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Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 195619, 10 pages
http://dx.doi.org/10.1155/2011/195619
Research Article

Oscillation of Certain Second-Order Sub-Half-Linear Neutral Impulsive Differential Equations

School of Mathematics, University of Jinan, Shandong, Jinan 250022, China

Received 17 May 2011; Accepted 30 June 2011

Academic Editor: Garyfalos Papaschinopoulos

Copyright © 2011 Yuangong Sun. 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.

Abstract

By introducing auxiliary functions, we investigate the oscillation of a class of second-order sub-half-linear neutral impulsive differential equations of the form [𝑟(𝑡)𝜙𝛽(𝑧(𝑡))]+𝑝(𝑡)𝜙𝛼(𝑥(𝜎(𝑡)))=0,𝑡𝜃𝑘,Δ𝜙𝛽(𝑧(𝑡))|𝑡=𝜃𝑘+𝑞𝑘𝜙𝛼(𝑥(𝜎(𝜃𝑘)))=0,Δ𝑥(𝑡)|𝑡=𝜃𝑘=0, where 𝛽>𝛼>0,𝑧(𝑡)=𝑥(𝑡)+𝜆(𝑡)𝑥(𝜏(𝑡)). Several oscillation criteria for the above equation are established in both the case 0𝜆(𝑡)1 and the case 1<𝜇𝜆(𝑡)0, which generalize and complement some existing results in the literature. Two examples are also given to illustrate the effect of impulses on the oscillatory behavior of solutions to the equation.

1. Introduction

Impulsive differential equations appear as a natural description of observed evolution phenomena of several real-world problems involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics, and frequency modulates systems [15]. In recent years, impulsive differential equations have received a lot of attention.

We are here concerned with the following second-order sub-half-linear neutral impulsive differential equation: 𝑟(𝑡)𝜙𝛽𝑧(𝑡)+𝑝(𝑡)𝜙𝛼(𝑥(𝜎(𝑡)))=0,𝑡𝜃𝑘,Δ𝜙𝛽𝑧(𝑡)𝑡=𝜃𝑘+𝑞𝑘𝜙𝛼𝑥𝜎𝜃𝑘=0,Δ𝑥(𝑡)𝑡=𝜃𝑘=0,(1.1) where 𝛽>𝛼>0, 𝑧(𝑡)=𝑥(𝑡)+𝜆(𝑡)𝑥(𝜏(𝑡)),𝑡𝑡0 and 𝜃𝑘𝑡0 for some 𝑡0𝑅, {𝜃𝑘}𝑘=1 is a strictly increasing unbounded sequence of real numbers, 𝜙𝛾(𝑢)=|𝑢|𝛾1𝑢 for 𝛾>0, and Δ𝑢(𝑡)𝑡=𝜃𝑘𝜃=𝑢+𝑘𝜃𝑢𝑘𝜃,𝑢±𝑘=lim𝑡𝜃±𝑘𝑢(𝑡).(1.2)

Let 𝑃𝐿𝐶(𝐽,𝑅) denote the set of all real-valued functions 𝑢(𝑡) defined on 𝐽[𝑡0,) such that 𝑢(𝑡) is continuous for all 𝑡𝐽 except possibly at 𝑡=𝜃𝑘 where 𝑢(𝜃±𝑘) exists and 𝑢(𝜃𝑘)=𝑢(𝜃𝑘).

We assume throughout this paper that(a)𝑟(𝑡)𝐶1([𝑡0,),𝑅), 𝑟(𝑡)>0 and 𝑡0[𝑟(𝑡)]1/𝛽𝑑𝑡=;(b)𝜆(𝑡)𝐶2([𝑡0,),𝑅), 0𝜆(𝑡)1 or 1<𝜇𝜆(𝑡)0;(c)𝑝(𝑡)𝑃𝐿𝐶([𝑡0,),𝑅), 𝑝(𝑡)0;(d)𝑞𝑘 is a sequence of nonnegative real numbers;(e)𝜏(𝑡),𝜎(𝑡)𝐶([𝑡0,),𝑅,0𝜏(𝑡),𝜎(𝑡)𝑡,lim𝑡𝜏(𝑡)=, and lim𝑡𝜎(𝑡)=.

