Open Access

A Survey on Oscillation of Impulsive Ordinary Differential Equations

Advances in Difference Equations20102010:354841

DOI: 10.1155/2010/354841

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

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 [16]. 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq1_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq2_HTML.gif denote the amounts of drug at time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq3_HTML.gif in the gastrointestinal tract and apparent volume of distribution, respectively, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq4_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq5_HTML.gif be the relevant rate constants. For simplicity, assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq6_HTML.gif The dynamic description of this model is then given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ1_HTML.gif
(1.1)
In [8], the authors postulate the following control problem. At discrete instants of time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq7_HTML.gif , the drug is ingested in amounts https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq8_HTML.gif This imposes the following boundary conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ2_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq9_HTML.gif say, during the time interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq10_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq11_HTML.gif . Thus, we have the constraint
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ3_HTML.gif
(1.3)

It is also assumed that only nonnegative amounts of the drug can be given. Then, a control vector is a point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq12_HTML.gif in the nonnegative orthant of Euclidean space of dimension https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq13_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq14_HTML.gif . Finally, the biological cost function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq15_HTML.gif minimizes both the side effects and the cost of the drug. The problem is to find https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq16_HTML.gif 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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ4_HTML.gif
(1.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ5_HTML.gif
(1.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq17_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq18_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq19_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq20_HTML.gif for some fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq22_HTML.gif be a sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq23_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq24_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq25_HTML.gif

By https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq26_HTML.gif we denote the set of all functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq27_HTML.gif which are continuous for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq28_HTML.gif and continuous from the left with discontinuities of the first kind at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq29_HTML.gif Similarly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq30_HTML.gif is the set of functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq31_HTML.gif having derivative https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq32_HTML.gif . One has   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq33_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq34_HTML.gif , or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq35_HTML.gif . In case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq36_HTML.gif , we simply write https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq37_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq38_HTML.gif . As usual, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq39_HTML.gif denotes the set of continuous functions from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq40_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq41_HTML.gif .

Consider the system of first-order impulsive ordinary differential equations having impulses at fixed moments of the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ6_HTML.gif
(2.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq42_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq43_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ7_HTML.gif
(2.2)

with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq44_HTML.gif . The notation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq45_HTML.gif in place of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq46_HTML.gif is also used. For simplicity, it is usually assumed that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq47_HTML.gif .

The qualitative theory of impulsive ordinary differential equations of the form (2.1) can be found in [16, 12].

Definition 2.1.

A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq48_HTML.gif is said to be a solution of (2.1) in an interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq49_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq50_HTML.gif satisfies (2.1) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq51_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq52_HTML.gif , we may impose the initial condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ8_HTML.gif
(2.3)

Each solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq53_HTML.gif of (2.1) which is defined in the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq54_HTML.gif and satisfying the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq55_HTML.gif is said to be a solution of the initial value problem (2.1)-(2.3).

Note that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq56_HTML.gif then the solution of the initial value problem (2.1)-(2.3) coincides with the solution of
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ9_HTML.gif
(2.4)

on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq57_HTML.gif .

Definition 2.2.

A real-valued function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq58_HTML.gif , 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq59_HTML.gif Then, the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ10_HTML.gif
(2.5)
is oscillatory if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ11_HTML.gif
(2.6)
and nonoscillatory if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ12_HTML.gif
(2.7)

Theorem 2.4 (see [14]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq61_HTML.gif be continuous functions and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq62_HTML.gif . If
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ13_HTML.gif
(2.8)
then the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ14_HTML.gif
(2.9)

is oscillatory.

Theorem 2.5 (see [15]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq63_HTML.gif be a positive and continuously differentiable function for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq64_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq65_HTML.gif If
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ15_HTML.gif
(2.10)
then the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ16_HTML.gif
(2.11)

has nonoscillatory solutions, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq66_HTML.gif is an integer.

Theorem 2.6 (see [15]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq67_HTML.gif be a positive and continuous function for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq68_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq69_HTML.gif an integer. Then every solution of (2.11) is oscillatory if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ17_HTML.gif
(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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ18_HTML.gif
(3.1)
together with the corresponding inequalities:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ19_HTML.gif
(3.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ20_HTML.gif
(3.3)

The following theorems are proved in [1].

Theorem 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq70_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq71_HTML.gif Then the following assertions are equivalent.
  1. (1)

    The sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq72_HTML.gif has infinitely many negative terms.

     
  2. (2)

    The inequality (3.2) has no eventually positive solution.

     
  3. (3)

    The inequality (3.3) has no eventually negative solution.

     
  4. (4)

    Each nonzero solution of (3.1) is oscillatory.

     

Proof.

(1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq73_HTML.gif (2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq74_HTML.gif Let the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq75_HTML.gif have infinitely many negative terms. Let us suppose that the assertion ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq76_HTML.gif ) is not true; that is, the inequality (3.2) has an eventually positive solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq77_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq78_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq79_HTML.gif Then, it follows from (3.2) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ21_HTML.gif
(3.4)

which is a contradiction.

(2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq80_HTML.gif (3). The validity of this relation follows from the fact that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq81_HTML.gif is a solution of the inequality (3.2), then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq82_HTML.gif is a solution of the inequality (3.3) and vice versa.

(2) and (3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq83_HTML.gif (4) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq84_HTML.gif 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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq85_HTML.gif (1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq86_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq87_HTML.gif is an oscillatory solution of (3.3), then it follows from the equality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ22_HTML.gif
(3.5)

that the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq88_HTML.gif has infinitely many negative terms.

The following theorem can be proved similarly.

Theorem 3.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq89_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq90_HTML.gif Then the following assertions are equivalent.
  1. (1)

    The sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq91_HTML.gif has finitely many negative terms.

     
  2. (2)

    The inequality (3.2) has an eventually positive solution.

     
  3. (3)

    The inequality (3.3) has an eventually negative solution.

     
  4. (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 [1618], 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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ23_HTML.gif
(3.6)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ24_HTML.gif
(3.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq92_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq93_HTML.gif are continuous for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq94_HTML.gif and they have a discontinuity of the first kind at the points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq95_HTML.gif 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. (1)
    Equation (3.7) has a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq96_HTML.gif such that
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ25_HTML.gif
    (3.8)
     
  1. (2)
    The following inequalities are valid:
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ26_HTML.gif
    (3.9)
     
  1. (3)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq97_HTML.gif in a subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq98_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq99_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq100_HTML.gif

     

Then (3.6) has no positive solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq101_HTML.gif defined on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq102_HTML.gif

Proof.

