The purpose of this paper is to obtain some integral inequalities with impulses by using the method of Stieltjes derivatives, and we use our results in the study of Lyapunov stability of solutions of a certain nonlinear impulsive integrodifferential equation.
1. INTRODUCTION
In this paper, we discuss various integral inequalities with impulses.
Differential equations with impulses arise in various real world phenomena in mathematical physics, mechanics, engineering, biology and so on. We refer to the monograph of Samoilenko and Perestyuk
[6]
. Also integral inequalities are very useful tools in global existence, uniqueness, stability and other properties of the solutions of various nonlinear di®erential equations, see,e.g.,
[5]
.
To obtain our results in the paper we need some preliminaries. Now we state them.
Assume that [
a
,
b
], [
c
,
d
]⊂
R
are bounded intervals, where
R
is the set of all real numbers.
A function
f
: [
a
,
b
]→
R
is called
regulated
on [
a
,
b
] if both
exist for every point
s
∈ [
a
,
b
]. As a convention we define
f
(
a
) =
f
(
a
) and
f
(
b
+) =
f
(
b
). Let
G
[
a
,
b
] be the set of all regulated functions on [
a
,
b
]. If we let for
f
∈
G
[
a
,
b
], ∥
f
∥ = sup
_{s}
_{∈}
_{[a, b]}
｜
f
(
s
)｜, then (
G
[
a
,
b
], ∥ ᐧ ∥) becomes a Banach space. For regulated functions, see
[1
,
2]
.
For a closed interval
I
= [
c
,
d
], we define
f
(
I
) =
f
(
d
)
f
(
c
). A function
f
: [
a
,
b
] →
R
is
of bounded variation
on [
a
,
b
] if
where the supremum is taken over all partitions
a
=
t
_{0}
<
t
_{1}
< ᐧ ᐧ ᐧ <
t
_{n1}
<
t
_{n}
=
b
.
Let
BV
[
a
,
b
] be the set of all functions of bounded variation on [
a
,
b
]. We use the following notations for the convenience:
A
tagged
interval (τ, [
c
,
d
]) in [
a
,
b
] consists of an interval [
c
,
d
] ⊂ [
a
,
b
] and a point τ ∈ [
c
,
d
]. Let
I
_{i}
= [
c
_{i}
,
d
_{i}
] ⊂ [
a
,
b
]. A finite collection {(τ
_{i}
, [
c
_{i}
,
d
_{i}
]) :
i
= 1,2, ...,
m
} of pairwise nonoverlapping tagged intervals is called a
tagged partition of
[
a
,
b
] if
=[
a
,
b
]. A positive function 𝛿 on [
a
,
b
] is called a
gauge
on [
a
,
b
].
Definition 1.1
(
[4
,
7]
). Let 𝛿 be a gauge on [
a
,
b
]. A tagged partition
P
={(τ
_{i}
, [
t
_{i1}
,
t
_{i}
]) :
i
= 1, 2,...,
m
} of [
a
,
b
] is said to be 𝛿fine if for every
i
= 1, ...,
m
we have
τ
_{i}
∈ [
t
_{i1}
,
t
_{i}
] ⊂ (τ
_{i}
𝛿(τ
_{i}
), τ
_{i}
+ 𝛿(τ
_{i}
)).
If moreover a 𝛿fine partition
P
satisfies the implications
τ
_{i}
=
t
_{i1}
⇒
i
= 1, τ
_{i}
=
t
_{i}
⇒
i
=
m
,
then it is called a 𝛿
^{*}

fine partition of
[
a
,
b
].
The following lemma implies that for a gauge 𝛿 on [
a
,
b
] there exists a 𝛿
^{*}
fine partition of [
a
,
b
]. This also implies the existence of a 𝛿fine partition of [
a
,
b
].
Lemma 1.2
(
[4]
).
Let
𝛿
be a gauge on
[
a
,
b
]
and a dense subset
Ω ⊂(
a
,
b
)
be given
.
Then there exists a
𝛿
^{*}

fine partition P
= {(τ
_{i}
, [
t
_{i1}
,
t
_{i}
]) :
i
= 1, 2,...,
m
} of [
a
,
b
]
such that
t
_{i}
∈Ω
for
i
= 1, ...,
m
1.
We are now ready to give a formal definition of both types of the Kurzweil integral.
Definition 1.3
([4
,
7]
). Assume that
f
,
g
: [
a
,
b
] →
R
are given. We say that
fdg
is
Kurzweil integrable
(or shortly,
Kintegrable
) on [
a
,
b
] and
v
∈
R
is its integral if for every ε > 0 there exists a gauge 𝛿 on [
a
,
b
] such that for
we have
｜
S
(
fdg
,
P
) 
v
｜ ≤ ε,
provided
P
= {(τ
_{i}
,
I
_{i}
):
i
=1,...,
n
} is a 𝛿fine tagged partition of [
a
,
b
]. In this case we denote
(or, shortly,
)
If, in the above definition, 𝛿fine is replaced by 𝛿
^{*}
fine, then we say that
fdg
is Kurzweil* integrable(or, shortly, K*integrable) on [
a
,
b
] and we denote
.