By a solution of (1.1) we mean a function 𝑥(𝑡) defined on [𝑇𝑥,) with 𝑇𝑥𝑡0 such that 𝑥,𝑧,𝑧𝑃𝐿𝐶([𝑡0,),𝑅) and 𝑥 satisfies (1.1). It is tacitly assumed that such solutions exist. Note the assumption Δ𝑥(𝑡)𝑡=𝜃𝑘=0; we have that each solution of (1.1) is continuous on [𝑡0,). As usual, a nontrivial solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros and nonoscillatory otherwise. Equation (1.1) is said to be oscillatory if its every nontrivial solution is oscillatory.

Compared to equations without impulses, little has been known about the oscillation problem for impulsive differential equations due to difficulties caused by impulsive perturbations [617].

When 𝛽=1, 𝑟(𝑡)1, and 𝜆(𝑡)0, (1.1) reduces to the following sublinear impulsive delay equation: 𝑥(𝑡)+𝑝(𝑡)𝜙𝛼(𝑥(𝜎(𝑡)))=0,𝑡𝜃𝑘,Δ𝑥(𝑡)𝑡=𝜃𝑘+𝑞𝑘𝜙𝛼𝑥𝜎𝜃𝑘=0,Δ𝑥(𝑡)𝑡=𝜃𝑘=0,(1.3) which has received a lot of attention in the literature. However, for the general sub-half-linear neutral equation (1.1) under the impulse condition given in this paper, little has been known about the oscillation of (1.1) to the best of our knowledge, especially for the case when 1<𝜇𝜆(𝑡)0.

The main objective of this paper is to establish oscillation criteria for the sub-half-linear impulsive differential equation (1.1) in both the case 0𝜆(𝑡)1 and the case 1<𝜇𝜆(𝑡)0. By introducing an auxiliary function 𝑔𝐶1[𝑡0,) and a function 𝐻(𝑡,𝑠) defined below, we establish some new oscillation criteria for (1.1) which complement the oscillation theory of impulsive differential equations. Examples are also given to show the effect of impulses on oscillation of solutions of (1.1).

2. Main Results

Theorem 2.1. Let 0𝜆(𝑡)1. If there exists a positive function 𝑔𝐶1[𝑡0,) such that 𝑔(𝑡)𝛼𝑔(𝑡)𝛽𝑟(𝑡),𝑟(𝑡)(2.1)[]1𝜆(𝜎(𝑡))𝛼𝑅𝛽(𝑡)𝛼𝑔(𝑡)𝑝(𝑡)𝑑𝑡+𝜎𝜃1𝜆𝑘𝛼𝑔𝜃𝑘𝑟𝜃𝑘𝑅𝛽𝜃𝑘𝛼𝑞𝑘=,(2.2) where 𝑅𝛽(𝑡)=𝑟1/𝛽(𝑡)𝑡𝜎(𝑡)0[𝑟(𝑠)]1/𝛽𝑑𝑠, then (1.1) is oscillatory.