Assume that (3.6) has a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq103_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq104_HTML.gif Then from the relation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ27_HTML.gif
(3.10)
an integration yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ28_HTML.gif
(3.11)
From (3.6), (3.7), condition ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq105_HTML.gif ), and the above inequality, we conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ29_HTML.gif
(3.12)

But, from conditions ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq106_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq107_HTML.gif ), 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. (1)
    Equation (3.7) has a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq108_HTML.gif such that
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ30_HTML.gif
    (3.13)
     
  1. (2)
    The following inequalities are valid:
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ31_HTML.gif
    (3.14)
     
  1. (3)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq109_HTML.gif in some subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq110_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq111_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq112_HTML.gif

     

Then, each solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq113_HTML.gif of (3.6) has at least one zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq114_HTML.gif

Corollary 3.5.

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

    Each solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq115_HTML.gif of (3.6) for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq116_HTML.gif has at least one zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq117_HTML.gif

     
  2. (2)

    Each solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq118_HTML.gif of (3.6) has at least one zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq119_HTML.gif

     

Corollary 3.6 (Oscillation Theorem).

Suppose the following.
  1. (1)
    There exists a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq120_HTML.gif of (3.7) and a sequence of disjoint intervals https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq121_HTML.gif such that
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ32_HTML.gif
    (3.15)
     
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq122_HTML.gif
  1. (2)
    The following inequalities are valid for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq123_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq124_HTML.gif ;
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ33_HTML.gif
    (3.16)
     

Then all solutions of (3.6) are oscillatory, and moreover, they change sign in each interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq125_HTML.gif

Corollary 3.7 (Comparison Theorem).

Let the inequalities https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq126_HTML.gif hold for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq127_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq128_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq129_HTML.gif are two consecutive zeros of one of the solutions, then the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq130_HTML.gif 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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ34_HTML.gif
(3.17)

Theorem 3.10 (see [1, 10]).

Suppose the following.
  1. (1)

    The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq131_HTML.gif is such that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq132_HTML.gif is a continuous function for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq133_HTML.gif having a piecewise continuous derivative https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq134_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq135_HTML.gif then the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq136_HTML.gif is piecewise continuous for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq137_HTML.gif

     
  2. (2)
    The following inequalities are valid:
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ35_HTML.gif
    (3.18)
     

Then every solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq138_HTML.gif of (3.17) defined for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq139_HTML.gif 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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ36_HTML.gif
(3.19)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ37_HTML.gif
(3.20)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq140_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq141_HTML.gif are real sequences, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq142_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq143_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq144_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq145_HTML.gif

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq146_HTML.gif be a nondegenerate subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq147_HTML.gif . In what follows we shall make use of the following condition:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ38_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq148_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq149_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq150_HTML.gif . These conditions simply mean that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq151_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq152_HTML.gif are in the domain of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq153_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq154_HTML.gif , respectively. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq155_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq156_HTML.gif , then we may define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ39_HTML.gif
(3.21)
For clarity, we suppress the variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq157_HTML.gif . Clearly,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ40_HTML.gif
(3.22)
In view of (3.19) and (3.20) it is not difficult to see from (3.22) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ41_HTML.gif
(3.23)
Employing the identity
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ42_HTML.gif
(3.24)

the following Picone's formula is easily obtained.

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

Let (H) be satisfied. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq158_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq159_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq160_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq161_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq162_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq163_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ43_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq164_HTML.gif of (3.19) such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq165_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq166_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ44_HTML.gif
(3.26)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ45_HTML.gif
(3.27)

Corollary 3.13.

If there exists an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq167_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq168_HTML.gif then every solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq169_HTML.gif of (3.19) has a zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq170_HTML.gif .

Corollary 3.14.

Suppose that for a given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq171_HTML.gif there exists an interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq172_HTML.gif and a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq173_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq174_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq175_HTML.gif of (3.19). If (H) is satisfied with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq176_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ46_HTML.gif
(3.28)

then every solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq177_HTML.gif of (3.20) must have at least one zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq178_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq179_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq180_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq181_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq182_HTML.gif are solutions of (3.19) and (3.20), respectively, we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq183_HTML.gif . Employing Picone's formula (3.25), we see that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ47_HTML.gif
(3.29)
The functions under integral sign are all integrable, and regardless of the values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq184_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq185_HTML.gif , the left-hand side of (3.29) tends to zero as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq186_HTML.gif . Clearly, (3.29) results in
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ48_HTML.gif
(3.30)

which contradicts (3.28).

Corollary 3.16 (Sturm-Picone type comparison).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq187_HTML.gif be a solution of (3.19) having two consecutive zeros https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq188_HTML.gif . Suppose that (H) holds, and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ49_HTML.gif
(3.31)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ50_HTML.gif
(3.32)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq189_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ51_HTML.gif
(3.33)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq190_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq191_HTML.gif .

If either (3.31) or (3.32) is strict in a subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq192_HTML.gif or (3.33) is strict for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq193_HTML.gif , then every solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq194_HTML.gif of (3.20) must have at least one zero on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq195_HTML.gif .

Corollary 3.17.

Suppose that conditions (3.31)-(3.32) are satisfied for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq196_HTML.gif for some integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq197_HTML.gif , and that (3.33) is satisfied for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq198_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq199_HTML.gif . If one of the inequalities (3.31)–(3.33) is strict, then (3.20) is oscillatory whenever any solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq200_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq201_HTML.gif there exists an interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq202_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq203_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ52_HTML.gif
(3.34)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ53_HTML.gif
(3.35)
It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ54_HTML.gif
(3.36)
Assuming that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq204_HTML.gif , the choice of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq205_HTML.gif yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ55_HTML.gif
(3.37)

Then, we have the following result.

Theorem 3.19 (Device of Picard [17]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq206_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq207_HTML.gif be a solution of (3.19) having two consecutive zeros https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq208_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq209_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq210_HTML.gif . Suppose that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ56_HTML.gif
(3.38)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ57_HTML.gif
(3.39)
are satisfied for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq211_HTML.gif , and that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ58_HTML.gif
(3.40)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq212_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq213_HTML.gif .

