A Survey on Oscillation of Impulsive Ordinary Differential Equations

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

(1.1)

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:

(1.2)

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

(1.3)

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

(1.4)

(1.5)

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

(2.1)

where , , and

(2.2)

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

(2.3)

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

(2.4)

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

(2.5)

is oscillatory if

(2.6)

and nonoscillatory if

(2.7)

Theorem 2.4 (see [14]).

Let and be continuous functions and . If

(2.8)

then the equation

(2.9)

is oscillatory.

Theorem 2.5 (see [15]).

Let be a positive and continuously differentiable function for and let If

(2.10)

then the equation

(2.11)

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

(2.12)

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

(3.1)

together with the corresponding inequalities:

(3.2)

(3.3)

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

(3.4)

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

(3.5)

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

(3.6)

(3.7)

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

  

  (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

(3.10)

an integration yields

(3.11)

From (3.6), (3.7), condition (), and the above inequality, we conclude that

(3.12)

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:

(3.17)

Theorem 3.10 (see [1, 10]).

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

(3.19)

(3.20)

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:

(H)

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

(3.21)

For clarity, we suppress the variable . Clearly,

(3.22)

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

(3.23)

Employing the identity

(3.24)

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

(3.25)

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

(3.26)

where

(3.27)

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

(3.28)

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

(3.29)

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

(3.30)

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

(3.31)

(3.32)

for all , and

(3.33)

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

(3.34)

Let

(3.35)

It follows that

(3.36)

Assuming that , the choice of yields

(3.37)

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

(3.38)

(3.39)

are satisfied for all , and that

(3.40)

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

(3.41)

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

(3.42)

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

(3.43)

where , and are two known sequences of real numbers, and

(3.44)

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:

(3.45)

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

(3.46)

Thus, is continuous on Furthermore, for we have

(3.47)

For it can be shown that

(3.48)

Thus, is continuous if we define the value of at as

(3.49)

Now, we have for

(3.50)

and for

(3.51)

Thus, we obtain

(3.52)

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

(3.53)

and so

(3.54)

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

(3.55)

and nonoscillatory if

(3.56)

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

(3.57)

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:

(3.58)

where

Theorem 3.29 (see [19]).

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

(3.59)

is oscillatory, where

Proof.

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

(3.60)

Then, it can be shown that is continuous and satisfies

(3.61)

Next, we define

(3.62)

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

(3.63)

where Then, (3.43) is oscillatory.

Example 3.31 (see [19]).

Consider the equation

(3.64)

If for some integer then it is easy to see that

(3.65)

where denotes the greatest integer function, and

(3.66)

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

(3.67)

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:

(3.68)

where

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

(3.69)

is investigated, where

(3.70)

for all and is the -function, that is,

(3.71)

for all being continuous at Before giving the main result, we need the following lemmas. For each define the sequence inductively by

(3.72)

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

(3.73)

Hence,

The next lemma can also be proved by induction.

Lemma 3.33.

Suppose that and for all Define, by induction,

(3.74)

If for all then

(3.75)

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:

(3.76)

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,

(3.77)

In particular, if is a continuous random variable having probability density function then

(3.78)

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

(3.79)

Definition 3.37.

The exponential distribution is a continuous random variable with the probability density function:

(3.80)

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:

(3.81)

Lemma 3.39.

The function is a solution of (3.76) if and only if

(3.82)

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

(3.83)

Proof.

If is a solution of system (3.81), for any we have

(3.84)

It can be seen that

(3.85)

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

(3.86)

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

(3.87)

where is a solution of (3.81). Hence,

(3.88)

Further, it can be seen that

(3.89)

So,

(3.90)

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

(3.91)

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

(3.92)

One has

(3.93)

Then for sufficiently large either or holds, where

(A) one has

(3.94)

(B) one has

(3.95)

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

(3.96)

in particular,

(3.97)

Integrating from to we obtain

(3.98)

By induction, for any natural number we have

(3.99)

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

(3.100)

Integrating by parts of the above inequality and considering and , we have the following inequality:

(3.101)

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:

(4.1)

(4.2)

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

(4.3)

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

(4.4)

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

(4.5)

(4.6)

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

(4.7)

we get

(4.8)

and so

(4.9)

Moreover,

(4.10)

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

(4.11)

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

(4.12)

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

(4.13)

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:

(4.14)

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

(4.15)

Theorem 4.8 (see [27]).