Proof. Suppose to the contrary that (1.1) has a nonoscillatory solution 𝑥(𝑡). Without loss of generality, we may assume that 𝑥(𝜏(𝑡))>0 and 𝑥(𝜎(𝑡))>0 for 𝑡𝑡1𝑡0. The case 𝑥(𝑡) being eventually negative can be similarly discussed. From (1.1), we have that 𝑧(𝑡)>0,𝑟(𝑡)𝜙𝛽𝑧(𝑡)0,𝑡𝑡1,𝑡𝜃𝑘.(2.3) Based on the impulsive condition Δ𝜙𝛽(𝑧(𝑡))|𝑡=𝜃𝑘0, we can deduce that 𝑟(𝑡)𝜙𝛽(𝑧(𝑡)) is nonincreasing on [𝑡1,). We may claim that 𝑧(𝑡)>0 holds eventually. Otherwise, there exists 𝑡𝑡1 such that 𝑧(𝑡)<0. Noting that 𝑧(𝑡) is continuous on [𝑡,), we have that 𝑡𝑧(𝑡)=𝑧+𝑡𝑡𝑟1/𝛽(𝑠)𝑟1/𝛽(𝑡𝑠)𝑧(𝑠)𝑑𝑠𝑧+𝑟1/𝛽𝑡𝑧𝑡𝑡𝑡𝑟1/𝛽(𝑠)𝑑𝑠,𝑡𝑡,(2.4) which implies that 𝑧(𝑡) is eventually negative since 𝑟1/𝛽(𝑠)𝑑𝑠=. This is a contradiction. Without loss of generality, say 𝑧(𝑡)>0 for 𝑡𝑡1. Choose sufficiently large 𝑡2𝑡1 such that 𝜏(𝑡)𝑡1 for 𝑡𝑡2, and 𝑡𝑡1𝑟1/𝛽(1𝑠)𝑑𝑠2𝑡𝑡0𝑟1/𝛽(𝑠)𝑑𝑠,𝑡𝑡2,(2.5) which is always possible because 𝑟1/𝛽(𝑠)𝑑𝑠=. Thus, we have 𝑧(𝑡)𝑟1/𝛽(𝑡)𝑧(𝑡)𝑡𝑡1𝑟1/𝛽(1𝑠)𝑑𝑠2𝑟1/𝛽(𝑡)𝑧(𝑡)𝑡𝑡0𝑟1/𝛽(𝑠)𝑑𝑠,𝑡𝑡2.(2.6) By choosing 𝑡3 sufficiently large such that 𝜎(𝑡)𝑡2 for 𝑡𝑡3 and using (2.6) and the nonincreasing character of 𝑟1/𝛽(𝑡)𝑧(𝑡), we have 1𝑧(𝜎(𝑡))2𝑟1/𝛽(𝑡)𝑧(𝑡)𝑡𝜎(𝑡)0𝑟1/𝛽𝑅(𝑠)𝑑𝑠=𝛽(𝑡)2𝑧(𝑡),𝑡𝑡3.