Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 405397 8 pageshttpdxdoiorg1011552013405397

Research ArticleOscillation Criteria of Third-Order Nonlinear ImpulsiveDifferential Equations with Delay

Xiuxiang Liu

School of Mathematical Sciences South China Normal University Guangzhou Guangdong 510631 China

Correspondence should be addressed to Xiuxiang Liu liuxxscnueducn

Received 14 November 2012 Accepted 10 January 2013

Academic Editor Norio Yoshida

Copyright copy 2013 Xiuxiang LiuThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper deals with the oscillation of third-order nonlinear impulsive equations with delayThe results in this paper improve andextend some results for the equations without impulses Some examples are given to illustrate the main results

1 Introduction

In this paper we are concerned with oscillation of the third-order nonlinear impulsive equations with delay

119909101584010158401015840

(119905) + 119891 (119905 119909 (119905) 119909 (119905 minus 120590)) = 0 119905 ge 1199050 119905 = 120591

119896

119909 (120591+

119896) = 119886119896119909 (120591119896) 119909

1015840(120591+

119896) = 1198871198961199091015840(120591119896)

11990910158401015840(120591+

119896) = 11988811989611990910158401015840(120591119896) 119896 = 1 2

(1)

where 120590 gt 0 is the delay 120591119896 is the sequence of impulsive

moments which satisfies 0 le 1199050

lt 1205911lt sdot sdot sdot lt 120591

119896lt sdot sdot sdot and

lim119896rarrinfin

120591119896= infin 120591

119896+1minus 120591119896ge 120590

Throughout this paper we will assume that the followingassumptions are satisfied

(H1) 119891(119905 119906 119907) is continuous in [1199050

minus 120590infin) times R times R119906119891(119905 119906 119907) gt 0 for 119906119907 gt 0

(H2) 119891(119905 119906 119907)120593(119907) ge 119901(119905) for 119907 = 0 where 119901(119905) is continu-ous in [119905

0minus120590infin) 119901(119905) ge 0( equiv 0) 120593(119909)119909 ge 120583 gt 0 for

all 119909 = 0(H3) 119886

119896 119887119896 and 119888

119896are positive constants

Our attention is restricted to those solutions of (1) whichexist on half line [119905

0infin) and satisfy sup|119909(119905) 119905 gt 119879 gt

0 for any 119879 ge 119905119909 For the general theory of impulsive

differential equations withwithout delay we refer the readersto monographs or papers [1ndash4] A solution of (1) is saidto be oscillatory if it has arbitrarily large zeros otherwise

it is nonoscillatory It is well known that there is a drasticdifference in the behavior of solutions between differen-tial equations with impulses and those without impulsesSome differential equations are nonoscillatory but they maybecome oscillatory if some proper impulse controls areadded to them see [5] and Example 13 in Section 4 In therecent years the oscillation theory and asymptotic behaviorof impulsive differential equations and their applicationshave been and still are receiving intensive attention Forcontribution we refer to the recent survey paper by Agarwalet al [6] and the references cited therein But to the best ofthe authorsrsquo knowledge it seems that little has been done foroscillation of third-order impulsive differential equations [7]

When 119886119896

= 119887119896

= 119888119896

= 1 (1) reduces third-order delayequation withwithout delay which oscillatory theory hasbeen studied by many researchers see [8ndash12]

Our aim in this paper is to establish some new sufficientconditions which ensure that the solutions of (1) oscillate orconverge to a finite limit as 119905 tends to infinity In particular weextend some results in [9 11] to impulsive delay differentialequations The results in this paper are more general com-pared by those obtained by Mao and Wan [7] and improvesome of the results in [7] (see Example 13 in Section 4) Thenew results will be proved by making use of the techniquesused in [9 11]

The paper is organized as follows In Section 2 we provesome lemmas which play important roles in the proof ofthe main results In Section 3 some new sufficient condi-tions which guarantee that the solution of (1) oscillates or

2 Abstract and Applied Analysis

converges to a finite limit are established In Section 4 twoexamples are given to illustrate the main results

2 Preliminary Results

In this section we state and prove some lemmas which wewill need in the proofs of the main results First of all weintroduce the following notations R+ and N are the sets ofreal numbers and positive integer numbers respectively PC1is defined by

PC1 (R+R)

= 119909 R+997888rarr R 119909 (119905) is differentiable for

119905 ge 0 and 119905 = 120591119896 119909 (120591+

119896) and 119909

1015840(120591+

119896) exist

and 119909 (120591minus

119896) = 119909 (120591

119896) 1199091015840(120591minus

119896) = 1199091015840(120591119896)

(2)

The following lemma is from Lakshmikantham et al [3 Page32 Theorem 141]

Lemma 1 Assume that

(i) 120591119896119896isinN is the impulse moments sequence with 0 le 119905

0lt

1205911lt sdot sdot sdot lt 120591

119896lt sdot sdot sdot lim

119896rarrinfin120591119896= infin

(ii) 119898 isin PC1(R+R) and for 119905 ge 1199050 119896 isin N it holds that

1198981015840

(119905) le 119906 (119905)119898 (119905) + 119907 (119905) 119905 = 120591119896

119898 (120591+

119896) le 119889119896119898(120591119896) + 119890119896

(3)

where 119906 119907 isin 119862(R+R) 119889119896ge 0 and 119890

119896are real constantsThen

119898(119905) le 119898 (1199050) prod

1199050lt120591119896lt119905

119889119896exp(int

119905

1199050

119906 (119904)d119904)

+ int

119905

1199050

prod

119904lt120591119896lt119905

119889119896exp(int

119905

119904

119906 (120590)d120590) 119907 (119904)d119904

+ sum

1199050lt120591119896lt119905

prod

120591119896lt120591119895lt119905

119889119895exp(int

119905

120591119896

119906 (119904)d119904) 119890119896

(4)

Motivated by the ideas of Chen and Feng [5] we presentthe following key lemma which determines the sign of 1199091015840(119905)and 119909

10158401015840(119905) of the nonoscillation solution 119909(119905) of (1)

Lemma 2 Suppose that 119909(119905) is an eventually positive solutionof (1) and

lim119905rarrinfin

int

119905

1199050

prod

1199050lt120591119896lt119904

119887119896

119886119896

d119904 = infin (5)

lim119905rarrinfin

int

119905

1199050

prod

1199050lt120591119896lt119904

119888119896

119887119896

d119904 = infin (6)

Then it holds that one of the following two cases for sufficientlylarge 119879

(i) 11990910158401015840(120591+

119896) gt 0 11990910158401015840(119905) gt 0 and 119909

1015840(120591+

119896) gt 0 119909

1015840(119905) gt 0

(ii) 11990910158401015840(120591+

119896) gt 0 11990910158401015840(119905) gt 0 and 119909

1015840(120591+

119896) lt 0 119909

1015840(119905) lt 0

with 119905 isin (120591119896 120591119896+1

] and 120591119896ge 119879

Proof Assume that 119909(119905) is an eventually positive solution of(1)Wemay assume that there exists 119905

1gt 1199050such that 119909(119905) gt 0

and 119909(119905 minus 120590) gt 0 for 119905 ge 1199051 First we assert that 11990910158401015840(120591

