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A Survey on Oscillation of Impulsive Ordinary Differential Equations
Advances in Difference Equations volume 2010, Article number: 354841 (2010)
Abstract
This paper summarizes a series of results on the oscillation of impulsive ordinary differential equations. We consider linear, half-linear, super-half-linear, and nonlinear equations. Several oscillation criteria are given. The Sturmian comparison theory for linear and half linear equations is also included.
1. Introduction
Impulsive differential equations, that is, differential equations involving impulse effect, appear as a natural description of observed evolution phenomena of several real world problems. There are many good monographs on the impulsive differential equations [1–6]. It is known that many biological phenomena, involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics, and frequency modulates systems, do exhibit impulse effects. Let us describe the Kruger-Thiemer model [7] for drug distribution to show how impulses occur naturally. It is assumed that the drug, which is administered orally, is first dissolved into the gastrointestinal tract. The drug is then absorbed into the so-called apparent volume of distribution and finally eliminated from the system by the kidneys. Let and
denote the amounts of drug at time
in the gastrointestinal tract and apparent volume of distribution, respectively, and let
and
be the relevant rate constants. For simplicity, assume that
The dynamic description of this model is then given by

In [8], the authors postulate the following control problem. At discrete instants of time , the drug is ingested in amounts
This imposes the following boundary conditions:

To achieve the desired therapeutic effect, it is required that the amount of drug in the apparent volume of distribution never goes below a constant level or plateau say, during the time interval
where
. Thus, we have the constraint

It is also assumed that only nonnegative amounts of the drug can be given. Then, a control vector is a point in the nonnegative orthant of Euclidean space of dimension
. Hence,
. Finally, the biological cost function
minimizes both the side effects and the cost of the drug. The problem is to find
subject to (1.1)–(1.3).
The first investigation on the oscillation theory of impulsive differential equations was published in 1989 [9]. In that paper Gopalsamy and Zhang consider impulsive delay differential equations of the form


where as
and
is a positive real number. Sufficient conditions are obtained for the asymptotic stability of the zero solution of (1.4) and existence of oscillatory solutions of (1.5). However, it seems that the problem of oscillation of ordinary differential equations with impulses has received attention much later [10]. Although, the theory of impulsive differential equations has been well established, the oscillation theory of such equations has developed rather slowly. To the best of our knowledge, except one paper [11], all of the investigations have been on differential equations subject to fixed moments of impulse effect. In [11], second-order differential equations with random impulses were dealt with, and there are no papers on the oscillation of differential equations with impulses at variable times.
In this survey paper, our aim is to present the results (within our reach) obtained so far on the oscillation theory of impulsive ordinary differential equations. The paper is organized as follows. Section 2 includes notations, definitions, and some well-known oscillation theorems needed in later sections. In Section 3, we are concerned with linear impulsive differential equations. In Section 4, we deal with nonlinear impulsive differential equations.
2. Preliminaries
In this section, we introduce notations, definitions, and some well-known results which will be used in this survey paper.
Let for some fixed
and
be a sequence in
such that
and
By we denote the set of all functions
which are continuous for
and continuous from the left with discontinuities of the first kind at
Similarly,
is the set of functions
having derivative
. One has
,
, or
. In case
, we simply write
for
. As usual,
denotes the set of continuous functions from
to
.
Consider the system of first-order impulsive ordinary differential equations having impulses at fixed moments of the form

where ,
, and

with . The notation
in place of
is also used. For simplicity, it is usually assumed that
.
The qualitative theory of impulsive ordinary differential equations of the form (2.1) can be found in [1–6, 12].
Definition 2.1.
A function is said to be a solution of (2.1) in an interval
if
satisfies (2.1) for
.
For , we may impose the initial condition

Each solution of (2.1) which is defined in the interval
and satisfying the condition
is said to be a solution of the initial value problem (2.1)-(2.3).
Note that if then the solution of the initial value problem (2.1)-(2.3) coincides with the solution of

on .
Definition 2.2.
A real-valued function , not necessarily a solution, is said to be oscillatory, if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory. A differential equation is called oscillatory if all its solutions are oscillatory.
For our purpose we now state some well-known results on oscillation of second-order ordinary differential equations without impulses.
Theorem 2.3 (see [13]).
Let Then, the equation

is oscillatory if

and nonoscillatory if

Theorem 2.4 (see [14]).
Let and
be continuous functions and
. If

then the equation

is oscillatory.
Theorem 2.5 (see [15]).
Let be a positive and continuously differentiable function for
and let
If

then the equation

has nonoscillatory solutions, where is an integer.
Theorem 2.6 (see [15]).
Let be a positive and continuous function for
and
an integer. Then every solution of (2.11) is oscillatory if and only if

