# A Survey on Oscillation of Impulsive Ordinary Differential Equations

- RaviP Agarwal
^{1, 2}Email author, - Fatma Karakoç
^{3}and - Ağacık Zafer
^{4}

**2010**:354841

https://doi.org/10.1155/2010/354841

© Ravi P. Agarwal et al. 2010

**Received: **1 December 2009

**Accepted: **3 March 2010

**Published: **14 April 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

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).

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 .

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 .

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).

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]).

Theorem 2.4 (see [14]).

is oscillatory.

Theorem 2.5 (see [15]).

has nonoscillatory solutions, where is an integer.

Theorem 2.6 (see [15]).

## 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

The following theorems are proved in [1].

Theorem 3.1.

- (1)
- (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.

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.

that the sequence has infinitely many negative terms.

The following theorem can be proved similarly.

Theorem 3.2.

- (1)
- (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.

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.

Then (3.6) has no positive solution defined on

Proof.

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).

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

Corollary 3.5.

- (1)
- (2)

Corollary 3.6 (Oscillation Theorem).

- (1)

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.

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

where and are real sequences, with and for all

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.

the following Picone's formula is easily obtained.

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

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

Theorem 3.12 (Wirtinger type inequality [17]).

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]).

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

Proof.

which contradicts (3.28).

Corollary 3.16 (Sturm-Picone type comparison).

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.

Then, we have the following result.

Theorem 3.19 (Device of Picard [17]).

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]).

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]).

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].

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.

This shows that is the solution of (3.45).

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.

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.

Theorem 3.29 (see [19]).

Proof.

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]).

is nonoscillatory by Theorem 2.3.

where provided and provided Let

Lemma 3.32.

Proof.

The next lemma can also be proved by induction.

Lemma 3.33.

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

Theorem 3.34.

- (i)
- (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.

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

Definition 3.35.

*expectation of*and is denoted by that is,

Definition 3.36.

*sample path solution to*(3.76) with the initial condition if for any sample value of then satisfies

Definition 3.37.

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

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.*

Lemma 3.39.

Proof.

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

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.

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].

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.

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

where and are real sequences, and with and

The lemma below can be found in [26].

Lemma 4.1.

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.

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).

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).

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

where denotes the impulse moments sequence with

Assume that the following conditions hold.

() is a constant, is a continuous function, .

() and are two intervals such that with for and for for

Theorem 4.8 (see [27]).

for Then every solution of (4.14) has at least one zero in

Proof.

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]).

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.

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.

where in explicit form and The functions and are both even and positive definite.

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]).

holds for every Then every solution of (4.32) is nonoscillatory.

Theorem 4.14 (see [29]).

holds for every Then every solution of (4.32) is oscillatory.

Proof.

Since and the right side of the above inequality tends to infinity as we have a contradiction.

In the special case we have the following necessary and sufficient condition.

Theorem 4.15 (see [29]).

Remark 4.16.

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

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.

then (4.44) is oscillatory.

Proof.

where dependence is suppressed for clarity.

Clearly, if , then we have . Thus, with our convention that , (4.51) holds for .

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

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.

where is a positive real number.

Note that if there is no impulse then the above integrals are negative, and therefore no conclusion can be drawn.

Corollary 4.19.

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

then (4.60) is oscillatory.

Corollary 4.20.

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

then (4.62) is oscillatory.

Corollary 4.21.

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

then (4.64) is oscillatory.

Example 4.22.

where is a positive real number.

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 .

Theorem 4.23.

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

Example 4.24.

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

- (i)
- (ii)

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.

then every solution of (4.73) is oscillatory.

Proof.

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.

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.

then every solution of (4.73) is oscillatory.

Corollary 4.29.

If then every solution of (4.73) is oscillatory.

Example 4.30.

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.

then every solution of (4.73) is oscillatory.

Proof.

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.

Then, every solution of (4.73) is oscillatory.

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.

- (i)
- (ii)
- (iii)
- (iv)

In order to prove the results the following well-known inequality is needed [26].

Lemma 4.33.

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]).

where , then (4.96) is oscillatory.

Proof.

which contradicts (4.100). This completes the proof.

As a corollary to Theorem 4.34, we have the following result.

Corollary 4.35.

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]).

### 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].

Let the following conditions hold.

for for , where is positive and continuous on for

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]).

then every bounded solution of (4.116) is oscillatory.

Theorem 4.38 (see [37]).

then every solution of (4.116) is oscillatory.

Theorem 4.39 (see [37]).

then every solution of (4.116) is oscillatory.

Corollary 4.40.

then every solution of (4.116) is oscillatory.

Corollary 4.41.

then every solution of (4.116) is oscillatory.

Example 4.42.

It is easy to verify conditions of Theorem 4.38. So every solution of (4.124) is oscillatory.

Example 4.43.

In this case, it can be show that conditions of Corollary 4.41 are satisfied. Thus, every solution of (4.125) is oscillatory.

- (i)
- (ii)

Theorem 4.44 (see [40]).

where . Then every solution of (4.126) is oscillatory.

Theorem 4.45 (see [40]).

Then every solution of (4.126) is oscillatory.

Example 4.46.

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

## Declarations

### 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.

## Authors’ Affiliations

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