119896) gt 0 for

any 119896 isin N Suppose not there exists some 120591119895ge 1199051such that

11990910158401015840(120591119895) le 0 By

119909101584010158401015840

(119905) = minus119891 (119905 119909 (119905) 119909 (119905 minus 120590)) le 0 equiv 0 (7)

we have 11990910158401015840(119905)monotonically decreasing in (120591119894 120591119894+1

] 119894 = 119895 119895+

1 Thus 11990910158401015840(120591+119894) = 11988811989411990910158401015840(120591119894) lt 0 119894 = 119895+1 119895+2 Consider

the impulsive differential inequalities

119909101584010158401015840

(119905) lt 0 119905 ge 120591119895+1

119905 = 120591119896

11990910158401015840(120591+

119896) le 11988811989611990910158401015840(120591119896) 119896 = 119895 + 1 119895 + 2

(8)

By Lemma 1 we have

11990910158401015840

(119905) le 11990910158401015840(120591+

119895+1) prod

120591119895+1lt120591119896lt119905

119888119896= minus120572 prod

120591119895+1lt120591119896lt119905

119888119896lt 0 (9)

There are two cases of the sign of 1199091015840(120591119896)

Case 1 If there exists some 120591119899

ge 120591119895+1

such that 1199091015840(120591119899) le 0

since 11990910158401015840(119905) lt 0 then 119909

1015840(120591119899+1

) lt 1199091015840(120591+

119899) le 0 and 119909

1015840(120591+

119899+1) =

120573 = 119887119899+1

1199091015840(120591119899+1

) lt 0 By induction it easily show 1199091015840(120591119896) lt 0

and hence 1199091015840(120591+

119896) le 1198871198961199091015840(120591119896) lt 0 for 119896 = 119899 + 1 119899 + 2 So

we obtain the following impulsive differential inequalities

11990910158401015840

(119905) le 0 119905 ge 120591119899+1

119905 = 120591119896

1199091015840(120591+

119896) le 119887lowast

1198961199091015840(120591119896) 119896 = 119899 + 1 119899 + 2

(10)

which follows from Lemma 1 that

1199091015840

(119905) le 1199091015840(120591+

119899+1) prod

120591119899+1lt120591119896lt119905

119887119896= minus120573 prod

120591119899+1lt120591119896lt119905

119887119896 (11)

From (11) and applying Lemma 1 noting that 119909(119905) gt 0 for119905 gt 1199051and 119909(120591

+

119896) le 119886119896119909(120591119896) for any 119896 isin N we have

119909 (119905) le 119909 (120591+

119899+1) prod

120591119899+1lt120591119896lt119905

119886119896minus 120573int

119905

120591119899+1

prod

119904lt120591119896lt119905

119886119896

prod

120591119899+1lt120591119896lt119904

119887119896d119904

= prod

120591119899+1lt120591119896lt119905

119886119896(119909 (120591

+

119899+1) minus 120573int

119905

120591119899+1

prod

120591119899+1lt120591119896lt119904

119887119896

119886119896

d119904)

(12)

Thus by (5) we have 119909(119905) lt 0 for 119905 sufficiently large which isa contradiction

Case 2 If 1199091015840(120591119896) gt 0 for any 119896 ge 119895 noting that 11990910158401015840(119905) lt 0 for

119905 isin (120591119895+1

120591119895+2

] we have 1199091015840(119905) gt 119909

1015840(120591119895+1

) gt 0 By induction

Abstract and Applied Analysis 3

we get that 1199091015840(119905) gt 0 for any 119905 isin (120591119896 120591119896+1

] 119896 = 119895 + 1 119895 + 2 So the following impulsive differential inequalities hold

11990910158401015840

(119905) le minus120572 prod

120591119895+1lt120591119896lt119905

119888119896 119905 gt 120591

119895+1 119905 = 120591

119896

1199091015840(120591+

119896) le 119887119896119909 (120591119896) 119896 = 119895 + 1 119895 + 2

(13)

According to Lemma 1 we get

1199091015840

(119905) le 1199091015840(120591+

119895+1) prod

120591119895+1lt120591119896lt119905

119887119896minus 120572int

119905

120591119895+1

prod

119904lt120591119896lt119905

119887119896

prod

120591119895+1lt120591119896lt119904

119888119896d119904

= prod

120591119895+1lt120591119896lt119905

119887119896(1199091015840(120591+

119895+1) minus 120572int

119905

120591119895+1

prod

120591119895+1lt120591119896lt119904

119888119896

119887119896

d119904)

(14)

Hence the condition (6) implies that 1199091015840(119905) lt 0 when 119905 is

sufficiently large which contradicts to 1199091015840(119905) gt 0 for 119905 gt 120591

119895+1

again In terms of the above discussion we see that11990910158401015840(120591119896) gt 0

for any 120591119896gt 119879 with sufficiently large 119879 Consequently noting

that 119909101584010158401015840(119905) lt 0 for any 119905 isin (120591

119896 120591119896+1

] we have 11990910158401015840(119905) gt

11990910158401015840(120591119896+1

) gt 0Next if there exists a 120591

119895gt 119879 such that 1199091015840(120591

119895) ge 0 then

1199091015840(120591+

119895) = 119887119895(119909(120591+

119895) ge 0 1199091015840(120591

119895+1) gt 119909

1015840(120591+

119895) ge 0 Therefore by

induction we have 1199091015840(119905) gt 1199091015840(120591+

119894) gt 0 for 119905 isin (120591

119894 120591119894+1

] 119894 = 119895 +

1 119895 + 2 So case (i) is satisfied Otherwise if 1199091015840(120591119896) lt 0 for

all 120591119896ge 119879 then 119909

1015840(120591+

119896) = 119887119896119909(120591+

119896) lt 0 Thus for 119905 isin (120591

119896 120591119896+1

]using 119909

10158401015840(119905) gt 0 we have 1199091015840(119905) lt 119909

1015840(120591119896+1

) lt 0 hence case (ii)is satisfied This completes the proof

Remark 3 Suppose that 119909(119905) is an eventually negative solu-tion of (1) If (5) and (6) hold one can prove it holds that oneof the following two cases in a similar way as Lemma 2

(i) 11990910158401015840(120591+

119896) lt 0 11990910158401015840(119905) lt 0 and 119909

1015840(120591+

119896) lt 0 1199091015840(119905) lt 0

(ii) 11990910158401015840(120591+

119896) lt 0 11990910158401015840(119905) lt 0 and 119909

1015840(120591+

119896) gt 0 1199091015840(119905) gt 0 with

119905 isin (120591119896 120591119896+1

] and 120591119896ge 119879

Lemma 4 Let 119909(119905) be a piecewise continuous function oncup119896isinN(120591119896 120591119896+1] which is continuous at 119905 = 120591

119896and is left contin-

uous at 119905 = 120591119896 If

(1) 119909(119905) ge 0 (le 0) for 119905 ge 1199050

(2) 119909(119905) is monotone nonincreasing (monotone nonde-creasing) on (120591

119896 120591119896+1

] (120591119896ge 119879) for 119879 large enough

(3) suminfin

119896=1[119909(120591+

119896) minus 119909(120591

119896)] converges

then lim119905rarrinfin

119909(119905) = 119886 ge 0 (le 0)

The proof of Lemma 4 is similar to that of [13 Theorem5] and hence is omitted

Lemma 5 Assume that 119909(119905) is a solution of (1) which satisfiescase (ii) in Lemma 2 In addition if

infin

sum

119896=1

1003816100381610038161003816119886119896minus 1

1003816100381610038161003816lt infin

infin

prod

119896=1

119886119896119894119904 119887119900119906119899119889119890119889 (15)

then lim119905rarrinfin

119909(119905) exists (finite)

