx(t − τ)v(t) + [θ(σ − τ)2 − η + δ] tanh2 x(t − τ) − δ tanh2 x(t − σ)

+2b tanh x(t − τ)u(t) + 2b2 tanh x(t − τ)v(t) − γu2(t) − θv2(t)

= [2(α − a) + β + γτ2 + η]x2(t)

+[−2α + 2p(α − f(x(t)) − g(x(t − τ)))]x(t)x(t − τ)

+2bx(t) tanh x(t − τ) + 2(α − f(x(t)) − g(x(t − τ)))x(t)u(t)

+2b(α − f(x(t)) − g(x(t − τ)))x(t)v(t)

−(2pα + β2)x2(t − τ) + 2pbx(t − τ) tanh x(t − τ) − 2αx(t − τ)u(t)

−2αbx(t − τ)v(t) + [θ(σ − τ)2 − η + δ] tanh2 x(t − τ) − δ tanh2 x(t − σ)

+2b tanh x(t − τ)u(t) + 2b2 tanh x(t − τ)v(t) − γu2(t) − θv2(t).

Since the terms on the right hand side of d
dt

V0 represent a specific quadratic form, we can rearrange

the terms given on the right hand side of d
dt

V0 as the following:

d

dt
V0 ≤ ζ

T (t) Ω2ζ(t),



CUBO
16, 2 (2014)

On the uniform asymptotic stability to certain first order neutral . . . 117

where

Ω2 =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

Ω11 Ω12 b Ω14 Ω15 0

∗ −β − 2pα pb −α −bα 0

∗ ∗ θ(σ − τ)2 − η + δ b b2 0

∗ ∗ ∗ −γ 0 0

∗ ∗ ∗ ∗ −θ 0

∗ ∗ ∗ ∗ ∗ −δ

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

.

ζT (t) =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

x(t)

x(t − τ)

tanh x(t − τ)

u(t)

v(t)

tanh x(t − σ)

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

T

, ζ(t) =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

x(t)

x(t − τ)

tanh x(t − τ)

u(t)

v(t)

tanh x(t − σ)

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

.

Making use of the assumption Ω < 0, we have

d

dt
V0 ≤ ζ

T (t)Ω2ζ(t) < 0.

Therefore, there exists a positive constant λ such that

d

dt
V0 ≤ −λ{∥x(t)∥

2
+ ∥x(t − τ)∥

2
+ ∥tanh x(t − τ)∥

2

+ ∥u(t)∥
2
+ ∥v(t)∥

2
+ ∥tanh x(t − σ)∥

2
}.

By a straightforward calculation for the time derivative of V7, it follows that

d

dt
V7 = 2ε[x(t) + px(t − τ)] [−f(x(t))x(t) − g(x(t − τ))x(t) + b tanh x(t − σ)]

= −2εf(x(t))x2(t) − 2εg(x(t − σ))x2(t) + 2bεx(t) tanh x(t − σ)

−2εpf(x(t))x(t)x(t − τ) − 2εpg(x(t − τ))x(t)x(t − τ)

+2εpbx(t − τ)) tanh x(t − σ).

Taking into account the assumptions f(x) ≥ a1 , g(x) ≥ a2, a = a1 + a2, and Lemma 2.1, we find

d

dt
V7 ≤ −2ε(a1 + a2)x

2(t) + 2bεx(t) tanh x(t − σ)

+2εp{f(x(t)) + g(x(t − τ))} |x(t)| |x(t − τ)|

+2εpbx(t − τ)) tanh x(t − σ)

≤ −2εax2(t) + 2bεx(t) tanh x(t − σ) + 2εap |x(t)| |x(t − τ)|

+2εpbx(t − τ)) tanh x(t − σ)

≤ ε{−2a + |b|
2
+ |ap|

2
}x2(t) + ε{1 + |bp|

2
}x2(t − τ) + 2ε tanh2 x(t − σ).



118 Cemil Tunç CUBO
16, 2 (2014)

Let us choose the constant ε as

ε =

{
2−1λ min{(1 + |bp|

2
)−1, 2−1}, if − 2a + |b|

2
+ |ap|

2
≤ 0

2−1λ min{(1 + |b|
2
+ |ap|

2
)−1, (1 + |bp|

2
)−1, 2−1}, if − 2a + |b|

2
+ |ap|

2
> 0.

Hence, we can obtain
d

dt
V < −

λ

2
∥x(t)∥

2
.

Further, we have V ≥ εD2
2
(xt), and D2(xt) is stable due to |p| < 1.

The proof of Theorem (2.2) is now completed.

3 Conclusion

A non-linear neutral differential equation of the first order is considered. We establish certain suffi-

cient conditions which guarantee that the zero solution of this equation is uniformly asymptotically

stable. In proving our main result, we employ the Lyapunov functional approach as a basic tool.

This paper has a contribution to the subject in the literature, and it may be useful for researchers

work on the qualitative behaviors of solutions.

Received: December 2012. Revised: May 2013.

References

[1] Agarwal, R. P.; Grace, S. R., Asymptotic stability of certain neutral differential equations,

Math. Comput. Modelling, 31 (2000), no. 8-9, 9–15.

[2] Gu, K., An integral inequality in the stability problem of time-delay system systems. In: Pro-

ceedings of 39th IEEE CDC, Sydney, Australia, 2000, pp. 2805–2810.

[3] Hale, Jack K.; Verduyn Lunel, Sjoerd M., Introduction to functional-differential equations.

Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993.

[4] Nam, P. T.; Phat, V. N., An improved stability criterion for a class of neutral differential

equations, Appl. Math.Lett., 22 (2009), no. 1, 31–35.

[5] Park, J. H., Delay-dependent criterion for asymptotic stability of a class of neutral equations,

Appl. Math. Lett., 17 (2004), no. 10, 1203–1206.

[6] Sun, Y. G.; Wang, L., Note on asymptotic stability of a class of neutral differential equations,

Appl. Math. Lett., 19 (2006), no. 9, 949–953.

[7] Tunç, C., Asymptotic stability of nonlinear neutral differential equations with constant delays:

a descriptor system approach, Ann. Differential Equations, 27 (2011), no. 1, 1–8.



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On the uniform asymptotic stability to certain first order neutral . . . 119

[8] Tunç, C., Exponential stability to a neutral differential equation of first order with delay. Ann.

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[9] Tunç, C.; Sirma, A., Stability analysis of a class of generalized neutral equations, J. Comput.

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