A Survey on Oscillation of Impulsive Ordinary Differential Equations

  • RaviP Agarwal1, 2Email author,

    Affiliated with

    • Fatma Karakoç3 and

      Affiliated with

      • Ağacık Zafer4

        Affiliated with

        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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq1_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq4_HTML.gif and http://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 http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq7_HTML.gif , the drug is ingested in amounts http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq8_HTML.gif This imposes the following boundary conditions:
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq9_HTML.gif say, during the time interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq10_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq11_HTML.gif . Thus, we have the constraint
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq13_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq14_HTML.gif . Finally, the biological cost function http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ4_HTML.gif
        (1.4)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ5_HTML.gif
        (1.5)

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

        By http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq27_HTML.gif which are continuous for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq29_HTML.gif Similarly, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq30_HTML.gif is the set of functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq31_HTML.gif having derivative http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq32_HTML.gif . One has   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq33_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq34_HTML.gif , or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq35_HTML.gif . In case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq36_HTML.gif , we simply write http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq37_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq38_HTML.gif . As usual, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq40_HTML.gif to http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ6_HTML.gif
        (2.1)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq42_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq43_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ7_HTML.gif
        (2.2)

        with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq44_HTML.gif . The notation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq45_HTML.gif in place of http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq49_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq50_HTML.gif satisfies (2.1) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq51_HTML.gif .

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

        Each solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq54_HTML.gif and satisfying the condition http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ9_HTML.gif
        (2.4)

        on http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq59_HTML.gif Then, the equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ10_HTML.gif
        (2.5)
        is oscillatory if
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ11_HTML.gif
        (2.6)
        and nonoscillatory if
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq60_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq61_HTML.gif be continuous functions and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq62_HTML.gif . If
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ13_HTML.gif
        (2.8)
        then the equation
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq64_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq65_HTML.gif If
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ15_HTML.gif
        (2.10)
        then the equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ16_HTML.gif
        (2.11)

        has nonoscillatory solutions, where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq68_HTML.gif and http://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
        http://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
        http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ19_HTML.gif
        (3.2)
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq70_HTML.gif and http://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 http://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) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq73_HTML.gif (2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq74_HTML.gif Let the sequence http://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 ( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq77_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq78_HTML.gif be such that http://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
        http://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) http://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 http://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 http://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) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq83_HTML.gif (4) http://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) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq85_HTML.gif (1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq86_HTML.gif If http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ22_HTML.gif
        (3.5)

        that the sequence http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq89_HTML.gif and http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ23_HTML.gif
        (3.6)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ24_HTML.gif
        (3.7)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq92_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq93_HTML.gif are continuous for http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq96_HTML.gif such that
          http://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:
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ26_HTML.gif
          (3.9)
           
        1. (3)

          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq97_HTML.gif in a subinterval of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq98_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq99_HTML.gif for some http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq101_HTML.gif defined on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq103_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq104_HTML.gif Then from the relation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ27_HTML.gif
        (3.10)
        an integration yields
        http://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 ( http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ29_HTML.gif
        (3.12)

        But, from conditions ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq106_HTML.gif ) and ( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq108_HTML.gif such that
          http://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:
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ31_HTML.gif
          (3.14)
           
        1. (3)

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

           

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

           
        2. (2)

          Each solution http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq121_HTML.gif such that
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ32_HTML.gif
          (3.15)
           
        for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq123_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq124_HTML.gif ;
          http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq126_HTML.gif hold for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq127_HTML.gif and http://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 http://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 http://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:
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq131_HTML.gif is such that if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq132_HTML.gif is a continuous function for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq133_HTML.gif having a piecewise continuous derivative http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq134_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq135_HTML.gif then the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq136_HTML.gif is piecewise continuous for http://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:
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ35_HTML.gif
          (3.18)
           

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

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

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq146_HTML.gif be a nondegenerate subinterval of http://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:
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq148_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq149_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq150_HTML.gif . These conditions simply mean that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq151_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq152_HTML.gif are in the domain of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq153_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq154_HTML.gif , respectively. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq155_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq156_HTML.gif , then we may define
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq157_HTML.gif . Clearly,
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ41_HTML.gif
        (3.23)
        Employing the identity
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq158_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq159_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq160_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq161_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq162_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq163_HTML.gif then
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq164_HTML.gif of (3.19) such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq165_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq166_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ44_HTML.gif
        (3.26)
        where
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq167_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq168_HTML.gif then every solution http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq171_HTML.gif there exists an interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq172_HTML.gif and a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq173_HTML.gif for which http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq176_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ46_HTML.gif
        (3.28)

        then every solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq178_HTML.gif .