Proof First we claim thatsuminfin119896=1

[119909(120591+

119896)minus119909(120591

119896)] is convergence

In fact since 119909(119905) is decreasing on (120591119896 120591119896+1

] (120591119896ge 119879 and 119879 is

defined in Lemma 2) then

119909 (120591119896+1

) le 119909 (120591+

119896) 119909 (120591

+

119896+1) le 119886119896+1

119909 (120591119896+1

) le 119886119896+1

119909 (120591+

119896)

(16)

Obviously by induction we can get

119909 (120591119896+119899

) le 119886119896+119899minus1

sdot sdot sdot 119886119896+1

119909 (120591+

119896)

119909 (120591+

119896+119899) le 119886119896+119899

sdot sdot sdot 119886119896+1

119909 (120591+

119896)

(17)

Since prodinfin

119896=1119886119896

is bounded we conclude that 119909(120591119896) is

bounded which follows that there exists1198721gt 0 such that

1003816100381610038161003816119909 (120591+

119896) minus 119909 (120591

119896)1003816100381610038161003816=

1003816100381610038161003816119886119896minus 1

1003816100381610038161003816119909 (120591119896) le 119872

1

1003816100381610038161003816119886119896minus 1

1003816100381610038161003816 (18)

Hencesuminfin119896=1

[119909(120591+

119896)minus119909(120591

119896)] is convergence sincesuminfin

119896=1|119886119896minus1| is

convergence which follows from Lemma 4 that lim119905rarrinfin

119909(119905)

exists The proof is complete

3 Main Results

In this section we establish some sufficient conditions whichguarantee that every solution 119909(119905) of (1) either oscillates orhas a finite limit Occasionally we will make the additionalassumption

lim sup119905rarrinfin

int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) d120579 d119904 d119906 = infin (19)

where here it is understood that

int

infin

1199050

119901 (119905) d119905 lt infin int

infin

1199050

int

infin

119906

119901 (120579) d120579 d119906 lt infin (20)

Now we are ready to state and prove the main results inthis paper The results will be proved by making use of thetechnique in [11]

Theorem 6 Assume that (5) (6) and (19) hold and 119909(119905) is asolution of (1) Furthermore assume that 119886

119896le 1 119887

119896ge 1 and

119888119896le 1 for 119896 isin N If there exists a positive differentiable function

119903 such that

lim119905rarrinfin

int

119905

1199050

[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)

)] prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896d119904 = infin

(21)

where 119896(119904) = max119896 120591119896lt 119904 and

119860 (119904 120591119896(119904)

)

=

119888119896(119904)[1199031015840(119904)]2

4120583119903 (119904) (119904 minus 120591119896(119904))

119904 isin (120591119896 120591119896+ 120590]

[1199031015840(119904)]2

4120583119903 (119904) (119904 minus 120591119896(119904)minus ((119888119896(119904)minus 1) 119888

119896(119904)) 120590)

119904 isin (120591119896+ 120590 120591119896+1]

(22)

Then 119909(119905) is oscillatory

4 Abstract and Applied Analysis

Proof Let 119909(119905) be a nonoscillatory solution of (1) withoutloss of generality we may assume that 119909(119905) gt 0 eventually(119909(119905) lt 0 eventually can be achieved in the similar way)By Lemma 2 either case (i) or case (ii) in Lemma 2 holdsAssume that 119909(119905) satisfies case (i) then 119909

10158401015840(119905) gt 0 1199091015840(119905) gt 0

for 119905 isin (120591119895 120591119895+1

] 120591119895ge 119879 (119879 is defined in Lemma 2) Define the

Riccati transformation 119906 by

119906 (119905) =

119903 (119905) 11990910158401015840(119905)

120593 (119909 (119905 minus 120590))

119905 ge 120591119895 119905 = 120591

119896 (23)

Thus 119906(119905) gt 0 for 119905 isin (120591119896 120591119896+1

] 119896 = 119895 119895 + 1 and

1199061015840

(119905) =

119903 (119905)

120593 (119909 (119905 minus 120590))

119909101584010158401015840

(119905)

+

1199031015840(119905) 120593 (119909 (119905 minus 120590)) minus 119903 (119905) 120593

1015840(119909 (119905 minus 120590)) 119909

1015840(119905 minus 120590)

1205932(119909 (119905 minus 120590))

times 11990910158401015840

(119905)

le minus 119901 (119905) 119903 (119905) +

1199031015840(119905)

119903 (119905)

119906 (119905)

minus

119903 (119905) 1205831199091015840(119905 minus 120590)

1205932(119909 (119905 minus 120590))

11990910158401015840

(119905)

(24)

If 119905 isin (120591119896 120591119896+120590] sub (120591

119896 120591119896+1

] namely 119905 minus120590 le 120591119896lt 119905 11990910158401015840(119905)

is decreasing in (119905 minus 120590 120591119896] and (120591

119896 119905] respectively In view of

the following

11990910158401015840

(119905 minus 120590) gt 11990910158401015840(120591119896) ge

11990910158401015840(120591+

119896)

119888119896

gt

11990910158401015840(119905)

119888119896

(25)

we have

1199091015840

(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int

119905minus120590

120591119896minus120590

11990910158401015840

(119904) d119904

gt 11990910158401015840

(119905 minus 120590) (119905 minus 120591119896) gt

11990910158401015840(119905)

119888119896

(119905 minus 120591119896)

(26)

Thus

1199061015840

(119905) le minus 119901 (119905) 119903 (119905) +

1199031015840(119905)

119903 (119905)

119906 (119905) minus

120583119903 (119905) (119905 minus 120591119896)

1198881198961205932(119909 (119905 minus 120590))

[11990910158401015840

(119905)]

2

= minus 119901 (119905) 119903 (119905) +

1199031015840(119905)

119903 (119905)

119906 (119905) minus

120583 (119905 minus 120591119896)

119888119896119903 (119905)

[119906 (119905)]2

= minus (radic

120583 (119905 minus 120591119896)

119888119896119903 (119905)

119906 (119905) minus radic

119888119896

4120583119903 (119905) (119905 minus 120591119896)

1199031015840

(119905))

2

minus[

[

119901 (119905) 119903 (119905) minus

119888119896[1199031015840(119905)]

2

4120583119903 (119905) (119905 minus 120591119896)

]

]

le minus[

[

119901 (119905) 119903 (119905) minus

119888119896[1199031015840(119905)]

2

4120583119903 (119905) (119905 minus 120591119896)

]

]

(27)

If 119905 isin (120591119896+120590 120591119896+1

] sub (120591119896 120591119896+1

] that is 120591119896lt 119905minus120590 lt 119905 le 120591

119896+1

then

1199091015840

(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int

120591119896

120591119896minus120590

11990910158401015840

(119904) d119904 + int

119905minus120590

120591119896

11990910158401015840

(119904) d119904

ge 11990910158401015840(120591119896) 120590 + 119909

10158401015840

(119905 minus 120590) (119905 minus 120591119896minus 120590)

ge

11990910158401015840(119905)

119888119896

120590 + 11990910158401015840

(119905) (119905 minus 120591119896minus 120590)

ge 11990910158401015840

(119905) (119905 minus 120591119896minus

119888119896minus 1

119888119896

120590)

