SOME OPIAL-TYPE INEQUALITIES APPLICABLE TO DIFFERENTIAL EQUATIONS INVOLVING IMPULSES

The Pure and Applied Mathematics.
2015.
Nov,
22(4):
315-331

- Received : March 09, 2015
- Accepted : September 07, 2015
- Published : November 30, 2015

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The purpose of this paper is to obtain Opial-type inequalities that are useful to study various qualitative properties of certain differential equations involving impulses. After we obtain some Opial-type inequalities, we apply our results to certain differential equations involving impulses.
R
be the set of all real numbers. Assume that [
a
,
b
] ⊂
R
is a bounded interval. A function
f
: [
a
,
b
] →
R
is called regulated on [
a
,
b
] if both
exist for every point
s
∈ [
a
,
b
),
t
∈ (
a
,
b
], respectively. Let
G
([
a
,
b
]) be the set of all regulated functions on [
a
,
b
]. For
f
∈
G
([
a
,
b
]) we define
f
(
a
−) =
f
(
a
);
f
(
b
+) =
f
(
b
). For convenience we define
Remark 2.1.
Let
f
∈
G
([
a
,
b
]). Since both
f
(
s
+) and
f
(
s
−) exist for every
s
∈ [
a
,
b
] it is obvious that
f
is bounded on [
a
,
b
], and since
f
is the uniform limit of step functions,
f
is Borel measurable (see [3, Theorem 3.1.]).
For a closed interval
I
= [
c
,
d
], we define
f
(
I
) =
f
(
d
) −
f
(
c
).
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
],
i
= 1, …,
m
. We say that the intervals
I_{i}
are pairwise non-overlapping if
for
i
≠
j
where
int
(
I
) denotes the interior of an interval
I
.
A finite collection {(
τ_{i}
,
I_{i}
) :
i
= 1, 2, …,
m
} of pairwise non-overlapping tagged intervals is called a
tagged partition of
[
a
,
b
] if
. A positive function
δ
on [
a
,
b
] is called a
gauge
on [
a
,
b
].
From now on we use notation
.
Definition 2.2
(
[6
,
9]
)
.
Let
δ
be a gauge on [
a
,
b
]. A tagged partition
of [
a
,
b
] is said to be
δ
−
fine
if for every
we have
Moreover if a
δ
−fine partition
P
satisfies the implications
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
δ
− partition of [
a
,
b
].
Lemma 2.3
(
[6]
)
.
Let δ be a gauge on
[
a
,
b
]
and a dense subset
Ω⊂ (
a
,
b
)
be given
.
Then there exists a δ*−fine partition
of
[
a
,
b
]
such that
t_{i}
∈ Ω
for
We now give a formal definition of two types of the Kurzweil integrals.
Definition 2.4
(
[6
,
9]
)
.
Assume that
f
,
g
: [
a
,
b
] →
R
are given. We say that
f
d
g
is
Kurzweil integrable
(or shortly, K-
integrable
) on [
a
,
b
] and
v
∈
R
is its integral if for every
ε
> 0 there exists a gauge
δ
on [
a
,
b
] such that
provided
is a
δ
−fine tagged partition of [
a
,
b
]. In this case we define
(or, shortly,
).
If, in the above definition,
δ
−fine is replaced by
δ
*−fine, then we say that
f
d
g
is
Kurzweil* integrable
(or, shortly, K*-
integrable
) on [
a
,
b
] and we define
.
Remark 2.5.
By the above definition it is obvious that K-integrability implies K*-integrability.
The following results are needed in this paper. For other properties of the K-integrals, see, e.g.,
[2
,
7
,
9
,
10]
.
In this paper
BV
([
a
,
b
]) denotes the set of all functions that are of bounded variation on [
a
,
b
].
Theorem 2.6
([
11
, 2.15. Theorem])
.
Assume that f
∈
G
([
a
,
b
])
and g
∈
BV
([
a
,
b
]).
Then both
f
d
g
and
g
d
f
are K-integrable on
[
a
,
b
]
and
Remark 2.7.
In the above theorem, the sum Σ
_{t∈[a,b]}
[Δ
^{−}
f
(
t
)Δ
^{−}
g
(
t
)−Δ
^{+}
f
(
t
)Δ
+
g
(
t
)] is actually a countable sum because every regulated function has only countable dis-continuities.
Theorem 2.8
([
10
, p. 40, 4.25. Theorem])
.
Let h
∈
BV
([
a
,
b
]),
g
: [
a
,
b
] →
R
and
f
: [
a
,
b
] →
R
.
If the integral
g dh exists and f is bounded on
[
a
,
b
],
then the integral
exists if and only if the integral
exists and in this case we have
Theorem 2.9
([
10
, p. 34, 4.13. Corollary])
