In this paper, we investigate bounds for solutions of nonlinear functional differential systems using the notion of
t
_{∞}
similarity.
1. INTRODUCTION
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 is continuous to be an effective tool to study sophisticated problems such as stability, boundedness, and uniqueness of solutions. The behavior of solutions of a perturbed system is determined in terms of the behavior of solutions of an unperturbed system. There are two 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.
The notion of
h
stability (hS) was introduced by Pinto
[13
,
14]
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. He obtained a general variational
h
stability and some properties about asymptotic behavior of solutions of differential systems called
h
systems. Choi and Ryu
[3]
, and Choi et al.
[4]
investigated
h
stability and bounds for solutions of the perturbed functional differential systems. Also, Goo
[6
,
7
,
8]
and Goo et al.
[9]
studied boundedness of solutions for the perturbed functional differential systems.
The aim of this paper is to obtain boundedness for solutions of nonlinear functional differential systems under suitable conditions on perturbed term.
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 functional 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
[14]
.
Definition 2.1.
The system (2.1) (the zero solution
x
= 0 of (2.1)) 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
Definition 2.2.
The system (2.1) (the zero solution
x
= 0 of (2.1)) is called (hS)
hstable
if there exists
δ
> 0 such that (2.1) is an
h
system for 
x
_{0}
 ≤
δ
and
h
is bounded.
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 2.3.
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
,
10]
.
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 2.4
(
[14]
)
.
The linear system
where
A
(
t
)
is an
n
×
n
continuous matrix, is an
h

system
(
respectively
h

stable
)
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 2.5
(
[2]
)
.
Let x and y be a solution of
(2.1)
and
(2.8),
respectively
.
If
y
_{0}
∈ ℝ
^{n}
,
then for all
t
≥
t
_{0}
such that
x
(
t
,
t
_{0}
,
y
_{0}
) ∈ ℝ
^{n}
,
y
(
t
,
t
_{0}
,
y
_{0}
) ∈ ℝ
^{n}
,
Theorem 2.6
(
[3]
)
.
If the zero solution of
(2.1)
is hS
,
then the zero solution of
(2.3)
is hS
.
Theorem 2.7
(
[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 2.8.
(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 2.9
(
[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,
Then
,
where
W
,
W
^{−1}
are the same functions as in Lemma
2.8,
and
We obtain the following corollary from Lemma 2.9.
Corollary 2.10.
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
2.8,
and
3. MAIN RESULTS
In this section, we investigate boundedness for solutions of the nonlinear perturbed differential systems via
t
_{∞}
similarity.
For the proof we need the following lemma.
Lemma 3.1.
Let
u
,
λ
_{1}
,
λ
_{2}
,
λ
_{3}
,
λ
_{4}
,
λ
_{5}
,
λ
_{6}
,
λ
_{7}
∈
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
t
_{0}
≤
t
<
b
_{1}
,
where W
,
W
^{−1}
are the same functions as in Lemma
2.8
and
Proof
. Defining
then we have
z
(
t
_{0}
) =
c
and
t
≥
t
_{0}
, since
z
(
t
) and
w
(
u
) are nondecreasing,
u
≤
w
(
u
), and
u
(
t
) ≤
z
(
t
). Therefore, by integrating on [
t
_{0}
,
t
], the function
z
satisfies
It follows from Lemma 2.8 that (3.2) yields the estimate (3.1). ☐
To obtain the bounded result, the following assumptions are needed:

(H1)fx(t, 0) ist∞similar tofx(t,x(t,t0,x0)) fort≥t0≥ 0 and x0 ≤δfor some constantδ> 0.

(H2) The solutionx= 0 of (1.1) is hS with the increasing functionh.

