In this section, we consider the oscillation problem for first-, second-, and higher-order linear impulsive differential equations. Moreover, the Sturm type comparison theorems for second-order linear impulsive differential equations are included.

### 3.2. Sturmian Theory for Second-Order Linear Equations

It is well-known that Sturm comparison theory plays an important role in the study of qualitative properties of the solutions of both linear and nonlinear equations. The first paper on the Sturm theory of impulsive differential equations was published in 1996. In [10], Bainov et al. derived a Sturmian type comparison theorem, a zeros-separation theorem, and a dichotomy theorem for second-order linear impulsive differential equations. Recently, the theory has been extended in various directions in [16–18], with emphasis on Picone's formulas, Wirtinger type inequalities, and Leighton type comparison theorems.

We begin with a series of results contained in [

1,

10]. The second-order linear impulsive differential equations considered are

where
and
are continuous for
and they have a discontinuity of the first kind at the points
where they are continuous from the left.

The main result is the following theorem, which is also valid for differential inequalities.

Theorem 3.3 (see [1, 10]).

Suppose the following.

- (1)
Equation (3.7) has a solution

such that

- (2)
The following inequalities are valid:

Then (3.6) has no positive solution
defined on

Proof.

Assume that (3.6) has a solution

such that

Then from the relation

From (3.6), (3.7), condition (

), and the above inequality, we conclude that

But, from conditions (
) and (
), it follows that the right side of the above inequality is negative, which leads to a contradiction. This completes the proof.

The following corollaries follow easily from Theorem 3.3.

Corollary 3.4 (Comparison Theorem).

Suppose the following.

- (1)
Equation (3.7) has a solution

such that

- (2)
The following inequalities are valid:

Then, each solution
of (3.6) has at least one zero in

Corollary 3.5.

If conditions ( 1) and ( 2) of Corollary 3.4 are satisfied, then one has the following.

- (1)

- (2)
Each solution
of (3.6) has at least one zero in

Corollary 3.6 (Oscillation Theorem).

Suppose the following.

- (1)
There exists a solution

of (3.7) and a sequence of disjoint intervals

such that

for

- (2)
The following inequalities are valid for

and

;

Then all solutions of (3.6) are oscillatory, and moreover, they change sign in each interval

Corollary 3.7 (Comparison Theorem).

Let the inequalities
hold for
and
Then, all solutions of (3.7) are nonoscillatory if (3.6) has a nonoscillatory solution.

Corollary 3.8 (Separation Theorem).

The zeros of two linearly independent solutions of (3.6) separate one another; that is, the two solutions have no common zeros, and if
are two consecutive zeros of one of the solutions, then the interval
contains exactly one zero of the other solution.

Corollary 3.9 (Dichotomy Theorem).

All solutions of (3.6) are oscillatory or nonoscillatory.

We can use Corollary 3.7, to deduce the following oscillation result for the equation:

Theorem 3.10 (see [1, 10]).

Suppose the following.

- (1)

- (2)
The following inequalities are valid:

Then every solution
of (3.17) defined for
is oscillatory if (3.7) has an oscillatory solution.

Recently, by establishing a Picone's formula and a Wirtinger type inequality, Özbekler and Zafer [

17] have obtained similar results for second-order linear impulsive differential equations of the form

where
and
are real sequences,
with
and
for all

Let

be a nondegenerate subinterval of

. In what follows we shall make use of the following condition:

It is well-known that condition (H) is crucial in obtaining a Picone's formula in the case when impulses are absent. If (H) fails to hold, then Wirtinger, Leighton, and Sturm-Picone type results require employing a so-called "device of Picard." We will show how this is possible for impulsive differential equations as well.

Let (H) be satisfied. Suppose that

such that

and

. These conditions simply mean that

and

are in the domain of

and

, respectively. If

for any

, then we may define

For clarity, we suppress the variable

. Clearly,

In view of (3.19) and (3.20) it is not difficult to see from (3.22) that

the following Picone's formula is easily obtained.

Theorem 3.11 (Picone's formula [17]).