(2.7) Since 𝑧(𝑡)>0 for 𝑡𝑡1 and 𝑧(𝑡) is continuous, we have []𝑥(𝑡)=𝑧(𝑡)𝜆(𝑡)𝑥(𝜏(𝑡))𝑧(𝑡)𝜆(𝑡)𝑧(𝜏(𝑡))1𝜆(𝑡)𝑧(𝑡),𝑡𝑡2.(2.8) By (1.1), (2.7), and (2.8), we get 𝑟(𝑡)𝜙𝛽𝑧(𝑡)+2𝛼[]𝑝(𝑡)1𝜆(𝜎(𝑡))𝛼𝑅𝛽(𝑡)𝛼𝑧(𝑡)𝛼0,𝑡𝑡3,𝑡𝜃𝑘,(2.9) which implies 𝑔(𝑡)𝑟(𝑡)𝜙𝛽𝑧(𝑡)+2𝛼[]𝑔(𝑡)𝑝(𝑡)1𝜆(𝜎(𝑡))𝛼𝑅𝛽(𝑡)𝛼𝑧(𝑡)𝛼0,𝑡𝑡3,𝑡𝜃𝑘.(2.10) From (2.1), we get 𝑔𝑟(𝑡)(𝑡)𝜙𝛽𝑧(𝑡)[𝑧](𝑡)𝛼=𝑧𝛽𝑔(𝑡)𝑟(𝑡)(𝑡)𝛽1𝑧+𝑔(𝑡)𝑟𝑧(𝑡)(𝑡)𝛽[𝑧](𝑡)𝛼𝑧𝛽𝑔(𝑡)𝑟(𝑡)(𝑡)𝛽1𝑧[]𝑧+(𝛽/(𝛽𝛼))𝑔(𝑡)𝑟(𝑡)(𝑡)𝛽[𝑧](𝑡)𝛼=𝛽𝛽𝛼𝑔(𝑡)𝑟(𝑡)(𝑧(𝑡))𝛽𝛼.(2.11) Multiplying (2.10) by ((𝛽𝛼)/𝛽)[𝑧(𝑡)]𝛼, we get 𝑧𝑔(𝑡)𝑟(𝑡)(𝑡)𝛽𝛼+2𝛼[]𝑔(𝑡)𝑝(𝑡)1𝜆(𝜎(𝑡))𝛼𝑅𝛽(𝑡)𝛼0,𝑡𝑡3,𝑡𝜃𝑘.(2.12) Integrating (2.12) from 𝑡3 to 𝑡, we have that 𝑡3𝜃𝑘<𝑡𝑔𝜃𝑘𝑟𝜃𝑘𝑧𝜃𝑘𝛽𝛼𝑧𝜃+𝑘𝛽𝛼𝑧+𝑔(𝑡)𝑟(𝑡)(𝑡)𝛽𝛼𝑡𝑔3𝑟𝑡3𝑧𝑡3𝛽𝛼+(𝛽𝛼)2𝛼𝛽𝑡𝑡3[]1𝜆(𝜎(𝑠))𝛼𝑅𝛽(𝑠)𝛼𝑔(𝑠)𝑝(𝑠)𝑑𝑠0,𝑡𝑡3,(2.13) which implies that 𝑡3𝜃𝑘<𝑡𝑔𝜃𝑘𝑟𝜃𝑘𝑧𝜃𝑘𝛽𝛼𝑧𝜃+𝑘𝛽𝛼+(𝛽𝛼)2𝛼𝛽𝑡𝑡3[]1𝜆(𝜎(𝑠))𝛼𝑅𝛽(𝑠)𝛼𝑡𝑔(𝑠)𝑝(𝑠)𝑑𝑠𝑔3𝑟𝑡3𝑧𝑡3𝛽𝛼.(2.14) On the other hand, by the given impulsive condition, we get 𝑧𝜃𝑘𝛽𝛼𝑧𝜃+𝑘𝛽𝛼=𝑧𝜃𝑘𝛽𝛼𝑧𝜃𝑘𝛽𝑞𝑘𝑥𝜎𝜃𝑘𝛼(𝛽𝛼)/𝛽=𝑧𝜃𝑘𝛽𝛼11𝑢𝑘(𝛽𝛼)/𝛽,(2.15) where 𝑢𝑘=𝑞𝑘𝑥𝜎𝜃𝑘𝛼𝑧𝜃𝑘𝛽.(2.16) Note that 0<(𝛽𝛼)/𝛽<1, 1(1𝑢𝑘)(𝛽𝛼)/𝛽((𝛽𝛼)/𝛽)𝑢𝑘 for 1𝑢𝑘0. Consequently, we see from (2.7), (2.8), and (2.15) that 𝑧𝜃𝑘𝛽𝛼𝑧𝜃+𝑘𝛽𝛼𝛽𝛼𝛽𝑞𝑘𝑥𝜎𝜃𝑘𝛼𝑧𝜃𝑘𝛼(𝛽𝛼)2𝛼𝛽𝑞𝑘𝜎𝜃1𝜆𝑘𝛼𝑅𝛽𝜃𝑘𝛼.(2.17) Substituting (2.17) into (2.14) yields []1𝜆(𝜎(𝑡))𝛼𝑅𝛽(𝑡)𝛼𝑝(𝑡)𝑑𝑡+𝜎𝜃1𝜆𝑘𝛼𝑔𝜃𝑘𝑟𝜃𝑘𝑅𝛽𝜃𝑘𝛼𝑞𝑘<,(2.18) which contradicts (2.2). This completes the proof.