3. Linear Equations
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.
3.1. Oscillation of First-Order Linear Equations
Let us consider the linear impulsive differential equation

together with the corresponding inequalities:


The following theorems are proved in [1].
Theorem 3.1.
Let and
Then the following assertions are equivalent.
-
(1)
The sequence
has infinitely many negative terms.
-
(2)
The inequality (3.2) has no eventually positive solution.
-
(3)
The inequality (3.3) has no eventually negative solution.
-
(4)
Each nonzero solution of (3.1) is oscillatory.
Proof.
(1)(2)
Let the sequence
have infinitely many negative terms. Let us suppose that the assertion (
) is not true; that is, the inequality (3.2) has an eventually positive solution
Let
be such that
Then, it follows from (3.2) that

which is a contradiction.
(2)(3). The validity of this relation follows from the fact that if
is a solution of the inequality (3.2), then
is a solution of the inequality (3.3) and vice versa.
(2) and (3) (4)
In fact, if (3.1) has neither an eventually positive nor an eventually negative solution, then each nonzero solution of (3.1) is oscillatory.
(4)(1)
If
is an oscillatory solution of (3.3), then it follows from the equality

that the sequence has infinitely many negative terms.
The following theorem can be proved similarly.
Theorem 3.2.
Let and
Then the following assertions are equivalent.
-
(1)
The sequence
has finitely many negative terms.
-
(2)
The inequality (3.2) has an eventually positive solution.
-
(3)
The inequality (3.3) has an eventually negative solution.
-
(4)
Each nonzero solution of (3.1) is nonoscillatory.
It is known that (3.1) without impulses has no oscillatory solutions. But (3.1) (with impulses) can have oscillatory solutions. So, impulse actions determine the oscillatory properties of first-order linear differential equations.
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.
Suppose the following.
-
(1)
Equation (3.7) has a solution
such that
(3.8)
-
(2)
The following inequalities are valid:
(3.9)
-
(3)
in a subinterval of
or
for some
Then (3.6) has no positive solution defined on
Proof.
Assume that (3.6) has a solution such that
Then from the relation

an integration yields

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
(3.13)
-
(2)
The following inequalities are valid:
(3.14)
-
(3)
in some subinterval of
or
for some
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)
Each solution
of (3.6) for which
has at least one zero in
-
(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
(3.15)
for
-
(2)
The following inequalities are valid for
and
;
(3.16)
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:

Suppose the following.
-
(1)
The function
is such that if
is a continuous function for
having a piecewise continuous derivative
for
then the function
is piecewise continuous for
-
(2)
The following inequalities are valid:
(3.18)
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

Employing the identity

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

where

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

Let

It follows that

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

Thus, we obtain

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

and so

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

and nonoscillatory 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

Next, we define

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.
Assume that

where Then, (3.43) is oscillatory.
Example 3.31 (see [19]).
Consider the equation

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

is investigated, where

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)
There is
such that
for all
-
(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 exponential 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

It can be seen that

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

So,

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.
One has

One has

Then for sufficiently large either
or
holds, where
(A) one has

(B) one has

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

in particular,

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].
4. Nonlinear Equations
In this section we present several oscillation theorems known for super-liner, half-linear, super-half-linear, and fully nonlinear impulsive differential equations of second and higher-orders. We begin with Sturmian and Leighton type comparison theorems for half-linear equations.
4.1. Sturmian Theory for Half-Linear Equations
Consider the second-order half linear impulsive differential equations of the form:


where and
are real sequences, and
with
and
The lemma below can be found in [26].
Lemma 4.1.
Let and
be a constant; then

where equality holds if and only if .
The results of this section are from [16].
Theorem 4.2 (Sturm-Picone type comparison).
Let be a solution of (4.1) having two consecutive zeros
and
in
. Suppose that
and
are satisfied for all
, and that
for all
for which
. If either
or
or
, then any solution
of (4.2) must have at least one zero in
.
Proof.
Assume that never vanishes on
. Define

where the dependence on of the solutions
and
is suppressed. It is not difficult to see that