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq179_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq180_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq181_HTML.gif and http://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 http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq184_HTML.gif or http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq186_HTML.gif . Clearly, (3.29) results in
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq188_HTML.gif . Suppose that (H) holds, and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ49_HTML.gif
        (3.31)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ50_HTML.gif
        (3.32)
        for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq189_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ51_HTML.gif
        (3.33)

        for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq190_HTML.gif for which http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq193_HTML.gif , then every solution http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq196_HTML.gif for some integer http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq198_HTML.gif for which http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq201_HTML.gif there exists an interval http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq203_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ52_HTML.gif
        (3.34)
        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ53_HTML.gif
        (3.35)
        It follows that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ54_HTML.gif
        (3.36)
        Assuming that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq204_HTML.gif , the choice of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq205_HTML.gif yields
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq206_HTML.gif and let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq208_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq209_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq210_HTML.gif . Suppose that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ56_HTML.gif
        (3.38)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ57_HTML.gif
        (3.39)
        are satisfied for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq211_HTML.gif , and that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ58_HTML.gif
        (3.40)

        for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq212_HTML.gif for which http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq215_HTML.gif , then any solution http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq218_HTML.gif for some integer http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq220_HTML.gif for which http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq221_HTML.gif . If http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq224_HTML.gif . If there exists a solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq225_HTML.gif of (3.19) such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ59_HTML.gif
        (3.41)

        then every solution http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq228_HTML.gif there exists an interval http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq230_HTML.gif of (3.20) such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq231_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq232_HTML.gif , then for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq233_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq234_HTML.gif
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq235_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq236_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq237_HTML.gif then every solution http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq240_HTML.gif there exists an interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq241_HTML.gif and a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq242_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq243_HTML.gif for which http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ61_HTML.gif
        (3.43)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq245_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq246_HTML.gif and http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq248_HTML.gif for all large http://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 http://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:
        http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq253_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq254_HTML.gif Then, for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq255_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ64_HTML.gif
        (3.46)
        Thus, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq256_HTML.gif is continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq257_HTML.gif Furthermore, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq258_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ65_HTML.gif
        (3.47)
        For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq259_HTML.gif it can be shown that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ66_HTML.gif
        (3.48)
        Thus, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq261_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq262_HTML.gif as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ67_HTML.gif
        (3.49)
        Now, we have for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq263_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ68_HTML.gif
        (3.50)
        and for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq264_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ69_HTML.gif
        (3.51)
        Thus, we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ70_HTML.gif
        (3.52)

        This shows that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq267_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq268_HTML.gif Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq269_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq270_HTML.gif Furthermore, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq271_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ71_HTML.gif
        (3.53)
        and so
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ72_HTML.gif
        (3.54)

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

        When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq274_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq276_HTML.gif on any interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq277_HTML.gif and let http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ75_HTML.gif
        (3.57)

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

        Now, let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq282_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq283_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq285_HTML.gif for http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ76_HTML.gif
        (3.58)

        where http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ77_HTML.gif
        (3.59)

        is oscillatory, where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq289_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq290_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq291_HTML.gif Now, define
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq292_HTML.gif is continuous and satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ79_HTML.gif
        (3.61)
        Next, we define
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ80_HTML.gif
        (3.62)

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

        where http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ82_HTML.gif
        (3.64)
        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq295_HTML.gif for some integer http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ83_HTML.gif
        (3.65)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq297_HTML.gif denotes the greatest integer function, and
        http://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
        http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ86_HTML.gif
        (3.68)

        where http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ87_HTML.gif
        (3.69)
        is investigated, where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ88_HTML.gif
        (3.70)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq299_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq300_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq301_HTML.gif is the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq302_HTML.gif -function, that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ89_HTML.gif
        (3.71)
        for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq303_HTML.gif being continuous at http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq305_HTML.gif define the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq306_HTML.gif inductively by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ90_HTML.gif
        (3.72)

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

        Lemma 3.32.

