This paper shows that the solutions to the perturbed nonlinear functional differential system
y^{′}
=
f
(
t
,
y
) +
,
Ty
(
s
))
ds
,
f
(
t
, 0) = 0,
g
(
t
, 0, 0) = 0
go to zero as
t
goes to infinity. To show asymptotic property, we impose conditions on the perturbed part
,
Ty
(
s
))
ds
and the fundamental matrix of the unperturbed system
y^{′}
=
f
(
t
,
y
).
AMS Mathematics Subject Classification : 34D05.
1. Introduction
Elaydi and Farran
[8]
introduced the notion of exponential asymptotic stabilit
y
(EAS) which is a stronger notion than that of ULS. They investigated some analytic criteria for an autonomous differential system and its perturbed systems to be EAS. Pachpatte
[13]
investigated the stability and asymptotic behavior of solutions of the functional differential equation. Gonzalez and Pinto
[9]
proved theorems which relate the asymptotic behavior and boundedness of the solutions of nonlinear differential systems. Choi et al.
[6
,
7]
examined Lipschitz and exponential asymptotic stability for nonlinear functional systems. Also, Goo
[11]
and Choi and Goo
[2
,
4]
investigated Lipschitz and asymptotic stability for perturbed differential systems.
In this paper we will obtain some results on asymptotic property for nonlinear perturbed differential systems. We will employ the theory of integral inequalities to study asymptotic property for solutions of the nonlinear differential systems. The method incorporating integral inequalities takes an important place among the methods developed for the qualitative analysis of solutions to linear and nonlinear system of differential equations.
2. Preliminaries
We consider the nonlinear nonautonomous differential system
where
f
∈
C
(ℝ
^{+}
× ℝ
^{n}
, ℝ
^{n}
), ℝ
^{+}
= [0,∞) and ℝ
^{n}
is the Euclidean
n
space. We assume that the Jacobian matrix
f_{x}
= ∂
f
/∂
x
exists and is continuous on ℝ
^{+}
× ℝ
^{n}
and
f
(
t
, 0) = 0. Also, we consider the perturbed differential system of (1)
where
g
∈
C
(ℝ
^{+}
× ℝ
^{n}
× ℝ
^{n}
, ℝ
^{n}
),
g
(
t
, 0, 0) = 0, and
T
:
C
(ℝ
^{+}
, ℝ
^{n}
) →
C
(ℝ
^{+}
, ℝ
^{n}
) is a continuous operator .
For
x
∈ ℝ
^{n}
, let
. For an
n
×
n
matrix
A
, define the norm 
A
 of A by 
A
 = sup
_{x≤1}

Ax
.
Let
x
(
t
,
t
_{0}
,
x
_{0}
) denote the unique solution of (1) with
x
(
t
_{0}
,
t
_{0}
,
x
_{0}
) =
x
_{0}
, existing on [
t
_{0}
,∞). Then we can consider the associated variational systems around the zero solution of (1) and around
x
(
t
), respectively,
and
The fundamental matrix Φ(
t
,
t
_{0}
,
x
_{0}
) of (4) is given by
and Φ(
t
,
t
_{0}
, 0) is the fundamental matrix of (3).
Before giving further details, we give some of the main definitions that we need in the sequel
[8]
.
Definition 2.1.
The system (1) (the zero solution
x
= 0 of (1)) is called (S)
stable
if for any
ϵ
> 0 and
t
_{0}
≥ 0, there exists
δ
=
δ
(
t
_{0}
,
ϵ
) > 0 such that if 
x
_{0}
 <
δ
, then 
x
(
t
) <
ϵ
for all
t
≥
t
_{0}
≥ 0,
(AS)
asymptotically stable
if it is stable and if there exists
δ
=
δ
(
t
_{0}
) > 0 such that if 
x
_{0}
 <
δ
, then 
x
(
t
) → 0 as
t
→ ∞,
(ULS)
uniformly Lipschitz stable
if there exist
M
> 0 and
δ
> 0 such that 
x
(
t
) ≤
M

