Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 475019, 16 pages
doi:10.1155/2010/475019
Research Article
A Generalized Halanay Inequality for
Stability of Nonlinear Neutral Functional
Differential Equations
Wansheng Wang
School of Mathematics and Computational Science, Changsha University of Science and Technology,
Changsha 410114, China
Correspondence should be addressed to Wansheng Wang,
Received 22 March 2010; Accepted 18 July 2010
Academic Editor: Kun quan Q. Lan
Copyright q 2010 Wansheng Wang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper is devoted to generalize Halanay’s inequality which plays an important rule in study
of stability of differential equations. By applying the generalized Halanay inequality, the stability
results of nonlinear neutral functional differential equations NFDEs and nonlinear neutral delay
integrodifferential equations NDIDEs are obtained.
1. Introduction
In 1966, in order to discuss the stability of the zero solution of
u
t
−Au
t
v
t
≤ ce
−κt−t
0
, for t ≥ t
0
,
1.3
and hence vt → 0 as t →∞.
2 Journal of Inequalities and Applications
In 1996, in order to investigate analytical and numerical stability of an equation of the
type
u
t
f
t, u
t
,u
η
,t≤ t
0
,φbounded and continuous for t ≤ t
0
,
1.4
Baker and Tang 2 give a generalization of Halanay inequality as Lemma 1.2 which can be
used for discussing the stability of solutions of some general Volterra functional differential
equations.
Lemma 1.2 see 2. Suppose vt > 0, t ∈ −∞, ∞, and
v
t
≤−A
t
v
t
B
t
sup
t−τ
, At, Bt > 0 for t ∈ t
0
, ∞, τt ≥ 0, and
t − τt → ∞ as t → ∞. If there exists p>0 such that
−A
t
B
t
≤−p<0, for t ≥ t
0
, 1.6
then
i
v
t
≤ sup
t∈
−∞,t
0
t
≤−A
t
u
t
B
t
max
s∈
t−τ,t
u
s
C
t
max
s∈
s∈
t−τ,t
w
s
,
t ≥ t
0
, 2.1
Journal of Inequalities and Applications 3
where At, Bt, Ct, Dt, Gt, and Ht are nonnegative continuous functions on t
0
, ∞, and
the notation
denotes the conventional derivative or the one-sided derivatives. Suppose that
A
t
≥ A
0
> 0,H
t
≤ H
. 2.2
Then for any ε>0, one has
u
t
<
1 ε
Ue
ν
∗
t−t
0
,w
t
<
1 ε
We
ν
∗
t−t
0
t
G
t
e
−2ντ
1 − H
t
e
−ντ
0.
2.4
Obviously, ν is different for different t, that is to say, ν is a function of t. Then we define ν
∗
as
ν
∗
: sup
t≥t
0
{
ν
t
W are constants) to systems
u
t
−A
t
u
t
B
t
u
t − τ
C
t
w
t − τ
ν
∗
t−t
0
, wt
We
ν
∗
t−t
0
, then ν
∗
is
obviously a nonnegative root of the characteristic equation 2.4. Conversely, if characteristic
equation 2.4 has nonnegative root ν for any fixed t, then ut
Ue
ν
∗
t−t
0
and wt
We
ν
∗
Using condition 2.2, we have
0 ν
t
∗
A
t
∗
− B
t
∗
−
C
t
∗
G
t
∗
1 − H
t
∗
< 0.
Case 2 τ>0. In this case, obviously, for any fixed t, 0 is not a root of 2.4.If2.4 has a
positive root ν at a certain fixed t, then it follows from 2.2 and 2.4 that
B
t
C
t
G
t
1 − H
t
<B
t
e
−ντ
C
t
G
t
e
−2ντ
1 − H
t
e
−ντ
.
2.9
After simply calculating, we have Ht > 1 which contradicts the assumption. Thus, 2.4
does not have any nonnegative root.
To prove that 2.4 has a negative root ν for any fixed t,wesetν
0
τ
−1
ln Ht and
define
H
ν
ν A
0
> 0, lim
ν →ν
0
H
ν
−∞.
2.11
Journal of Inequalities and Applications 5
On the other hand, when ν ∈ ν
0
, 0, we have
H
ν
1 B
t
τe
−ντ
2C
G
t
e
−2ντ
H
t
τe
−ντ
1 − H
t
e
−ντ
2
> 0,
2.12
which implies that Hν is a strictly monotone increasing function. Therefore, for any fixed t
the characteristic equation 2.4 has a negative root ν ∈ ν
0
, 0.