Remark 1.4.
By the above definition it is obvious that Kintegrability implies K*integrability.
The integrals have the following properties. For the proofs, see, e.g.,
[7
,
8]
.
Theorem 1.5.
Assume that f, f_{1}, f_{2}, g
: [
a, b
] →
R and that f
_{1}
d
g
and f
_{2}
d
g
are integrable in the sense of Kurzweil or Kurzweil
*
on
[
a
,
b
].
Let
k
_{1}
,
k
_{2}
∈
R
. Then we have
If for c
∈ [
a
,
b
],
integrals
,
exist
,
then
exists also and we have
For the integrability we have the following fundamental result.
Theorem 1.6.
Assume that f
∈
G
[
a
,
b
]
and g
∈
BV
[
a
,
b
].
Then fdg is Kintegrable on
[
a
,
b
].
Theorem 1.7.
Assume that f, g
: [
a
,
b
] →
R
and that fdg is Kintegrable. If g is a regulated function on
[
a
,
b
],
then we have
2. THE STIELTJES DERIVATIVES
In this section we state the results in
[3]
that are essential to verify our main results.
Throughout this section, we assume that
f
∈
G
[
a
,
b
] and
g
is a nondecreasing function on [
a
,
b
].
A
neighborhood
of
t
∈ [
a
,
b
] is an open interval containing
t
. We say that the function
g is not locally
constant
at
t
∈ (
a
,
b
) if there exists 𝜂 > 0 such that
g
is not constant on (
t
ε,
t
+ε) for every ε < 𝜂. We also say that the function
g is not locally constant
at
a
and
b
, respectively if there exists 𝜂 > 0 such that
g
is not constant on [
a
,
a
+ε), (
b
 ε,
b
], respectively for every ε < 𝜂.
Definition 2.1.
If
g
is not locally constant at
t
∈ (
a
,
b
), we define
provided that the limit exists. If
g
is not locally constant at
t
=
a
and
t
=
b
respectively, we define
respectively. Sometimes we use
instead of
If both
f
and
g
are constant on some neighborhood of t, we define
= 0.
Remark 2.2.
It is obvious that if
g
is not continuous at
t
then
exists. Thus if
does not exist then g is continuous at
t
.
is called the Stieltjes derivative.
K*integrals recover Stieltjes derivatives.
Theorem 2.3.
Assume that if g is constant on some neighborhood of t then f is also constant there
.
Suppose that
exists at every
t
∈ [
a
,
b
] {
c
_{1}
,
c
_{2}
, ...},
where f is continuous at every
t
∈ {
c
_{1}
,
c
_{2}
, ...}.
Then we have
3. MAIN RESULTS
In this section we will state and prove our results.
Let
0 <
t
_{1}
<
t
_{2}
< ᐧ ᐧ ᐧ <
t
_{m}
< 1,
and let 0 <
a
< 1: Two sorts of Heaviside functions
H
_{a}
,
: [0, 1] → {0, 1}are defined respectively by
Using the Heaviside functions
H
_{a}
,
a we define functions 𝜙, 𝜓 : [0, 1] → [0,∞) by
respectively.
It is obvious that the functions 𝜙, 𝜓 are strictly increasing and of bounded variation on [0, 1].
From now on, we assume that
c
≥ 0 and that all the functions
u
,
f
,
g
,
g
_{i}
,
i
= 1, . . . ,
n
are nonnegative functions defined on [0, 1] that are regulated on [0, 1] and continuous at every
t
≠
t
_{k}
,
k
=
where
= 1, . . . ,
m
.
Lemma 3.1.
Assume that f ’(t) exists for t ≠= t_{k}, k
=
.
Then we have
(b)
If a leftcontinuous function f is positive, nondecreasing, and differentiable at t
≠=
t_{k}, k
=
, then
(c)
Proof
. (a) By definition, for
t
_{k}
<
t
<
t
_{k+1}
and for sufficiently small 𝛿 and 𝜂 we have
so we have
And
This implies
Similarly we can verify
And
This completes the proof for (a).
