In this paper, we investigate bounds for solutions of the nonlinear perturbed functional differential systems using the notion of
t
_{∞}
similarity.
AMS Mathematics Subject Classification : 34C11, 34D10.
1. Introduction and 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 nonlinear perturbed functional differential systems 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).
We recall some notions of
h
stability
[15]
.
Definition 1.1.
The system (1) (the zero solution
x
= 0 of (1)) is called an
hsystem
if there exist a constant
c
≥ 1, and a positive continuous function
h
on ℝ
^{+}
such that
for
t
≥
t
_{0}
≥ 0 and 
x
_{0}
 small enough (here
)
Definition 1.2.
The system (1) (the zero solution
x
= 0 of (1)) is called
hstable
(hS) if there exists
δ
> 0 such that (1) is an
h
system for 
x
_{0}
 ≤
δ
and
h
is bounded.
Integral inequalities play a vital role in the study of boundedness and other qualitative properties of solutions of differential equations. In particular, Bihari’s integral inequality continues to be an effective tool to study sophisticated problems such as stability, boundedness, and uniqueness of solutions.
The notion of
h
stability (hS) was introduced by Pinto
[14
,
15]
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
[15]
. Choi and Koo
[2]
, Choi and Ryu
[3]
, and Choi et al.
[4]
investigated
h
stability and bounds of solutions for the perturbed functional differential systems. Also, Goo
[6
,
7
,
8
,
9]
and Goo et al.
[10]
studied
h
stability and boundedness of solutions for the perturbed functional differential systems.
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
[5]
.
Definition 1.3.
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
[5
,
11]
.
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 1.1
(
[15]
).
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 (6)
.
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 (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 1.2.
If y
_{0}
∈ ℝ
^{n}
,
then for all t such that x
(
t
,
t
_{0}
,
y
_{0}
) ∈ ℝ
^{n}
,
Theorem 1.3
(
[3]
).
If the zero solution of (1) is hS, then the zero solution of (3) is hS
.
Theorem 1.4
(
[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 (3) is hS, then the solution z
= 0
of (4) is hS
.
Lemma 1.5.
(
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
,
W
^{−1}
(
u
)
is the inverse of W
(
u
)
and
Lemma 1.6
(
[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 W and W
^{−1}
are the same functions as in Lemma 1.5, and
Lemma 1.7
(
[6]
).
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
0 ≤
t
_{0}
≤
t
,
Then
where W and W
^{−1}
are the same functions as in Lemma 1.5, and
2. Main results
In this section, we investigate boundedness for solutions of the nonlinear perturbed functional differential systems via
t
_{∞}
similarity.
Lemma 2.1.
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
t
_{0}
≤
t
<
b
_{1}
,
where W and W
^{−1}
are the same functions as in Lemma 1.5, and
Proof
. Define a function
v
(
t
) by the right member of (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, (11) yields the estimate (10). □
Theorem 2.2.
Let a
,
b
,
c
,
k
,
q
,
u
,
w
∈
C
(ℝ
^{+}
)
and 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 (1) is hS with the increasing function h, and g in (2) satisfies
and
where
.
Then, any solution y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
of (2) is bounded on
[
t
_{0}
,∞)
and it satisfies
t
_{0}
≤
t
<
b
_{1}
,
where W and W
^{−1}
are the same functions as in Lemma 1.5, and
Proof
. Using the nonlinear variation of constants formula of Alekseev
[1]
, any solution
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) of (2) passing through (
t
_{0}
,
y
_{0}
) is given by
By Theorem 1.3, since the solution
x
= 0 of (1) is hS, the solution
v
= 0 of (3) is hS. Therefore, by Theorem 1.4, the solution
z
= 0 of (4) is hS. By Lemma 1.1 the hS condition of
x
= 0 of (1), (12), (13), and (14), we have
Set
u
(
t
) = 
y
(
t
)
h
(
t
)
^{−1}
. Then, an application of Lemma 2.1 yields
where
c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
. Thus, any solution
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) of (2) is bounded on [
t
_{0}
,∞), and so the proof is complete. □
Remark 2.1.
Letting w
(
u
) =
u and c
(
t
) = 0
in Theorem 2.2, we obtain the same result as that of Theorem 3.1 in [9]
.