Theorem 2.2. Let 1<𝜇𝜆(𝑡)0. If there exists a positive function 𝑔𝐶1[𝑡0,) such that (2.1) holds and 𝑅𝛽(𝑡)𝛼𝑔(𝑡)𝑝(𝑡)𝑑𝑡+𝑔𝜃𝑘𝑟𝜃𝑘𝑅𝛽𝜃𝑘𝛼𝑞𝑘=,(2.19) where 𝑅𝛽(𝑡) is defined as in Theorem 2.1, then every solution of (1.1) is either oscillatory or tends to zero.

Proof. Suppose to the contrary that there is a solution 𝑥(𝑡) of (1.1) which is neither oscillatory nor tends to zero. Without loss of generality, we may let 𝑥(𝜏(𝑡))>0 and 𝑥(𝜎(𝑡))>0 for 𝑡𝑡1𝑡0. Thus, 𝑟(𝑡)𝜙𝛽(𝑧(𝑡)) is nonincreasing for 𝑡𝑡1. As a result, 𝑧(𝑡) and 𝑧(𝑡) are eventually of constant sign. Now, we consider the following two cases: (i) 𝑧(𝑡)>0 eventually; (ii) 𝑧(𝑡)<0 eventually. For the case (i), similar to the analysis as in the proof of Theorem 2.1, we have 𝑧(𝑡)>0 eventually and (2.6) holds. Notice that 𝑥(𝑡)=𝑧(𝑡)𝑝(𝑡)𝑥(𝜏(𝑡))𝑧(𝑡) because 𝑝(𝑡)0; from (1.1) and (2.6), we get 𝑟(𝑡)𝜙𝛽𝑧(𝑡)+2𝛼𝑅𝛽(𝑡)𝛼𝑧𝑝(𝑡)(𝑡)𝛼0.(2.20) Following the similar arguments as in the proof of Theorem 2.1, we can get a contradiction with (2.19).
For the case (ii), assume that 𝑧(𝑡)<0 for 𝑡𝑡2𝑡1. It must now hold that 𝜏(𝑡)<𝑡 for 𝑡𝑡2. Let us consider two cases: (a) 𝑥(𝑡) is unbounded; (b) 𝑥(𝑡) is bounded. If 𝑥(𝑡) is unbounded, then we have 𝑥(𝑡)=𝑧(𝑡)𝑝(𝑡)𝑥(𝜏(𝑡))<𝑝(𝑡)𝑥(𝜏(𝑡))<𝑥(𝜏(𝑡)),𝑡𝑡2.(2.21) On the other hand, there exists a sequence {𝑇𝑛} satisfying lim𝑛𝑇𝑛=, lim𝑛𝑥(𝑡)=, and max𝑇1𝑡𝑇𝑛𝑥(𝑡)=𝑥(𝑇𝑛). Let 𝑡𝑛 be sufficiently large such that 𝑇𝑛>𝑡2 and 𝜏(𝑇𝑛)>𝑇1. Then, we have max𝜏(𝑇𝑛)𝑡𝑇𝑛𝑥(𝑡)=𝑥(𝑇𝑛) which contradicts (2.21). If 𝑥(𝑡) is bounded, then we can prove that lim𝑡𝑥(𝑡)=0. In fact, 0limsup𝑡𝑧(𝑡)=limsup𝑡[]𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡))limsup𝑡𝑥(𝑡)+limsup𝑡𝑝(𝑡)𝑥(𝜏(𝑡))limsup𝑡𝑥(𝑡)𝜇limsup𝑡𝑥(𝜏(𝑡))(1𝜇)limsup𝑡𝑥(𝑡),(2.22) which implies that lim𝑡𝑥(𝑡)=0 since 1𝜇>0. This is a contradiction. The proof of Theorem 2.2 is complete.