If either (3.38) or (3.39) is strict in a subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq214_HTML.gif or (3.40) is strict for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq215_HTML.gif , then any solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq216_HTML.gif of (3.20) must have at least one zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq217_HTML.gif .

Corollary 3.20.

Suppose that (3.38)-(3.39) are satisfied for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq218_HTML.gif for some integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq219_HTML.gif , and that (3.40) is satisfied for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq220_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq221_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq222_HTML.gif and one of the inequalities (3.38)–(3.40) is strict, then (3.20) is oscillatory whenever any solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq223_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq224_HTML.gif . If there exists a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq225_HTML.gif of (3.19) such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ59_HTML.gif
(3.41)

then every solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq226_HTML.gif of (3.20) must have at least one zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq227_HTML.gif .

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

Corollary 3.22.

Suppose that for a given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq228_HTML.gif there exists an interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq229_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq230_HTML.gif of (3.20) such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq231_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq232_HTML.gif , then for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq233_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq234_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ60_HTML.gif
(3.42)

Corollary 3.24.

If there exists an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq235_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq236_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq237_HTML.gif then every solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq238_HTML.gif of (3.20) must have at least one zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq239_HTML.gif .

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

Corollary 3.25.

Suppose that for a given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq240_HTML.gif there exists an interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq241_HTML.gif and a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq242_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq243_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq244_HTML.gif . 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, 1921].

Let us consider the second-order linear differential equation with impulses
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ61_HTML.gif
(3.43)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq245_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq246_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq247_HTML.gif are two known sequences of real numbers, and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ62_HTML.gif
(3.44)
For (3.43), it is clear that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq248_HTML.gif for all large https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq249_HTML.gif then (3.43) is oscillatory. So, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq250_HTML.gif 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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ63_HTML.gif
(3.45)

Theorem 3.26 (see [19]).

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq251_HTML.gif Then the oscillation of all solutions of (3.43) is equivalent to the oscillation of all solutions of (3.45).

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq252_HTML.gif be any solution of (3.43). Set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq253_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq254_HTML.gif Then, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq255_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ64_HTML.gif
(3.46)
Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq256_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq257_HTML.gif Furthermore, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq258_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ65_HTML.gif
(3.47)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq259_HTML.gif it can be shown that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ66_HTML.gif
(3.48)
Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq260_HTML.gif is continuous if we define the value of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq261_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq262_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ67_HTML.gif
(3.49)
Now, we have for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq263_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ68_HTML.gif
(3.50)
and for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq264_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ69_HTML.gif
(3.51)
Thus, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ70_HTML.gif
(3.52)

This shows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq265_HTML.gif is the solution of (3.45).

Conversely, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq266_HTML.gif is the continuous solution of (3.45), we set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq267_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq268_HTML.gif Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq269_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq270_HTML.gif Furthermore, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq271_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ71_HTML.gif
(3.53)
and so
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ72_HTML.gif
(3.54)

Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq272_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq273_HTML.gif . Then, (3.43) is oscillatory if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ73_HTML.gif
(3.55)
and nonoscillatory if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ74_HTML.gif
(3.56)

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq274_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq275_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq276_HTML.gif on any interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq277_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq278_HTML.gif be an eventually positive solution of (3.43). If
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ75_HTML.gif
(3.57)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq279_HTML.gif then, eventually https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq280_HTML.gif

Now, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq281_HTML.gif be an eventually positive solution of (3.43) such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq282_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq283_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq284_HTML.gif Under conditions of Lemma 3.28, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq285_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq286_HTML.gif Then, (3.43) leads to an impulsive Riccati equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ76_HTML.gif
(3.58)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq287_HTML.gif

Theorem 3.29 (see [19]).

Equation (3.43) is oscillatory if the second-order self-adjoint differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ77_HTML.gif
(3.59)

is oscillatory, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq288_HTML.gif

Proof.

Assume, for the sake of contradiction, that (3.43) has a nonoscillatory solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq289_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq290_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq291_HTML.gif Now, define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ78_HTML.gif
(3.60)
Then, it can be shown that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq292_HTML.gif is continuous and satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ79_HTML.gif
(3.61)
Next, we define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ80_HTML.gif
(3.62)

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq293_HTML.gif 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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ81_HTML.gif
(3.63)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq294_HTML.gif Then, (3.43) is oscillatory.

Example 3.31 (see [19]).

Consider the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ82_HTML.gif
(3.64)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq295_HTML.gif for some integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq296_HTML.gif then it is easy to see that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ83_HTML.gif
(3.65)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq297_HTML.gif denotes the greatest integer function, and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ84_HTML.gif
(3.66)
Thus, by Corollary 3.30, (3.64) is oscillatory. We note that the corresponding differential equation without impulses
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ85_HTML.gif
(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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ86_HTML.gif
(3.68)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq298_HTML.gif

In [21], the oscillatory and nonoscillatory properties of the second-order linear impulsive differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ87_HTML.gif
(3.69)
is investigated, where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ88_HTML.gif
(3.70)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq299_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq300_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq301_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq302_HTML.gif -function, that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ89_HTML.gif
(3.71)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq303_HTML.gif being continuous at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq304_HTML.gif Before giving the main result, we need the following lemmas. For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq305_HTML.gif define the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq306_HTML.gif inductively by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ90_HTML.gif
(3.72)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq307_HTML.gif provided https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq308_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq309_HTML.gif provided https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq310_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq311_HTML.gif

Lemma 3.32.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq312_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq313_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq314_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq315_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq316_HTML.gif

Proof.

By induction and in view of the fact that the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq317_HTML.gif is increasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq318_HTML.gif it can be seen that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ91_HTML.gif
(3.73)

Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq319_HTML.gif

The next lemma can also be proved by induction.

Lemma 3.33.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq320_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq321_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq322_HTML.gif Define, by induction,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ92_HTML.gif
(3.74)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq323_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq324_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ93_HTML.gif
(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.
  1. (i)

    There is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq325_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq326_HTML.gif

     
  2. (ii)

    There is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq327_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq328_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq329_HTML.gif

     
  3. (iii)

    Equation (3.69) is nonoscillatory.

