Alexseev’s formula generalizes the variation of constants formula and permits the study of a nonlinear perturbation of a system with certain stability properties. In this paper, we investigate bounds for solutions of the functional nonlinear perturbed differential systems using the two notion of
h
stability and
t
_{∞}
similarity.
1. INTRODUCTION
The behavior of solutions of a perturbed system is determined in terms of the behavior of solutions of an unperturbed system. There are three useful methods for showing the qualitative behavior of the solutions of perturbed nonlinear system : the use of integral inequalities, the method of variation of constants formula, and Lyapunov’s second method.
The notion of
h
stability (hS) was introduced by Pinto
[15
,
16]
with the intention of obtaining results about stability for a weakly stable system (at least, weaker than those given exponential asymptotic stability) under some perturbations. That is, Pinto extended the study of exponential asymptotic stability to a variety of reasonable systems called
h
systems. Also, he studied some general results about asymptotic integration and gave some important examples in
[15]
. Choi and Koo
[2]
, Choi and Ryu
[3]
, and Choi et al.
[4
,
5
,
6]
investigated
h
stability and bounds of solutions for the perturbed functional differential systems. Also, Goo
[8
,
9
,
10]
studied the boundedness of solutions for the perturbed differential systems.
The main conclusion to be drawn from this paper is that the use of inequalities provides a powerful tool for obtaining bounds for solutions.
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 functional nonlinear perturbed differential systems of (2.1)
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 (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).
We recall some notions of
h
stability
[15]
.
Definition 2.1.
The system (2.1) (the zero solution
x
= 0 of (2.1)) is called (hS)
hstable
if there exist a constant
c
≥ 1, and a positive bounded continuous function
h
on ℝ
^{+}
such that
x
(
t
) ≤
c

x
_{0}

h
(
t
)
h
(
t
_{0}
)
^{−1}
for
t
≥
t
_{0}
≥ 0 and 
x
_{0}
 ≤
δ
(here
).
Let
M
denote the set of all
n
×
n
continuous matrices
A
(
t
) defined on ℝ
^{+}
and
N
be the subset of
M
consisting of those nonsingular matrices
S
(
t
) that are of class
C
^{1}
with the property that
S
(
t
) and
S
^{−1}
(
t
) are bounded. The notion of
t
_{∞}
similarity in
M
was introduced by Conti
[7]
.
Definition 2.2.
A matrix
A
(
t
) ∈
M
is
t
_{∞}

similar
to a matrix
B
(
t
) ∈
M
if there exists an
n
×
n
matrix
F
(
t
) absolutely integrable over ℝ
^{+}
, i.e.,
such that
for some
S
(
t
) ∈
N
.
The notion of
t
_{∞}
similarity is an equivalence relation in the set of all
n
×
n
continuous matrices on ℝ
^{+}
, and it preserves some stability concepts
[4
,
12]
.
In this paper, we investigate bounds for solutions of the nonlinear differential systems using the notion of
t
_{∞}
similarity.
We give some related properties that we need in the sequal.
Lemma 2.3
(
[16]
)
.
The linear system
where A(t) is an n
×
n continuous matrix, is an hsystem (respectively hstable) if and only if there exist c
≥ 1
and a positive and continuous (respectively bounded) function h defined on
ℝ
^{+}
such that
for t
≥
t
_{0}
≥ 0,
where φ
(
t
,
t
_{0}
)
is a fundamental matrix of (2.6)
.
We need Alekseev formula to compare between the solutions of (2.1) 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 (2.8) 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 2.4.
Let x
(
t
) =
x
(
t
,
t
_{0}
,
y
_{0}
)
and y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
be solutions of (2.1) and (2.8), respectively. If
y
_{0}
∈ ℝ
^{n}
,
then for all t such that x
(
t
,
t
_{0}
,
y
_{0}
) ∈ ℝ
^{n}
,
Theorem 2.5
(
[3]
)
.
If the zero solution of (2.1) is hS, then the zero solution of (2.3) is hS.
Theorem 2.6
(
[4])
.
Suppose that f_{x}
(
t
, 0)
is t
_{∞}