Let (H) be satisfied. Suppose that

such that

and

. If

for any

and

then

In a similar manner one may derive a Wirtinger type inequality.

Theorem 3.12 (Wirtinger type inequality [17]).

If there exists a solution

of (3.19) such that

on

, then

Corollary 3.13.

If there exists an
such that
then every solution
of (3.19) has a zero in
.

Corollary 3.14.

Suppose that for a given
there exists an interval
and a function
for which
. Then (3.19) is oscillatory.

Next, we give a Leighton type comparison theorem.

Theorem 3.15 (Leighton type comparison [17]).

Suppose that there exists a solution

of (3.19). If (H) is satisfied with

and

then every solution
of (3.20) must have at least one zero in
.

Proof.

Let

and

. Since

and

are solutions of (3.19) and (3.20), respectively, we have

. Employing Picone's formula (3.25), we see that

The functions under integral sign are all integrable, and regardless of the values of

or

, the left-hand side of (3.29) tends to zero as

. Clearly, (3.29) results in

which contradicts (3.28).

Corollary 3.16 (Sturm-Picone type comparison).

Let

be a solution of (3.19) having two consecutive zeros

. Suppose that (H) holds, and

for all

, and

for all
for which
.

If either (3.31) or (3.32) is strict in a subinterval of
or (3.33) is strict for some
, then every solution
of (3.20) must have at least one zero on
.

Corollary 3.17.

Suppose that conditions (3.31)-(3.32) are satisfied for all
for some integer
, and that (3.33) is satisfied for all
for which
. If one of the inequalities (3.31)–(3.33) is strict, then (3.20) is oscillatory whenever any solution
of (3.19) is oscillatory.

As a consequence of Theorem 3.15 and Corollary 3.16, we have the following oscillation result.

Corollary 3.18.

Suppose for a given
there exists an interval
for which that condition of either Theorem 3.15 or Corollary 3.16 are satisfied. Then (3.20) is oscillatory.

If (H) does not hold, we introduce a setting, which is based on a device of Picard, leading to different versions of Corollary 3.16.

Indeed, for any

we have

Assuming that

, the choice of

yields

Then, we have the following result.

Theorem 3.19 (Device of Picard [17]).

Let

and let

be a solution of (3.19) having two consecutive zeros

and

in

. Suppose that

are satisfied for all

, and that

for all
for which
.

If either (3.38) or (3.39) is strict in a subinterval of
or (3.40) is strict for some
, then any solution
of (3.20) must have at least one zero in
.

Corollary 3.20.

Suppose that (3.38)-(3.39) are satisfied for all
for some integer
, and that (3.40) is satisfied for all
for which
. If
and one of the inequalities (3.38)–(3.40) is strict, then (3.20) is oscillatory whenever any solution
of (3.19) is oscillatory.

As a consequence of Theorem 3.19, we have the following Leighton type comparison result which is analogous to Theorem 3.15.

Theorem 3.21 (Leighton type comparison [17]).

Let

. If there exists a solution

of (3.19) such that

then every solution
of (3.20) must have at least one zero in
.

As a consequence of Theorems 3.19 and 3.21, we have the following oscillation result.

Corollary 3.22.

Suppose that for a given
there exists an interval
for which conditions of either Theorem 3.19 or Theorem 3.21 are satisfied. Then (3.20) is oscillatory.

Moreover, it is possible to obtain results for (3.20) analogous to Theorem 3.12 and Corollary 3.13.

Theorem 3.23 (Wirtinger type inequality [17]).

If there exists a solution

of (3.20) such that

on

, then for

and for all

Corollary 3.24.

If there exists an
with
such that
then every solution
of (3.20) must have at least one zero in
.

As an immediate consequence of Corollary 3.24, we have the following oscillation result.

Corollary 3.25.

Suppose that for a given
there exists an interval
and a function
with
for which
. Then (3.20) is oscillatory.

### 3.3. Oscillation of Second-Order Linear Equations

The oscillation theory of second-order impulsive differential equations has developed rapidly in the last decade. For linear equations, we refer to the papers [11, 19–21].