     
  4. (iv)

    Equation (3.69) has a nonoscillatory solution.

     

Applying Theorem 3.34, the nonoscillation and oscillation of (3.69), in the case of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq330_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq331_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq332_HTML.gif be a random variable defined in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq333_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq334_HTML.gif be a constant. Consider the second-order linear differential equation with random impulses:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ94_HTML.gif
(3.76)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq335_HTML.gif are Lebesque measurable and locally essentially bounded functions, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq336_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq337_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq338_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq339_HTML.gif

Definition 3.35.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq340_HTML.gif be a real-valued random variable in the probability space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq341_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq342_HTML.gif is the sample space, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq343_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq344_HTML.gif -field, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq345_HTML.gif is the probability measure. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq346_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq347_HTML.gif is called the expectation of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq348_HTML.gif and is denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq349_HTML.gif that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ95_HTML.gif
(3.77)
In particular, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq350_HTML.gif is a continuous random variable having probability density function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq351_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ96_HTML.gif
(3.78)

Definition 3.36.

A stochastic process https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq352_HTML.gif is said to be a sample path solution to(3.76) with the initial condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq353_HTML.gif if for any sample value https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq354_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq355_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq356_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ97_HTML.gif
(3.79)

Definition 3.37.

The exponential distribution is a continuous random variable with the probability density function:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ98_HTML.gif
(3.80)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq357_HTML.gif is a parameter.

Definition 3.38.

A solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq358_HTML.gif of (3.76) is said to be nonoscillatory in mean if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq359_HTML.gif is either eventually positive or eventually negative. Otherwise, it is called oscillatory.

Consider the following auxiliary differential equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ99_HTML.gif
(3.81)

Lemma 3.39.

The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq360_HTML.gif is a solution of (3.76) if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ100_HTML.gif
(3.82)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq361_HTML.gif is a solution of (3.81) with the same initial conditions for (3.76), and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq362_HTML.gif is the index function, that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ101_HTML.gif
(3.83)

Proof.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq363_HTML.gif is a solution of system (3.81), for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq364_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ102_HTML.gif
(3.84)
It can be seen that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ103_HTML.gif
(3.85)

which imply that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq365_HTML.gif satisfies (3.76), that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq366_HTML.gif is a sample path solution of (3.76). If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq367_HTML.gif is a sample path solution of (3.76), then it is easy to check that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq368_HTML.gif is a solution of (3.81). This completes the proof.

Theorem 3.40 (see [11]).

Let the following condition hold.

(C) Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq370_HTML.gif be exponential distribution with parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq371_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq372_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq373_HTML.gif be independent of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq374_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq375_HTML.gif

If there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq376_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ104_HTML.gif
(3.86)

does not change sign for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq377_HTML.gif then all solutions of (3.76) are oscillatory in mean if and only if all solutions of (3.81) are oscillatory.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq378_HTML.gif be any sample path solution of (3.76); then Lemma 3.39 implies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ105_HTML.gif
(3.87)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq379_HTML.gif is a solution of (3.81). Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ106_HTML.gif
(3.88)
Further, it can be seen that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ107_HTML.gif
(3.89)
So,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ108_HTML.gif
(3.90)

By assumption, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq380_HTML.gif has the same sign as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq381_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq382_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq383_HTML.gif is finite, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq384_HTML.gif , then the following result can be proved.

Theorem 3.41 (see [11]).

Let condition (C) hold, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq385_HTML.gif be finite for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq386_HTML.gif . Further assume that there are a finite number of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq387_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq388_HTML.gif 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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ109_HTML.gif
(3.91)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq389_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq390_HTML.gif is not always zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq391_HTML.gif for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq392_HTML.gif

The following lemma is a generalization of Lemma https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq393_HTML.gif in [25].

Lemma 3.42 (see [22]).

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq394_HTML.gif is a solution of (3.91) and there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq395_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq396_HTML.gif Let the following conditions be fulfilled.

One has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ110_HTML.gif
(3.92)
One has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ111_HTML.gif
(3.93)

Then for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq399_HTML.gif either https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq400_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq401_HTML.gif holds, where

(A) one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ112_HTML.gif
(3.94)
(B) one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ113_HTML.gif
(3.95)

Theorem 3.43 (see [22]).

Assume that conditions of Lemma 3.42 are fulfilled and for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq404_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq405_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq406_HTML.gif Moreover, assume that the sequence of numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq407_HTML.gif has a positive lower bound, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq408_HTML.gif converges, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq409_HTML.gif holds. Then every bounded solution of (3.91) either oscillates or tends asymptotically to zero with fixed sign.

Proof.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq410_HTML.gif is a bounded nonoscillatory solution of (3.91) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq411_HTML.gif According to Lemma 3.42, either https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq412_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq413_HTML.gif is satisfied. We claim that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq414_HTML.gif does not hold. Otherwise, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq415_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq416_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq417_HTML.gif it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq418_HTML.gif is monotonically increasing for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq419_HTML.gif For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq420_HTML.gif ,   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq421_HTML.gif By induction, it can be seen that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ114_HTML.gif
(3.96)
in particular,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ115_HTML.gif
(3.97)
Integrating https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq422_HTML.gif from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq423_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq424_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ116_HTML.gif
(3.98)
By induction, for any natural number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq425_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ117_HTML.gif
(3.99)
Considering the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq426_HTML.gif in Lemma 3.42 and the sequence of numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq427_HTML.gif has a positive lower bound, we conclude that the inequality above leads to a contradiction that the right side tends to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq428_HTML.gif while https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq429_HTML.gif is bounded. Therefore, case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq430_HTML.gif holds. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq431_HTML.gif implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq432_HTML.gif is strictly monotonically decreasing. From the facts that the series https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq433_HTML.gif converges and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq434_HTML.gif is bounded, it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq435_HTML.gif converges and there exists limit https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq436_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq437_HTML.gif Now, we claim that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq438_HTML.gif Otherwise, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq439_HTML.gif and there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq440_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq441_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq442_HTML.gif From (3.91) and the last inequality, we can deduce
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ118_HTML.gif
(3.100)
Integrating by parts of the above inequality and considering https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq443_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq444_HTML.gif , we have the following inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ119_HTML.gif
(3.101)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq445_HTML.gif and the series https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq446_HTML.gif converges, the above inequality contradicts the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq447_HTML.gif is bounded, hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq448_HTML.gif , 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq449_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq450_HTML.gif Moreover, assume that the sequence of numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq451_HTML.gif is bounded above, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq452_HTML.gif converges, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq453_HTML.gif 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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ120_HTML.gif
(4.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ121_HTML.gif
(4.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq454_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq455_HTML.gif are real sequences, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq456_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq457_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq458_HTML.gif

The lemma below can be found in [26].