Clearly, the last term of (4.5) is integrable over if
and
. Moreover,
in this case. Suppose that
. The case
is similar. Since
and

we get

and so

Moreover,

Integrating (4.5) from to
and using (4.6), we see that

where we have used Lemma 4.1 with ,
, and
. It is clear that (4.11) is not possible under our assumptions, and hence
must have a zero in
.
Corollary 4.3 (Separation Theorem).
The zeros of two linearly independent solutions and
of (4.1) separate each other.
Corollary 4.4 (Comparison Theorem).
Suppose that and
are satisfied for all
for some
, and that
for all
for which
. If either
or
or
, then every solution
of (4.2) is oscillatory whenever a solution
of (4.1) is oscillatory.
Corollary 4.5 (Dichotomy Theorem).
The solutions of (4.1) are either all oscillatory or all nonoscillatory.
Theorem 4.6 (Leighton-type Comparison).
Let be a solution of (4.1) having two consecutive zeros
and
in
. Suppose that

Then any nontrivial solution of (4.2) must have at least one zero in
.
Proof.
Assume that has no zero in
. Define the function
as in (4.4).
Clearly, (4.5) and (4.6) hold. It follows that

which is a contradiction. Therefore, must have a zero on
.
If , then we may conclude that either
has a zero in
or
is a constant multiple of
.
As a consequence of Theorems 4.2 and 4.6, we have the following oscillation result.
Corollary 4.7.
Suppose for a given there exists an interval
for which either conditions of Theorem 4.2 or Theorem 4.6 are satisfied, then every solution
of (4.2) is oscillatory.
4.2. Oscillation of Second-Order Superlinear and Super-Half-Linear Equations
Let us consider the forced superlinear second-order differential equation of the following form:

where denotes the impulse moments sequence with
Assume that the following conditions hold.
() is a constant,
is a continuous function,
.
() are constants,
() and
are two intervals such that
with
for
and
for
for
Interval oscillation criteria for (4.14) are given in [27]. Denote and for
let

Theorem 4.8 (see [27]).
Assume that conditions (A1)–(A3) hold, and that there exists such that

where for
and

for Then every solution of (4.14) has at least one zero in
Proof.
Let be a solution of (4.14). Suppose that
does not have any zero in
Without loss of generality, we may assume that
for
Define

Then, by Hölder's inequality, for and
we have

For we obtain

If then all impulsive moments are in
Multiplying both sides of (4.19) by
and integrating on
and using the hypotheses, we get

On the other hand, for it follows that

which implies that is nonincreasing on
So, for any
one has

It follows from the above inequality that

Making a similar analysis on we obtain

From (4.21)–(4.25) and we get

which contradicts (4.16). If then
and there are no impulse moments in
Similarly to the proof of (4.21), we get

which again contradicts (4.16).
In the case one can repeat the above procedure on the subinterval
in place of
This completes the proof.
Corollary 4.9.
Assume that conditions (A1) and (A2) hold. If for any there exist
satisfying (A3) with
and
satisfying (4.16),
then (4.14) is oscillatory.
The proof of following theorem is similar to that of Theorem 4.8.
Theorem 4.10 (see [27]).
Assume that conditions (A1)–(A3) hold, and there exists a such that

where for
and

for Then every solution of (4.14) has at least one zero in
Corollary 4.11.
Assume that conditions (A1) and (A2) hold. If for any there exist
satisfying (A3) with
and
satisfying (4.28),
then (4.14) is oscillatory.
Example 4.12.
Consider the following superlinear impulsive differential equation:

It can be seen that if

then, conditions of Corollary 4.9 are satisfied; here is the gamma function, and
and
satisfy (
2). So, every solution of (4.30) is oscillatory.
In [28–30], the authors have used an energy function approach to obtain conditions for the existence of oscillatory or nonoscillatory solutions of the half-linear impulsive differential equations of the following form:

where for
Define the energy functional

where in explicit form and
The functions
and
are both even and positive definite.
The function is constant along the solutions of the nonimpulsive equation

The change in the energy along the solutions of (4.32) is given by

We see that these impulsive perturbations increase the energy. If the energy increases slowly, then we expect the solutions to oscillate. On the other hand, if the energy increases too fast, the solutions become nonoscillatory. Let be a solution of (4.32),
and
Calculating
in terms of
we obtain