(b) By (a) if
t
≠=
t
_{k}
then it is obvious that
and by the Mean Value Theorem and since
f
is nondecreasing and leftcontinuous we have
(c) By Theorem 1.7, we have for
t
_{k}
<
t
Through the same process, we can obtain that
By the same method we can easily verify that
Since for
t
_{k}
>
t
,
H_{tk}
(
s
) = 0 =
(
s
) for every
s
∈ [0,
t
] we have
Using the above results and the definition and properties of Kintegral we get
Considering
we get
The proof is complete. □
Now we define functions
A
,
B
_{i}
: [0, 1] → [0,∞) as follows:
The following theorem is a GronwallBellman type integral inequality with impulses.
Theorem 3.2
(
[6]
).
Let
a
_{k}
≥ 0,
k
=
,
If
then we have
Proof
. Define a function
z
(
t
) by the right side of (3.1), then we observe that
z
(0) =
c
,
u
(
t
)≤
z
(
t
) and for
t
≠
t
_{k}
,
k
=
, we have by Lemma 3.1
So, we have
By Lemma 3.1 this implies
By setting
t
=
s
in (3.2) and integrating it with respect to 𝜙 from 0 to t then by Theorem 2.3 and Lemma 3.1 we get
Since
z
(0) =
c
we get
This completes the proof. □
A generalization of Theorem 3.2 is the following result.
Theorem 3.3.
Let
0 <
m
_{1}
<
m
_{2}
< ⋯ <
m
_{n}
and let a
_{k}
,
a
_{ik}
≥ 0,
i
=
k
=
,
If
then
where
provided that M(t
) +
N(t)
< 1.
Proof
. Inequality (3.3) is written as:
By applying Theorem 3.2, we get
Then for every
m
_{j}
,
j
=
, we have
Multiplying the last inequality by a negative term 
m
_{n}
g
_{j}
(
t
), we have
By summing the inequality for
j
=
, we obtain
This implies that for
t
≠=
t
_{k}
,
k
=
And by Lemma 3.1 we have
By the Mean Value Theorem we get for some
So we conclude that
By (3.6) and (3.7) we obtain
Integrating from 0 to
t
with respect to 𝜙 we get by Theorem 2.3
This implies that
So inequality (3.5) becomes
The proof is complete. □
From now on a function
:[0, 1]→ [0,∞) is defined by
Theorem 3.4.
Let
1 <
p and let
a
_{k}
,
b
_{k}
≥ 0,
k
=
.
If
and
1
M
(
t
)
N
(
t
) > 0,
where
then we have
where
Proof
. Define a function
z
(
t
) by the right side of (3.8), then we observe that
z
(0) =
c
,
u
(
t
) ≤
z
(
t
) and for
t
≠=
t
_{k}
,
k
=
we have
and
This implies that
Define a function
v(t)
by
and then for
t
≠
t
_{k}
,
k
=
, by Lemma 3.1 and (3.9)
and
and
Thus we have
This implies that
Since
z
(
t
) is leftcontinuous we have
So we have
Thus
By Theorem 3.3 we have
And
v
(
t
_{k}
) ≤
a
_{k}
v
(
t
_{k}
) +
b
_{k}
v
^{p}
(
t
_{k}
)≤
c
[
a
_{k}
W
(
t
_{k}
) +
b
_{k}
c
^{p1}
W
^{p}
(
t
_{k}
)].
Thus we get
So we have by Theorem 2.3,
This completes the proof. □
4. AN EXAMPLE
There are many applications of the inequalities obtained in Section 3. Here we shall give an example which is su±cient to show the usefulness of our results.
Consider the following impulsive integrodifferential equation
where 0<
t
_{1}
< ⋯ <
t
_{k}
< ⋯ <
t
_{m}
< 1, where a function
F
: [0, 1] ×
R
^{2}
→
R
is continuous on [0, 1] ×
R
^{2}
and satisfies
｜
F
(
s
,
x
,
y
)｜ ≤
f
(
s
)(｜
x
｜ + ｜
y
｜)
for some continuous function
f
: [0, 1] → [0,∞), and a function
G
: [0, 1]×
R
→
R
is continuous on [0, 1] ×
R
and satisfies
｜
G
(
s
,
x
)｜ ≤
g
(
s
)｜
x
｜
^{p}
for some continuous function
g
: [0, 1] → [0,∞) and
p
> 1. Then we have
Assume that
F
(
t
, 0, 0) = 0 and that the function
I
_{k}
:
R
→
R
is continuous and ｜
I
_{k}
(
x
)｜ ≤
a
_{k}
｜
x
｜ +
b
_{k}
｜
x
｜
^{p}
,
a
_{k}
,
b
_{k}
≥ 0,
k
=
.
Let
q
=
p
 1 and suppose that
M
(
t
) +
N
(
t
) < 1,where
Then we have
Applying Theorem 3.4 to the above inequality, we get
Since the function
K
(
t
) is bounded on [0, 1], the above inequality implies that the zero solution of equation (4.1) is Lyapunov stable. □
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