Lemma 4.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq459_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq460_HTML.gif be a constant; then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ122_HTML.gif
(4.3)

where equality holds if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq461_HTML.gif .

The results of this section are from [16].

Theorem 4.2 (Sturm-Picone type comparison).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq462_HTML.gif be a solution of (4.1) having two consecutive zeros https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq463_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq464_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq465_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq466_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq467_HTML.gif are satisfied for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq468_HTML.gif , and that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq469_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq470_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq471_HTML.gif . If either https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq472_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq473_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq474_HTML.gif , then any solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq475_HTML.gif of (4.2) must have at least one zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq476_HTML.gif .

Proof.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq477_HTML.gif never vanishes on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq478_HTML.gif . Define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ123_HTML.gif
(4.4)
where the dependence on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq479_HTML.gif of the solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq480_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq481_HTML.gif is suppressed. It is not difficult to see that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ124_HTML.gif
(4.5)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ125_HTML.gif
(4.6)
Clearly, the last term of (4.5) is integrable over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq482_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq483_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq484_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq485_HTML.gif in this case. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq486_HTML.gif . The case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq487_HTML.gif is similar. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq488_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ126_HTML.gif
(4.7)
we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ127_HTML.gif
(4.8)
and so
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ128_HTML.gif
(4.9)
Moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ129_HTML.gif
(4.10)
Integrating (4.5) from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq489_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq490_HTML.gif and using (4.6), we see that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ130_HTML.gif
(4.11)

where we have used Lemma 4.1 with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq491_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq492_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq493_HTML.gif . It is clear that (4.11) is not possible under our assumptions, and hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq494_HTML.gif must have a zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq495_HTML.gif .

Corollary 4.3 (Separation Theorem).

The zeros of two linearly independent solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq496_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq497_HTML.gif of (4.1) separate each other.

Corollary 4.4 (Comparison Theorem).

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq498_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq499_HTML.gif are satisfied for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq500_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq501_HTML.gif , and that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq502_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq503_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq504_HTML.gif . If either https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq505_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq506_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq507_HTML.gif , then every solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq508_HTML.gif of (4.2) is oscillatory whenever a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq509_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq510_HTML.gif be a solution of (4.1) having two consecutive zeros https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq511_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq512_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq513_HTML.gif . Suppose that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ131_HTML.gif
(4.12)

Then any nontrivial solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq514_HTML.gif of (4.2) must have at least one zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq515_HTML.gif .

Proof.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq516_HTML.gif has no zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq517_HTML.gif . Define the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq518_HTML.gif as in (4.4).

Clearly, (4.5) and (4.6) hold. It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ132_HTML.gif
(4.13)

which is a contradiction. Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq519_HTML.gif must have a zero on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq520_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq521_HTML.gif , then we may conclude that either https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq522_HTML.gif has a zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq523_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq524_HTML.gif is a constant multiple of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq525_HTML.gif .

As a consequence of Theorems 4.2 and 4.6, we have the following oscillation result.

Corollary 4.7.

Suppose for a given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq526_HTML.gif there exists an interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq527_HTML.gif for which either conditions of Theorem 4.2 or Theorem 4.6 are satisfied, then every solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq528_HTML.gif 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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ133_HTML.gif
(4.14)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq529_HTML.gif denotes the impulse moments sequence with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq530_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq531_HTML.gif

Assume that the following conditions hold.

()  https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq533_HTML.gif is a constant, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq534_HTML.gif is a continuous function, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq535_HTML.gif .

()  https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq537_HTML.gif are constants, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq538_HTML.gif

()  https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq540_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq541_HTML.gif are two intervals such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq542_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq543_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq544_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq545_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq546_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq547_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq548_HTML.gif

Interval oscillation criteria for (4.14) are given in [27]. Denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq549_HTML.gif and for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq550_HTML.gif let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq551_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ134_HTML.gif
(4.15)

Theorem 4.8 (see [27]).

Assume that conditions (A1)–(A3) hold, and that there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq552_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ135_HTML.gif
(4.16)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq553_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq554_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ136_HTML.gif
(4.17)

for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq555_HTML.gif Then every solution of (4.14) has at least one zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq556_HTML.gif

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq557_HTML.gif be a solution of (4.14). Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq558_HTML.gif does not have any zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq559_HTML.gif Without loss of generality, we may assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq560_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq561_HTML.gif Define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ137_HTML.gif
(4.18)
Then, by Hölder's inequality, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq562_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq563_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ138_HTML.gif
(4.19)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq564_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ139_HTML.gif
(4.20)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq565_HTML.gif then all impulsive moments are in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq566_HTML.gif Multiplying both sides of (4.19) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq567_HTML.gif and integrating on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq568_HTML.gif and using the hypotheses, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ140_HTML.gif
(4.21)
On the other hand, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq569_HTML.gif it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ141_HTML.gif
(4.22)
which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq570_HTML.gif is nonincreasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq571_HTML.gif So, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq572_HTML.gif one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ142_HTML.gif
(4.23)
It follows from the above inequality that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ143_HTML.gif
(4.24)
Making a similar analysis on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq573_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ144_HTML.gif
(4.25)
From (4.21)–(4.25) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq574_HTML.gif we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ145_HTML.gif
(4.26)
which contradicts (4.16). If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq575_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq576_HTML.gif and there are no impulse moments in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq577_HTML.gif Similarly to the proof of (4.21), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ146_HTML.gif
(4.27)

which again contradicts (4.16).

In the case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq578_HTML.gif one can repeat the above procedure on the subinterval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq579_HTML.gif in place of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq580_HTML.gif This completes the proof.

Corollary 4.9.