(28)

Similarly we have

1199061015840

(119905) le minus[

[

119901 (119905) 119903 (119905) minus

[1199031015840(119905)]

2

4120583119903 (119905) (119905 minus 120591119896minus ((119888119896minus 1) 119888

119896) 120590)

]

]

(29)

Thus we obtain

1199061015840

(119905) le minus [119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)

)] for 119905 isin (120591119896 120591119896+1

]

(30)

On the other hand

119906 (120591+

119896) =

119903 (120591119896) 11990910158401015840(120591+

119896)

120593 (119909 (120591119896minus 120590))

le 119888119896119906 (120591119896) (31)

Observing that 120593(119906) ge 120583119906 we have

119906 (120591+

119896+ 120590) =

119903 (120591119896+ 120590) 119909

10158401015840(120591119896+ 120590)

120593 (119909 (120591+

119896))

le

119903 (120591119896+ 120590) 119909

10158401015840(120591119896+ 120590)

120593 (119886119896119909 (120591119896))

le

119906 (120591119896+ 120590)

120583119886119896

(32)

Applying Lemma 1 it follows from (30) (31) and (32) that

119906 (119905) le 119906 (120591+

119895) prod

120591119895lt120591119896lt119905

119888119896

prod

120591119895lt120591119896+120590lt119905

1

120583119886119896

minus int

119905

120591119895

[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)

)]

sdot prod

119904lt120591119896lt119905

119888119896

prod

119904lt120591119896+120590lt119905

1

120583119886119896

d119904

le prod

120591119895lt120591119896lt119905

119888119896

prod

120591119895lt120591119896+120590lt119905

1

120583119886119896

times (119906 (120591+

119895) minus 120583int

119905

120591119895

[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)

)]

sdot prod

120591119895lt120591119896lt119904

1

119888119896

prod

120591119895lt120591119896+120590lt119904

119886119896d119904)

(33)

Abstract and Applied Analysis 5

which yields 119906(119905) lt 0 for all large 119905 This is contrary to 119906(119905) gt

0 and so case (i) in Lemma 2 is not possibleIf 119909(119905) satisfies the case (ii) in Lemma 2 that is 11990910158401015840(120591+

119896) gt

0 11990910158401015840(119905) gt 0 and 1199091015840(120591+

119896) lt 0 1199091015840(119905) lt 0 which proves that the

solution 119909(119905) is positive and decreasing Integrating (1) from119904 to 119905 (119905 ge 119904 ge 119879) we obtain

11990910158401015840

(119905) minus sum

119904lt120591119896lt119905

(119888119896minus 1) 119909

10158401015840(120591119896) minus 11990910158401015840

(119904)

+ int

119905

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0

(34)

Noting 119888119896le 1 and 119909

10158401015840(119905) gt 0 then it holds that

11990910158401015840

(119905) minus 11990910158401015840

(119904) + int

119905

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (35)

which leads to

minus11990910158401015840

(119904) + int

119905

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (36)

and hence

minus11990910158401015840

(119904) + int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (37)

Integrating the above inequality again from 119906 to 119905 (119905 ge 119906 ge 119879)one has

minus 1199091015840

(119905) + sum

119904lt120591119896lt119905

(119887119896minus 1) 119909

1015840(120591119896) + 1199091015840

(119906)

+ int

119905

119906

int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0

(38)

Using 1199091015840(119905) lt 0 and 119887

119896ge 1 we have

1199091015840

(119906) + int

infin

119906

int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0 (39)

Now we integrate the last inequality from 120578 to 119905 (119905 ge 120578 ge 119879)to obtain

119909 (119905) minus sum

120578lt120591119896lt119905

(119886119896minus 1) 119909 (120591

119896) minus 119909 (120578)

+ int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 d119906 le 0

(40)

Since 119886119896

le 1 (119896 isin N) and 119909(119905) is decreasing then for 119905 isin

(120591119896 120591119896+1

] 119909(120591119896+1

) le 119909(119905) le 119909(120591+

119896) = 119886119896119909(120591119896) le 119909(120591

119896) 119896 isin N

Thus we get

120583119909 (120578) int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) d120579 d119904 d119906 le minus119909 (119905) + 119909 (120578) le 119909 (120578)

(41)

and then

int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) d120579 d119904 d119906 le

1

120583

(42)

which contradicts the condition (19) The proof is complete

Replace the condition (19) with (15) we may obtain thefollowing asymptotic results

Theorem 7 Assume that (5) (6) and (15) hold and 119909(119905) is asolution of (1) If there exists a positive differentiable function 119903

such that (21) hold then 119909(119905) is either oscillatory or has a finitelimit

Proof By the proof of Theorem 6 we know the case (i) inLemma 2 is not possible too since the condition (19) is notrequired to prove it So it suffices to show if there is a solutionsatisfying case (ii) in Lemma 2 that is if

11990910158401015840(120591+

119896) gt 0 119909

10158401015840

(119905) gt 0 1199091015840(120591+

119896) lt 0 119909

1015840

(119905) lt 0 (43)

with 119905 isin (120591119896 120591119896+1

] and 120591119896ge 119879 then lim

119905rarrinfin119909(119905) exists This

is obtained by applying Lemma 5 which leads to lim119905rarrinfin

119909(119905)

exists The proof is complete

Corollary 8 In addition to the assumption of Theorem 7assume that

lim119905rarrinfin

int

119905

1199050

prod

1199050lt120591119896lt119904

1

119888119896

119901 (119904) d119904 = infin (44)

Then solution 119909(119905) of (1) either oscillates or satisfieslim119905rarrinfin

119909(119905) = 0

Proof By the proof ofTheorem 7 lim119905rarrinfin

119909(119905) exists and wedefine it by lim

119905rarrinfin119909(119905) = 120574 ge 0 We now show 120574 = 0 If not

then 120574 gt 0 So lim119905rarrinfin

120593(119909(119905 minus 120590)) = 120593(120574) = 120581 gt 0 Hencethere exists 120591

119895gt 119879 such that 120593(119909(119905minus120590)) gt 1205812 for 119905 gt 120591

119895Then

119909101584010158401015840

(119905) = minus119891 (119905 119909 (119905) 119909 (119905 minus 120590))

le minus119901 (119905) 120593 (119909 (119905 minus 120590)) le minus

120581

2

119901 (119905) 119905 ge 120591119895

(45)

and note that 11990910158401015840(120591+119896) le 11988811989611990910158401015840(120591119896) since 11990910158401015840(119905) gt 0 which imply

that

11990910158401015840

(119905) le 11990910158401015840(120591+

119895) prod

120591119895lt120591119896lt119905

119888119896minus

120581

2

int

119905

120591119895

prod

119904lt120591119896lt119905

119888119896119901 (119904) d119904

le prod

120591119895lt120591119896lt119905

119888119896[

[

11990910158401015840(120591+

119895) minus

120581

2

int

119905

120591119895

prod

120591119895lt120591119896lt119904

1

119888119896

119901 (119904) d119904]

]

(46)

Thus in virtue of (44) it holds that 11990910158401015840(119905) lt 0 and contradicts11990910158401015840(119905) gt 0 for 119905 large enough the proof is complete

Remark 9 Theorem 6 and Corollary 8 extend the results in[11 Theorem 31] and [9 Corollary 1] respectively In factwhen 119886

