The present paper is concerned with the notions of Lipschitz and asymptotic stability for perturbed nonlinear differential system knowing the corresponding stability of nonlinear differential system. We investigate Lipschitz and asymtotic stability for perturbed nonlinear differential systems. The main tool used is integral inequalities of the Biharitype, in special some consequences of an extension of Bihari's result to Pinto and Pachpatte, and all that sort of things.
1. INTRODUCTION
The notion of uniformly Lipschitz stability (ULS) was introduced by Dannan and Elaydi
[8]
. For linear systems, the notions of uniformly Lipschitz stability and that of uniformly stability are equivalent. However, for nonlinear systems, the two notions are quite distinct. In fact, uniformly Lipschitz stability lies somewhere between uniformly stability on one side and the notions of asmptotic stability in variation of Brauer
[4]
and uniformly stability in variation of Brauer and Strauss
[3]
on the other side. Gonzalez and Pinto
[9]
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 nonlinear 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
and
is the Euclidean
n
space. We assume that the Jacobian matrix
f
_{x}
= ∂
f
/ ∂
x
exists and is continuous on
and
f
(
t
, 0) = 0. Also, consider the perturbed differential system of (2.1)
where
,
g
(
t
, 0) = 0. For
, 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 (2.1) 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 (2.1) and around
x
(
t
), respectively,
and
The fundamental matrix Φ(
t
,
t
_{0}
,
x
_{0}
) of (2.4) is given by
and Φ(
t
,
t
_{0}
, 0) is the fundamental matrix of (2.3).
Before giving further details, we give some of the main definitions that we need in the sequel
[8]
.
Definition 2.1.
The system (2.1) (the zero solution
x
= 0 of (2.1)) 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 Φ(
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
x(t) ≤ K x_{0}e^{c(tt0)}, 0 ≤ t_{0} ≤ t
provided that 
x
_{0}
 < 𝛿,
(EASV)
exponentially asymptotically stable in variation
if there exist constants
K
> 0 and
c
> 0 such that
Φ(t,t_{0},x_{0}) ≤ K e^{c(tt0)}, 0 ≤ t_{0} ≤ t
provided that 
x
_{0}
 < ∞.
We give some related properties that we need in the sequel.
We need Alekseev formula to compare between the solutions of (2.1) and the solutions of perturbed nonlinear system
where
and
g
(
t
, 0) = 0. Let
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) denote the solution of (2.5) passing through the point (
t
_{0}
,
y
_{0}
) in
×
.
The following is a generalization to nonlinear system of the variation of constants formula due to Alekseev
[1]
.
Lemma 2.2.
Let x and y be a solution of
(
.
1
)
and
(
.
5
),
respectively. If
then for all t such that
Lemma 2.3
(
[7]
)
.
Let u
, λ
_{1}
, λ
_{2}
,
w
(
u
)
be nondecreasing in u and
w
(
u
) ≤ w(
)
for some
v
> 0.
If , for some c
> 0,
then
where
u
> 0,
u
_{0}
> 0
W
^{1}
(
u
)
is the inverse of W
(
u
)
and
Lemma 2.4
(
[10]
)
.
Let u, p, q,w, and r
∈
C
(
)
and suppose that, for some c
≥ 0,
we have
Then
Lemma 2.5
(
[15]
)
.
Let u
(
t
),
f
(
t
),
and g
(
t
)
be realvalued nonnegative continuous functions defined on
,
for which the inequality
holds, where u
_{0}
is a nonnegative constant. Then,
Lemma 2.6
(
[12]
)
.
Let u
, λ
_{1}
, λ
_{2}
, λ
_{3}
∈
C
(
),
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 2.3 and
Lemma 2.7
(
[13]
)
.
Let u, p, q,w, r
∈
C
(
),
w
∈
C
((0,∞))
and w
(
u
)
be nondecreasing in u. Suppose that for some c
≥ 0,
Then
where
and
Lemma 2.8
(
[14]
)
.
Let the following condition hold for functions
u
(
t
),
v
(
t
) ∈
C
[[
t
_{0}
,∞)
)
and
k
(
t, u
) ∈
C
[[
t
_{0}
,∞) ×
,
)
:
t
≥
t
_{0}
and
k
(
s
,
u
)
is strictly increasing in u for each fixed s
≥ 0.
If u
(
t
_{0}
) <
v
(
t
_{0}
),
then u
(
t
) <
v
(
t
),
t
≥
t
_{0}
≥ 0.
Lemma 2.9
(
[5]
)
.
Let u, λ
_{1}
, λ
_{2}
, λ
_{3}
∈
C
(
),
w
∈
C
((0,∞))
and w
(
u
)
be nondecreasing in u. Suppose that for some c
> 0,
Then
where
u
> 0,
u
_{0}
> 0,
W
^{1}
(
u
)
is the inverse of
W
(
u
)
and
3. MAIN RESULTS
In this section, we investigate Lipschitz and asymptotic stability for solutions of the nonlinear perturbed differential systems.
Theorem 3.1.
Assume that x
= 0
of
(
.
1) is ULS. Let the following condition hold for
(
.
):
where W
(
t
,
u
) ∈
C
(
×
,
)
is monotone nondecreasing in u with W
(
t
, 0) = 0.
Suppose that u
(
t
)
is any solution of the scalar differential equation
existing on
such that m
(
t
_{0}
) <
u
(
t
_{0}
).
If u
= 0
of (3.1) is ULS, then y
= 0
of
(
.
)
is also ULS whenever M