Assume that conditions (A1) and (A2) hold. If for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq581_HTML.gif there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq582_HTML.gif satisfying (A3) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq583_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq584_HTML.gif satisfying (4.16), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq585_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq586_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ147_HTML.gif
(4.28)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq587_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq588_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ148_HTML.gif
(4.29)

for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq589_HTML.gif Then every solution of (4.14) has at least one zero in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq590_HTML.gif

Corollary 4.11.

Assume that conditions (A1) and (A2) hold. If for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq591_HTML.gif there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq592_HTML.gif satisfying (A3) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq593_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq594_HTML.gif satisfying (4.28), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq595_HTML.gif then (4.14) is oscillatory.

Example 4.12.

Consider the following superlinear impulsive differential equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ149_HTML.gif
(4.30)
It can be seen that if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ150_HTML.gif
(4.31)

then, conditions of Corollary 4.9 are satisfied; here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq596_HTML.gif is the gamma function, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq597_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq598_HTML.gif satisfy ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq599_HTML.gif 2). So, every solution of (4.30) is oscillatory.

In [2830], 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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ151_HTML.gif
(4.32)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq600_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq601_HTML.gif

Define the energy functional
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ152_HTML.gif
(4.33)

where in explicit form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq602_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq603_HTML.gif The functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq604_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq605_HTML.gif are both even and positive definite.

The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq606_HTML.gif is constant along the solutions of the nonimpulsive equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ153_HTML.gif
(4.34)
The change in the energy along the solutions of (4.32) is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ154_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq607_HTML.gif be a solution of (4.32), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq608_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq609_HTML.gif Calculating https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq610_HTML.gif in terms of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq611_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ155_HTML.gif
(4.36)
To simplify the notation, we introduce the function
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ156_HTML.gif
(4.37)

The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq612_HTML.gif gives the jump in the quantity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq613_HTML.gif Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq614_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq615_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq616_HTML.gif is monotone increasing with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq617_HTML.gif and decreasing with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq618_HTML.gif

Theorem 4.13 (see [29]).

Assume that there exist a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq619_HTML.gif and a sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq620_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq621_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ157_HTML.gif
(4.38)

holds for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq622_HTML.gif Then every solution of (4.32) is nonoscillatory.

Theorem 4.14 (see [29]).

Assume that there exist a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq623_HTML.gif and a sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq624_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq625_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq626_HTML.gif such that for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq627_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ158_HTML.gif
(4.39)

holds for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq628_HTML.gif Then every solution of (4.32) is oscillatory.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq629_HTML.gif be a nontrivial solution of (4.32). It suffices to show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq630_HTML.gif cannot hold on any interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq631_HTML.gif Assume that to the contrary, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq632_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq633_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq634_HTML.gif be defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq635_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq636_HTML.gif It follows from (4.39) that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq637_HTML.gif Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ159_HTML.gif
(4.40)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq638_HTML.gif and the right side of the above inequality tends to infinity as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq639_HTML.gif we have a contradiction.

Now, assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq640_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq641_HTML.gif It can be shown that the integral
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ160_HTML.gif
(4.41)
takes its maximum in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq642_HTML.gif at
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ161_HTML.gif
(4.42)

In the special case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq643_HTML.gif we have the following necessary and sufficient condition.

Theorem 4.15 (see [29]).

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq644_HTML.gif Then every solution of (4.32) is nonoscillatory if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ162_HTML.gif
(4.43)

Remark 4.16.

Equation (4.32) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq645_HTML.gif was studied in [30].

Finally, we consider the second-order impulsive differential equation of the following form:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ163_HTML.gif
(4.44)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq646_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq647_HTML.gif are real constants, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq648_HTML.gif is a strictly increasing unbounded sequence of real numbers, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq649_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq650_HTML.gif are real sequences, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq651_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq652_HTML.gif

All results given in the remainder of this section are from [31].

Theorem 4.17.

Suppose that for any given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq653_HTML.gif , there exist intervals https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq654_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq655_HTML.gif , such that
  1. (a)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq656_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq657_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq658_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq659_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq660_HTML.gif ;

     
  2. (b)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq661_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq662_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq663_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq664_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq665_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq666_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq667_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq668_HTML.gif   for all   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq669_HTML.gif

     
If there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq670_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ164_HTML.gif
(4.45)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ165_HTML.gif
(4.46)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ166_HTML.gif
(4.47)

then (4.44) is oscillatory.

Proof.

Suppose that there exists a nonoscillatory solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq671_HTML.gif of (4.44) so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq672_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq673_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq674_HTML.gif . Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ167_HTML.gif
(4.48)
It follows that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq675_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ168_HTML.gif
(4.49)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq676_HTML.gif dependence is suppressed for clarity.

Define a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq677_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ169_HTML.gif
(4.50)
It is not difficult to see that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq678_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ170_HTML.gif
(4.51)

Clearly, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq679_HTML.gif , then we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq680_HTML.gif . Thus, with our convention that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq681_HTML.gif , (4.51) holds for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq682_HTML.gif .

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq683_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq684_HTML.gif . Choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq685_HTML.gif and consider the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq686_HTML.gif . From (b), we see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq687_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq688_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq689_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq690_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq691_HTML.gif . Applying (4.51) to the terms in the parenthesis in (4.49) we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ171_HTML.gif
(4.52)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq692_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq693_HTML.gif are defined by (4.46) and (4.47), respectively.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq694_HTML.gif . Multiplying (4.52) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq695_HTML.gif and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq696_HTML.gif give
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ172_HTML.gif
(4.53)
In view of (4.52) and the assumption https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq697_HTML.gif , employing the integration by parts formula in the last integral we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ173_HTML.gif
(4.54)
We use Lemma 4.1 with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ174_HTML.gif
(4.55)
to obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ175_HTML.gif
(4.56)

which obviously contradicts (4.45).

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq698_HTML.gif is eventually negative then we can consider https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq699_HTML.gif and reach a similar contradiction. This completes the proof.

Example 4.18.

Consider
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ176_HTML.gif
(4.57)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq700_HTML.gif is a positive real number.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq701_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq702_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq703_HTML.gif . For any given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq704_HTML.gif we may choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq705_HTML.gif sufficiently large so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq706_HTML.gif . Then conditions (a)-(b) are satisfied. It is also easy to see that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq707_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq708_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ177_HTML.gif
(4.58)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq709_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq710_HTML.gif . It follows from Theorem 4.17 that (4.57) is oscillatory if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ178_HTML.gif
(4.59)

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

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq711_HTML.gif , then (4.44) reduces to forced half-linear impulsive equation with damping
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ179_HTML.gif
(4.60)

Corollary 4.19.