It remains to prove that ν
∗
Then there exists t
∗
≥ t
0
such that 0 >νt
∗
> −. Since
e
τ
H
t
∗
≤ H
0
e
τ
≤ H
0
p
1 − H
0
p H
0
1/2
A
t
∗
− B
t
∗
e
−νt
∗
τ
−
C
t
∗
G
t
∗
e
−2νt
∗
τ
∗
e
2τ
1 − H
t
∗
e
τ
> − A
t
∗
−
e
2τ
1 − H
0
1 − H
0
e
τ
B
0
1 − H
0
e
τ
pA
t
∗
> − A
t
∗
− pA
t
∗
−
1 − p
A
t
∗
,
ν
∗
< 0, then for any ε>0, any nontrivial solution ut, wt of 2.1 satisfies 2.3.
Proof. The required result follows at once when t ∈ t
0
− τ,t
0
. If there exists t
∗
such that when
t<t
∗
,
u
t
<
1 ε
Ue
ν
∗
t−t
0
,w
−t
0
, then for t ≤ t
∗
, we can find that
u
t
≤ e
−
t
t
0
Axdx
u
t
0
t
t
0
e
−
dr
<e
−
t
t
0
Axdx
1 ε
U
t
t
0
e
−
t
r
Axdx
B
r
1 ε
∗
t−t
0
,
w
t
<G
t
max
s∈
t−τ,t
1 ε
Ue
ν
∗
s−t
0
H
t
2.17
a contradiction proving the lemma.
Proof of Theorem 2.1. By Lemma 2.3, we can find that for any fixed t, characteristic equation
2.4 only has negative root and ν
∗
< 0. Thus from Lemma 2.2 we know that systems 2.6
have not nontrivial solution with the form ut
Ue
ν
∗
t−t
0
, wt
We
ν
∗
t−t
0
, t ≥ t
0
, ν
∗
≥ 0.
However, it is easily verified that systems 2.6 have nontrivial solution ut
Ue
t
0
−τ,t
0
u
s
,w
t
≤ max
s∈
t
0
−τ,t
0
w
s
;
ii
lim
0
Gt, and H sup
t≥t
0
Ht. Then when
A>0,H<1, −A B
CG
1 − H
< 0,
2.19
equation 2.3 holds for any ε>0,whereν
∗
< 0 is defined by
ν
∗
: max
ν : H
ν
ν A − Be
−ντ
−
CGe
−2ντ
1 − He
−ντ
0
f
t, y
t
,y
t
, ˙y
t
,t≥ t
0
,
y
t
0
φ, ˙y
t
0
˙
φ,
3.1
where the derivative
·
is the conventional derivative, y
t
θyt θ, −τ ≤ θ ≤ 0, τ ≥ 0
and t
f
λ, t, y
1
,y
2
,χ,ψ
, ∀λ ≥ 0,t≥ t
0
,y
1
,y
2
∈ X,χ,ψ∈ C
X
−τ,0
,
3.2
f
t, y
1
,χ
1
,ψ
χ
1
− χ
2
t−τ,t
γ
t
ψ
1
− ψ
2
t−τ,t
,
∀t ≥ t
0
,y
1
,y
2
∈ X,χ
1
− λ
f
t, y
1
,χ,ψ
− f
t, y
2
,χ,ψ
,
∀λ ∈ R,t≥ t
0
,y
1
,y
2
∈ X,χ,ψ∈ C
X
−τ,0
,
3.4
0
˙ϕ,
3.5
where we assume the initial function ϕt is also a given continuously differentiable mapping,
but it may be different from φt in problem 3.1.
To prove our main results in this section, we need the following lemma.
Lemma 3.1 cf. Li 14. If the abstract function ωt : R → X has a left-hand derivative at point
t t
∗
, then the function ωt also has the left-hand derivative at point t t
∗
, and the left-hand
derivative is
D
−
ω
t
∗
lim
ξ →−0
ω
t
ω
t
∗
lim
ξ → 0
ω
t
∗
ξω
t
∗
0
−
ω
t
∗
t
−
1 − γ
t
α
t
≤ p<1, ∀t ≥ t
0
.