When there is no impulse, (1.1) reduces to 𝑟(𝑡)𝜙𝛽𝑧(𝑡)+𝑝(𝑡)𝜙𝛼(𝑥(𝜎(𝑡)))=0,𝑡𝑡0.(2.23) The following oscillation results for (2.23) are immediate.

Corollary 2.3. Let 0𝜆(𝑡)1. If there exists a positive function 𝑔𝐶1[𝑡0,) such that (2.1) holds and []1𝜆(𝜎(𝑡))𝛼𝑅𝛽(𝑡)𝛼𝑔(𝑡)𝑝(𝑡)𝑑𝑡=,(2.24) where 𝑅𝛽(𝑡) is the same as in Theorem 2.1, then (2.23) is oscillatory.

Corollary 2.4. Let 1<𝜇𝜆(𝑡)0. If there exists a positive function 𝑔𝐶1[𝑡0,) such that (2.1) holds and 𝑅𝛽(𝑡)𝛼𝑔(𝑡)𝑝(𝑡)𝑑𝑡=,(2.25) where 𝑅𝛽(𝑡) is the same as in Theorem 2.1, then every solution of (2.23) is either oscillatory or tends to zero.

Next, we introduce the function defined in [18] to further study oscillation of (1.1). Say that 𝐻(𝑡,𝑠) defined on 𝐷={(𝑡,𝑠)𝑡𝑠𝑡0} belongs to the function class 𝒳 if 𝜕𝐻/𝜕𝑠𝐿loc(𝐷,𝑅), 𝐻(𝑡,𝑡)=0, 𝐻(𝑡,𝑠)0, and (𝜕𝐻/𝜕𝑠)(𝑡,𝑠)0 for (𝑡,𝑠)𝐷.

Theorem 2.5. Let 0𝜆(𝑡)1. If there exist a positive function 𝑔𝐶1[𝑡0,) and 𝐻𝒳 such that (2.1) holds and 1𝐻𝑡,𝑡0𝑡𝑡0[]𝐻(𝑡,𝑠)1𝜆(𝜎(𝑠))𝛼𝑅𝛽(𝑠)𝛼+1𝑔(𝑠)𝑝(𝑠)𝑑𝑠𝐻𝑡,𝑡0𝑡0𝜃𝑘<𝑡𝐻𝑡,𝜃𝑘𝜎𝜃1𝜆𝑘𝛼𝑔𝜃𝑘𝑟𝜃𝑘𝑅𝛽𝜃𝑘𝛼𝑞𝑘=,(2.26) where 𝑅𝛽(𝑡) is defined as in Theorem 2.1, then (1.1) is oscillatory.