x
_{0}
 whenever 
x
_{0}
 ≤
δ
and
t
≥
t
_{0}
≥ 0,
(EAS)
exponentially asymptotically stable
if there exist constants
K
> 0 ,
c
> 0, and
δ
> 0 such that
provided that 
x
_{0}
 <
δ
,
(EASV)
exponentially asymptotically stable in variation
if there exist constants
K
> 0 and
c
> 0 such that
provided that 
x
_{0}
 < ∞.
Remark 2.1
(
[9]
). The last definition implies that for 
x
_{0}
 ≤
δ
We give some related properties that we need in the sequel. We need Alekseev formula to compare between the solutions of (1) and the solutions of perturbed nonlinear system
where
g
∈
C
(ℝ
^{+}
× ℝ
^{n}
, ℝ
^{n}
) and
g
(
t
, 0) = 0. Let
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) denote the solution of (5) passing through the point (
t
_{0}
,
y
_{0}
) in ℝ
^{+}
× ℝ
^{n}
.
The following is a generalization to nonlinear system of the variation of constants formula due to Alekseev
[1]
.
Lemma 2.1.
Let x and y be a solution of (1) and (5), respectively. If y
_{0}
∈ ℝ
^{n}, then for all t
≥
t
_{0}
such that x
(
t
,
t
_{0}
,
y
_{0}
) ∈ ℝ
^{n}
,
y
(
t
,
t
_{0}
,
y
_{0}
) ∈ ℝ
^{n}
,
Lemma 2.2
(Biharitype inequality).
Let u
, λ ∈
C
(ℝ
^{+}
)
, w
∈
C
((0,∞))
and w
(
u
)
be nondecreasing in u. Suppose that, for some c
> 0,
Then
where t
_{0}
≤
t
<
b
_{1}
,
,
W
^{−1}
(
u
)
is the inverse of W
(
u
)
, and
Lemma 2.3
(
[10]
).
Let u
, λ
_{1}
, λ
_{2}
, λ
_{3}
, λ
_{4}
∈
C
(ℝ
^{+}
),
w
∈
C
((0,∞))
, and w
(
u
)
be nondecreasing in u, u
≤
w
(
u
).
Suppose that for some c
> 0
and
0 ≤
t
_{0}
≤
t
,
Then
where t
_{0}
≤
t
<
b
_{1}
,
W
,
W
^{−1}
are the same functions as in Lemma 2.2, and
Lemma 2.4
(
[3]
).
Let u
, λ
_{1}
, λ
_{2}
, λ
_{3}
, λ
_{4}
, λ
_{5}
, λ
_{6}
, λ
_{7}
∈
C
(ℝ
^{+}
),
w
∈
C
((0,∞)),
and w
(
u
)
be nondecreasing in u
,
u
≤
w
(
u
).
Suppose that for some c
> 0
and
0 ≤
t
_{0}
≤
t
,
Then
where t
_{0}
≤
t
<
b
_{1}
,
W
,
W
^{−1}
are the same functions as in Lemma 2.2, and
Proof
. Define a function
v
(
t
) by the right member of (6). Then, we have
v
(
t
_{0}
) =
c
and
t
≥
t
_{0}
, since
v
(
t
) is nondecreasing,
u
≤
w
(
u
), and
u
(
t
) ≤
v
(
t
). Now, by integrating the above inequality on [
t
_{0}
,
t
] and
v
(
t
_{0}
) =
c
, we have
It follows from Lemma 2.2 that (8) yields the estimate (7). □
For the proof we need the following corollary from Lemma 2.4.
Corollary 2.5.
Let u
, λ
_{1}
, λ
_{2}
, λ
_{3}
, λ
_{4}
, λ
_{5}
, λ
_{6}
∈
C
(ℝ
^{+}
),
w
∈
C
((0,∞))
, and w
(
u
)
be nondecreasing in u, u
≤
w
(
u
).
Suppose that for some c
> 0
and
0 ≤
t
_{0}
≤
t
,
Then
where t
_{0}
≤
t
<
b
_{1}
,
W
,
W
^{−1}
are the same functions as in Lemma 2.2, and
Lemma 2.6
(
[5]
).
Let u
, λ
_{1}
, λ
_{2}
, λ
_{3}
, λ
_{4}
, λ
_{5}
, λ
_{6}
∈
C
(ℝ
^{+}
),
w
∈
C
((0,∞))
and w
(
u
)
be nondecreasing in u, u
≤
w
(
u
).
Suppose that for some c
> 0,
Then
where t
_{0}
≤
t
<
b
_{1}
,
W
,
W
^{−1}
are the same functions as in Lemma 2.2, and
Proof
. Define a function
z
(
t
) by the right member of (9). Then, we have
z
(
t
_{0}
) =
c
and
since
z
(
t
) and
w
(
u
) are nondecreasing,
u
≤
w
(
u
), and
u
(
t
) ≤
z
(
t
). Therefore, by integrating on [
t
_{0}
,
t
], the function
z
satisfies
It follows from Lemma 2.2 that (11) yields the estimate (10). □
We prepare two corollaries from Lemma 2.6 that are used in proving the theorems.
Corollary 2.7.
Let u
, λ
_{1}
, λ
_{2}
, λ
_{3}
∈
C
(ℝ
^{+}
),
w
∈
C
((0,∞))
and w
(
u
)
be nondecreasing in u, u
≤
w
(
u
).
Suppose that for some c
> 0,
Then
where t
_{0}
≤
t
<
b
_{1}
,
W
,
W
^{−1}
are the same functions as in Lemma 2.2, and
Corollary 2.8.
Let u
, λ
_{1}
, λ
_{2}
, λ
_{3}
, λ
_{4}
∈
C
(ℝ
^{+}
),
w
∈
C
((0,∞))
and w
(
u
)
be nondecreasing in u, u
≤
w
(
u
).
Suppose that for some c
> 0,
Then
where t
_{0}
≤
t
<
b
_{1}
,
W
,
W
^{−1}
are the same functions as in Lemma 2.2, and
3. Main results
In this section, we investigate asymptotic property for solutions of perturbed nonlinear functional differential systems.
Theorem 3.1.
Let the solution x
= 0 of ( 1)
be EASV. Suppose that the perturbing term g
(
t
,
y
,
Ty
) satisfies
and
where α
> 0,
a
,
b
,
k
,
w
∈
C
(ℝ
^{+}
),
a
,
b
,
k
,
w
∈
L
_{1}
(ℝ
^{+}
),
w
(
u
)
is nondecreasing in u, and u
≤
w
(
u
).
If
where t
≥
t
_{0}
and c
= 
y
_{0}