y
_{0}
 <
u
_{0}
.
Proof. Let x
(
t
) =
x
(
t
,
t
_{0}
,
y
_{0}
) and
y
(
t
) = y(
t
,
t
_{0}
,
y
_{0}
) be solutions of (2.1) and (2.2), respectively. Using the variation of constants formula, we have
where Φ(
t
,
t
_{0}
,
y
_{0}
) is the fundemental matrix of (2.4). Since
x
= 0 of (2.1) is ULS, it is ULSV by Corollary 3.6
[5]
. Thus there exist
M
> 0 and 𝛿 > 0 such that Φ(
t
,
t
_{0}
,
y
_{0}
) ≤
M
for
t
≥
t
_{0}
≥ 0. Therefore, by the assmption, we have
Hence 
y
(
t
) <
u
(
t
) by Lemma 2.8. Since
u
= 0 of (3.1) is ULS, it easily follows that
y
= 0 of (2.2) is ULS.
Corollary 3.2.
Assume that x
= 0
of
(
.
1
)
is ULS. Consider the scalar differential
equation
where u
_{0}
≥ 1,
K
≥ 1
and a
,
k
∈
C
(
)
satisfy the conditions
(
a
)
where
(
b
)
Then y
= 0
of
(
.
)
is ULS.
Proof.
Let
u
(
t
) =
u
(
t
,
t
_{0}
,
x
_{0}
) be any solution of (3.2). Then, by Lemma 2.5 , we have
Hence
u
= 0 of (3.2) is ULS. This implies that the solution
y
= 0 of (2.2) is ULS by Theorem 3.1.
Remark 3.3.
In Corollary 3.2, it is needed that
b
_{1}
= ∞. The condition
W
(∞) = ∞ is too strong and it represents situations which are not stable. For example, if
w
(
u
) =
u
^{𝛼}
, then only 𝛼 ≤ 1 satisfies
W
(∞) = ∞ and 𝛼 < 1 is not stable. See
[18]
.
Corollary 3.4.
Assume that x
= 0
of
(
.
1
)
is ULS. Consider the scalar differential equation
where u
_{0}
≥ 1,
K
≥ 1,
u
,
w
∈
C
(
),
w
(
u
)
be nondecreasing in u and
w
(
u
)≤
w
(
)
for some v
> 0,
and
a
,
k
∈
C
(
)
satisfy the conditions
(
a
)
where
(
b
)
and a, k
∈
L
_{1}
(
).
Then y
= 0
of
(
.
)
is ULS.
Proof.
Let
u
(
t
) =
u
(
t
,
t
_{0}
,
x
_{0}
) be any solution of (3.3). Then, by Lemma 2.3, we have
Hence
u
= 0 of (3.3) is ULS. By Theorem 3.1, the solution
y
= 0 of (2.2) is ULS.
Corollary 3.5.
Assume that x
= 0
of
(
.
1
)
is ULS. Consider the scalar differential equation
where w
∈
C
((0,∞),
w
(
u
)
is nondecreasing on u and u
≤
w
(
u
),
u
_{0}
≥ 1,
K
≥ 1
and
a, b,
k
∈
C
(
)
satisfy the conditions
(
a
)
where
(
b
)
L
_{1}
(
).
Then y
= 0
of
(
.
)
is ULS.
Proof.
Let
u
(
t
) =
u
(
t
,
t
_{0}
,
x
_{0}
) be any solution of (3.4). Then, Lemma 2.6, we have
Hence
u
= 0 of (3.4) is ULS, and so by Theorem 3.1, the solution
y
= 0 of (2.2) is ULS. □
Theorem 3.6.
For the perturbed
(
.
),
we asssume that
where a, b, k
∈
C
(
),
a, b, k
∈
L
_{1}
(
),
w
∈
C
((0,∞),
and
w
(
u
)
is nondecreasing in u,u
≤
w
(
u
),
and
w
(
u
) ≤
w
(
)
for some v
> 0,
where
M
(
t
_{0}
) < ∞
and
b
_{1}
= ∞.
Then the zero solution of
(
.
)
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 (2.1) and (2.2), respectively. Since
x
= 0 of (2.1) is ULSV, it is ULS by Theorem 3.3
[8]
. Applying Lemma 2.2, we have
Set
u
(
t
) = 
y
(
t
)
y
_{0}