Suppose that for any given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq712_HTML.gif , there exist intervals https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq713_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq714_HTML.gif for which (a)-(b) hold.

If there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq715_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ180_HTML.gif
(4.61)

then (4.60) is oscillatory.

Taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq716_HTML.gif in (4.44), we have the forced superlinear impulsive equation with damping
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ181_HTML.gif
(4.62)

Corollary 4.20.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq717_HTML.gif . Suppose that for any given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq718_HTML.gif , there exist intervals https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq719_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq720_HTML.gif , such that (a)-(b) hold for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq721_HTML.gif .

If there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq722_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ182_HTML.gif
(4.63)

then (4.62) is oscillatory.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq723_HTML.gif in (4.62). Then we have the forced linear equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ183_HTML.gif
(4.64)

Corollary 4.21.

Suppose that for any given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq724_HTML.gif , there exist intervals https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq725_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq726_HTML.gif , such that (a)-(b) hold for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq727_HTML.gif .

If there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq728_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ184_HTML.gif
(4.65)

then (4.64) is oscillatory.

Example 4.22.

Consider
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ185_HTML.gif
(4.66)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq729_HTML.gif is a positive real number.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq730_HTML.gif . For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq731_HTML.gif , choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq732_HTML.gif sufficiently large so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq733_HTML.gif and set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq734_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq735_HTML.gif . Clearly, (a)-(b) are satisfied for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq736_HTML.gif . It is easy to see that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq737_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq738_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ186_HTML.gif
(4.67)
Thus (4.65) holds if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ187_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq739_HTML.gif .

Finally we state a generalization of Theorem 4.17 for a class of more general type impulsive equations. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq740_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq741_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq742_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq743_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq744_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq745_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq746_HTML.gif be as above, and consider
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ188_HTML.gif
(4.69)
where the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq747_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq748_HTML.gif satisfy
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ189_HTML.gif
(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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ190_HTML.gif
(4.71)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ191_HTML.gif
(4.72)

Clearly if we take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq749_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq750_HTML.gif , then (4.70) holds with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq751_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq752_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq753_HTML.gif . Further, we see that all conditions of Theorem 4.17 are satisfied if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq754_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq755_HTML.gif ; see Example 4.18. Therefore we may deduce from Theorem 4.23 that (4.71) is oscillatory if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq756_HTML.gif .

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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ192_HTML.gif
(4.73)
Assume that the following conditions hold.
  1. (i)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq757_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq758_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq759_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq760_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq761_HTML.gif

     
  2. (ii)
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq762_HTML.gif , and there exist positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq763_HTML.gif such that
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ193_HTML.gif
    (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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq764_HTML.gif be a solution of (4.73). Suppose that there exists some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq765_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq766_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq767_HTML.gif If conditions (i) and (ii) are satisfied, and

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq769_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq770_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq771_HTML.gif holds, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq772_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq773_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq774_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq775_HTML.gif

Theorems 4.26–4.32 are obtained in [25]. For some improvements and/or generalizations, see [3235].

Theorem 4.26.

Assume that conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq776_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq777_HTML.gif of Lemma 4.25 hold, and there exists a positive integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq778_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq779_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq780_HTML.gif If
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ194_HTML.gif
(4.75)

then every solution of (4.73) is oscillatory.

Proof.

Without loss of generality, we can assume https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq781_HTML.gif If (4.73) has a nonoscillatory solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq782_HTML.gif we might as well assume https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq783_HTML.gif From Lemma 4.25, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq784_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq785_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq786_HTML.gif Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ195_HTML.gif
(4.76)
Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq787_HTML.gif Using condition (i) in (4.73), we get for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq788_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ196_HTML.gif
(4.77)
Using condition (ii) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq789_HTML.gif yield
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ197_HTML.gif
(4.78)
From the above inequalities, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ198_HTML.gif
(4.79)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq790_HTML.gif Taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq791_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq792_HTML.gif we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ199_HTML.gif
(4.80)
By induction, for any natural number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq793_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ200_HTML.gif
(4.81)

Since, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq794_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq795_HTML.gif of Lemma 4.25 hold and there exists a positive integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq796_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq797_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq798_HTML.gif If
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ201_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq799_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq800_HTML.gif of Lemma 4.25 hold and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq801_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq802_HTML.gif If
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ202_HTML.gif
(4.83)

then every solution of (4.73) is oscillatory.

Corollary 4.29.

Assume that conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq803_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq804_HTML.gif of Lemma 4.25 hold and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq805_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq806_HTML.gif Suppose that there exist a positive integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq807_HTML.gif and a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq808_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ203_HTML.gif
(4.84)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq809_HTML.gif then every solution of (4.73) is oscillatory.

Example 4.30.

Consider the superlinear equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ204_HTML.gif
(4.85)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq810_HTML.gif is a natural number. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq811_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq812_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq813_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq814_HTML.gif It is easy to see that conditions (i), (ii), and (iii) are satisfied. Moreover
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ205_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq815_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq816_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq817_HTML.gif If
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ206_HTML.gif
(4.87)

then every solution of (4.73) is oscillatory.

Proof.

Without loss of generality, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq818_HTML.gif If (4.1) has a nonoscillatory solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq819_HTML.gif assume https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq820_HTML.gif By Lemma 4.25, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq821_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq822_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ207_HTML.gif
(4.88)
It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq823_HTML.gif is monotonically nondecreasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq824_HTML.gif Now (4.73) yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ208_HTML.gif
(4.89)
Hence, from the above inequality and condition (ii), we find that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ209_HTML.gif
(4.90)
Generally, for any natural number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq825_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ210_HTML.gif
(4.91)
By (4.89) and (4.91), noting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq826_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq827_HTML.gif for any natural number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq828_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ211_HTML.gif
(4.92)
Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq829_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq830_HTML.gif is nondecreasing. Dividing the above inequality by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq831_HTML.gif and then integrating from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq832_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq833_HTML.gif we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ212_HTML.gif
(4.93)
Since (4.88) holds, the above inequality yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ213_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq834_HTML.gif of Lemma 4.25 hold, and there exists a positive integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq835_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq836_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq837_HTML.gif Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq838_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq839_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ214_HTML.gif
(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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ215_HTML.gif
(4.96)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq840_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq841_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq842_HTML.gif .