Me
^{αt0}
, then all solutions of (2) approach zero as t
→ ∞.
Proof
. Let
x
(
t
) =
x
(
t
,
t
_{0}
,
y
_{0}
) and
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) be solutions of (1) and (2), respectively. Since the solution
x
= 0 of (1) is EASV, it is EAS by remark 2.1.
Using Lemma 2.1, (12), and (13), we have
Then, we obtain
since
w
is nondecreasing. Set
u
(
t
) = 
y
(
t
)
e^{αt}
. By Lemma 2.3 and (14) we have
where
c
=
M

y
_{0}

e
^{αt0}
. The above estimation yields the desired result. □
Remark 3.1.
Letting
b
(
t
) = 0 in Theorem 3.1, we obtain the similar result as that of Theorem 3.5 in
[4]
.
Theorem 3.2.
Let the solution x
= 0
of (1) be EASV. Suppose that the perturbing term g
(
t
,
y
,
Ty
)
satisfies
and
where α
> 0,
a
,
b
,
c
,
k
,
w
∈
C
(ℝ
^{+}
),
a
,
b
,
c
,
k
,
w
∈
L
_{1}
(ℝ
^{+}
),
w
(
u
)
is nondecreasing in u
,
u
≤
w
(
u
).
If
where t
≥
t
_{0}
and c
= 
y
_{0}

