In this paper, we investigate Lipschitz and asymptotic stability for perturbed functional differential systems
AMS Mathematics Subject Classification : 34D10.
1. Introduction
The notion of uniformly Lipschitz stability (ULS) was introduced by Dannan and Elaydi
[9]
. This notion of ULS lies somewhere between uniformly stability on one side and the notions of asymptotic stability in variation of Brauer
[4]
and uniformly stability in variation of Brauer and Strauss
[3]
on the other side. An important feature of ULS is that for linear systems, the notion of uniformly Lipschitz stability and that of uniformly stability are equivalent. However, for nonlinear systems, the two notions are quite distinct. Also, Elaydi and Farran
[10]
introduced the notion of exponential asymptotic stability(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. Gonzalez and Pinto
[11]
proved theorems which relate the asymptotic behavior and boundedness of the solutions of nonlinear differential systems.
In this paper, we investigate Lipschitz and asymptotic stability for solutions of the functional differential systems. To do this we need some integral inequalities. 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. In the presence the method of integral inequalities is as efficient as the direct Lyapunov’s method.
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, consider the perturbed differential system of (1)
where
g
∈
C
(ℝ
^{+}
× ℝ
^{n}
, ℝ
^{n}
),
h
∈
C
[ℝ
^{+}
× ℝ
^{n}
× ℝ
^{n}
, ℝ
^{n}
] ,
g
(
t
, 0) = 0,
h
(
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 Փ(
t
,
t
_{0}
,
x
_{0}
) of (4) is given by
and Փ(
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
[9]
.
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,
(US)
uniformly stable
if the
δ
in (S) is independent of the time
t
_{0}
,
(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
(ULSV)
uniformly Lipschitz stable in variation
if there exist
M
> 0 and
δ
> 0 such that Փ(
t
,
t
_{0}
,
x
_{0}
) ≤
M
for 
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
(
[11]
). 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 such that x
(
t
,
t
_{0}
,
y
_{0}
) ∈ ℝ
^{n}
,
Lemma 2.2
(
[8]
).
Let u
,
λ
_{1}
,
λ
_{1}
,
w
∈
C
(ℝ
^{+}
),
w
(
u
)
be nondecreasing in u and
for some v
> 0.
If, for some c
> 0,
then
where
is the inverse of W
(
u
),
and
Lemma 2.3
(
[15]
).
Let u, p, q, w, and r
∈
C
(ℝ
^{+}
)
and suppose that, for some c
≥ 0,
we have
Then
Lemma 2.4
(
[13]
).
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 W , W
^{−1}
are the same functions as in Lemma 2.2, and
Lemma 2.5
(
[13]
).
Let u, p, q, w, r
∈
C
(ℝ
^{+}
),
w
∈
C
((0, ∞))
and w
(
u
)
be nondecreasing in u, u
≤
w
(
u
).
Suppose that for some c
≥ 0,
Then
where W , W
^{−1}
are the same functions as in Lemma 2.2, and
Lemma 2.6
(
[6]
).
Let u,
λ
_{1}
,
λ
_{2}
,
λ
_{3}
∈
C
(ℝ
^{+}
),
w
∈
C
((0, ∞))
and w
(
u
)
be nondecreasing in u. Suppose that for some c
> 0,
Then
where W , W
^{−1}
are the same functions as in Lemma 2.2, and
Lemma 2.7
(
[5]
).
Let u,
λ
_{1}
,
λ
_{2}
,
λ
_{3}
,
λ
_{4}
,
w
∈
C
(ℝ
^{+}
),
w
∈
C
((0, ∞))
and w
(
u
)
be nondecreasing in u, u
≤
w
(
u
).
Suppose that for some c
≥ 0,
for t
≥
t
_{0}
≥ 0
and for some c
≥ 0.
Then
for t
_{0}
≤
t
<
b
_{1}
,
where W, W
^{−1}
are the same functions as in Lemma 2.2, and
3. Main results
In this section, we investigate Lipschitz and asymptotic stability for solutions of the perturbed functional differential systems.
Theorem 3.1.
For the perturbed (2), we asssume that
and
where a, b, c, k
∈
C
(ℝ
^{+}
),
a, b, c, k
∈
L
_{1}
(ℝ
^{+}
),
w
∈
C
((0, ∞),
and w
(
u
)
is nondecreasing in
where M
(
t
_{0}
) < ∞
and b
_{1}
= ∞.
Then the zero solution of (2) is ULS whenever the zero solution of (1) is ULSV.
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
x
= 0 of (1) is ULSV, it is ULS (
[9]
,Theorem 3.3). Applying Lemma 2.1, condition (6), and condition (7), we have
Set
u
(
t
) = 
y
(
t
)
y
_{0}