3.8
Then for any ε>0, one have
y
t
− z
t
,
˙y
t
− ˙z
t
<
1 ε
max
s∈
t
0
−τ,t
0
˙
φ
−
γ
t
L
t
β
t
e
−2ντ
1 − γ
t
e
−ντ
0.
3.10
Since ν is a function of t, then one defines ν
#
as ν
#
: sup
s
,
˙y
t
− ˙z
t
≤ max
s∈
t
0
−τ,t
0
˙
φ
t
− ˙z
t
0.
3.11
Proof. Let us define Ytyt − zt and
Yt ˙yt − ˙zt. By means of
y
t
− z
t
− λ
˙y
t
− ˙z
− f
t, z
t
,y
t
, ˙y
t
− λ
β
t
y − z
t−τ,t
γ
t
˙y
t
− ˙z
t
−
y
t
− z
t
−λ
≤ lim
λ → 0
G
f
≤ lim
λ → 0
1 −
1 − α
t
λ
G
f
0
λ
β
t
y − z
t−τ,t
γ
˙y − ˙z
t−τ,t
.
3.13
10 Journal of Inequalities and Applications
On the other hand, it is easily obtained from 3.3 that
Y
t
≤ L
t
Y
t
β
t
∂y
f
t,
1 − θ
z
t
θy
t
,y
t
, ˙y
t
,θ∈
0, 1
,t≥ t
0
,
3.15
we have
D
−
y
t
− z
t
λ
≤ lim
λ → 0
1
λ
I λ
1
0
t
β
t
y − z
t−τ,t
γ
t
˙y − ˙z
t−τ,t
≤ lim
λ → 0
1
λ
β
t
y − z
t−τ,t
γ
t
˙y − ˙z
t−τ,t
≤ μ
1
0
M
t
Y
t
β
t
y − z
t−τ,t
γ
t
˙y − ˙z
t−τ,t
,
3.16
where I denotes the identity matrix, and μ· denotes the logarithmic norm induced by ·, ·.
Remark 3.4. From 3.9, we know that yt − zt and ˙yt − ˙zt have an exponential
− y
2
, ∀t ≥ t
0
,y
1
,y
2
∈ X,χ,ψ∈ C
X
−τ,0
.
3.17
Journal of Inequalities and Applications 11
Example 3.5. Consider neutral delay differential equations with maxima see 15
˙y
t
f
t, y
t
s
,
t − h ≤ η
i
t
,ζ
i
t
≤ t, i 0, 1,
t ≥
0,T
y
t
φ
t
, ˙y
t
t
0.1sin y
2
η
1
t
sin t
t
t−1
0.3˙y
1
θ
1 ˙y
2
1
θ
dθ, t ≥ 0,
˙y
2
1 ˙y
2
2
θ
dθ, t ≥ 0,
y
1
t
φ
1
t
,y
2
t
φ
2
t
,t≤ 0,
3.19
where there exists a constant τ such that t − τ ≤ η
,
t
t−τt
K
t, θ, y
θ
dθ
,t≥ t
0
,
y
t
φ
t
, ˙y
t
˙
0,t,y
1
,y
2
,u,v,w
≤
G
f
λ, t, y
1
,y
2
,u,v,w
, ∀λ ≥ 0,t≥ t
0
,y
1
,y
2
,u,v,w ∈ X,
3.21
12 Journal of Inequalities and Applications
t
y
1
− y
2
β
t
u
1
− u
2
γ
t
v
1
− v
2
K
t, θ, y
1
− K
t, θ, y
2
≤ L
K
t
y
1
− y
2
,
t, θ
t, y
1
,u,v,w
−
f
t, y
2
,u,v,w
,
∀λ ∈ R,t≥ t
0
,y
1
,y
2
,u,v,w ∈ X.
3.24
Then if
α
t
t
−
1 − γ
t
α
t
≤ p<1, ∀t ≥ t
0
,
3.26
one has 3.9 and 3.11.
Our main objective in this subsection is to apply Corollary 2.5 to 3.20 and give
another sufficient condition for the asymptotical stability of the solution to 3.20. We will
assume that 3.21 and 3.23 are satisfied. We also assume that the continuous mapping
f in
3.20 satisfies
f
t
w
1
− w
2
, ∀t ≥ t
0
, y,u,v
1
,v
2
,w
1
,w
2
∈ X,
F
t, y, u
1
,v,w,r,s
−F
t, y, u
f
t, y, u,
f
t − τ
t
,u,v,w,r
,s
. 3.28
Journal of Inequalities and Applications 13
The mappings η
ν
t, ν 1, 2, , which are frequently used in that following analysis,
are defined recursively by
η
1
t
η
t
t − τ
η
ν−1
t
.