similar to f_{x}
(
t
,
x
(
t
,
t
_{0}
,
x
_{0}
))
for t
≥
t
_{0}
≥ 0
and

x
_{0}
 ≤
δ for some constant δ
> 0.
If the solution v
= 0
of (2.3) is hS, then the solution z
= 0
of (2.4) is hS.
Lemma 2.7
(
[6)
.
(Bihari − type inequality)
Let u, λ
∈
C
(ℝ
^{+}
),
w
∈
C
((0, ∞))
and w
(
u
)
be nondecreasing in u. Suppose that, for some c
> 0,
Then
where
,
W
^{−1}
(
u
)
is the inverse of W
(
u
)
and
Lemma 2.8
(
[5]
)
.
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 2.7, and
Lemma 2.9
(
[9]
)
.
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 W, W
^{−1}
are the same functions as in Lemma 2.7, and
3. MAIN RESULTS
In this section, we investigate boundedness for solutions of the functional nonlinear perturbed differential systems via
t
_{∞}
similarity.
Theorem 3.1.
Let a, b, c, k, u, w
∈
C
(ℝ
^{+}
),
w
(
u
)
be nondecreasing in u such that
for some v
> 0.
Suppose that f_{x}
(
t
, 0)
is t
_{∞}

similar to
f_{x}
(
t
,
x
(
t
,
t
_{0}
,
x
_{0}
))
for t
≥
t
_{0}
≥ 0
and

x
_{0}
 ≤
δ for some constant δ
> 0,
the solution x
= 0
of (2.1) is hS with the increasing function h, and g in (2.2) satisfies
and
where
and
Then, any solution y
(
t)
=
y
(
t
,
t
_{0}
,
y
_{0}
)
of (2.2) is bounded on
[
t
_{0}
,∞)
and it satisfies
where c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
,
W
,
W
^{−1}
are the same functions as in Lemma 2.7, and
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. By Theorem 2.5, since the solution
x
= 0 of (2.1) is hS, the solution
v
= 0 of (2.3) is hS. Therefore, by Theorem 2.6, the solution
z
= 0 of (2.4) is hS. Applying Lemmma 2.3, Lemma 2.4, the increasing property of the function
h
, (3.1), and (3.2), we have
Defining
u
(
t
) = 
y
(
t
)
h
(
t
)
^{−1}
, then, by Lemma 2.8, we have
t
_{0}
≤
t
<
b
_{1}
, where
c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
. Thus, any solution
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) of (2.2) is bounded on [
t
_{0}
, ∞). This completes the proof. □
Remark 3.2.
Letting
c
(
t
) = 0 in Theorem 3.1, we obtain the same result as that of Theorem 3.2 in
[10]
.
Theorem 3.3.
Let a, b, c, k, u, w
∈
C
(ℝ
^{+}
),
w
(
u
)
be nondecreasing in u such that u
≤
w
(
u
)
and
for some v
> 0.
Suppose that f_{x}
(
t
, 0)
is
t
_{∞}

similar to f_{x}
(
t
,
x
(
t
,
t
_{0}
,
x
_{0}
))
for t
≥
t
_{0}
≥ 0
and

x
_{0}
 ≤
δ for some constant δ
> 0,
the solution
x
= 0 of (2.1)
is hS with the increasing function h, and g in (2.2) satisfies
and
where
and
Then, any solution y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
of (2.2) is bounded on
[
t
_{0}
,∞)
and it satisfies
where c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
,
W
,
W
^{−1}
are the same functions as in Lemma 2.7, and
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. By Theorem 2.5, since the solution
x
= 0 of (2.1) is hS, the solution
v
= 0 of (2.3) is hS. Therefore, by Theorem 2.6, the solution
z
= 0 of (2.4) is hS. Using the nonlinear variation of constants formula and the hS condition of
x
= 0 of (2.1), (3.3), and (3.4), we have
Set
u
(
t
) = 
y
(
t
)
h
(
t
)
^{−1}
. Then, an application of Lemma 2.9 yields
t
_{0}
≤
t
<
b
_{1}
, where
c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
. The above estimation yields the desired result since the function
h
is bounded. Thus, the theorem is proved. □
Remark 3.4.
Letting
w
(
u
) =
u
and
b
(
s
) =
c
(
s
) = 0 in Theorem 3.3, we obtain the same result as that of Theorem 3.3 in
[11]
.
Lemma 3.5.
Let u
,
λ
_{1}
,
λ
_{2}
,
λ
_{3}
,
λ
_{4}
,
λ
_{5}
∈
C
(ℝ
^{+}
),
w
∈
C
((0, ∞))
and w
(
u
)
be nondecreasing in u. Suppose that for some c
> 0
and
0 ≤
t
_{0}
≤
t
,
Then
where W, W
^{−1}
are the same functions as in Lemma 2.7, and
Proof
. Setting
then we have
z
(
t
_{0}
) =
c
and
since
z
(
t
) and
w
(
u
) are nondecreasing and
u
(
t
) ≤
z
(
t
). Therefore, by integrating on [
t
_{0}
,
t
], the function
z
satisfies
It follows from Lemma 2.7 that (3.6) yields the estimate (3.5). □
Theorem 3.6.
Let a, b, c, k, u, w
∈
C
(ℝ
^{+}
),
w
(
u
)
be nondecreasing in u such that
for some v
> 0.
Suppose that f_{x}
(
t
, 0)
is t
_{∞}