To simplify the notation, we introduce the function

The function gives the jump in the quantity
Note that
for
and
is monotone increasing with respect to
and decreasing with respect to
Theorem 4.13 (see [29]).
Assume that there exist a constant and a sequence
with
such that

holds for every Then every solution of (4.32) is nonoscillatory.
Theorem 4.14 (see [29]).
Assume that there exist a constant and a sequence
with
and
such that for every

holds for every Then every solution of (4.32) is oscillatory.
Proof.
Let be a nontrivial solution of (4.32). It suffices to show that
cannot hold on any interval
Assume that to the contrary,
for
Let
be defined by
where
It follows from (4.39) that
Hence,

Since and the right side of the above inequality tends to infinity as
we have a contradiction.
Now, assume that for every
It can be shown that the integral

takes its maximum in at

In the special case we have the following necessary and sufficient condition.
Theorem 4.15 (see [29]).
Assume that Then every solution of (4.32) is nonoscillatory if and only if

Remark 4.16.
Equation (4.32) with was studied in [30].
Finally, we consider the second-order impulsive differential equation of the following form:

where ,
are real constants,
is a strictly increasing unbounded sequence of real numbers,
and
are real sequences,
, and
All results given in the remainder of this section are from [31].
Theorem 4.17.
Suppose that for any given , there exist intervals
,
, such that
-
(a)
for all
and
for all
for which
;
-
(b)
,
,
,
;
,
,
,
for all
If there exists such that

where


then (4.44) is oscillatory.
Proof.
Suppose that there exists a nonoscillatory solution of (4.44) so that
for all
for some
. Let

It follows that for ,

where dependence is suppressed for clarity.
Define a function by

It is not difficult to see that if , then

Clearly, if , then we have
. Thus, with our convention that
, (4.51) holds for
.
Suppose that for all
. Choose
and consider the interval
. From (b), we see that
on
and
for all
for which
. Applying (4.51) to the terms in the parenthesis in (4.49) we obtain

where and
are defined by (4.46) and (4.47), respectively.
Let . Multiplying (4.52) by
and integrating over
give

In view of (4.52) and the assumption , employing the integration by parts formula in the last integral we have

We use Lemma 4.1 with

to obtain

which obviously contradicts (4.45).
If is eventually negative then we can consider
and reach a similar contradiction. This completes the proof.
Example 4.18.
Consider

where is a positive real number.
Let ,
, and
. For any given
we may choose
sufficiently large so that
. Then conditions (a)-(b) are satisfied. It is also easy to see that, for
and
,

where and
. It follows from Theorem 4.17 that (4.57) is oscillatory if

Note that if there is no impulse then the above integrals are negative, and therefore no conclusion can be drawn.
When , then (4.44) reduces to forced half-linear impulsive equation with damping

Corollary 4.19.
Suppose that for any given , there exist intervals
,
for which (a)-(b) hold.
If there exists such that

then (4.60) is oscillatory.
Taking in (4.44), we have the forced superlinear impulsive equation with damping

Corollary 4.20.
Let . Suppose that for any given
, there exist intervals
,
, such that (a)-(b) hold for all
.
If there exists such that

then (4.62) is oscillatory.
Let in (4.62). Then we have the forced linear equation:

Corollary 4.21.
Suppose that for any given , there exist intervals
,
, such that (a)-(b) hold for all
.
If there exists such that

then (4.64) is oscillatory.
Example 4.22.
Consider

where is a positive real number.
Let . For any
, choose
sufficiently large so that
and set
and
. Clearly, (a)-(b) are satisfied for all
. It is easy to see that for
and
,

Thus (4.65) holds if

which by Corollary 4.21 is sufficient for oscillation of (4.66).
Note that if the impulses are removed, then (4.66) becomes nonoscillatory with a nonoscillatory solution .
Finally we state a generalization of Theorem 4.17 for a class of more general type impulsive equations. Let ,
,
,
,
,
, and
be as above, and consider

where the functions and
satisfy

Theorem 4.23.
In addition to conditions of Theorem 4.17, if (4.70) holds then (4.69) is oscillatory.
Example 4.24.
Consider

where

Clearly if we take and
, then (4.70) holds with
,
and
. Further, we see that all conditions of Theorem 4.17 are satisfied if
and
; see Example 4.18. Therefore we may deduce from Theorem 4.23 that (4.71) is oscillatory if
.
4.3. Oscillation of Second-Order Nonlinear Equations
In this section, we first consider the second-order nonlinear impulsive differential equations of the following form:

Assume that the following conditions hold.
-
(i)
and
where
,
and
-
(ii)
, and there exist positive numbers
such that
(4.74)
In most of the investigations about oscillation of nonlinear impulsive differential equations, the following lemma is an important tool.
Lemma 4.25 (see [25]).
Let be a solution of (4.73). Suppose that there exists some
such that
for
If conditions (i) and (ii) are satisfied, and
(iii)
holds, then
and
for
where
Theorems 4.26–4.32 are obtained in [25]. For some improvements and/or generalizations, see [32–35].
Theorem 4.26.
Assume that conditions and
of Lemma 4.25 hold, and there exists a positive integer
such that
for
If

then every solution of (4.73) is oscillatory.
Proof.
Without loss of generality, we can assume If (4.73) has a nonoscillatory solution
we might as well assume
From Lemma 4.25,
for
where
Let

Then, Using condition (i) in (4.73), we get for

Using condition (ii) and yield

From the above inequalities, we have

where Taking
and
we get

By induction, for any natural number we obtain

Since, above inequality and the hypothesis lead to a contradiction. So, every solution of (4.73) oscillatory.
From Theorem 4.26, the following corollary is immediate.
Corollary 4.27.
Assume that conditions of Lemma 4.25 hold and there exists a positive integer
such that
for
If

then every solution of (4.73) is oscillatory.
The proof of the following theorem is similar to that of Theorem 4.26.
Theorem 4.28.
Assume that conditions and
of Lemma 4.25 hold and
for any
If

then every solution of (4.73) is oscillatory.
Corollary 4.29.
Assume that conditions and
of Lemma 4.25 hold and
for any
Suppose that there exist a positive integer
and a constant
such that

If then every solution of (4.73) is oscillatory.
Example 4.30.
Consider the superlinear equation:

where is a natural number. Since
,
,
and
It is easy to see that conditions (i), (ii), and (iii) are satisfied. Moreover

Hence, by Corollary 4.29, we find that every solution of (4.85) is oscillatory. On the other hand, by Theorem 2.6, it follows that (4.85) without impulses is nonoscillatory.
Theorem 4.31.
Assume that conditions (i), (ii), and (iii) of Lemma 4.25 hold, and there exists a positive integer such that
for
If

then every solution of (4.73) is oscillatory.
Proof.
Without loss of generality, let If (4.1) has a nonoscillatory solution
assume
By Lemma 4.25,
Since
we have

It is easy to see that is monotonically nondecreasing in
Now (4.73) yields

Hence, from the above inequality and condition (ii), we find that

Generally, for any natural number we have

By (4.89) and (4.91), noting and
for any natural number
we obtain

Note that and
is nondecreasing. Dividing the above inequality by
and then integrating from
to
we get

Since (4.88) holds, the above inequality yields

The above inequality and the hypotheses lead to a contradiction. So, every solution of (4.73) is oscillatory.
The proof of the following theorem is similar to that of Theorem 4.31.
Theorem 4.32.
Assume that conditions of Lemma 4.25 hold, and there exists a positive integer
such that
for
Suppose that
for any
and

Then, every solution of (4.73) is oscillatory.
In [36], the author studied the second-order nonlinear impulsive differential equations of the following form:

where and
,
.
Sufficient conditions are obtained for oscillation of (4.96) by using integral averaging technique. In particular the Philos type oscillation criteria are extended to impulsive differential equations.
It is assumed that
-
(i)
is a constant;
-
(ii)
is a strictly increasing unbounded sequence of real numbers;
is a real sequence;
-
(iii)
,
;
-
(iv)
,
with
,
for
and
(4.97)
is satisfied; is a constant.
In order to prove the results the following well-known inequality is needed [26].
Lemma 4.33.
If ,
are nonnegative numbers, then

and the equality holds if and only if .
The following theorem is one of the main results of this study.
Theorem 4.34 (see [36]).
Let and
. Assume that
,
satisfy the following conditions.
-
(i)
for
and
on
.
-
(ii)
has a continuous and nonpositive partial derivative on
with respect to the second variable.
-
(iii)
One has
(4.99)
If

where , then (4.96) is oscillatory.
Proof.
Let be a nonoscillatory solution of (4.96). We assume that
on
for some sufficiently large
. Define