Let us consider the second-order linear differential equation with impulses

where

,

and

are two known sequences of real numbers, and

For (3.43), it is clear that if

for all large

then (3.43) is oscillatory. So, we assume that

The following theorem gives the relation between the existence of oscillatory solutions of (3.43) and the existence of oscillatory solutions of second-order linear nonimpulsive differential equation:

Theorem 3.26 (see [19]).

Assume that
Then the oscillation of all solutions of (3.43) is equivalent to the oscillation of all solutions of (3.45).

Proof.

Let

be any solution of (3.43). Set

for

Then, for all

we have

Thus,

is continuous on

Furthermore, for

we have

For

it can be shown that

Thus,

is continuous if we define the value of

at

as

Now, we have for

and for

This shows that
is the solution of (3.45).

Conversely, if

is the continuous solution of (3.45), we set

for

Then,

and

Furthermore, for

we have

Thus,
is the solution of (3.43). This completes the proof.

By Theorems 3.26 and 2.3, one may easily get the following corollary.

Corollary 3.27.

Assume that

. Then, (3.43) is oscillatory if

When
and
oscillation criteria for (3.43) can be obtained by means of a Riccati technique as well. First, we need the following lemma.

Lemma 3.28.

Assume that

on any interval

and let

be an eventually positive solution of (3.43). If

where
then, eventually

Now, let

be an eventually positive solution of (3.43) such that

and

for

Under conditions of Lemma 3.28, let

for

Then, (3.43) leads to an impulsive Riccati equation:

where

Theorem 3.29 (see [19]).

Equation (3.43) is oscillatory if the second-order self-adjoint differential equation

is oscillatory, where

Proof.

Assume, for the sake of contradiction, that (3.43) has a nonoscillatory solution

such that

for

Now, define

Then, it can be shown that

is continuous and satisfies

Then
is a solution of (3.59). This completes the proof.

By Theorems 3.29 and 2.4, we have the following corollary.

Corollary 3.30.

where
Then, (3.43) is oscillatory.

Example 3.31 (see [19]).

If

for some integer

then it is easy to see that

where

denotes the greatest integer function, and

Thus, by Corollary 3.30, (3.64) is oscillatory. We note that the corresponding differential equation without impulses

is nonoscillatory by Theorem 2.3.

In [

20], Luo and Shen used the above method to discuss the oscillation and nonoscillation of the second-order differential equation:

where

In [

21], the oscillatory and nonoscillatory properties of the second-order linear impulsive differential equation

for all

and

is the

-function, that is,

for all

being continuous at

Before giving the main result, we need the following lemmas. For each

define the sequence

inductively by

where
provided
and
provided
Let

Lemma 3.32.

If
for some
then
and
for all

Proof.

By induction and in view of the fact that the function

is increasing in

it can be seen that

Hence,

The next lemma can also be proved by induction.

Lemma 3.33.

Suppose that

and

for all

Define, by induction,

If

for all

then

The following theorem is the main result of [21]. The proof uses the above two lemmas and the induction principle.

Theorem 3.34.

The following statements are equivalent.

- (i)
There is
such that

- (ii)

- (iii)
Equation (3.69) is nonoscillatory.

- (iv)
Equation (3.69) has a nonoscillatory solution.

Applying Theorem 3.34, the nonoscillation and oscillation of (3.69), in the case of
and
are investigated in [21].

In all the publications mentioned above, the authors have considered differential equations with fixed moments of impulse actions. That is, it is assumed that the jumps happen at fixed points. However, jumps can be at random points as well. The oscillation of impulsive differential equations with random impulses was investigated in [11]. Below we give the results obtained in this case.

Let

be a random variable defined in

and let

be a constant. Consider the second-order linear differential equation with random impulses:

where
are Lebesque measurable and locally essentially bounded functions,
,
for all
and

Definition 3.35.

Let

be a real-valued random variable in the probability space

where

is the sample space,

is the

-field, and

is the probability measure. If

, then

is called the

*expectation of*
and is denoted by

that is,

In particular, if

is a continuous random variable having probability density function

then

Definition 3.36.