119896= 119887119896

= 119888119896

= 1 for 119896 isin N which implies that theimpulses in (1) disappear In such a case (5) and (6) holdnaturally and (21) and (44) are reduced to

lim119905rarrinfin

int

119905

1199050

[

[

119901 (119904) 119903 (119904) minus

[1199031015840(119904)]

2

4120583119903 (119904) (119904 minus 119879)

]

]

d119904 = infin

lim119905rarrinfin

int

119905

1199050

119901 (119904) d119904 = infin

(47)

6 Abstract and Applied Analysis

which are similar to those in [11 Theorem 31] and [9Corollary 1] respectively

Next we present some new oscillation results for (1) byusing an integral averaging condition of Kamenevrsquos type

Theorem 10 Assume (5) (6) and (19) hold Furthermore119886119896

le 1120583 le 1 119887119896

ge 1 and 119888119896

le 1 119896 isin N If there exists apositive differentiable function 119903 such that

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 = infin

(48)

where 119860(119904 120591119896(119904)

) is defined by (21) and 119898 ge 1 Then everysolution of (1) is oscillatory

Proof We choose 119879 large enough such that Lemma 2 holdsBy Lemma 2 there are two possible cases First if the case (i)holds proceeding as in the proof of Theorem 6 we will endup with (32) By (30) we have

119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)

) le minus1199061015840

(119905) 119905 ge 119879 119905 = 120591119896 (49)

If 119905 isin (120591119896+ 120590 120591119896+1

] sub (120591119896 120591119896+1

] for 120591119895ge 119879 we obtain

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904

le minusint

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904(50)

An integration by parts of the right-hand side leads to

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904

= (int

120591119895+120590

120591119895

+int

120591119895+1

120591119895+120590

+ sdot sdot sdot + int

120591119896+120590

120591119896

+int

119905

120591119896+120590

) (119905 minus 119904)1198981199061015840

(119904) d119904

= int

119905

120591119895

119906 (119904)

119898

(119905 minus 119904)119898minus1d119904

+

119896

sum

119894=119895

[119905 minus (120591119894+ 120590)]119898

[119906 (120591119894+ 120590) minus 119906 (120591

+

119894+ 120590)]

+

119896

sum

119894=119895+1

(119905 minus 120591119894)119898

[119906 (120591119894) minus 119906 (120591

+

119894)] minus (119905 minus 120591

119895)

119898

119906 (120591+

119895)

(51)

Take into account (31) (32) 119886119896le 1120583 and 119888

119896le 1 we have

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904

ge

119896

sum

119894=119895+1

(119905 minus 120591119894)119898

(1 minus 119888119894) 119906 (120591119894) minus (119905 minus 120591

119895)

119898

119906 (120591+

119895)

ge minus(119905 minus 120591119895)

119898

119906 (120591+

119895)

(52)

If 119905 isin (120591119896 120591119896+ 120590] similarly we also get

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904 ge minus(119905 minus 120591119895)

119898

119906 (120591+

119895) (53)

So it yields

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 le (119905 minus 120591

119895)

119898

119906 (120591+

119895)

(54)

which follows that1

119905119898

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904

le (

119905 minus 120591119895

119905

)

119898

119906 (120591+

119895)

(55)

Hence

lim sup119905rarrinfin

1

119905119898

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 le 119906 (120591

+

119895)

(56)

which is a contradiction of (48)If case (ii) holds then as a manner with case (ii) in

Theorem 6 it is not possible too The proof is complete

Corollary 11 Assume (19) holds and 119886119896

= 119887119896

= 119888119896

= 1 for119896 isin N If there exists a positive differential function 119903 such that

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898[

[

119901 (119904) 119903 (119904) minus

[1199031015840(119904)]

2

4120583119903 (119904) (119904 minus 119879)

]

]

d119904 = infin

(57)

where 119879 is large enough such that Lemma 2 holds Then everysolution of (1) is oscillatory

Remark 12 Corollary 11 is an extension of [11 Theorem 32]into impulsive case Especially let 119903(119905) equiv 1 in (48) it reducesto

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898119901 (119904) d119904 = infin (58)

naturally which can be considered as the extension ofKamenev-type oscillation criteria for third-order impulsivedifferential equations with delay (see [8 14 15])

4 Examples

Example 13 Consider the third-order impulsive differentialequation with delay

119909101584010158401015840

(119905) + (1 + 1205721199092

(119905)) 119909 (119905 minus 120590) = 0 119905 gt 1199050 119905 = 120591

119896

119909 (120591+

119896) = 119886119896119909 (120591119896) 119909

1015840(120591+

119896) = 1198871198961199091015840(120591119896)

11990910158401015840(120591+

119896) = 11988811989611990910158401015840(120591119896) 119896 isin N

(59)

where 120590 gt 0 120572 ge 0 are constants 120591119896minus 120591119896minus1

gt 120590 for any 119896 isin N

Abstract and Applied Analysis 7

When 119886119896

= 119887119896

= 119888119896

= 1 for any 119896 isin N the impulses in(59) disappear by [16 Theorem 4] (59) is nonoscillatory if120590 lt e3 and 120572 = 0 However we may change its oscillation byproper impulsive control In fact let 120590 lt e3 and 120572 = 0 and120591119896= 1199050+ 119896120590 (119896 isin N) in (59) choose 120593(119909) = 119909 119901(119905) equiv 1 and

119903(119905) = 1 a simple calculation leads to

int

120591119899

1199050

prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896119901 (119904) d119904

= [int

1205911

1199050

+(int

1205911+120590

1205911

+int

1205912

1205911+120590

) + + (int

120591119899minus1+120590

120591119899minus1

+int

120591119899

120591119899minus1+120590

)]

sdot prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896d119904

= (1205911minus 1199050) + [

120590

1198881

+

1198861

1198881

(1205912minus 1205911minus 120590)] + sdot sdot sdot +

11988611198862sdot sdot sdot 119886119899minus2

120590

11988811198882sdot sdot sdot 119888119899minus1

+

11988611198862sdot sdot sdot 119886119899minus1

11988811198882sdot sdot sdot 119888119899minus1

(120591119899minus 120591119899minus1

minus 120590)

= 120590(1 +

1

1198881

+

1

1198882

+ sdot sdot sdot +

1

119888119899minus1

)

(60)

Then let

119886119896= 119888119896=

119896

119896 + 1

119887119896= 1 119896 isin N (61)

Obviously (5) (6) and (19) hold and

int

120591119899

1199050

prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896119901 (119904) d119904

= 120590 (1 +

2

1

+

3

2

+ sdot sdot sdot +

119899

119899 minus 1

) 997888rarr infin (119899 997888rarr infin)

(62)

Thus (21) is also satisfied By Theorem 6 every solution of(59) is oscillatory

If we let

119886119896= 1 +

1

1198962 119887119896= 119888119896=

119896 + 1

119896

119896 isin N (63)

In this case it is easily to verify (5) (6) (15) (44) and(21) hold By Corollary 8 every solution of (59) is eitheroscillatory or tends to zero

Remark 14 It is easy to verify that in [7 Theorems 1 2 and3] cannot be applied to (59) On the other hand Theorem 7is not applicable for the condition (61) sincesuminfin