Differentiating (4.101) and making use of (4.96) and (4.97), we obtain


Replacing by
in (4.102) and multiplying the resulting equation by
and integrating from
to
, we get

Integrating by parts and using (4.103), we find

Combining (4.104) and (4.105), we obtain

Using inequality (4.98) with

we find

From (4.106) and (4.108), we obtain

for all . In the above inequality we choose
, to get

Thus, it follows that

which contradicts (4.100). This completes the proof.
As a corollary to Theorem 4.34, we have the following result.
Corollary 4.35.
Let condition (4.100) in Theorem 4.34 be replaced by

then (4.96) is oscillatory.
Note that in the special case of half-linear equations, for and
, the condition (4.97) is satisfied with
.
The proof of the following theorem can be accomplished by using the method developed for the nonimpulsive case and similar arguments employed in the proof of Theorem 4.34.
Theorem 4.36 (see [36]).
Let the functions and
be defined as in Theorem 4.34. Moreover, Suppose that

If there exists a function such that

and for every

where , then (4.96) is oscillatory.
4.4. Higher-Order Nonlinear Equations
There are only a very few works concerning the oscillation of higher-order nonlinear impulsive differential equations [37–40].
In [37] authors considered even order impulsive differential equations of the following form

where

Let the following conditions hold.
for
for
, where
is positive and continuous on
for
, and there exist positive numbers
such that

A function is said to be a solution of (4.116), if (i)
; (ii) for
and
satisfies
; (iii)
is left continuous on
and
The first two theorems can be considered as modifications of Theorems 3.43 and 4.26, respectively.
Theorem 4.37 (see [37]).
If conditions and
hold,
,
, and if

then every bounded solution of (4.116) is oscillatory.
Theorem 4.38 (see [37]).
If conditions and
hold,
and

then every solution of (4.116) is oscillatory.
Theorem 4.39 (see [37]).
If conditions (A) and (B) hold,
,
, and for any
,

then every solution of (4.116) is oscillatory.
Corollary 4.40.
Assume that conditions (A) and (B) hold, and that ,
. If

then every solution of (4.116) is oscillatory.
Corollary 4.41.
Assume that conditions (A) and (B) hold, and that there exists a positive number , such that
,
. If

then every solution of (4.116) is oscillatory.
Example 4.42.
Consider the impulsive differential equation:

where ,
,
,
,
,
,
,
.
It is easy to verify conditions of Theorem 4.38. So every solution of (4.124) is oscillatory.
Example 4.43.
Consider the impulsive differential equation:

where ,
,
,
,
,
,
,
.
In this case, it can be show that conditions of Corollary 4.41 are satisfied. Thus, every solution of (4.125) is oscillatory.
In [40], the authors considered the impulsive differential equations with piecewise constant argument of the following form:

where is the set of all positive integers,
,
and
are given positive constants,
denotes the set of maximum integers, and
for all
. It is assumed that
-
(i)
;
-
(ii)
for any
and all
,
(4.127)
-
(iii)
there exists
such that
(4.128)
Theorem 4.44 (see [40]).
Assume that conditions ,
, and
hold. Moreover, suppose that for any
, there exists a
such that

where . Then every solution of (4.126) is oscillatory.
Theorem 4.45 (see [40]).
Assume that conditions and
hold. Moreover, suppose that for any
, there exists a
such that

Then every solution of (4.126) is oscillatory.
Example 4.46.
Consider the impulsive differential equation:

It is easy to verify that conditions of Theorem 4.45 are all satisfied. Therefore every solution of (4.131) is oscillatory.
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Acknowledgments
This work was done when the second author was on academic leave, visiting Florida Institute of Technology. The financial support of The Scientific and Technological Research Council of Turkey (TUBITAK) is gratefully acknowledged.
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Agarwal, R., Karakoç, F. & Zafer, A. A Survey on Oscillation of Impulsive Ordinary Differential Equations. Adv Differ Equ 2010, 354841 (2010). https://doi.org/10.1155/2010/354841
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DOI: https://doi.org/10.1155/2010/354841