        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq312_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq313_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq314_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq315_HTML.gif for all http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq317_HTML.gif is increasing in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq318_HTML.gif it can be seen that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ91_HTML.gif
        (3.73)

        Hence, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq320_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq321_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq322_HTML.gif Define, by induction,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ92_HTML.gif
        (3.74)
        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq323_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq324_HTML.gif then
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq325_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq326_HTML.gif

           
        2. (ii)

          There is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq327_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq328_HTML.gif for all http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq330_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq332_HTML.gif be a random variable defined in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq333_HTML.gif and let http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ94_HTML.gif
        (3.76)

        where http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq336_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq337_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq338_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq339_HTML.gif

        Definition 3.35.

        Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq341_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq342_HTML.gif is the sample space, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq343_HTML.gif is the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq344_HTML.gif -field, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq345_HTML.gif is the probability measure. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq346_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq347_HTML.gif is called the expectation of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq348_HTML.gif and is denoted by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq349_HTML.gif that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ95_HTML.gif
        (3.77)
        In particular, if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq351_HTML.gif then
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq353_HTML.gif if for any sample value http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq354_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq355_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq356_HTML.gif satisfies
        http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ98_HTML.gif
        (3.80)

        where http://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 http://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 http://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:
        http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ100_HTML.gif
        (3.82)
        where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq362_HTML.gif is the index function, that is,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ101_HTML.gif
        (3.83)

        Proof.

        If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq364_HTML.gif we have
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ103_HTML.gif
        (3.85)

        which imply that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq365_HTML.gif satisfies (3.76), that is, http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq370_HTML.gif be exponential distribution with parameter http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq371_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq372_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq373_HTML.gif be independent of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq374_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq375_HTML.gif

        If there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq376_HTML.gif such that
        http://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 http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ105_HTML.gif
        (3.87)
        where http://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,
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ107_HTML.gif
        (3.89)
        So,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ108_HTML.gif
        (3.90)

        By assumption, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq380_HTML.gif has the same sign as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq381_HTML.gif for all http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq383_HTML.gif is finite, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq385_HTML.gif be finite for all http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq387_HTML.gif such that http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ109_HTML.gif
        (3.91)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq389_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq390_HTML.gif is not always zero in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq391_HTML.gif for sufficiently large http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq395_HTML.gif such that for any http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ110_HTML.gif
        (3.92)
        One has
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ111_HTML.gif
        (3.93)

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

        (A) one has
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ112_HTML.gif
        (3.94)
        (B) one has
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq404_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq405_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq407_HTML.gif has a positive lower bound, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq408_HTML.gif converges, and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq411_HTML.gif According to Lemma 3.42, either http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq412_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq413_HTML.gif is satisfied. We claim that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq414_HTML.gif does not hold. Otherwise, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq415_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq416_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq417_HTML.gif it follows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq418_HTML.gif is monotonically increasing for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq419_HTML.gif For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq420_HTML.gif ,   http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ114_HTML.gif
        (3.96)
        in particular,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ115_HTML.gif
        (3.97)
        Integrating http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq422_HTML.gif from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq423_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq424_HTML.gif we obtain
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq425_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ117_HTML.gif
        (3.99)
        Considering the condition http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq428_HTML.gif while http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq429_HTML.gif is bounded. Therefore, case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq430_HTML.gif holds. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq431_HTML.gif implies that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq433_HTML.gif converges and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq434_HTML.gif is bounded, it follows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq435_HTML.gif converges and there exists limit http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq436_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq437_HTML.gif Now, we claim that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq438_HTML.gif Otherwise, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq439_HTML.gif and there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq440_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq441_HTML.gif for http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq443_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq444_HTML.gif , we have the following inequality:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ119_HTML.gif
        (3.101)

        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq445_HTML.gif and the series http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq447_HTML.gif is bounded, hence http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq449_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq451_HTML.gif is bounded above, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq452_HTML.gif converges, and http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ120_HTML.gif
        (4.1)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ121_HTML.gif
        (4.2)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq454_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq455_HTML.gif are real sequences, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq456_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq457_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq459_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq460_HTML.gif be a constant; then
        http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq463_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq464_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq465_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq466_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq467_HTML.gif are satisfied for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq468_HTML.gif , and that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq469_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq470_HTML.gif for which http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq471_HTML.gif . If either http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq472_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq473_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq474_HTML.gif , then any solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq476_HTML.gif .

