In this paper, we investigate bounds for solutions of nonlinear perturbed differential systems.
AMS Mathematics Subject Classification : 34D10.
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 : Lyapunov’s second method, the use of integral inequalities, and the method of variation of constants formula. 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.
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. Using this notion, Choi and Ryu
[3
,
5]
investigated bounds of solutions for nonlinear perturbed systems and nonlinear functional differential systems. Also, Goo et al.
[8]
studied the boundedness of solutions for nonlinear perturbed systems.
In this paper, we obtain some results on boundedness of solutions of nonlinear perturbed differential systems under suitable conditions on perturbed term. To do this we need some integral inequalities.
2. Preliminaries
We are interested in the relations of the unperturbed system
and the solutions of the perturbed system
where
x
,
y
,
f
and
g
are elements of ℝ
^{n}
, an
n
dimensional real Euclidean space.
We assume that
f
,
g
∈
C
(ℝ
^{+}
× ℝ
^{n}
,ℝ
^{n}
), ℝ
^{+}
= [0,∞), and that
f
is continuously differentiable with respect to the components of
x
on ℝ
^{+}
× ℝ
^{n}
,
f
(
t
, 0) = 0 for all
t
∈ ℝ
^{+}
. The symbol ｜·｜ will be used to denote arbitrary vector norm in ℝ
^{n}
.
Let
x
(
t
,
t
_{0}
,
x
_{0}
) denote the unique solutions of (1) and (2), satisfying the initial conditions
x
(
t
_{0}
,
t
_{0}
,
x
_{0}
) =
x
_{0}
, and
y
(
t
_{0}
,
t
_{0}
,
y
_{0}
) =
y
_{0}
, existing on [
t
_{0}
,∞), respectively. Then we can consider the associated variational systems around the zero solution of (1) and around
x
(
t
), respectively,
and
Here,
f_{x}
(
t
,
x
) is the matrix whose element in the
i
th row,
j
th column is the partial derivative of the
i
th component of
f
with respect to the
j
th component of
x
. 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 2.1.
The system (1)(the zero solution
x
= 0 of (1)) is called an
h

system
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 2.2.
The system (1) (the zero solution
x
= 0 of (1)) is called
h

