In this paper, we investigate bounds for solutions of the nonlinear functional differential systems
AMS Mathematics Subject Classification : 34D10.
1. Introduction
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. 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 investigating the qualitative behavior of the solutions of perturbed nonlinear system of differential systems: the method of variation of constants formula, Lyapunov’s second method, and the use of integral inequalities. In the presence the method of integral inequalities is as efficient as the direct Lyapunov’s method.
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. Also, he studied some general results about asymptotic integration and gave some important examples in
[13]
. Choi and Ryu
[3]
, Choi, Koo
[5]
, and Choi et al.
[4]
investigated bounds of solutions for nonlinear perturbed systems and nonlinear functional differential systems. Also, Goo
[9
,
10]
studied the boundedness of solutions for nonlinear functional perturbed systems.
In this paper, we investigate bounds of solutions of the nonlinear functional perturbed differential systems.
2. Preliminaries
We consider the nonlinear functional differential equation
where
t
∈ ℝ
^{+}
= [0,
_{∞}
),
x
∈ ℝ
^{n}
,
f
∈
C
(ℝ
^{+}
× ℝ
^{n}
, ℝ
^{n}
),
f
(
t
, 0) = 0, the derivative
f_{x}
∈
C
(ℝ
^{+}
× ℝ
^{n}
, ℝ
^{n}
),
g
∈
C
((ℝ
^{+}
× ℝ
^{n}
, ℝ
^{n}
),
g
(
t
, 0, 0) = 0 and
T
is a continuous operator mapping from
C
(ℝ
^{+}
,ℝ
^{n}
) into
C
(ℝ
^{+}
,ℝ
^{n}
). The symbol  ·  will be used to denote arbitrary vector norm in ℝ
^{n}
. We assume that for any two continuous functions
u, v
×
C
(
I
) where
I
is the closed interval, the operator
T
satisfies the following property:
implies
Tu
(
t
) ≤
Tv
(
t
), 0 ≤
t
≤
t
_{1}
, and 
Tu
 ≤
T

u
.
Equation (1) can be considered as the perturbed equation of
Let
x
(
t
,
t
_{0}
,
x
_{0}
) be denoted by the unique solution of (2) passing through the point (
t
_{0}
,
x
_{0}
) ∈ ℝ
^{+}
× ℝ
^{n}
such that
x
(
t
_{0}
,
t
_{0}
,
x
_{0}
) =
x
_{0}
. Also, we can consider the associated variational systems around the zero solution of (2) and around
x
(
t
), respectively,
and
The fundamental matrix Φ(
t
,
t
_{0}
,
x
_{0}
) of (4) is given by
and Φ(
t
,
t
_{0}
, 0) is the fundamental matrix of (3).
We recall some notions of
h
stability
[13]
.
Definition 2.1.
The system (2) (the zero solution
x
= 0 of (2)) 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) (the zero solution
x
= 0 of (2)) is called (hS)
hstable
if there exists
δ
> 0 such that (2) 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
[6]
.
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
.
We give some related properties that we need in the sequal.
Lemma 2.1
(
[14]
).
The linear system
where A
(
t
)
is an n
×
n continuous matrix, is an hsystem ( hstable, respectively,) if and only if there exist c
≥ 1
and a positive and continuous ( bounded, respectively,) 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 (2) 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 2.2.
If y
_{0}
∈ ℝ
^{n}
,
then for all t such that x
(
t
,
t
_{0}
,
y
_{0}
) ∈ ℝ
^{n}
,
Theorem 2.3
(
[3]
).
If the zero solution of (2) is hS, then the zero solution of (3) is hS.
Theorem 2.4
(
[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 2.5
(
[5]
).
Let u, λ
_{1}
,
λ
_{2}
,
w
∈
C
(ℝ
^{+}
)
and w
(
u
)
be nondecreasing in u such that
for some v
> 0.