119899=1|119886119896minus1| does

not convergence

Example 15 Consider the third-order impulsive differentialequation with delay

119909101584010158401015840

(119905) + e119905 cosh (|119909 (119905)|120572minus1

119909 (119905)) 119909 (119905 minus 1) = 0

119905 ge 0 119905 = 2119896

119909 (120591+

119896) = 119886119896119909 (120591119896) 119909

1015840(120591+

119896) = 1198871198961199091015840(120591119896)

11990910158401015840(120591+

119896) = 11988811989611990910158401015840(120591119896) 120591

119896= 2119896 119896 isin N

(64)

where 120572 gt 0 119886119896= 1 119887119896= 119888119896= 119896(119896 + 1) 119896 isin N

Let 120593(119909) = 119909 119901(119905) = e119905 it is easy to verify that (5) (6)and (19) hold Choose 119903(119905) equiv 1 and119898 = 1 we have

1

119905

int

119905

119879

(119905 minus 119904) e119904d119904 =

e119905

119905

+ (

119879 minus 1

119905

minus 1) e119879 rarr infin (119905 rarr infin)

(65)

Acknowledgments

Theauthor is very grateful to ProfessorH Saker who presentsthe references [8ndash12 17] and gives many helpful suggestionswhich leads to an improvement of this paper The workis supported in part by the NSF of Guangdong province(S2012010010034)

References

[1] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificand Technical 1993

[2] G Ballinger and X Liu ldquoExistence uniqueness and bounded-ness results for impulsive delay differential equationsrdquo Applica-ble Analysis vol 74 no 1-2 pp 71ndash93 2000

[3] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing New Jersey NJ USA 1989

[4] A M Samoılenko and N A Perestyuk Impulsive DifferentialEquations vol 14 World Scientific Publishing New Jersey NJUSA 1995

[5] Y S Chen andW Z Feng ldquoOscillations of second order nonlin-ear ODE with impulsesrdquo Journal of Mathematical Analysis andApplications vol 210 no 1 pp 150ndash169 1997

[6] R P Agarwal F Karakoc and A Zafer ldquoA survey on oscilla-tion of impulsive ordinary differential equationsrdquo Advances inDifferential Equations vol 2010 pp 1ndash52 2010

[7] W-H Mao and A-HWan ldquoOscillatory and asymptotic behav-ior of solutions for nonlinear impulsive delay differential equa-tionsrdquo Acta Mathematicae Applicatae Sinica vol 22 EnglishSeries no 3 pp 387ndash396 2006

[8] L Erbe A Peterson and S H Saker ldquoAsymptotic behavior ofsolutions of a third-order nonlinear dynamic equation on timescalesrdquo Journal of Computational and Applied Mathematics vol181 no 1 pp 92ndash102 2005

[9] L Erbe A Peterson and S H Saker ldquoOscillation and asymp-totic behavior of a third-order nonlinear dynamic equationrdquoThe Canadian Applied Mathematics Quarterly vol 14 no 2 pp124ndash147 2006

8 Abstract and Applied Analysis

[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007

[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006

[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010

[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997

[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003

[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978

[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984

[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008

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Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

converges to a finite limit are established In Section 4 twoexamples are given to illustrate the main results

2 Preliminary Results

In this section we state and prove some lemmas which wewill need in the proofs of the main results First of all weintroduce the following notations R+ and N are the sets ofreal numbers and positive integer numbers respectively PC1is defined by

PC1 (R+R)

= 119909 R+997888rarr R 119909 (119905) is differentiable for

119905 ge 0 and 119905 = 120591119896 119909 (120591+

119896) and 119909

1015840(120591+

119896) exist

and 119909 (120591minus

119896) = 119909 (120591

119896) 1199091015840(120591minus

119896) = 1199091015840(120591119896)

(2)

The following lemma is from Lakshmikantham et al [3 Page32 Theorem 141]

Lemma 1 Assume that

(i) 120591119896119896isinN is the impulse moments sequence with 0 le 119905

0lt

1205911lt sdot sdot sdot lt 120591

119896lt sdot sdot sdot lim

119896rarrinfin120591119896= infin

(ii) 119898 isin PC1(R+R) and for 119905 ge 1199050 119896 isin N it holds that

1198981015840

(119905) le 119906 (119905)119898 (119905) + 119907 (119905) 119905 = 120591119896

119898 (120591+

119896) le 119889119896119898(120591119896) + 119890119896

(3)

where 119906 119907 isin 119862(R+R) 119889119896ge 0 and 119890

119896are real constantsThen

119898(119905) le 119898 (1199050) prod

1199050lt120591119896lt119905

119889119896exp(int

119905

1199050

119906 (119904)d119904)

+ int

119905

1199050

prod

119904lt120591119896lt119905

119889119896exp(int

119905

119904

119906 (120590)d120590) 119907 (119904)d119904

+ sum

1199050lt120591119896lt119905

prod

120591119896lt120591119895lt119905

119889119895exp(int

119905

120591119896

119906 (119904)d119904) 119890119896

(4)

Motivated by the ideas of Chen and Feng [5] we presentthe following key lemma which determines the sign of 1199091015840(119905)and 119909

10158401015840(119905) of the nonoscillation solution 119909(119905) of (1)

Lemma 2 Suppose that 119909(119905) is an eventually positive solutionof (1) and

lim119905rarrinfin

int

119905

1199050

prod

1199050lt120591119896lt119904

119887119896

119886119896

d119904 = infin (5)

lim119905rarrinfin

int

119905

1199050

prod

1199050lt120591119896lt119904

119888119896

119887119896

d119904 = infin (6)

Then it holds that one of the following two cases for sufficientlylarge 119879

(i) 11990910158401015840(120591+

119896) gt 0 11990910158401015840(119905) gt 0 and 119909

1015840(120591+

119896) gt 0 119909

1015840(119905) gt 0

(ii) 11990910158401015840(120591+

119896) gt 0 11990910158401015840(119905) gt 0 and 119909

1015840(120591+

119896) lt 0 119909

1015840(119905) lt 0

with 119905 isin (120591119896 120591119896+1

] and 120591119896ge 119879

Proof Assume that 119909(119905) is an eventually positive solution of(1)Wemay assume that there exists 119905

1gt 1199050such that 119909(119905) gt 0

and 119909(119905 minus 120590) gt 0 for 119905 ge 1199051 First we assert that 11990910158401015840(120591

119896) gt 0 for

any 119896 isin N Suppose not there exists some 120591119895ge 1199051such that

11990910158401015840(120591119895) le 0 By

119909101584010158401015840

(119905) = minus119891 (119905 119909 (119905) 119909 (119905 minus 120590)) le 0 equiv 0 (7)

we have 11990910158401015840(119905)monotonically decreasing in (120591119894 120591119894+1

] 119894 = 119895 119895+

1 Thus 11990910158401015840(120591+119894) = 11988811989411990910158401015840(120591119894) lt 0 119894 = 119895+1 119895+2 Consider

the impulsive differential inequalities

119909101584010158401015840

(119905) lt 0 119905 ge 120591119895+1

119905 = 120591119896

11990910158401015840(120591+

119896) le 11988811989611990910158401015840(120591119896) 119896 = 119895 + 1 119895 + 2

(8)

By Lemma 1 we have

11990910158401015840

(119905) le 11990910158401015840(120591+

119895+1) prod

120591119895+1lt120591119896lt119905

119888119896= minus120572 prod

120591119895+1lt120591119896lt119905

119888119896lt 0 (9)