        Proof.

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq477_HTML.gif never vanishes on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq478_HTML.gif . Define
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ123_HTML.gif
        (4.4)
        where the dependence on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq479_HTML.gif of the solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq480_HTML.gif and http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ124_HTML.gif
        (4.5)
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq482_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq483_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq484_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq485_HTML.gif in this case. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq486_HTML.gif . The case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq487_HTML.gif is similar. Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq488_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ126_HTML.gif
        (4.7)
        we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ127_HTML.gif
        (4.8)
        and so
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ128_HTML.gif
        (4.9)
        Moreover,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ129_HTML.gif
        (4.10)
        Integrating (4.5) from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq489_HTML.gif to http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq491_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq492_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq494_HTML.gif must have a zero in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq496_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq498_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq499_HTML.gif are satisfied for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq500_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq501_HTML.gif , and that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq502_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq503_HTML.gif for which http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq504_HTML.gif . If either http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq505_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq506_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq507_HTML.gif , then every solution http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq511_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq512_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq513_HTML.gif . Suppose that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ131_HTML.gif
        (4.12)

        Then any nontrivial solution http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq515_HTML.gif .

        Proof.

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq516_HTML.gif has no zero in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq517_HTML.gif . Define the function http://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
        http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq519_HTML.gif must have a zero on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq520_HTML.gif .

        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq521_HTML.gif , then we may conclude that either http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq522_HTML.gif has a zero in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq523_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq524_HTML.gif is a constant multiple of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq526_HTML.gif there exists an interval http://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 http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ133_HTML.gif
        (4.14)

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

        Assume that the following conditions hold.

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

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

        ()  http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq540_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq541_HTML.gif are two intervals such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq542_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq543_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq544_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq545_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq546_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq547_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq549_HTML.gif and for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq550_HTML.gif let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq551_HTML.gif
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq552_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ135_HTML.gif
        (4.16)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq553_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq554_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ136_HTML.gif
        (4.17)

        for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq556_HTML.gif

        Proof.

        Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq558_HTML.gif does not have any zero in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq560_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq561_HTML.gif Define
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq562_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq563_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ138_HTML.gif
        (4.19)
        For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq564_HTML.gif we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ139_HTML.gif
        (4.20)
        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq565_HTML.gif then all impulsive moments are in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq567_HTML.gif and integrating on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq568_HTML.gif and using the hypotheses, we get
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq569_HTML.gif it follows that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ141_HTML.gif
        (4.22)
        which implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq570_HTML.gif is nonincreasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq571_HTML.gif So, for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq572_HTML.gif one has
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq573_HTML.gif we obtain
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq574_HTML.gif we get
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq575_HTML.gif then http://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 http://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
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq579_HTML.gif in place of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq581_HTML.gif there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq582_HTML.gif satisfying (A3) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq583_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq584_HTML.gif satisfying (4.16), http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq586_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ147_HTML.gif
        (4.28)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq587_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq588_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ148_HTML.gif
        (4.29)

        for http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq591_HTML.gif there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq592_HTML.gif satisfying (A3) with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq593_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq594_HTML.gif satisfying (4.28), http://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:
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq596_HTML.gif is the gamma function, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq597_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq598_HTML.gif satisfy ( http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ151_HTML.gif
        (4.32)

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

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

        where in explicit form http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq602_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq603_HTML.gif The functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq604_HTML.gif and http://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 http://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
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq607_HTML.gif be a solution of (4.32), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq608_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq609_HTML.gif Calculating http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq610_HTML.gif in terms of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq611_HTML.gif we obtain
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ156_HTML.gif
        (4.37)