3.29
Theorem 3.8. Let the continuous mapping
f in 3.20 satisfy 3.21, 3.23, and 3.27. Suppose
that 3.25 and
σ
t
τμ
t
L
K
t
−
1 − γ
t
f
t, y
1
,u,v,w
−
f
t, y
2
,u,v,w
≤ L
y
1
− y
2
, ∀t ≥ t
f
t, z
t
,y
η
t
, ˙y
η
t
,
t
ηt
t
ηt
K
t, s, z
s
ds
.
3.34
Then it follows that
Y
t
≤ α
t
Y
η
t
,
f
η
t
,y
η
t
,y
η
2
t
, ˙y
t, s, y
s
ds
−
f
t, z
t
,y
η
t
,
f
η
t
K
t, s, y
s
ds
,
t
ηt
K
t, s, y
s
ds
≤ σ
K
t, s, y
s
− K
t, s, z
s
≤ σ
t
Y
η
t
γ
t
Φ
η
t
σ
t
μ
t
τL
K
t
max
s∈
t−τ,t
Y
y
t
− z
t
Φ
t
,t≥ t
0
, 3.38
the last assertion follows.
3.3. Comparison with the Existing Results
i In 2004, Wang and Li 16 were among the first who studied IVP in nonlinear NDDEs with
a single delay τt in a finite dimensional space C
n
,thatis,
˙z
t
f
φ
t
,t≤ t
0
.
3.39
They obtained the asymptotic stability result 3.31 for the cases of 3.25, 3.26 and 3.25,
and 3.30 under the following assumptions:
a there exists a constant τ
0
> 0 such that
τ
t
≥ τ
0
, ∀t ≥ t
0
; 3.40
Journal of Inequalities and Applications 15
b t − τt is a strictly increasing function on the interval t
0
, ∞;
c lim
t → ∞
t − τt∞.
From Theorems 3.7 and 3.8 of the present paper, we can obtain the asymptotic stability
dθ
,t≥ t
0
,
y
t
φ
t
,t∈
t
0
− τ, t
0
,
3.41
in finite-dimensional space for the case of
β τμL
K
−α
≤ p<1,
3.42
where α sup
t≥t
39–55, Gordon and Breach, Amsterdam, The Netherlands, 2000.
3 E. Liz and S. Trofimchuk, “Existence and stability of almost periodic solutions for quasilinear delay
systems and the Halanay inequality,” Journal of Mathematical Analysis and Applications, vol. 248, no. 2,
pp. 625–644, 2000.
4 H. Tian, “Numerical and analytic dissipativity of the θ-method for delay differential equations with a
bounded variable lag,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering,
vol. 14, no. 5, pp. 1839–1845, 2004.
5 S. Q. Gan, “Dissipativity of θ-methods for nonlinear Volterra delay-integro-differential equations,”
Journal of Computational and Applied Mathematics, vol. 206, no. 2, pp. 898–907, 2007.
16 Journal of Inequalities and Applications
6 L. P. Wen, Y. X. Yu, and W. S. Wang, “Generalized Halanay inequalities for dissipativity of Volterra
functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 347, no. 1, pp.
169–178, 2008.
7 L. P. Wen, W. S. Wang, and Y. X. Yu, “Dissipativity and asymptotic stability of nonlinear neutral delay
integro-differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp.
1746–1754, 2010.
8 J. K. Hale and S. M. Lunel, Introduction to Functional-Differential Equations,vol.99ofApplied
Mathematical Sciences, Springer, New York, NY, USA, 1993.
9 M. I. Gil’, Stability of Finite- and Infinite-Dimensional Systems, The Kluwer International Series in
Engineering and Computer Science no. 455, Kluwer Academic Publishers, Boston, Mass, USA, 1998.
10 V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional-Differential
Equations, vol. 463 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The
Netherlands, 1999.
11 J P. Richard, “Time-delay systems: an overview of some recent advances and open problems,”
Automatica, vol. 39, no. 10, pp. 1667–1694, 2003.
12 K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems,Birkh
¨
auser, Boston, Mass, USA,
2003.
13 A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical Mathematics
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