Proof. Suppose to the contrary that (1.1) has a nonoscillatory solution 𝑥(𝑡). Without loss of generality, we may assume that 𝑥(𝜏(𝑡))>0 and 𝑥(𝜎(𝑡))>0 for 𝑡𝑡1𝑡0. Similar to the proof of Theorem 2.1, we have that (2.12) holds. Multiplying 𝐻(𝑡,𝑠) on both sides of (2.12) and integrating it from 𝑡3 to 𝑡, we get 𝑡3𝜃𝑘<𝑡𝐻𝑡,𝜃𝑘𝑔𝜃𝑘𝑟𝜃𝑘𝑧𝜃𝑘𝛽𝛼𝑧𝜃+𝑘𝛽𝛼𝑡𝑡3𝜕𝐻(𝑡,𝑠)𝑧𝜕𝑠𝑔(𝑠)𝑟(𝑠)(𝑠)𝛽𝛼𝑑𝑠𝐻𝑡,𝑡3𝑔𝑡3𝑟𝑡3𝑧𝑡3𝛽𝛼+(𝛽𝛼)2𝛼𝛽𝑡𝑡3[]𝐻(𝑡,𝑠)1𝜆(𝜎(𝑠))𝛼𝑅𝛽(𝑠)𝛼𝑔(𝑠)𝑝(𝑠)𝑑𝑠0,𝑡𝑡3,(2.27) which implies that 𝑡3𝜃𝑘<𝑡𝐻𝑡,𝜃𝑘𝑔𝜃𝑘𝑟𝜃𝑘𝑧𝜃𝑘𝛽𝛼𝑧𝜃+𝑘𝛽𝛼+(𝛽𝛼)2𝛼𝛽𝑡𝑡3[]𝐻(𝑡,𝑠)1𝜆(𝜎(𝑠))𝛼𝑅𝛽(𝑠)𝛼𝑔(𝑠)𝑝(𝑠)𝑑𝑠𝐻𝑡,𝑡3𝑔𝑡3𝑟𝑡3𝑧𝑡3𝛽𝛼.(2.28) Therefore, 𝑡0𝜃𝑘<𝑡𝐻𝑡,𝜃𝑘𝑔𝜃𝑘𝑟𝜃𝑘𝑧𝜃𝑘𝛽𝛼𝑧𝜃+𝑘𝛽𝛼+(𝛽𝛼)2𝛼𝛽𝑡𝑡0[]𝐻(𝑡,𝑠)1𝜆(𝜎(𝑠))𝛼𝑅𝛽(𝑠)𝛼𝑔(𝑠)𝑝(𝑠)𝑑𝑠𝐻𝑡,𝑡0𝑔𝑡3𝑟𝑡3𝑧𝑡3𝛽𝛼+𝑡0𝜃𝑘<𝑡3𝐻𝑡,𝜃𝑘𝑔𝜃𝑘𝑟𝜃𝑘𝑧𝜃𝑘𝛽𝛼𝑧𝜃+𝑘𝛽𝛼+(𝛽𝛼)2𝛼𝛽𝑡3𝑡0[]𝐻(𝑡,𝑠)1𝜆(𝜎(𝑠))𝛼𝑅𝛽(𝑠)𝛼𝑔(𝑠)𝑝(𝑠)𝑑𝑠.(2.29) Proceeding as in the proof of Theorem 2.1, we get a contradiction with (2.26). This completes the proof.

For the case 1𝜇𝜆(𝑡)0, we have the following oscillation result. Since the proof is similar to that of Theorem 2.2, we omit it here.

Theorem 2.6. Let 1𝜇𝜆(𝑡)0. If there exist a positive function 𝑔𝐶1[𝑡0,) and 𝐻𝒳 such that (2.1) holds and 1𝐻𝑡,𝑡0𝑡𝑡0𝑅𝐻(𝑡,𝑠)𝛽(𝑠)𝛼𝑔(𝑠)𝑝(𝑠)𝑑𝑠+𝑡0𝜃𝑘<𝑡𝐻𝑡,𝜃𝑘𝑔𝜃𝑘𝑟𝜃𝑘𝑅𝛽𝜃𝑘𝛼𝑞𝑘=,(2.30) where 𝑅𝛽(𝑡) is defined as in Theorem 2.1, then every solution of (1.1) is either oscillatory or tends to zero.

The following two corollaries for (2.23) are immediate.

Corollary 2.7. Let 0𝜆(𝑡)1. If there exist a positive function 𝑔𝐶1[𝑡0,) and 𝐻𝒳 such that (2.1) holds and 1𝐻𝑡,𝑡0𝑡𝑡0[]𝐻(𝑡,𝑠)1𝜆(𝜎(𝑠))𝛼𝑅𝛽(𝑠)𝛼𝑔(𝑠)𝑝(𝑠)𝑑𝑠=,(2.31) where 𝑅𝛽(𝑡) is defined as in Theorem 2.1, then (2.23) is oscillatory.