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
  1. (i)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq843_HTML.gif is a constant;

     
  2. (ii)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq844_HTML.gif is a strictly increasing unbounded sequence of real numbers; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq845_HTML.gif is a real sequence;

     
  3. (iii)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq846_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq847_HTML.gif ;

     
  4. (iv)
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq848_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq849_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq850_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq851_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq852_HTML.gif and
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ216_HTML.gif
    (4.97)
     

is satisfied; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq853_HTML.gif is a constant.

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

Lemma 4.33.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq854_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq855_HTML.gif are nonnegative numbers, then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ217_HTML.gif
(4.98)

and the equality holds if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq856_HTML.gif .

The following theorem is one of the main results of this study.

Theorem 4.34 (see [36]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq857_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq858_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq859_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq860_HTML.gif satisfy the following conditions.
  1. (i)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq861_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq862_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq863_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq864_HTML.gif .

     
  2. (ii)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq865_HTML.gif has a continuous and nonpositive partial derivative on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq866_HTML.gif with respect to the second variable.

     
  3. (iii)
    One has
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ218_HTML.gif
    (4.99)
     
If
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ219_HTML.gif
(4.100)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq867_HTML.gif , then (4.96) is oscillatory.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq868_HTML.gif be a nonoscillatory solution of (4.96). We assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq869_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq870_HTML.gif for some sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq871_HTML.gif . Define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ220_HTML.gif
(4.101)
Differentiating (4.101) and making use of (4.96) and (4.97), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ221_HTML.gif
(4.102)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ222_HTML.gif
(4.103)
Replacing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq872_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq873_HTML.gif in (4.102) and multiplying the resulting equation by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq874_HTML.gif and integrating from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq875_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq876_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ223_HTML.gif
(4.104)
Integrating by parts and using (4.103), we find
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ224_HTML.gif
(4.105)
Combining (4.104) and (4.105), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ225_HTML.gif
(4.106)
Using inequality (4.98) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq877_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ226_HTML.gif
(4.107)
we find
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ227_HTML.gif
(4.108)
From (4.106) and (4.108), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ228_HTML.gif
(4.109)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq878_HTML.gif . In the above inequality we choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq879_HTML.gif , to get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ229_HTML.gif
(4.110)
Thus, it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ230_HTML.gif
(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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ231_HTML.gif
(4.112)

then (4.96) is oscillatory.

Note that in the special case of half-linear equations, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq880_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq881_HTML.gif , the condition (4.97) is satisfied with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq882_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq883_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq884_HTML.gif be defined as in Theorem 4.34. Moreover, Suppose that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ232_HTML.gif
(4.113)
If there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq885_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ233_HTML.gif
(4.114)
and for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq886_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ234_HTML.gif
(4.115)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq887_HTML.gif , 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 [3740].

In [37] authors considered even order impulsive differential equations of the following form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ235_HTML.gif
(4.116)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ236_HTML.gif
(4.117)

Let the following conditions hold.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq889_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq890_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq891_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq892_HTML.gif is positive and continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq893_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq894_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq896_HTML.gif , and there exist positive numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq897_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ237_HTML.gif
(4.118)

A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq898_HTML.gif is said to be a solution of (4.116), if (i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq899_HTML.gif ; (ii) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq900_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq901_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq902_HTML.gif ; (iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq903_HTML.gif is left continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq904_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq905_HTML.gif

The first two theorems can be considered as modifications of Theorems 3.43 and 4.26, respectively.

Theorem 4.37 (see [37]).

If conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq906_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq907_HTML.gif hold, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq908_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq909_HTML.gif , and if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ238_HTML.gif
(4.119)

then every bounded solution of (4.116) is oscillatory.

Theorem 4.38 (see [37]).

If conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq910_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq911_HTML.gif hold, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq912_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ239_HTML.gif
(4.120)

then every solution of (4.116) is oscillatory.

Theorem 4.39 (see [37]).

If conditions (A) and (B) hold, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq913_HTML.gif     https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq914_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq915_HTML.gif , and for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq916_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ240_HTML.gif
(4.121)

then every solution of (4.116) is oscillatory.

Corollary 4.40.

Assume that conditions (A) and (B) hold, and that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq917_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq918_HTML.gif . If
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ241_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq919_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq920_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq921_HTML.gif . If
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ242_HTML.gif
(4.123)

then every solution of (4.116) is oscillatory.

Example 4.42.

Consider the impulsive differential equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ243_HTML.gif
(4.124)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq922_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq923_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq924_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq925_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq926_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq927_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq928_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq929_HTML.gif .

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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ244_HTML.gif
(4.125)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq930_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq931_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq932_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq933_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq934_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq935_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq936_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq937_HTML.gif .

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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ245_HTML.gif
(4.126)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq938_HTML.gif is the set of all positive integers, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq939_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq940_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq941_HTML.gif are given positive constants, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq942_HTML.gif denotes the set of maximum integers, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq943_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq944_HTML.gif . It is assumed that
  1. (i)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq945_HTML.gif ;

     
  2. (ii)
    for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq946_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq947_HTML.gif ,
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ246_HTML.gif
    (4.127)
     
  1. (iii)
    there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq948_HTML.gif such that
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ247_HTML.gif
    (4.128)
     

Theorem 4.44 (see [40]).

Assume that conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq949_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq950_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq951_HTML.gif hold. Moreover, suppose that for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq952_HTML.gif , there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq953_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ248_HTML.gif
(4.129)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq954_HTML.gif . Then every solution of (4.126) is oscillatory.

Theorem 4.45 (see [40]).

Assume that conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq955_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq956_HTML.gif hold. Moreover, suppose that for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq957_HTML.gif , there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq958_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ249_HTML.gif
(4.130)

Then every solution of (4.126) is oscillatory.

Example 4.46.

Consider the impulsive differential equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ250_HTML.gif
(4.131)

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

(1)
Department of Mathematical Sciences, FL Institute of Technology
(2)
Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals
(3)
Department of Mathematics, Faculty of Sciences, Ankara University
(4)
Department of Mathematics, Middle East Technical University

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© Ravi P. Agarwal et al. 2010

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