If ,for some c
> 0,
then
where
is the inverse of W
(
u
),
and
Lemma 2.6
(
[2]
).
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 2.5, and
Lemma 2.7
(
[10]
).
Let u, p, q, w, r
∈
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 2.5, and
3. Main results
In this section, we investigate the bounded property for the nonlinear functional differential systems.
Theorem 3.1.
Let a, c, u, w
∈
C
(ℝ
^{+}
),
w
(
u
)
be nondecreasing in 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) is hS with the increasing function h, and g in (1) satisfies
and
where
Then, any solution y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
of (1) is bounded on
[
t
_{0}
,
_{∞}
)
and it satisfies
where
W
,
W
^{−1}
are the same functions as in Lemma 2.5 , β
(
t
) =
c
_{2}
a
(
t
) ,
k is a positive constant, and
Proof
. Let
x
(
t
) =
x
(
t
,
t
_{0}
,
y
_{0}
) and
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) be solutions of (2) and (1), respectively. By Theorem 2.3, since the solution
x
= 0 of (2) is hS, the solution
v
= 0 of (3) is hS. Therefore, by Theorem 2.4, the solution
z
= 0 of (4) is hS. By Lemma 2.1, Lemma 2.2 and the increasing property of the function
h
, we have
Set
u
(
t
) = 
y
(
t
)
h
(
t
)
^{−1}
. Then, by Lemma 2.5, we obtain
where
k
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
and
β
(
t
) =
c
_{2}
a
(
t
). This completes the proof.
Remark 3.1.
Letting
c
(
τ
) = 0 in Theorem 3.1, we have the similar result as that of Theorem 3.3 in
[7]
.
Theorem 3.2.
Let a, b, c, u, w
∈
C
(ℝ
^{+}
),
w
(
u
)
be nondecreasing in u and
for some v
> 0.
Suppose that the solution x
= 0
of (2) is hS with a nondecreasing function h and the perturbed term g in (1) satisfies
and
where
Then any solution y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
of (1) is bounded on
[
t
_{0}
,
_{∞}
)
and it satisfies
where
W
,
W
^{−1}
are the same functions as in Lemma 2.5, k is a positive constant, and
Proof
. Let
x
(
t
) =
x
(
t
,
t
_{0}
,
y
_{0}
) and
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) be solutions of (2) and (1), respectively. By Theorem 2.2, we obtain
since
h
is nondecreasing. Set
u
(
t
) = 
y
(
t
)
h
(
t
)
^{−1}
. Then, by Lemma 2.6, we have
where
k
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
. Therefore, we obtain the result.
Remark 3.2.
Letting
c
(
τ
) = 0 in Theorem 3.2, we have the similar result as that of Theorem 3.1 in
[8]
.
Theorem 3.3.
Let a, b, c, u, w
∈
C
(ℝ
^{+}
),
w
(
u
)
be nondecreasing in 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) is hS with the increasing function h, and g in (1) satisfies
and
where
Then any solution y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
of (1) is bounded on
[
t
_{0}
,
_{∞}
)
and it satisfies
where W, W
^{−1}
are the same functions as in Lemma 2.5, k is a positive constant, and
Proof
. Let
x
(
t
) =
x
(
t
,
t
_{0}
,
y
_{0}
) and
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) be solutions of (2) and (1), respectively. By Theorem 2.3, since the solution
x
= 0 of (2) is hS, the solution
v
= 0 of (3) is hS. Therefore, by Theorem 2.4, the solution
z
= 0 of (4) is hS. By Lemma 2.1, Lemma 2.2 and the increasing property of the function
h
, we have
Set
u
(
t
) = 
y
(
t
)
h
(
t
)
^{−1}
. Then, by Lemma 2.6, we obtain
where
k
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
. Hence, the proof is complete.
Remark 3.3.
Letting
c
(
τ
) = 0 in Theorem 3.3, we have the similar result as that of Theorem 3.2 in
[8]
.
Theorem 3.4.