Assume that conditions (A1)–(A3) hold, and that there exists such that

(4.16)

where for and

(4.17)

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

(4.18)

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

(4.19)

For we obtain

(4.20)

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

(4.21)

On the other hand, for it follows that

(4.22)

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

(4.23)

It follows from the above inequality that

(4.24)

Making a similar analysis on we obtain

(4.25)

From (4.21)–(4.25) and we get

(4.26)

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

(4.27)

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

(4.28)

where for and

(4.29)

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:

(4.30)

It can be seen that if

(4.31)

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:

(4.32)

where for

Define the energy functional

(4.33)

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

(4.34)

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

(4.35)

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

(4.36)

To simplify the notation, we introduce the function

(4.37)

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

(4.38)

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

(4.39)

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,

(4.40)

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

(4.41)

takes its maximum in at

(4.42)

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

(4.43)

Remark 4.16.

Equation (4.32) with was studied in [30].

Finally, we consider the second-order impulsive differential equation of the following form:

(4.44)

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

(4.45)

where

(4.46)

(4.47)

then (4.44) is oscillatory.

Proof.

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

(4.48)

It follows that for ,

(4.49)

where dependence is suppressed for clarity.

Define a function by

(4.50)

It is not difficult to see that if , then

(4.51)

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

(4.52)

where and are defined by (4.46) and (4.47), respectively.

Let . Multiplying (4.52) by and integrating over give

(4.53)

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

(4.54)

We use Lemma 4.1 with

(4.55)

to obtain

(4.56)

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

(4.57)

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 ,

(4.58)

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

(4.59)

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

(4.60)

Corollary 4.19.

Suppose that for any given , there exist intervals , for which (a)-(b) hold.

If there exists such that

(4.61)

then (4.60) is oscillatory.

Taking in (4.44), we have the forced superlinear impulsive equation with damping

(4.62)

Corollary 4.20.

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

If there exists such that

(4.63)

then (4.62) is oscillatory.

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

(4.64)

Corollary 4.21.

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

If there exists such that

(4.65)

then (4.64) is oscillatory.

Example 4.22.

Consider

(4.66)

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 ,

(4.67)

Thus (4.65) holds if

(4.68)

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

(4.69)

where the functions and satisfy

(4.70)

Theorem 4.23.

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

Example 4.24.

Consider

(4.71)

where

(4.72)

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:

(4.73)

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

(4.75)

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

(4.76)

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

(4.77)

Using condition (ii) and yield

(4.78)

From the above inequalities, we have

(4.79)

where Taking and we get

(4.80)

By induction, for any natural number we obtain

(4.81)

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

(4.82)

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

(4.83)

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

(4.84)

If then every solution of (4.73) is oscillatory.

Example 4.30.

Consider the superlinear equation:

(4.85)

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

(4.86)

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

(4.87)

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

(4.88)

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

(4.89)

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

(4.90)

Generally, for any natural number we have

(4.91)

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

(4.92)

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

(4.93)

Since (4.88) holds, the above inequality yields

(4.94)

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

(4.95)

Then, every solution of (4.73) is oscillatory.

In [36], the author studied the second-order nonlinear impulsive differential equations of the following form:

(4.96)

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

(4.98)

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

(4.100)

where , then (4.96) is oscillatory.

Proof.

Let be a nonoscillatory solution of (4.96). We assume that on for some sufficiently large . Define

(4.101)

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

(4.102)

(4.103)

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

(4.104)

Integrating by parts and using (4.103), we find

(4.105)

Combining (4.104) and (4.105), we obtain

(4.106)

Using inequality (4.98) with

(4.107)

we find

(4.108)

From (4.106) and (4.108), we obtain

(4.109)

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

(4.110)

Thus, it follows that

(4.111)

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

(4.112)

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

(4.113)

If there exists a function such that

(4.114)

and for every

(4.115)

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

(4.116)

where

(4.117)

Let the following conditions hold.

for for , where is positive and continuous on for

, and there exist positive numbers such that

(4.118)

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

(4.119)

then every bounded solution of (4.116) is oscillatory.

Theorem 4.38 (see [37]).

If conditions and hold, and

(4.120)

then every solution of (4.116) is oscillatory.

Theorem 4.39 (see [37]).

If conditions (A) and (B) hold,    , , and for any ,

(4.121)

then every solution of (4.116) is oscillatory.

Corollary 4.40.

Assume that conditions (A) and (B) hold, and that , . If

(4.122)

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

(4.123)

then every solution of (4.116) is oscillatory.

Example 4.42.

Consider the impulsive differential equation:

(4.124)

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:

(4.125)

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:

(4.126)

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

(4.129)

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

(4.130)

Then every solution of (4.126) is oscillatory.

Example 4.46.

Consider the impulsive differential equation:

(4.131)

It is easy to verify that conditions of Theorem 4.45 are all satisfied. Therefore every solution of (4.131) is oscillatory.