        The function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq612_HTML.gif gives the jump in the quantity http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq613_HTML.gif Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq614_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq615_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq616_HTML.gif is monotone increasing with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq617_HTML.gif and decreasing with respect to http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq619_HTML.gif and a sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq620_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq621_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ157_HTML.gif
        (4.38)

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

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

        Since http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq640_HTML.gif for every http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ160_HTML.gif
        (4.41)
        takes its maximum in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq642_HTML.gif at
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ161_HTML.gif
        (4.42)

        In the special case http://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 http://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
        http://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 http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ163_HTML.gif
        (4.44)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq646_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq647_HTML.gif are real constants, http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq649_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq650_HTML.gif are real sequences, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq651_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq653_HTML.gif , there exist intervals http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq654_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq655_HTML.gif , such that
        1. (a)

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

           
        2. (b)

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

           
        If there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq670_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ164_HTML.gif
        (4.45)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ165_HTML.gif
        (4.46)
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq671_HTML.gif of (4.44) so that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq672_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq673_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq674_HTML.gif . Let
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ167_HTML.gif
        (4.48)
        It follows that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq675_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ168_HTML.gif
        (4.49)

        where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq677_HTML.gif by
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq678_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ170_HTML.gif
        (4.51)

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

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

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq692_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq694_HTML.gif . Multiplying (4.52) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq695_HTML.gif and integrating over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq696_HTML.gif give
        http://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 http://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
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ174_HTML.gif
        (4.55)
        to obtain
        http://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 http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ176_HTML.gif
        (4.57)

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

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq701_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq702_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq703_HTML.gif . For any given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq704_HTML.gif we may choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq705_HTML.gif sufficiently large so that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq707_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq708_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ177_HTML.gif
        (4.58)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq709_HTML.gif and http://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
        http://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 http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq712_HTML.gif , there exist intervals http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq713_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq715_HTML.gif such that
        http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ181_HTML.gif
        (4.62)

        Corollary 4.20.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq717_HTML.gif . Suppose that for any given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq718_HTML.gif , there exist intervals http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq719_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq721_HTML.gif .

        If there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq722_HTML.gif such that
        http://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 http://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:
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq724_HTML.gif , there exist intervals http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq725_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq727_HTML.gif .

        If there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq728_HTML.gif such that
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ185_HTML.gif
        (4.66)

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

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq730_HTML.gif . For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq731_HTML.gif , choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq732_HTML.gif sufficiently large so that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq733_HTML.gif and set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq734_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq737_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq738_HTML.gif ,
        http://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
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq740_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq741_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq742_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq743_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq744_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq745_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq746_HTML.gif be as above, and consider
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ188_HTML.gif
        (4.69)
        where the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq747_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq748_HTML.gif satisfy
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ190_HTML.gif
        (4.71)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ191_HTML.gif
        (4.72)

        Clearly if we take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq749_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq750_HTML.gif , then (4.70) holds with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq751_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq752_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq754_HTML.gif and http://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 http://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:
        http://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)

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

           
        2. (ii)
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq762_HTML.gif , and there exist positive numbers http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq763_HTML.gif such that
          http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq765_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq766_HTML.gif for http://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) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq769_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq770_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq771_HTML.gif holds, then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq772_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq773_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq774_HTML.gif where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq776_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq778_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq779_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq780_HTML.gif If
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq782_HTML.gif we might as well assume http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq783_HTML.gif From Lemma 4.25, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq784_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq785_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq786_HTML.gif Let
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ195_HTML.gif
        (4.76)
        Then, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq788_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ196_HTML.gif
        (4.77)
        Using condition (ii) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq789_HTML.gif yield
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ198_HTML.gif
        (4.79)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq790_HTML.gif Taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq791_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq792_HTML.gif we get
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq793_HTML.gif we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ200_HTML.gif
        (4.81)

        Since, http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq796_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq797_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq798_HTML.gif If
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq799_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq800_HTML.gif of Lemma 4.25 hold and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq801_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq802_HTML.gif If
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq803_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq804_HTML.gif of Lemma 4.25 hold and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq805_HTML.gif for any http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq807_HTML.gif and a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq808_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ203_HTML.gif
        (4.84)