.
Assume that f
∈
G
([
a
,
b
])
and g
∈
BV
([
a
,
b
]).
Then we have for every t
∈ [
a
,
b
]
The following result is the Hölder’s inequality for K-integral. In this paper we frequently use this inequality.
Theorem 2.10.
(Hölder’s inequality)
Assume that
f
,
g
∈
G
([
a
,
b
])
and h is a nondecreasing function defined on
[
a
,
b
].
Let
.
Then we have
Proof
. The proof of this theorem is very similar to the proof of the classical Hölder’s inequality. So we omit the proof. ☐
f
∈
G
([
a
,
b
]) and
g
is a nondecreasing function on [
a
,
b
].
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 0 <
ε
<
η
. We also say that the function
g
is
not locally constant
at
a
and
b
, respectively if there exist
η
,
η
* > 0 such that
g
is not constant on [
a
,
a
+
ε
), (
b
−
ε
*,
b
] respectively, for every
ε
∈ (0,
η
),
ε
*∈ (0,
η*
).
Definition 3.1
(
[4]
)
.
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, provided that the limits exist. Frequently we use
instead of
.
If both
f
and
g
are constant on some neighborhood of
t
, then we define
.
Remark 3.2.
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 a
Stieltjes derivative
of
f
with respect to
g
.
Theorem 3.3
(
[4]
)
.
Assume that if g is not locally constant at t
∈ [
a
,
b
].
If f is continuous at t or g is not continuous at t; then we have
K*-integrals recover Stieltjes derivatives.
Theorem 3.4
(
[4]
)
.
Assume that if g is constant on some neighborhood of t then there is a neighborhood of t where both f and g are constant. 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
Lemma 3.5
(
[4]
)
.
Assume that if g is constant on some neighborhood of t then there is a neighborhood of t such that both
f
_{1}
and
f
_{2}
are constant there. If both
and
exist and
f
_{1}
,
f
_{2}
∈
G
([
a
,
b
]), t
hen we have
Similarly to the Riemann integral we have the following integration by parts formula.
Theorem 3.6.
(Integration by Parts)
Assume that functions f, g, h
∈
G
([
a
,
b
])
are all left-continuous and h is nondecreasing. Suppose that both
and
exist for every t
∈ [
a
,
b
]
and
.
Then we have
Proof
. By Theorem 2.8 and Theorem 3.4 we have
So by Theorem 2.6 we get
This completes the proof. ☐
Let
The Heaviside function
H_{τ}
:
R
→ {0, 1} is defined by
Using the Heaviside function
H_{τ}
, we define function
ϕ
: [
a
,
b
] →
R
by
Remark 3.7.
It is obvious that the function
ϕ
is strictly increasing and of bounded variation on [
a
,
b
], and left-continuous on [
a
,
b
].
Lemma 3.8
(
[5]
)
.
Assume that f
∈
G
([
a
,
b
])
and
f'
(
t
)
exists for t
≠
t_{k}
,
Then we have
and that a function
α
: [
a
,
b
] →
R
is strictly increasing on [
a
,
b
], and continuous at
t
≠
t_{k}
, and Δ
α
(
t_{k}
) ≠ 0; for every
.
Remark 4.1.
Note that strictly increasing implies nondecreasing, and a nondecreasing function is regulated.
Let
PC
([
a
,
b
]) = {
u
∈
G
([
a
,
b
]) :
u
is continuous at every
t
≠
t_{k}
,
}.
From now on we always assume that
u
,
, and we define
The following result is an Opial-type inequality with Stieltjes derivatives.
Theorem 4.2.
Assume that u
(
a
) =
u
(
b
) = 0.
If both u and α are left-continuous on
[
a
,
b
],
then we have
where
K_{α}
= inf
_{h∈[a, b]}
max{
α
(
h
) −
α
(
a
),
α
(
b
) −
α
(
h
)}.
Proof
. Let for
t
∈[
a
,
b
],
By Theorem 2.9, the functions,
y
and
z
are left-continuous on [
a
,
b
]. Also by Theorem 3.3, we have
and we have by Theorem 3.4 and
u
(
a
) =
u
(
b
) = 0
for
t
∈ [
a
,
b
]. So by Theorem 3.4, Lemma 3.5, and using Hölder’s inequality, we get
and similarly we obtain
So we have
The proof is complete. ☐
A slightly more general result is as follows.
Theorem 4.3.
Assume that u
(
b
) = 0.
If both u and α are left-continuous on
[
a
,
b
],
then we have
Proof
. From (4.2) we have
So we get
This gives (4.3). The proof is complete. ☐