^{1}
. Now an application of Lemma 2.6 yields
Hence we have 
y
(
t
) ≤
M
(
t
_{0}
)
y
_{0}
 for some
M
(
t
_{0}
) > 0 whenever 
y
_{0}
 < 𝛿. This completes the proof. □
Theorem 3.7.
For the perturbed
(
.
),
we asssume that
where a, b, k
∈
C
(
),
a, b, k
∈
L
_{1}
(
),
w
∈
C
((0,∞),
and
w
(
u
)
is nondecreasing in u,u
≤
w
(
u
),
and
w
(
u
) ≤
w
(
)
for some v
> 0,
where
M
(
t
_{0}
) < ∞
and
b
_{1}
= ∞.
Then the zero solution of
(
.
)
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 (2.1) and (2.2), respectively. Using the nonlinear variation of constants formula and the ULSV condition of
x
= 0 of (2.1), we have
Set
u
(
t
) = 
y
(
t
)
y
_{0}

^{1}
. Now an application of Lemma 2.7 yields
Thus 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. □
Theorem 3.8.
Let the solution x
= 0 of (
.
1
)
be EAS. Suppose that the perturbing term g
(
t
,
y
)
satisfies
where
𝛼 > 0,
a, b, k
∈
C
(
),
a, b, k
∈
L
_{1}
(
),
w
(
u
)
is nondecreasing in u, and
w
(
u
) ≤
w
(
)
for some v
> 0.
If
where c
= 
y
_{0}

Me
^{𝛼t0}
,
then all solutions of
(
.
)
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 (2.1) and (2.2), respectively. Since the solution
x
= 0 of (2.1) is EAS, we have Φ(
t
,
t
_{0}
,
x
_{0}
) ≤
Me
^{𝛼(tt0)}
for some
M
> 0 and
c
> 0(Theorem 2
[2]
). Using Lemma 2.2, we have
since
e
^{𝛼t}
is increasing. Set
u
(
t
) = 
y
(
t
)
e
^{𝛼t}
. An application of Lemma 2.4 obtains
The above estimation yields the desired result. □
Theorem 3.9.
Let the solution x
= 0
of
(
.
1
)
be EAS. Suppose that the perturbing term g
(
t
,
y
)
satisfies
where
𝛼 > 0,
a, b, k
,
w
∈
C
(
),
a, b, k
∈
L
_{1}
(
)
and w
(
u
)
is nondecreasing in u. If
where c
=
M

y
_{0}

e
^{𝛼t0}
, then all solutions of
(
.
)
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 (2.1) and (2.2), respectively. Using Lemma 2.2 and the assmptions, we have
Set
u
(
t
) = 
y
(
t
)
e
^{𝛼t}
. Since
w
(
u
) is nondecreasing, an application of Lemma 2.9 obtains
where
c
=
M

y
_{0}

e
^{𝛼t0}
. From the above estimation, we obtains the desired result. □
Acknowledgements
The author is very grateful for the referee’s valuable comments.
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