^{−1}
. Now an application of Lemma 2.4 yields
By condition (8), we have 
y
(
t
) ≤
M
(
t
_{0}
)
y
_{0}
 for some
M
(
t
_{0}
) > 0 whenever 
y
_{0}
 <
δ
. This completes the proof. □
Remark 3.1.
Letting
c
(
t
) = 0 in Theorem 3.1, we obtain the same result as that of Theorem 3.6 in
[12]
.
Theorem 3.2.
For the perturbed (2), we asssume that
and
where a, b, c, k
∈
C
(ℝ
^{+}
),
a, b, c, k
∈
L
_{1}
(ℝ
^{+}
),
w
∈
C
((0, ∞),
and w
(
u
)
is nondecreasing in
where M
(
t
_{0}
) < ∞
and b
_{1}
= ∞.
Then the zero solution of (2) is ULS whenever the zero solution of (1) is ULSV.
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
x
= 0 of (1) is ULSV, it is ULS . Using the nonlinear variation of constants formula , condition (9), and condition (10), we have
Set
u
(
t
) = 
y
(
t
)
y
_{0}

^{−1}
. Then, it follows from Lemma 2.7 that
By condition (11), we have 
y
(
t
) ≤
M
(
t
_{0}
)
y
_{0}
 for some
M
(
t
_{0}
) > 0 whenever 
y
_{0}
 <
δ
, and so the proof is complete. □
Remark 3.2.
Letting
c
(
t
) = 0 in Theorem 3.2, we obtain the same result as that of Theorem 3.7 in
[12]
.
Theorem 3.3.
For the perturbed (2), we asssume that
and
where a, b, c, k
∈
C
(ℝ
^{+}
),
a, b, c, k
∈
L
_{1}
(ℝ
^{+}
),
w
∈
C
((0, ∞),
and w
(
u
)
is nondecreasing in
where M
(
t
_{0}
) < ∞
and b
_{1}
= ∞.
Then the zero solution of (2) is ULS whenever the zero solution of (1) is ULSV.
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
x
= 0 of (1) is ULSV, it is ULS. Using the nonlinear variation of constants formula, condition (12), and condition (13), we have
Set
u
(
t
) = 
y
(
t
)
y
_{0}

^{−1}
. Now an application of Lemma 2.5 yields
t
≥
t
_{0}
. From condition (14) we get 
y
(
t
) ≤
M
(
t
_{0}
)
y
_{0}
 for some
M
(
t
_{0}
) > 0 whenever 
y
_{0}
 <
δ
, and so the proof is complete. □
Remark 3.3.
Letting
c
(
s
) = 0 in Theorem 3.3, we obtain the same result as that of Theorem 3.7 in
[12]
.
Theorem 3.4.
Let the solution x
= 0
of (1) be EASV. Suppose that the perturbing term g(t, y) satisfies
and
where α
> 0,
a, b, c, k
∈
C
(ℝ
^{+}
),
a, b, c, k
∈
L
_{1}
(ℝ
^{+}
),
w
(
u
)
is nondecreasing in u. If
where c
= 
y
_{0}

Me
^{α}
^{t}
_{0}
,
then all solutions of (2) approch 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, by remark 2.1, it is EAS. Using Lemma 2.1, condition (15), and condition (16), we have
Set
u
(
t
) = 
y
(
t
)
e
^{αt}
. By Lemma 2.3, we obtain
where
c
=
M

y
_{0}

e
^{αt}
_{0}
. By condition (17), we have 
y
(
t
) ≤
ce
^{−}
^{αt}
M
(
t
_{0}
). This estimation yields the desired result. □
Remark 3.4.
Letting
c
(
s
) = 0 in Theorem 3.4, we obtain the same result as that of Theorem 3.8 in
[12]
.
Theorem 3.5.
Let the solution x
= 0
of (1) be EASV. Suppose that the perturbed term g
(
t, y
)
satisfies
and
where α
> 0,
a, b, c, k, w
∈
C
(ℝ
^{+}
),
a, b, c, k
∈
L
_{1}
(ℝ
^{+}
),
w
(
u
)
is nondecreasing in u, and
where c
=
M

y
_{0}

e
^{αt}
^{0}
,
then all solutions of (2) approch 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, condition (18), and condition (19), we have
Set
u
(
t
) = 
y
(
t
)
e
^{αt}
. Since
w
(
u
) is nondecreasing, an application of Lemma 2.6 obtains
where
c
=
M

y
_{0}

e
^{αt}
_{0}
. By condition (20), we have 
y
(
t
) ≤
e
^{−}
^{αt}
M
(
t
_{0}
). From this estimation, we obtain the desired result. □
Remark 3.5.
Letting
c
(
t
) = 0 in Theorem 3.5, we obtain the same result as that of Theorem 3.9 in
[12]
.
Acknowledgements
The author is very grateful for the referee’s valuable comments.
BIO
Sang Il Choi received the BS from Korea University and Ph.D at North Carolina State University under the direction of J. Silverstein . Since 1995 he has been at Hanseo University as a professor. His research interests focus on Analysis and Probability theory.
Department of Mathematics, Hanseo University, Seasan 356706, Republic of Korea.
email:schoi@hanseo.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 topological dynamical systems and differential equations.
Department of Mathematics, Hanseo University, Seasan 356706, Republic of Korea.
email:yhgoo@hanseo.ac.kr
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