More generally we have the following result.
Theorem 4.4.
Let p
≥ 0,
q
≥ 1,
r
≥ 0,
m
≥ 1
be real numbers and let f
∈
PC
([
a
,
b
])
be a positive function on
[
a
,
b
]
with
inf
_{s∈[a, b]}
f
(
s
) > 0.
Assume that both functions u and α are left-continuous on
[
a
,
b
].
If u
(
b
) = 0,
then we have
where
,
for m
≠ 1,
and
for m
= 1.
Proof
. Let for
t
∈ [
a
,
b
],
Then by Theorem 3.4, |
u
(
t
)| ≤
z
(
t
) and by Theorem 2.9,
z
is left-continuous, and non-increasing on [
a
,
b
].
If
t
≠
t_{k}
,
, Then by Theorem 3.3,
exists, and by Theorem 2.9,
z
is continuous at
t
. Using the Mean Value Theorem and by the definition of the Stieltjes derivatives, if
z
is not locally constant at
t
, then we have,
If
z
is constant on some neighborhood of
t
, then since
, the above equality is also true. If
t
=
t_{k}
,
, since
z
is non-increasing on [
a
,
b
], and
and by the Mean Value Theorem, and by the definition of the Stieltjes derivatives, we have,
where
z
(
t_{k}
+) ≤
ω
≤
z
(
t_{k}
) = z(
t_{k}
−): Thus we have
Let
. Then by hypotheses,
β
is strictly increasing on [
a
,
b
].
Since
we have by Theorem 3.4 and (4.5), since
z
(
b
) = 0 and
,
Using Hölder’s inequality with indices
m
,
, we have
Integrating (4.6) on [
a
,
b
] and using Hölder’s inequality with indices
q
,
, and considering
by Theorem 2.8, we get
If
, then
is obviously true, otherwise, dividing both sides of (4.7) by
and then taking the
q
th power on both sides of the resulting inequality we get also (4.8).
Using the Hölder’s inequality with indice
,
, we have, by (4.8),
Using Hölder’s inequality with indices
,
, we get by (4.9)
If
, then the inequality
is obviously true, otherwise, dividing both sides of (4.10) by
and then taking the
power on both sides of the resulting inequality we get also (4.11). Since |
u
| ≤
z
and
we have
This gives (4.4). The proof is complete. ☐
u
and
u'
are left- continuous on [
a
,
b
], and that
α
=
ϕ
(see (3.2)). Consider the following impulsive differential equation: for
,
where
q
_{1}
∈
PC
([
a
,
b
]): Now we define
Since by Lemma 3.8 for
the equation (5.1) implies the following equation:
where
We need the following result.
Lemma 5.1.
If the function u satisfies the equation (5.1) and c
∈ [
a
,
b
],
then we have
Proof
. In the proof, we frequently use Lemma 3.8,
.
, and
,
.
This gives (5.4). And
This gives (5.5). And
This gives (5.6). Also
This gives (5.7). The proof is complete. ☐
Theorem 5.2.
Assume that u satisfies the equation (5.1) and
u'
(
a
) = 0,
u
(
a
) ≠ 0.
If we have
where
,
then u
(
t
) ≠ 0
for every t
∈ [
a
,
b
].
Proof
. Assume that there is a number
c
∈ (
a
,
b
] with
u
(
c
) = 0. Then multiplying both sides of (5.2) by
u
and integrating we have
Using Theorem 3.3, Lemma 3.5 and Theorem 3.6, and
u
(
c
) =
Q
(
a
) = 0, we get, since, by Theorem 2.9 and Remark 3.7,
Q
is left-continuous on [
a
,
b
]; and Δ
α
(
t_{k}
) = Δ
^{+}
α
(
t_{k}
) = 1,
q
(
t_{k}
) = 0,
,
Since both
u
and
u'
are left-continuous
By Lemma 3.8 and Lemma 5.1, we get, since
u
(
c
) =
u'
(
a
) = 0,
By (5.9), (5.10) and (5.11), we have
Hence by Theorem 4.3 and Lemma 5.1, we get
If
then, since
,
and
u' _{α}
(
t_{k}
) =
u
(
t_{k}
+) −
u
(
t_{k}
) = 0. This implies that
u
is a constant on [
a
,
c
]. So
u
(
c
) =
u
(
a
) ≠0. But this is a contradiction to
u
(
c
) = 0. Hence we conclude that
.
In (5.12), canceling
, we get a contradiction to (5.8). This completes the proof. ☐
In the following result we apply Theorem 4.4.
Theorem 5.3.
Let q
∈
PC
([
a
,
b
])
and let α
=
ϕ
(see (3.2)). If
u
∈
PC
([
a
,
b
]) is left-continuous and a nontrivial solution of the following equation:
then we have
Proof
. Substituting
f
≡ 1,
p
= 0,
q
= 1,
r
= 0 into Theorem 4.4, then we have
So we have
Canceling
, we get (5.13). ☐

