The present paper is concerned with the notions of Lipschitz and asymptotic for perturbed functional differential system knowing the corresponding stability of functional differential system. We investigate Lipschitz and asymptotic stability for perturbed functional differential systems. The main tool used is integral inequalities of the Biharitype, and all that sort of things.
1. INTRODUCTION
Dannan and Elaydi introduced a new notion of uniformly Lipschitz stability (ULS)
[8]
. 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
[9]
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
[10]
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 et al.
[11
,
13]
investigated Lipschitz and asymptotic stability for perturbed differential systems.
In this paper, we investigate Lipschitz and asymptotic stability for solutions of the functional differential systems using 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 functional differential system of (2.1)
where
,
g
(
t
, 0, 0) =
h
(
t
, 0, 0) = 0 and
is a continuous operator .
For
. 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) ≤Kx0e−c(tt0), 0 ≤t0≤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,t0,x0) ≤Ke−c(t−t0), 0 ≤t0≤t
provided that 
x
_{0}
 < ∞.
Remark 2.2
(
[10]
). The last deffnition implies that for 
x
_{0}
 ≤
δ

x(t) ≤Kx0e−c(t−t0), 0 ≤t0≤t
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 2.3.
Let x and y be a solution of (2.1) and (2.5), respectively
.
If
then for all t such that
,
where
Φ(
t
,
s
,
y
(
s
))
is a fundamental matrix of (2.4)
.
Lemma 2.4
(
[14]
).
Let
and suppose that, for some c
≥ 0,
we have
Then
Lemma 2.5
(
[7]
). (Bihari – type Inequality)
Let
,
w
∈
C
((0, ∞))
and w
(
u
)
be nondecreasing in u. Suppose that, for some c
> 0,
Then
where
is the inverse of W
(
u
),
and
Lemma 2.6
(
[12]
).
Let
,
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 2.5, and
Lemma 2.7
(
[12]
).
Let
,
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 2.5, and
Lemma 2.8
(
[5]
).
Let
,
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 2.5, and
3. MAIN RESULTS
In this section, we investigate Lipschitz and asymptotic stability for solutions of the perturbed functional differential systems.
We need the lemma to prove the following theorem.
Lemma 3.1.
Let
,
w
∈
C
((0, ∞)),
and w
(
u
)
be nondecreasing in u, u
≤
w
(
u
).
Suppose that for some c
≥ 0,
for t
≥
t
_{0}
≥
and for some c
≥ 0.
Then
for t
_{0}
≤
t
≤
b
_{1}
,
where W
,
W
^{−1}
are the same functions as in Lemma 2.5, and
Proof
. Define a function
v
(
t
) by the right member of (3.1) . Then
which implies
since
v
and
w
are 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
Then, by the wellknown Biharitype inequality, (3.3) yields the estimate (3.2).
Theorem 3.2.
For the perturbed (2.2), we assume that
and
where
,
w
∈
C
((0, ∞)),
and w
(
u
)
is nondecreasing in u
,
u
≤
w
(
u
),
and
for some v
> 0,
where M
(
t
_{0}
) < ∞
and b
_{1}
= ∞.
Then the zero solution of (2.2) is ULS whenever the zero solution of (2.1) is ULSV
.
Proof
. Using the nonlinear variation of constants formula of Alekseev
[1]
, the solu tions of (2.1) and (2.2) with the same initial value are related by
Since
x
= 0 of (2.1) is ULSV, it is ULS(
[8]
,Theorem 3.3) . Using the ULSV condition of
x
= 0 of (2.1), (3.4), and (3.5), we have
Set u
(
t
) = 
y
(
t
)
y
_{0}

^{−1}
. Now an application of Lemma 3.1 yields
Thus, by (3.6), we have 
y
(
t
) ≤
M
(
t
_{0}
)
y
_{0}
 for some
M
(
t
_{0}
) > 0 whenever 
y
_{0}
 <
δ
. So, the proof is complete.
Remark 3.3.
Letting
c
(
t
) = 0 in Theorem 3.2, we obtain the same result as that of Theorem 3.6 in
[13]
.
Theorem 3.4.
For the perturbed (2.2), we assume that
and
where
,
w
∈
C
((0, ∞)),
and w
(
u
)
is nondecreasing in u
,
u
≤
w
(
u
),
and
for some v
> 0,
where M
(
t
_{0}
) < ∞
and b
_{1}
= ∞.
Then the zero solution of (2.2) is ULS whenever the zero solution of (2.1) 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 . Applying Lemma 2.3, (3.7), and (3.8), we have
Set
u
(
t
) = 
y
(
t
)
y
_{0}

^{−1}
. Now an application of Lemma 2.6 yields
Hence, by (3.9), we have 
y
(
t
) ≤
M
(
t
_{0}
)
y
_{0}
 for some
M
(
t
_{0}
) > 0 whenever 
y
_{0}
 <
δ
. This completes the proof.
Remark 3.5.
Letting
c
(
t
) = 0 in Theorem 3.4, we obtain the same result as that of Theorem 3.5 in
[13]
.
Theorem 3.6.
For the perturbed (2.2), we assume that
and
where
,
w
∈
C
((0, ∞)),
and w
(
u
)
is nondecreasing in u
,
u
≤
w
(
u
),
and
for some v
> 0,
where M
(
t
_{0}
) < ∞
and b
_{1}
= ∞.
Then the zero solution of (2.2) is ULS whenever the zero solution of (2.1) 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. Using the nonlinear variation of constants formula and the ULSV condition of
x
= 0 of (2.1), (3.10), and (3.11), we have
Set
u
(
t
) = 
y
(
t
)
y
_{0}

^{−1}
. Now an application of Lemma 2.7 and (3.12) yield
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.
Remark 3.7.
Letting
c
(
t
) = 0 in Theorem 3.6, we obtain the same result as that of Theorem 3.6 in
[13]
.
Theorem 3.8.
Let the solution x
= 0
of (2.1) be EASV. Suppose that the perturbing term g(t, y, Ty) satisfies
and
where α
> 0,
.
If
where c
= 
y
_{0}

Me
^{αt0}
,
then all solutions of (2.2) 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 (2.1) and (2.2), respectively. Since the solution
x
= 0 of (2.1) is EASV, it is EAS by remark 2.2. Using Lemma 2.3, (3.13), and (3.14), we have
Set
u
(
t
) = 
y
(
t
)
e
^{αt}
. An application of Lemma 2.4 and (3.15) obtain
t
≥
t
_{0}
, where
c
=
M

y
_{0}

e
^{αt0}
. Hence, all solutions of (2.2) approch zero as
t
→ ∞.
Theorem 3.9.
Let the solution x
= 0
of (2.1) be EASV. Suppose that the perturbed term g
(
t
,
y
,
Ty
)
satisfies
and
where α
> 0,
and w
(
u
)
is nondecreasing in u, and
for some v
> 0.
If
where c
=
M

y
_{0}

e
^{αt0}
,
then all solutions of (2.2) 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 (2.1) and (2.2), respectively. Since the solution
x
= 0 of (2.1) is EASV, it is EAS. Using Lemma 2.3, (3.16), and (3.17), we have
Set
u
(
t
) = 
y
(
t
)
e^{αt}
. Since
w
(
u
) is nondecreasing, an application of Lemma 2.8 and (3.18) obtain
where
c
=
M

y
_{0}

e
^{αt0}
. Therefore, all solutions of (2.2) approch zero as
t
→ ∞.
Acknowledgements
The authors are very grateful for the referee’s valuable comments.
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