Me
^{αt0}
, then all solutions of (2) approach zero as t
→ ∞.
Proof
. Let
x
(
t
) =
x
(
t
,
t
_{0}
,
y
_{0}
) and
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) be solutions of (1) and (2), respectively. Since the solution
x
= 0 of (1) is EASV, it is EAS by Remark 2.1. Applying Lemma 2.1, (15), and (16), we have
Since
w
is nondecreasing, we obtain
Set
u
(
t
) = 
y
(
t
)
e^{αt}
. By Corollary 2.5 and (17), we have
where
c
=
M

y
_{0}

e
^{αt0}
. From the above estimation, we obtain the desired result. □
Remark 3.2.
Letting
w
(
u
) =
u
,
b
(
t
) =
c
(
t
) = 0 in Theorem 3.2, we obtain the similar result as that of Corollary 3.6 in
[4]
.
Theorem 3.3.
Let the solution x
= 0
of (1) be EASV. Suppose that the perturbed term g
(
t
,
y
,
Ty
)
satisfies
and
where α
> 0,
a
,
b
,
k
,
w
∈
C
(ℝ
^{+}
),
a
,
b
,
k
,
w
∈
L
_{1}
(ℝ
^{+}
)
and w
(
u
)
is nondecreasing in u, u
≤
w
(
u
).
If
where b
_{1}
= ∞
and c
=
M

y
_{0}

e
^{αt0}
, then all solutions of (2) approach zero as t
→ ∞.
Proof
. Let
x
(
t
) =
x
(
t
,
t
_{0}
,
y
_{0}
) and
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) be solutions of (1) and (2), respectively. Since the solution
x
= 0 of (1) is EASV, it is EAS. By conditions, Lemma 2.1, (18), and (19), we have
Since
w
is nondecreasing, we have
Then, it follows from Corollary 2.7 with
u
(
t
) = 
y
(
t
)
e^{αt}
and (20) that
where
c
=
M

y
_{0}

e
^{αt0}
. Hence, all solutions of (2) approach zero as
t
→ ∞ , and so the proof is complete. □
Remark 3.3.
Letting
b
(
t
) = 0 in Theorem 3.3, we obtain the similar result as that of Theorem 3.7 in
[4]
.
Theorem 3.4.
Let the solution x
= 0
of (1) be EASV. Suppose that the perturbed term g
(
t
,
y
,
Ty
)
satisfies
and
where α
> 0,
a
,
b
,
c
,
k
,
w
∈
C
(ℝ
^{+}
),
a
,
b
,
c
,
k
,
w
∈
L
_{1}
(ℝ
^{+}
)
and w
(
u
)
is nondecreasing in u, u
≤
w
(
u
).
If
where b
_{1}
= ∞
and c
=
M

y
_{0}

e
^{αt0}
, then all solutions of (2) approach zero as t
→ ∞.
Proof
. Let
x
(
t
) =
x
(
t
,
t
_{0}
,
y
_{0}
) and
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) be solutions of (1) and (2), respectively. Since the solution
x
= 0 of (1) is EASV, it is EAS. By means of Lemma 2.1, (21), and (22), we have
Since
w
is nondecreasing, we have
Set
u
(
t
) = 
y
(
t
)
e^{αt}
. Then, it follows from Corollary 2.8 and (23) that
where
c
=
M

y
_{0}

e
^{αt0}
. From the above inequality, we obtain the desired result. □
Remark 3.4.
Letting
w
(
u
) =
u
,
b
(
t
) =
c
(
t
) = 0 in Theorem 3.4, we obtain the similar result as that of Corolary 3.8 in
[4]
.
Acknowledgements
The authors are very grateful for the referee’s valuable comments.
BIO
Dong Man Im received the BS and Ph.D at Inha University. Since 1982 he has been at Cheongju University as a professor. His research interests focus on Algebra and differential equations.
Department of Mathematics Education Cheongju University Cheongju Chungbuk 360764, Korea.
email: dmim@cheongju.ac.kr
Yoon Hoe Goo received the BS from Cheongju University and Ph.D at Chungnam National University under the direction of ChinKu Chu. Since 1993 he has been at Hanseo University as a professor. His research interests focus on topologival dynamical systems and differential equations.
Department of Mathematics, Hanseo University, Seosan 356706, Korea.
email: yhgoo@hanseo.ac.kr
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