There are two cases of the sign of 1199091015840(120591119896)

Case 1 If there exists some 120591119899

ge 120591119895+1

such that 1199091015840(120591119899) le 0

since 11990910158401015840(119905) lt 0 then 119909

1015840(120591119899+1

) lt 1199091015840(120591+

119899) le 0 and 119909

1015840(120591+

119899+1) =

120573 = 119887119899+1

1199091015840(120591119899+1

) lt 0 By induction it easily show 1199091015840(120591119896) lt 0

and hence 1199091015840(120591+

119896) le 1198871198961199091015840(120591119896) lt 0 for 119896 = 119899 + 1 119899 + 2 So

we obtain the following impulsive differential inequalities

11990910158401015840

(119905) le 0 119905 ge 120591119899+1

119905 = 120591119896

1199091015840(120591+

119896) le 119887lowast

1198961199091015840(120591119896) 119896 = 119899 + 1 119899 + 2

(10)

which follows from Lemma 1 that

1199091015840

(119905) le 1199091015840(120591+

119899+1) prod

120591119899+1lt120591119896lt119905

119887119896= minus120573 prod

120591119899+1lt120591119896lt119905

119887119896 (11)

From (11) and applying Lemma 1 noting that 119909(119905) gt 0 for119905 gt 1199051and 119909(120591

+

119896) le 119886119896119909(120591119896) for any 119896 isin N we have

119909 (119905) le 119909 (120591+

119899+1) prod

120591119899+1lt120591119896lt119905

119886119896minus 120573int

119905

120591119899+1

prod

119904lt120591119896lt119905

119886119896

prod

120591119899+1lt120591119896lt119904

119887119896d119904

= prod

120591119899+1lt120591119896lt119905

119886119896(119909 (120591

+

119899+1) minus 120573int

119905

120591119899+1

prod

120591119899+1lt120591119896lt119904

119887119896

119886119896

d119904)

(12)

Thus by (5) we have 119909(119905) lt 0 for 119905 sufficiently large which isa contradiction

Case 2 If 1199091015840(120591119896) gt 0 for any 119896 ge 119895 noting that 11990910158401015840(119905) lt 0 for

119905 isin (120591119895+1

120591119895+2

] we have 1199091015840(119905) gt 119909

1015840(120591119895+1

) gt 0 By induction

Abstract and Applied Analysis 3

we get that 1199091015840(119905) gt 0 for any 119905 isin (120591119896 120591119896+1

] 119896 = 119895 + 1 119895 + 2 So the following impulsive differential inequalities hold

11990910158401015840

(119905) le minus120572 prod

120591119895+1lt120591119896lt119905

119888119896 119905 gt 120591

119895+1 119905 = 120591

119896

1199091015840(120591+

119896) le 119887119896119909 (120591119896) 119896 = 119895 + 1 119895 + 2

(13)

According to Lemma 1 we get

1199091015840

(119905) le 1199091015840(120591+

119895+1) prod

120591119895+1lt120591119896lt119905

119887119896minus 120572int

119905

120591119895+1

prod

119904lt120591119896lt119905

119887119896

prod

120591119895+1lt120591119896lt119904

119888119896d119904

= prod

120591119895+1lt120591119896lt119905

119887119896(1199091015840(120591+

119895+1) minus 120572int

119905

120591119895+1

prod

120591119895+1lt120591119896lt119904

119888119896

119887119896

d119904)

(14)

Hence the condition (6) implies that 1199091015840(119905) lt 0 when 119905 is

sufficiently large which contradicts to 1199091015840(119905) gt 0 for 119905 gt 120591

119895+1

again In terms of the above discussion we see that11990910158401015840(120591119896) gt 0

for any 120591119896gt 119879 with sufficiently large 119879 Consequently noting

that 119909101584010158401015840(119905) lt 0 for any 119905 isin (120591

119896 120591119896+1

] we have 11990910158401015840(119905) gt

11990910158401015840(120591119896+1

) gt 0Next if there exists a 120591

119895gt 119879 such that 1199091015840(120591

119895) ge 0 then

1199091015840(120591+

119895) = 119887119895(119909(120591+

119895) ge 0 1199091015840(120591

119895+1) gt 119909

1015840(120591+

119895) ge 0 Therefore by

induction we have 1199091015840(119905) gt 1199091015840(120591+

119894) gt 0 for 119905 isin (120591

119894 120591119894+1

] 119894 = 119895 +

1 119895 + 2 So case (i) is satisfied Otherwise if 1199091015840(120591119896) lt 0 for

all 120591119896ge 119879 then 119909

1015840(120591+

119896) = 119887119896119909(120591+

119896) lt 0 Thus for 119905 isin (120591

119896 120591119896+1

]using 119909

10158401015840(119905) gt 0 we have 1199091015840(119905) lt 119909

1015840(120591119896+1

) lt 0 hence case (ii)is satisfied This completes the proof

Remark 3 Suppose that 119909(119905) is an eventually negative solu-tion of (1) If (5) and (6) hold one can prove it holds that oneof the following two cases in a similar way as Lemma 2

(i) 11990910158401015840(120591+

119896) lt 0 11990910158401015840(119905) lt 0 and 119909

1015840(120591+

119896) lt 0 1199091015840(119905) lt 0

(ii) 11990910158401015840(120591+

119896) lt 0 11990910158401015840(119905) lt 0 and 119909

1015840(120591+

119896) gt 0 1199091015840(119905) gt 0 with

119905 isin (120591119896 120591119896+1

] and 120591119896ge 119879

Lemma 4 Let 119909(119905) be a piecewise continuous function oncup119896isinN(120591119896 120591119896+1] which is continuous at 119905 = 120591

119896and is left contin-

uous at 119905 = 120591119896 If

(1) 119909(119905) ge 0 (le 0) for 119905 ge 1199050

(2) 119909(119905) is monotone nonincreasing (monotone nonde-creasing) on (120591

119896 120591119896+1

] (120591119896ge 119879) for 119879 large enough

(3) suminfin

119896=1[119909(120591+

119896) minus 119909(120591

119896)] converges

then lim119905rarrinfin

119909(119905) = 119886 ge 0 (le 0)

The proof of Lemma 4 is similar to that of [13 Theorem5] and hence is omitted

Lemma 5 Assume that 119909(119905) is a solution of (1) which satisfiescase (ii) in Lemma 2 In addition if

infin

sum

119896=1

1003816100381610038161003816119886119896minus 1

1003816100381610038161003816lt infin

infin

prod

119896=1

119886119896119894119904 119887119900119906119899119889119890119889 (15)

then lim119905rarrinfin

119909(119905) exists (finite)

Proof First we claim thatsuminfin119896=1

[119909(120591+

119896)minus119909(120591

119896)] is convergence

In fact since 119909(119905) is decreasing on (120591119896 120591119896+1

] (120591119896ge 119879 and 119879 is

defined in Lemma 2) then

119909 (120591119896+1

) le 119909 (120591+

119896) 119909 (120591

+

119896+1) le 119886119896+1

119909 (120591119896+1

) le 119886119896+1

119909 (120591+

119896)

(16)

Obviously by induction we can get

119909 (120591119896+119899

) le 119886119896+119899minus1

sdot sdot sdot 119886119896+1

119909 (120591+

119896)

119909 (120591+

119896+119899) le 119886119896+119899

sdot sdot sdot 119886119896+1

119909 (120591+

119896)