stable
(
hS
) if there exists
δ
> 0 such that (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
[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 sequel.
Lemma 2.1
(
[16]
). 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 continuous( bounded, repectively) function h defined on
ℝ
^{+}
such that
for t
×
t
_{0}
× 0,
where ϕ
(
t, t
_{0}
)
is a fundamental matrix of (6).
The following is a generalization to nonlinear system of the variation of constants formula due to Alekseev
[1]
.
Lemma 2.2.
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. 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 (1) 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
(
[13]
).
Let u, f, g
∈
C
(ℝ
^{+}
),
for which the inequality
holds, where u
_{0}
is a nonnegative constant. Then,
Lemma 2.6
(
[5]
).
Let u, λ
_{1}
,
λ
_{2}
,
w
∈
C
(ℝ
^{+}
),
w
(
u
)
be nondecreasing in u and
for some v
> 0.
If ,for some c
> 0,
then
where
is the inverse of W
(
u
)
and
Lemma 2.7
(
[11]
).
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
is the inverse of W
(
u
)
and
3. Main results
In this section, we investigate bounds for the nonlinear differential systems. Also, we examine the bounded property for the perturbed system of (1)
where
g
∈
C
(ℝ
^{+}
× ℝ
^{n}
,ℝ
^{n}
) and
g
(
t
, 0) = 0.
The generalization of a function
h
’s condition and the strong condition of a function
g
in Theorem 3.1
[10]
are the following result.
Theorem 3.1.
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 a positive continuous function h, and g in (9) satisfies
where
Then, the solution y
= 0
of (9) is hS.
Proof
. Using the nonlinear variation of Alekseev
[1]
, any solution
y
(
t
) =
y
(
t, t
_{0}
,
y
_{0}
) of (9) passing through (
t
_{0}
,
y
_{0}
) is given by
By Theorem 2.3, since the solution
x
= 0 of (1) 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 and (10) , we have
Set
u
(
t
) = 
y
(
t
)
h
(
t
)
^{−1}
. Then, by Lemma 2.5, we obtain
It follows that
y
= 0 of (9) is hS. Hence, the proof is complete.
Remark 3.1.
In the linear case, we can obtain that if the zero solution
x
= 0 of (6) is hS, then the perturbed system
is also hS under the same hypotheses in Theorem 3.1 except the condition of
t
_{∞}
similarity.
Remark 3.2.
Letting
k
(
t
) = 0 in Theorem 3.1, we obtain the same result as that of Theorem 3.3 in
[9]
.
The weak condition of a function
h
and the strong condition of a function
g
in Theorem 3.3
[8]
are the following result.
Theorem 3.2.
Let a, b, k, u, w
∈
C
(ℝ
^{+}
),
w
(
u
)
be nondecreasing in u, 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 a positive continuous function h, and g in (9) satisfies
where
Then, any solution y
(
t
) =
y
(
t, t
_{0}
,
y
_{0}
)
of (9) is bounded on
[
t
_{0}
,∞)
and it satisfies
where c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
and W, W
^{−1}
are the same functions as in Lemma 2.6 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 (9), respectively. By Theorem 2.3, since the solution
x
= 0 of (1) is hS, the solution
v
= 0 of (3) is hS. Therefore, by Theorem 2.4, the solution
z
= 0 of (4) is hS. Using Lemma 2.1 and (10), we have
Set
u
(
t
) = 
y
(
t
)
h
(
t
)
^{−1}
. Now an application of Lemma 2.6 yields
where
c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
. The above estimation yields the desired result since the function
h
is bounded, and the theorem is proved.
The generalization of a function
h
’s condition and a slight modification of a function
g
’s condition in Theorem 3.4
[11]
are the following result.
Theorem 3.3.
Let a, b, k, u, w
∈
C
(ℝ
^{+}
),
w
(
u
)
be nondecreasing in u, 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 positive continuous function h, and g in (9) satisfies
where
Then, any solution y
(
t
) =
y
(
t, t
_{0}
,
y
_{0}
)
of (9) is bounded on
[
t
_{0}
,∞)
and it satisfies
where W, W
^{1}
are the same functions as in Lemma 2.6 and
Proof
. It is known that the solution of (9) is represented by the integral equation(10). By Theorem 2.3, since the solution
x
= 0 of (1) is hS, the solution
v
= 0 of (3) is hS. Therefore, by Theorem 2.4, the solution
z
= 0 of (4) is hS. Using Lemma 2.1 and (10), we have
Set
u
(
t
) = 
y
(
t
)
h
(
t
)
^{−1}
. Now an application of Lemma 2.7 yields
where
c
=
c
_{1}

y
_{0}

h
(
t
_{0}
)
^{−1}
. The above estimation implies the boundedness of
y
(
t
), and the proof is complete.
Remark 3.3.
Letting
k
(
t
) = 0 in Theorem 3.5 and adding the increasing condition of the function
h
, we obtain the same result as that of Theorem 3.2 in
[8]
.
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
Alexse V. M.
(1961)
An estimate for the perturbations of the solutions of ordinary differential equations
Vestn. Mosk. Univ. Ser. I. Math. Mekh.
(Russian)
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 t∞similarity
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
Elaydi S.
,
Rao R. R. M.
(1988)
Lipschitz stability for nonlinear Volterra integrodifferential systems
Appl. Math. Computations
27
191 
199
DOI : 10.1016/00963003(88)90001X
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.
,
Ryu D. H.
(2010)
hstability of the nonlinear perturbed differential systems
J. Chungcheong Math. Soc.
23
827 
834
Goo Y. H.
(2012)
hstability of perturbed differential systems via t∞similarity
J. Appl. Math. and Informatics
30
511 
516
Goo Y. H.
(2013)
Boundedness in the perturbed differential systems
J. Korean Soc. Math. Edu. Ser.B: Pure Appl. Math.
20
137 
144
Lakshmikantham V.
,
Leela S.
1969
Differential and Integral Inequalities: Theory and Applications Vol. I
Academic Press
New York and London