        If http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ204_HTML.gif
        (4.85)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq810_HTML.gif is a natural number. Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq811_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq812_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq813_HTML.gif and http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq815_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq816_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq817_HTML.gif If
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq819_HTML.gif assume http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq820_HTML.gif By Lemma 4.25, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq821_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq822_HTML.gif we have
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq823_HTML.gif is monotonically nondecreasing in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq824_HTML.gif Now (4.73) yields
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq825_HTML.gif we have
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq826_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq827_HTML.gif for any natural number http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq828_HTML.gif we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ211_HTML.gif
        (4.92)
        Note that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq829_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq831_HTML.gif and then integrating from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq832_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq833_HTML.gif we get
        http://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
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq835_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq836_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq837_HTML.gif Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq838_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq839_HTML.gif and
        http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ215_HTML.gif
        (4.96)

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

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

           
        2. (ii)

          http://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; http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq845_HTML.gif is a real sequence;

           
        3. (iii)

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

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

        is satisfied; http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq854_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq855_HTML.gif are nonnegative numbers, then
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq857_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq858_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq859_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq860_HTML.gif satisfy the following conditions.
        1. (i)

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

           
        2. (ii)

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

        where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq869_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq870_HTML.gif for some sufficiently large http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq871_HTML.gif . Define
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ221_HTML.gif
        (4.102)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ222_HTML.gif
        (4.103)
        Replacing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq872_HTML.gif by http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq874_HTML.gif and integrating from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq875_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq876_HTML.gif , we get
        http://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
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq877_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ226_HTML.gif
        (4.107)
        we find
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ228_HTML.gif
        (4.109)
        for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq878_HTML.gif . In the above inequality we choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq879_HTML.gif , to get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ229_HTML.gif
        (4.110)
        Thus, it follows that
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq880_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq883_HTML.gif and http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq885_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ233_HTML.gif
        (4.114)
        and for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq886_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ234_HTML.gif
        (4.115)

        where http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ235_HTML.gif
        (4.116)
        where
        http://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.

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

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

        A function http://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) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq899_HTML.gif ; (ii) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq900_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq901_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq902_HTML.gif ; (iii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq903_HTML.gif is left continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq904_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq906_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq907_HTML.gif hold, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq908_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq909_HTML.gif , and if
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq910_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq911_HTML.gif hold, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq912_HTML.gif and
        http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq913_HTML.gif     http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq914_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq915_HTML.gif , and for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq916_HTML.gif ,
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq917_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq918_HTML.gif . If
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq919_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq920_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq921_HTML.gif . If
        http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ243_HTML.gif
        (4.124)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq922_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq923_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq924_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq925_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq926_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq927_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq928_HTML.gif , http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ244_HTML.gif
        (4.125)

        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq930_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq931_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq932_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq933_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq934_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq935_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq936_HTML.gif , http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ245_HTML.gif
        (4.126)
        where http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq939_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq940_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq941_HTML.gif are given positive constants, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq943_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq944_HTML.gif . It is assumed that
        1. (i)

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

           
        2. (ii)
          for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq946_HTML.gif and all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq947_HTML.gif ,
          http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ246_HTML.gif
          (4.127)
           
        1. (iii)
          there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq948_HTML.gif such that
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq949_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq950_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq951_HTML.gif hold. Moreover, suppose that for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq952_HTML.gif , there exists a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_IEq953_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F354841/MediaObjects/13662_2009_Article_1281_Equ248_HTML.gif
        (4.129)