Corollary 2.8. Let 1𝜇𝜆(𝑡)0. If there exist a positive function 𝑔𝐶1[𝑡0,) and 𝐻𝒳 such that (2.1) holds and 1𝐻𝑡,𝑡0𝑡𝑡0𝑅𝐻(𝑡,𝑠)𝛽(𝑠)𝛼𝑔(𝑠)𝑝(𝑠)𝑑𝑠=,(2.32) where 𝑅𝛽(𝑡) is defined as in Theorem 2.1, then every solution of (2.23) is either oscillatory or tends to zero.

3. Examples

We now present two examples to illustrate the effect of impulses on oscillation of solutions of (1.1).

Example 3.1. Consider the following impulsive delay differential equation: ||𝑧||𝑧(𝑡)(𝑡)𝑡+𝑡2||||𝑥(𝑡1)1/2Δ||𝑧𝑥(𝑡1)=0,𝑡𝑘,(||𝑧𝑡)(𝑡)𝑡=𝑘+𝑘1/2||||𝑥(𝑘1)1/2𝑥(𝑘1)=0,Δ𝑥(𝑡)𝑡=𝑘=0,(3.1) where 𝑧(𝑡)=𝑥(𝑡)+𝜆𝑥(𝑡1), 𝜆 is a constant, 𝑡2, and 𝑘2. We see that 𝜏(𝑡)=𝜎(𝑡)=𝑡1, 𝑟(𝑡)=1/𝑡, 𝛽=2, 𝛼=1/2, 𝑝(𝑡)=𝑡2, 𝑞𝑘=𝑘1/2,and𝜃𝑘=𝑘. Let 𝑔(𝑡)=1. A straightforward computation yields 𝑅𝛽(𝑡)=(2/3)𝑡1/2[(𝑡1)3/223/2]. Therefore, when 0<𝜆<1, it is not difficult to verify that (2.1) and (2.2) hold. Thus, (3.1) is oscillatory by Theorem 2.1. However, when there is no impulse in (3.1), Corollary 2.3 cannot guarantee the oscillation of (3.1) since condition (2.24) is invalid for this case. Therefore, the impulsive perturbations may greatly affect the oscillation of (3.1). If 1<𝜆<0, then we have that every solution of (3.1) is either oscillatory or tends to zero by Theorem 2.2. Such behavior of solutions of (3.1) is determined by the impulsive perturbations to a great extent, since Corollary 2.4 fails to apply for this case.

Example 3.2. Consider the following impulsive delay differential equation: 𝑡||𝑧||𝑧(𝑡)(𝑡)+𝑡2Δ||𝑧𝑥(𝑡1)=0,𝑡𝑘,||𝑧(𝑡)(𝑡)𝑡=𝑘+𝑘2𝑥(𝑘1)=0,Δ𝑥(𝑡)𝑡=𝑘=0,(3.2) where 𝑧(𝑡)=𝑥(𝑡)+𝜆𝑥(𝑡1), 𝜆 is a constant, 𝑡2, and 𝑘2. We see that 𝜏(𝑡)=𝜎(𝑡)=𝑡1, 𝑟(𝑡)=𝑡, 𝛽=2, 𝛼=1, 𝑝(𝑡)=𝑡2, 𝑞𝑘=𝑘2,and𝜃𝑘=𝑘. Let 𝑔(𝑡)=𝑡1/2. It is not difficult to verify that (2.1) and (2.2) hold if 0<𝜆<1, which implies that (3.2) is oscillatory by Theorem 2.1. We also can verify that and (2.1) and (2.19) hold if 1<𝜆<0. Thus, by Theorem 2.2, every solution of (3.2) is either oscillatory or tends to zero. However, Corollaries (1.1) and (2.1) do not apply for this case. Therefore, the impulsive perturbations play a key role in the oscillation problem of (3.2).

Acknowledgments

The author thanks the reviewers for their valuable comments and suggestions on this paper. This paper was supported by the Natural Science Foundation of Shandong Province under the Grant ZR2010AL002.

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