A stochastic process

is said to be a

*sample path solution to*(3.76) with the initial condition

if for any sample value

of

then

satisfies

Definition 3.37.

The e

*xponential distribution* is a continuous random variable with the probability density function:

where
is a parameter.

Definition 3.38.

A solution
of (3.76) is said to be *nonoscillatory in mean* if
is either eventually positive or eventually negative. Otherwise, it is called *oscillatory.*

Consider the following auxiliary differential equation:

Lemma 3.39.

The function

is a solution of (3.76) if and only if

where

is a solution of (3.81) with the same initial conditions for (3.76), and

is the index function, that is,

Proof.

If

is a solution of system (3.81), for any

we have

which imply that
satisfies (3.76), that is,
is a sample path solution of (3.76). If
is a sample path solution of (3.76), then it is easy to check that
is a solution of (3.81). This completes the proof.

Theorem 3.40 (see [11]).

Let the following condition hold.

(C) Let
be exponential distribution with parameter
,
and let
be independent of
if

If there exists

such that

does not change sign for all
then all solutions of (3.76) are oscillatory in mean if and only if all solutions of (3.81) are oscillatory.

Proof.

Let

be any sample path solution of (3.76); then Lemma 3.39 implies

where

is a solution of (3.81). Hence,

Further, it can be seen that

By assumption,
has the same sign as
for all
That is, all solutions of (3.76) are oscillatory in mean if and only if all solutions of (3.81) are oscillatory. This completes the proof.

When
is finite,
, then the following result can be proved.

Theorem 3.41 (see [11]).

Let condition (C) hold, and let
be finite for all
. Further assume that there are a finite number of
such that
Then all solutions of (3.76) are oscillatory in mean if and only if all solutions of (3.81) are oscillatory.

### 3.4. Oscillation of Higher-Order Linear Equations

Unlike the second-order impulsive differential equations, there are only very few papers on the oscillation of higher-order linear impulsive differential equations. Below we provide some results for third-order equations given in [22]. For higher-order liner impulsive differential equations we refer to the papers [23, 24].

Let us consider the third-order linear impulsive differential equation of the form

where
and
is not always zero in
for sufficiently large

The following lemma is a generalization of Lemma
in [25].

Lemma 3.42 (see [22]).

Assume that
is a solution of (3.91) and there exists
such that for any
Let the following conditions be fulfilled.

Then for sufficiently large
either
or
holds, where

Theorem 3.43 (see [22]).

Assume that conditions of Lemma 3.42 are fulfilled and for any
and
Moreover, assume that the sequence of numbers
has a positive lower bound,
converges, and
holds. Then every bounded solution of (3.91) either oscillates or tends asymptotically to zero with fixed sign.

Proof.

Suppose that

is a bounded nonoscillatory solution of (3.91) and

According to Lemma 3.42, either

or

is satisfied. We claim that

does not hold. Otherwise,

for some

Since

it follows that

is monotonically increasing for

For any

,

By induction, it can be seen that

Integrating

from

to

we obtain

By induction, for any natural number

we have

Considering the condition

in Lemma 3.42 and the sequence of numbers

has a positive lower bound, we conclude that the inequality above leads to a contradiction that the right side tends to

while

is bounded. Therefore, case

holds.

implies that

is strictly monotonically decreasing. From the facts that the series

converges and

is bounded, it follows that

converges and there exists limit

where

Now, we claim that

Otherwise,

and there exists

such that

for

From (3.91) and the last inequality, we can deduce

Integrating by parts of the above inequality and considering

and

, we have the following inequality:

Since
and the series
converges, the above inequality contradicts the fact that
is bounded, hence
, and the proof is complete.

The proof of the following theorem is similar.

Theorem 3.44 (see [22]).

Assume that conditions of Lemma 3.42 hold and for any
, and
Moreover, assume that the sequence of numbers
is bounded above,
converges, and
holds. Then every solution of (3.91) either oscillates or tends asymptotically to zero with fixed sign.

Some results similar to the above theorems have been obtained for fourth-order linear impulsive differential equations; see [24].