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

        References

        1. Bainov DD, Simeonov PS: Oscillation Theory of Impulsive Differential Equations. International Publications, Orlando, Fla, USA; 1998:ii+284.MATH
        2. Baĭnov DD, Simeonov PS: Impulsive Differential Equations: Asymptotic Properties of the Solutions, Series on Advances in Mathematics for Applied Sciences. Volume 28. World Scientific, Singapore; 1995:xii+230.
        3. Bainov D, Covachev V: Impulsive Differential Equations with a Small Parameter, Series on Advances in Mathematics for Applied Sciences. Volume 24. World Scientific, Singapore; 1994:x+268.View Article
        4. Bainov DD, Simeonov PS: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific and Technical, Harlow, UK; 1993.MATH
        5. Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Singapore; 1989:xii+273.View Article
        6. Samoilenko AM, Perestyuk NA: Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. Volume 14. World Scientific, Singapore; 1995:x+462.
        7. Kruger-Thiemer E: Formal theory of drug dosage regiments I. Journal of Theoretical Biology 1966, 13: 212-235. 10.1016/0022-5193(66)90018-XView Article
        8. Pierce JG, Schumitzky A: Optimal impulsive control of compartment models. I. Qualitative aspects. Journal of Optimization Theory and Applications 1976,18(4):537-554. 10.1007/BF00932661MathSciNetView ArticleMATH
        9. Gopalsamy K, Zhang BG: On delay differential equations with impulses. Journal of Mathematical Analysis and Applications 1989,139(1):110-122. 10.1016/0022-247X(89)90232-1MathSciNetView ArticleMATH
        10. Bainov DD, Domshlak YI, Simeonov PS: Sturmian comparison theory for impulsive differential inequalities and equations. Archiv der Mathematik 1996,67(1):35-49. 10.1007/BF01196165MathSciNetView ArticleMATH
        11. Wu S, Duan Y: Oscillation, stability, and boundedness of second-order differential systems with random impulses. Computers & Mathematics with Applications 2005,49(9-10):1375-1386. 10.1016/j.camwa.2004.12.009MathSciNetView ArticleMATH
        12. Agarwal RP, Grace SR, O'Regan D: Oscillation Theory for Second Order Dynamic Equations, Series in Mathematical Analysis and Applications. Volume 5. Taylor' Francis, London, UK; 2003:viii+404.View Article
        13. Hille E: Non-oscillation theorems. Transactions of the American Mathematical Society 1948, 64: 234-252. 10.1090/S0002-9947-1948-0027925-7MathSciNetView ArticleMATH
        14. Leighton W: On self-adjoint differential equations of second order. Journal of the London Mathematical Society 1952, 27: 37-47. 10.1112/jlms/s1-27.1.37MathSciNetView ArticleMATH
        15. Atkinson FV: On second-order non-linear oscillations. Pacific Journal of Mathematics 1955, 5: 643-647.MathSciNetView ArticleMATH
        16. Özbekler A, Zafer A: Sturmian comparison theory for linear and half-linear impulsive differential equations. Nonlinear Analysis: Theory, Methods & Applications 2005,63(5–7):289-297. 10.1016/j.na.2005.01.087View Article
        17. Özbekler A, Zafer A: Picone's formula for linear non-selfadjoint impulsive differential equations. Journal of Mathematical Analysis and Applications 2006,319(2):410-423. 10.1016/j.jmaa.2005.06.019MathSciNetView ArticleMATH
        18. Özbekler A, Zafer A: Forced oscillation of super-half-linear impulsive differential equations. Computers & Mathematics with Applications 2007,54(6):785-792. 10.1016/j.camwa.2007.03.003MathSciNetView ArticleMATH
        19. Shen J: Qualitative properties of solutions of second-order linear ODE with impulses. Mathematical and Computer Modelling 2004,40(3-4):337-344. 10.1016/j.mcm.2003.12.009MathSciNetView ArticleMATH
        20. Luo Z, Shen J: Oscillation of second order linear differential equations with impulses. Applied Mathematics Letters 2007,20(1):75-81. 10.1016/j.aml.2006.01.019MathSciNetView ArticleMATH
        21. Huang C: Oscillation and nonoscillation for second order linear impulsive differential equations. Journal of Mathematical Analysis and Applications 1997,214(2):378-394. 10.1006/jmaa.1997.5572MathSciNetView ArticleMATH