Theorem 2.3.
Let a
,
b
,
c
,
q
,
u
,
w
∈
C
(ℝ
^{+}
)
and 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 (1) is hS with the increasing function h, and g in (2) satisfies
and
where
.
Then, any solution y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
of (2) is bounded on
[
t
_{0}
,∞)
and it satisfies
where W and W
^{−1}
are the same functions as in Lemma 1.5, and
Proof
. It is known that the solution of (2) is represented by the integral equation (14). By Theorem 1.3, since the solution
x
= 0 of (1) is hS, the solution
v
= 0 of (3) is hS. Therefore, by Theorem 1.4, the solution
z
= 0 of (4) is hS. Using the nonlinear variation of constants formula (14), the hS condition of
x
= 0 of (1), (15), and (16), we have
Put
u
(
t
) = 
y
(
t
)
h
(
t
)
^{−1}
. Then, by Lemma 1.6, we have
where
c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
. From the above estimation, we obtain the desired result. Thus, the theorem is proved. □
Remark 2.2.
Letting b
(
t
) =
c
(
t
) = 0
in Theorem 2.3, we obtain the same result as that of Theorem 3.2 in [10]
.
Lemma 2.4.
Let u
,
λ
_{1}
,
λ
_{2}
,
λ
_{3}
,
λ
_{4}
,
λ
_{5}
,
λ
_{6}
,
λ
_{7}
∈
C
[ℝ
^{+}
, ℝ
^{+}
],
w
∈
C
((0,∞))
and w
(
u
)
be nondecreasing in u, u
≤
w
(
u
).
Suppose that, for some c
≥ 0,
we have
Then
Proof
. Define a function
v
(
t
) by the right member of (17). 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
Thus, (19) yields the estimate (18). □
Theorem 2.5.
Let a
,
b
,
c
,
k
,
q
,
u
,
w
∈
C
(ℝ
^{+}
)
and 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 (1) is hS with the increasing function h, and g in (2) satisfies
and
t
≥
t
_{0}
≥ 0,
where
.
Then, any solution y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
of (2) is bounded on on
[
t
_{0}
,∞)
and it satisfies
where W and W
^{−1}
are the same functions as in Lemma 1.5, and
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. By Theorem 1.3, since the solution
x
= 0 of (1) is hS, the solution
v
= 0 of (3) is hS. Therefore, by Theorem 1.4, the solution
z
= 0 of (4) is hS. Applying the nonlinear variation of constants formula (14), the hS condition of
x
= 0 of (1), (20), and (21), we have
Set
u
(
t
) = 
y
(
t
)
h
(
t
)
^{−1}
. Then, by Lemma 2.4, we have
where
c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
. The above estimation yields the desired result since the function
h
is bounded, and so the proof is complete. □
Remark 2.3.
Letting
w
(
u
) =
u
and
c
(
t
) = 0 in Theorem 2.5, we obtain the similar result as that of Theorem 3.1 in
[9]
.
Theorem 2.6.
Let a
,
b
,
c
,
q
,
u
,
w
∈
C
(ℝ
^{+}
)
and 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 (1) is hS with the increasing function h, and g in (2) satisfies
and
where
.
Then, any solution y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
of (2) is bounded on
[
t
_{0}
,∞)
and
where W and W
^{−1}
are the same functions as in Lemma 1.5, and
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. By Theorem 1.3, since the solution
x
= 0 of (1) is hS, the solution
v
= 0 of (3) is hS. Therefore, by Theorem 1.4, the solution
z
= 0 of (4) is hS. By Lemma 2.1, the hS condition of
x
= 0 of (1), (22), and (23), we have
Set
u
(
t
) = 
y
(
t
)
h
(
t
)
^{−1}
. Then, by Lemma 1.7, we have
where
c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
. Thus, any solution
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) of (2) is bounded on [
t
_{0}
,∞). This completes the proof. □
Remark 2.4.
Letting
b
(
t
) =
c
(
t
) = 0 in Theorem 2.6, we obtain the similar result as that of Theorem 3.5 in
[10]
.
Acknowledgements
The author is very grateful for the referee’s valuable comments.
BIO
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, Seasan 356706, Korea
email: yhgoo@hanseo.ac.kr
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