(H3)w(u) be nondecreasing inusuch thatu≤w(u) andfor somev> 0.
Theorem 3.2.
Let a
,
b
,
c
,
k
,
q
,
u
,
w
∈
C
(ℝ
^{+}
).
Suppose that
(
H
1), (
H
2), (
H
3),
and g in
(2.2)
satisfies
and
where t
≥
t
_{0}
≥ 0
and
a
,
b
,
c
,
k
,
q
∈
L
_{1}
(ℝ
^{+}
).
Then, any solution
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
of
(2.2)
is bounded on
[
t
_{0}
, ∞)
and
where W
,
W
^{−1}
are the same functions as in Lemma
2.8,
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 2.6, since the solution
x
= 0 of (2.1) is hS, the solution
v
= 0 of (2.3) is hS. Using (H1), by Theorem 2.7, the solution
z
= 0 of (2.4) is hS. By Lemma 2.4, Lemma 2.5, together with (3.3), and (3.4), we have
Applying (H2) and (H3), we obtain
Set
u
(
t
) = 
y
(
t
)‖
h
(
t
)
^{−1}
. Then, it follows from Lemma 3.1 that 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 3.3.
Letting
c
(
t
) = 0 in Theorem 3.2, we obtain the similar result as that of Theorem 3.3 in
[7]
.
Theorem 3.4.
Let a
,
b
,
c
,
q
,
u
,
w
∈
C
(ℝ
^{+}
).
Suppose that
(
H
1), (
H
2), (
H
3),
and g in
(2.2)
satisfies
and
where a
,
b
,
c
,
q
∈
L
_{1}
(ℝ
^{+}
).
Then, any solution
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
of
(2.2)
is bounded on
[
t
_{0}
, ∞)
and it satisfies
t
_{0}
≤
t
<
b
_{1}
,
where
W
,
W
^{−1}
are the same functions as in Lemma
2.8,
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 the same argument as in the proof in Theorem 3.2, the solution
z
= 0 of (2.4) is hS. Applying the nonlinear variation of constants formula, Lemma 2.4, together with (3.5), and (3.6), we have
By the assumptions (H2) and (H3), we obtain
Set
u
(
t
) = 
y
(
t
)‖
h
(
t
)
^{−1}
. Then, by Corollary 2.10, we have
where
c
=
c
_{1}

Y
_{0}

h
(
t
_{0}
)
^{−1}
. Thus, any solution
y
(
t
) =
y
(
t
,
t
_{0}
,
Y
_{0}
) of (2.2) is bounded on [
t
_{0}
, ∞). This completes the proof. ☐
Remark 3.5.
Letting
b
(
t
) =
c
(
t
) = 0 in Theorem 3.4, we obtain the same result as that of Theorem 3.2 in
[9]
.
We need the following lemma for the proof of Theorem 3.7.
Lemma 3.6.
Let u
,
λ
_{1}
,
λ
_{2}
,
λ
_{3}
,
λ
_{4}
,
λ
_{5}
,
λ
_{6}
,
λ
_{7}
∈
C
(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 (3.7). 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, by Lemma 2.8, (3.9) yields the estimate (3.8). ☐
Theorem 3.7.
Let a
,
b
,
c
,
k
,
q
,
u
,
w
∈
C
(ℝ
^{+}
).
Suppose that
(
H
1), (
H
2), (
H
3),
and g in
(2.2)
satisfies
and
where
t
≥
t
_{0}
≥ 0
and
a
,
b
,
c
,
k
,
q
∈
L
_{1}
(ℝ
^{+}
).
Then, any solution
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
of
(2.2)
is bounded on on
[
t
_{0}
, ∞)
and it satisfies
where
W
,
W
^{−1}
are the same functions as in Lemma
2.8,
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 the same argument as in the proof in Theorem 3.2, the solution
z
= 0 of (2.4) is hS. Applying the nonlinear variation of constants formula , Lemma 2.4, together with (3.10), and (3.11), we have
Using the assumptions (H2) and (H3), we obtain
Set
u
(
t
) = 
y
(
t
)‖
h
(
t
)
^{−1}
. Then, by Lemma 3.6, 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 3.8.
Letting
c
(
t
) = 0 in Theorem 3.7, we obtain the similar result as that of Theorem 3.7 in
[8]
.
Theorem 3.9.
Let a
,
b
,
c
,
k
,
q
,
u
,
w
∈
C
(ℝ
^{+}
).
Suppose that
(
H
1), (
H
2), (
H
3),
and g in
(2.2)
satisfies
and
where
s
≥
t
_{0}
≥ 0
and
a
,
b
,
c
,
k
,
q
∈
L
_{1}
(ℝ
^{+}
).
Then, any solution
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
of
(2.2)
is bounded on
[
t
_{0}
, ∞)
and it satisfies
Proof
. Using the nonlinear variation of constants formula of Alekseev
[1]
, any solution
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) of (2.2) passing through (
t
_{0}
,
y
_{0}
) is given by
By the same argument as in the proof in Theorem 3.2, the solution
z
= 0 of (2.4) is hS. Applying Lemma 2.4, together with (3.12), (3.13), and (3.14), we have
It follows from (H2) and (H3) that
Defining
u
(
t
) = 
y
(
t
)‖
h
(
t
)
^{−1}
, then, by Lemma 2.9, we have
where
t
_{0}
≤
t
<
b
_{1}
and
c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
. The above estimation yields the desired result since the function
h
is bounded. Hence, the proof is complete. ☐
Remark 3.10.
Letting
c
(
s
) =
b
(
s
) = 0 in Theorem 3.9, we obtain the same result as that of Theorem 3.2 in
[9]
.
Remark 3.11.
Letting
c
(
s
) = 0 in Theorem 3.9, we obtain the same result as that of Theorem 3.4 in
[8]
.
Acknowledgements
The author is very grateful for the referee’s valuable comments.
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