        22. Wan A, Mao W: Oscillation and asymptotic stability behavior of a third order linear impulsive equation. Journal of Applied Mathematics & Computing 2005,18(1-2):405-417. 10.1007/BF02936583MathSciNetView ArticleMATH
        23. Chen YS, Feng WZ: Oscillations of higher-order linear ODEs with impulses. Journal of South China Normal University. Natural Science Edition 2003, (3):14-19.
        24. Feng W: Oscillations of fourth order ODE with impulses. Annals of Differential Equations 2003,19(2):136-145.MathSciNetMATH
        25. Chen YS, Feng WZ: Oscillations of second order nonlinear ODE with impulses. Journal of Mathematical Analysis and Applications 1997,210(1):150-169. 10.1006/jmaa.1997.5378MathSciNetView ArticleMATH
        26. Hardy GH, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge, UK; 1934.
        27. Liu X, Xu Z: Oscillation of a forced super-linear second order differential equation with impulses. Computers & Mathematics with Applications 2007,53(11):1740-1749. 10.1016/j.camwa.2006.08.040MathSciNetView ArticleMATH
        28. Graef JR, Karsai J: Intermittant and impulsive effects in second order systems. Nonlinear Analysis: Theory, Methods & Applications 1997, 30: 1561-1571. 10.1016/S0362-546X(97)00029-1MathSciNetView ArticleMATH
        29. Graef JR, Karsai J: Oscillation and nonoscillation in nonlinear impulsive systems with increasing energy. Proceedings of the International Conference on Dynamical Systems and Differential Equations, 2000, Atlanta, Ga, USA
        30. Graef JR, Karsai J: On the oscillation of impulsively damped halflinear oscillators. In Proceedings of the 6th Colloquium on the Qualitative Theory of Differential Equations (Szeged, '99), 2000. Electronic Journal of Qualitative Theory of Differential Equations; 1-12.
        31. Özbekler A, Zafer A: Interval criteria for the forced oscillation of super-half-linear differential equations under impulse effects. Mathematical and Computer Modelling 2009,50(1-2):59-65. 10.1016/j.mcm.2008.10.020MathSciNetView ArticleMATH
        32. Luo J, Debnath L: Oscillations of second-order nonlinear ordinary differential equations with impulses. Journal of Mathematical Analysis and Applications 1999,240(1):105-114. 10.1006/jmaa.1999.6596MathSciNetView ArticleMATH
        33. Xiu-li W, Si-Yang C, Ji H: Oscillation of a class of second-order nonlinear ODE with impulses. Applied Mathematics and Computation 2003,138(2-3):181-188. 10.1016/S0096-3003(02)00073-5MathSciNetView ArticleMATH
        34. Luo J: Second-order quasilinear oscillation with impulses. Computers & Mathematics with Applications 2003,46(2–3):279-291. 10.1016/S0898-1221(03)90031-9MathSciNetView Article
        35. He Z, Ge W: Oscillations of second-order nonlinear impulsive ordinary differential equations. Journal of Computational and Applied Mathematics 2003,158(2):397-406. 10.1016/S0377-0427(03)00474-6MathSciNetView ArticleMATH
        36. Özbekler A: Oscillation criteria for second order nonlinear impulsive differential equations, further progress in analysis. In Proceedings of the 6th International ISAAC Congress, 2009, Ankara, Turkey. World Scientific; 545-554.
        37. Zhang C, Feng W: Oscillation for higher order nonlinear ordinary differential equations with impulses. Electronic Journal of Differential Equations 2006, (18):1-12.
        38. Pan LJ, Wang GQ, Cheng SS: Oscillation of even order nonlinear differential equations with impulses. Funkcialaj Ekvacioj 2007,50(1):117-131. 10.1619/fesi.50.117MathSciNetView ArticleMATH
        39. Dimitrova MB: Nonoscillation of the solutions of impulsive differential equations of third order. Journal of Applied Mathematics and Stochastic Analysis 2001,14(3):309-316. 10.1155/S1048953301000272MathSciNetView ArticleMATH
        40. Fu X, Li X: Oscillation of higher order impulsive differential equations of mixed type with constant argument at fixed time. Mathematical and Computer Modelling 2008,48(5-6):776-786. 10.1016/j.mcm.2007.11.006MathSciNetView ArticleMATH

        Copyright

        © Ravi P. Agarwal et al. 2010

        This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.