1. INTRODUCTION

Opial-type inequalities are very useful to study various qualitative properties of differential equations. For a good reference of the work on such inequalities together with various applications, we recommend the monograph
[1]
. In this paper we obtain some Opial-type inequalities that involve Stieltjes derivatives which are applicable to differential equations with impulses. Differential equations involving impulses arise in various real world phenomena, we refer to the monograph
[8]
.
2. PRELIMINARIES

To obtain our results in this paper we need some preliminaries.
Let
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3. STIELTJES DERIVATIVES

In this section we state the results in
[4
,
5]
that are essential to obtain our main results.
Throughout this section, we assume that
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4. OPIAL-TYPE INTEGRAL INEQUALITIES INVOLVING STIELTJES DERIVATIVES

In this section we obtain some Opial-type integral inequalities involving Stieltjes derivatives. The Opial-type inequalities have many interesting applications in the theory of differential equations(see, e.g.,
[1]
).
Throughout this paper we always assume that
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5. SOME APPLICATIONS TO CERTAIN DIFFERENTIAL EQUATIONS INVOLVING IMPULSES

In this section we always assume that both functions
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Agarwal R.P.
,
Pang P.Y.H.
1995
Opial inequalities with applications in differential anddifference equations
Kluwer Academic Publishers
Dordrecht

Henstock R.
1988
Lectures on the theory of integration
World Scientific
Singapore

Hönig C.S.
1973
Volterra Stieltjes-integral equations
North Holand and American Elsevier, Mathematics Studies
Amsterdam and New York

Kim Y.J.
(2011)
Stieltjes derivatives and its applications to integral inequalities of Stieltjes type
J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math.
18
(1)
63 -
78

Kim Y.J.
(2014)
Stieltjes derivative method for integral inequalities with impulses
J. Korean Soc. Math. Educ.Ser. B: Pure Appl. Math.
21
(1)
61 -
75

Krejčí P.
,
Kurzweil J.
(2002)
A nonexistence result for the Kurzweil integral
Math. Bohem.
127
571 -
580

Pfeffer W.F.
1993
The Riemann approach to integration: local geometric theory
Cambridge University Press

Samoilenko A.M.
,
Perestyuk N.A.
1995
Impulsive differential equations
World Scientific
Singapore

Schwabik Š.
1992
Generalized ordinary differential equations
World Scientific
Singapore

Schwabik Š.
,
Tvrdý M.
,
Vejvoda O.
1979
Differntial and integral equations: boundary value problems and adjoints
Academia and D. Reidel, Praha and Dordrecht

Tvrdý M.
(1989)
Regulated functions and the Perron-Stieltjes integral
Časopis pešt. mat.
114
(2)
187 -
209

Citing 'SOME OPIAL-TYPE INEQUALITIES APPLICABLE TO DIFFERENTIAL EQUATIONS INVOLVING IMPULSES
'

@article{ SHGHCX_2015_v22n4_315}
,title={SOME OPIAL-TYPE INEQUALITIES APPLICABLE TO DIFFERENTIAL EQUATIONS INVOLVING IMPULSES}
,volume={4}
, url={http://dx.doi.org/10.7468/jksmeb.2015.22.4.315}, DOI={10.7468/jksmeb.2015.22.4.315}
, number= {4}
, journal={The Pure and Applied Mathematics}
, publisher={Korean Society of Mathematical Education}
, author={KIM, YOUNG JIN}
, year={2015}
, month={Nov}