similar to f_{x}
(
t
,
x
(
t
,
t
_{0}
,
x
_{0}
))
for t
≥
t
_{0}
≥ 0
and

x
_{0}
 ≤
δ
for some constant δ
> 0,
the solution x
= 0
of (2.1) is hS with the increasing function h, and g in (2.2) satisfies
and
where
and
Then, any solution y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
of (2.2) is bounded on
[
t
_{0}
,∞)
and
t
_{0}
≤
t
<
b
_{1}
,
where c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
,
W, W
^{−1}
are the same functions as in Lemma 2.7, and
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. By Theorem 2.5, since the solution
x
= 0 of (2.1) is hS, the solution
v
= 0 of (2.3) is hS. Therefore, by Theorem 2.6, the solution
z
= 0 of (2.4) is hS. By Lemma 2.3, Lemma 2.4, the increasing property of the function
h
, (3.7), and (3.8), we have
Set
u
(
t
) = 
y
(
t
)
h
(
t
)
^{−1}
. Then, Lemma 3.5, we obtain
t
_{0}
≤
t
<
b
_{1}
, where
c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
. Thus, any solution
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) of (2.2) is bounded on [
t
_{0}
, ∞). This completes the proof. □
Lemma 3.7.
Let
u
,
λ
_{1}
,
λ
_{2}
,
λ
_{3}
,
λ
_{4}
,
λ
_{5}
∈
C
(ℝ
^{+}
),
w
∈
C
((0, ∞))
and w
(
u
)
be nondecreasing in u, u
≤
w
(
u
).
Suppose that for some c
> 0
and t
_{0}
≤ t <
b
_{1}
,
Then
where W
,
W
^{−1}
are the same functions as in Lemma 2.7, and
Proof
. Define a function
v
(
t
) by the right member of (3.9) . 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, (3.11) yields the estimate (3.10). □
Theorem 3.8.
Let a, b, c, k, u, w
∈
C
(ℝ
^{+}
),
w
(
u
)
be nondecreasing in u such that u
≤
w
(
u
)
and
for some v
> 0.
Suppose that f_{x}
(
t
, 0)
is t
_{∞}

similar to
f_{x}
(
t
,
x
(
t
,
t
_{0}
,
x
_{0}
))
for t
≥
t
_{0}
≥ 0
and

x
_{0}
 ≤
δ for some constant δ
> 0,
the solution
x
= 0
of
(2.1)
is hS with the increasing function h, and g in (2.2) satisfies
and
where
and
Then, any solution y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
of (2.2) is bounded on
[
t
_{0}
,∞)
and
t
_{0}
≤
t
<
b
_{1}
,
where c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
,
W, W
^{−1}
are the same functions as in Lemma 2.7, and
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. By Theorem 2.5, since the solution
x
= 0 of (2.1) is hS, the solution
v
= 0 of (2.3) is hS. Therefore, by Theorem 2.6, the solution
z
= 0 of (2.4) is hS. Using two Lemma 2.3, Lemma 2.4, the hS condition of
x
= 0 of (2.1), (3.12), and (3.13), we have
Using Lemma 3.7 with
u
(
t
) = 
y
(
t
)
h
(
t
)
^{−1}
, we have
t
_{0}
≤
t
<
b
_{1}
, where
c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
. The above estimation implies the boundedness of
y
(
t
), and so the proof is complete. □
Remark 3.9.
Letting
b
(
t
)
w
(
u
(
t
)) =
b
(
t
)
u
(
t
) in Theorem 3.8, we obtain the same result as that of Theorem 2.4 in
[9]
.
Acknowledgements
The author is very grateful for the referee’s valuable comments.
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