Let b, c, u, w
∈
C
(ℝ
^{+}
),
w
(
u
)
be nondecreasing in 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,
If the solution x
= 0
of (2) is an hsystem with a positive continuous function h and g in (1) satisfies
and
where a
: ℝ
^{+}
→ ℝ
^{+}
is continuous with
for all
t
_{0}
≥ 0,
then any solution y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
)
of (1) satisfies
,
t
_{0}
≤
t
<
b
_{1}
,
where W, W
^{−1}
are the same functions as in Lemma 2.5, k is a positive constant, and
Proof
. Let
x
(
t
) =
x
(
t
,
t
_{0}
,
y
_{0}
) and
y
(
t
) =
y
(
t
,
t
_{0}
,
y
_{0}
) be solutions of (2) and (1), respectively. By Theorem 2.3, since the solution
x
= 0 of (2) is an
h
system, the solution
v
= 0 of (3) is an
h
system. Therefore, by Theorem 2.4, the solution
z
= 0 of (4) is an
h
system. By Lemma 2.2, we have
Setting
u
(
t
) = 
y
(
t
)
h
(
t
)
^{−1}
and using Lemma 2.7, we obtain
,
t
_{0}
≤
t
<
b
_{1}
, where
k
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
. Hence, the proof is complete.
Remark 3.4.
Letting
c
(
τ
) = 0 in Theorem 3.4, we have the similar result as that of Theorem 3.5 in
[8]
.
Acknowledgements
The author is very grateful for the referee’s valuable comments.
BIO
Sang Il Choi received the BS from Korea University and Ph.D at North Carolina State University under the direction of J. Silverstein . Since 1995 he has been at Hanseo University as a professor. His research interests focus on Analysis and Probability theory.
Department of Mathematics, Hanseo University, Seasan 356706, Republic of Korea
email:schoi@hanseo.ac.kr
Dong Man Im received the BS and Ph.D at Inha University. Since 1982 he has been at Cheongju University as a professor. His research interests focus on Algebra and differential equations.
Department of Mathematics Education Cheongju University Cheongju Chungbuk 360764, Republic of Korea
email:dmim@cheongju.ac.kr
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 topological dynamical systems and differential equations.
Department of Mathematics, Hanseo University, Seasan 356706, Republic of Korea
email:yhgoo@hanseo.ac.kr
Alekseev V.M.
(1961)
An estimate for the perturbations of the solutions of ordinary differential equations
Vestn. Mosk. Univ. Ser. I. Math. Mech.
2
28 
36
Choi S.K.
,
Ryu H.S.
(1993)
hstability in differential systems
Bull. Inst. Math. Acad. Sinica
21
245 
262
Choi S.K.
,
Koo N.J.
,
Ryu H.S.
(1997)
hstability of differential systems via t1similarity
Bull. Korean. Math. Soc.
34
371 
383
Choi S.K.
,
Koo N.J.
,
Song S.M.
(1999)
Lipschitz stability for nonlinear functional differential systems
Far East J. Math. Sci(FJMS)I
5
689 
708
Conti R.
(1957)
Sulla t∞similitudine tra matricie l’equivalenza asintotica dei sistemi differenziali lineari
Rivista di Mat. Univ. Parma
8
43 
47
Goo Y.H.
,
Ryu D.H.
(2010)
hstability of the nonlinear perturbed differential systems
J. Chungcheong Math. Soc.
23
(4)
827 
834
Goo Y.H.
,
Park D.G.
,
Ryu D.H.
(2012)
Boundedness in perturbed differential systems
J. Appl. Math. and Informatics
30
279 
287
Goo Y.H.
(2013)
Boundedness in the perturbed nonlinear differential systems
Far East J. Math. Sci(FJMS)
79
205 
217
Lakshmikantham V.
,
Leela S.
1969
Differential and Integral Inequalities: Theory and Applications Vol. I
Academic Press
New York and London