(17)

Since prodinfin

119896=1119886119896

is bounded we conclude that 119909(120591119896) is

bounded which follows that there exists1198721gt 0 such that

1003816100381610038161003816119909 (120591+

119896) minus 119909 (120591

119896)1003816100381610038161003816=

1003816100381610038161003816119886119896minus 1

1003816100381610038161003816119909 (120591119896) le 119872

1

1003816100381610038161003816119886119896minus 1

1003816100381610038161003816 (18)

Hencesuminfin119896=1

[119909(120591+

119896)minus119909(120591

119896)] is convergence sincesuminfin

119896=1|119886119896minus1| is

convergence which follows from Lemma 4 that lim119905rarrinfin

119909(119905)

exists The proof is complete

3 Main Results

In this section we establish some sufficient conditions whichguarantee that every solution 119909(119905) of (1) either oscillates orhas a finite limit Occasionally we will make the additionalassumption

lim sup119905rarrinfin

int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) d120579 d119904 d119906 = infin (19)

where here it is understood that

int

infin

1199050

119901 (119905) d119905 lt infin int

infin

1199050

int

infin

119906

119901 (120579) d120579 d119906 lt infin (20)

Now we are ready to state and prove the main results inthis paper The results will be proved by making use of thetechnique in [11]

Theorem 6 Assume that (5) (6) and (19) hold and 119909(119905) is asolution of (1) Furthermore assume that 119886

119896le 1 119887

119896ge 1 and

119888119896le 1 for 119896 isin N If there exists a positive differentiable function

119903 such that

lim119905rarrinfin

int

119905

1199050

[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)

)] prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896d119904 = infin

(21)

where 119896(119904) = max119896 120591119896lt 119904 and

119860 (119904 120591119896(119904)

)

=

119888119896(119904)[1199031015840(119904)]2

4120583119903 (119904) (119904 minus 120591119896(119904))

119904 isin (120591119896 120591119896+ 120590]

[1199031015840(119904)]2

4120583119903 (119904) (119904 minus 120591119896(119904)minus ((119888119896(119904)minus 1) 119888

119896(119904)) 120590)

119904 isin (120591119896+ 120590 120591119896+1]

(22)

Then 119909(119905) is oscillatory

4 Abstract and Applied Analysis

Proof Let 119909(119905) be a nonoscillatory solution of (1) withoutloss of generality we may assume that 119909(119905) gt 0 eventually(119909(119905) lt 0 eventually can be achieved in the similar way)By Lemma 2 either case (i) or case (ii) in Lemma 2 holdsAssume that 119909(119905) satisfies case (i) then 119909

10158401015840(119905) gt 0 1199091015840(119905) gt 0

for 119905 isin (120591119895 120591119895+1

] 120591119895ge 119879 (119879 is defined in Lemma 2) Define the

Riccati transformation 119906 by

119906 (119905) =

119903 (119905) 11990910158401015840(119905)

120593 (119909 (119905 minus 120590))

119905 ge 120591119895 119905 = 120591

119896 (23)

Thus 119906(119905) gt 0 for 119905 isin (120591119896 120591119896+1

] 119896 = 119895 119895 + 1 and

1199061015840

(119905) =

119903 (119905)

120593 (119909 (119905 minus 120590))

119909101584010158401015840

(119905)

+

1199031015840(119905) 120593 (119909 (119905 minus 120590)) minus 119903 (119905) 120593

1015840(119909 (119905 minus 120590)) 119909

1015840(119905 minus 120590)

1205932(119909 (119905 minus 120590))

times 11990910158401015840

(119905)

le minus 119901 (119905) 119903 (119905) +

1199031015840(119905)

119903 (119905)

119906 (119905)

minus

119903 (119905) 1205831199091015840(119905 minus 120590)

1205932(119909 (119905 minus 120590))

11990910158401015840

(119905)

(24)

If 119905 isin (120591119896 120591119896+120590] sub (120591

119896 120591119896+1

] namely 119905 minus120590 le 120591119896lt 119905 11990910158401015840(119905)

is decreasing in (119905 minus 120590 120591119896] and (120591

119896 119905] respectively In view of

the following

11990910158401015840

(119905 minus 120590) gt 11990910158401015840(120591119896) ge

11990910158401015840(120591+

119896)

119888119896

gt

11990910158401015840(119905)

119888119896

(25)

we have

1199091015840

(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int

119905minus120590

120591119896minus120590

11990910158401015840

(119904) d119904

gt 11990910158401015840

(119905 minus 120590) (119905 minus 120591119896) gt

11990910158401015840(119905)

119888119896

(119905 minus 120591119896)

(26)

Thus

1199061015840

(119905) le minus 119901 (119905) 119903 (119905) +

1199031015840(119905)

119903 (119905)

119906 (119905) minus

120583119903 (119905) (119905 minus 120591119896)

1198881198961205932(119909 (119905 minus 120590))

[11990910158401015840

(119905)]

2

= minus 119901 (119905) 119903 (119905) +

1199031015840(119905)

119903 (119905)

119906 (119905) minus

120583 (119905 minus 120591119896)

119888119896119903 (119905)

[119906 (119905)]2

= minus (radic

120583 (119905 minus 120591119896)

119888119896119903 (119905)

119906 (119905) minus radic

119888119896

4120583119903 (119905) (119905 minus 120591119896)

1199031015840

(119905))

2

minus[

[

119901 (119905) 119903 (119905) minus

119888119896[1199031015840(119905)]

2

4120583119903 (119905) (119905 minus 120591119896)

]

]

le minus[

[

119901 (119905) 119903 (119905) minus

119888119896[1199031015840(119905)]

2

4120583119903 (119905) (119905 minus 120591119896)

]

]

(27)

If 119905 isin (120591119896+120590 120591119896+1

] sub (120591119896 120591119896+1

] that is 120591119896lt 119905minus120590 lt 119905 le 120591

119896+1

then

1199091015840

(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int

120591119896

120591119896minus120590

11990910158401015840

(119904) d119904 + int

119905minus120590

120591119896

11990910158401015840

(119904) d119904

ge 11990910158401015840(120591119896) 120590 + 119909

10158401015840

(119905 minus 120590) (119905 minus 120591119896minus 120590)

ge

11990910158401015840(119905)

119888119896

120590 + 11990910158401015840

(119905) (119905 minus 120591119896minus 120590)

ge 11990910158401015840

(119905) (119905 minus 120591119896minus

119888119896minus 1

119888119896

120590)

(28)

Similarly we have

1199061015840

(119905) le minus[

[

119901 (119905) 119903 (119905) minus

[1199031015840(119905)]

2

4120583119903 (119905) (119905 minus 120591119896minus ((119888119896minus 1) 119888

119896) 120590)

]

]

(29)

Thus we obtain

1199061015840

(119905) le minus [119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)

)] for 119905 isin (120591119896 120591119896+1

]

(30)

On the other hand

119906 (120591+

119896) =

119903 (120591119896) 11990910158401015840(120591+

119896)

120593 (119909 (120591119896minus 120590))

le 119888119896119906 (120591119896) (31)

Observing that 120593(119906) ge 120583119906 we have

119906 (120591+

119896+ 120590) =

119903 (120591119896+ 120590) 119909

10158401015840(120591119896+ 120590)

120593 (119909 (120591+

119896))

le

119903 (120591119896+ 120590) 119909

10158401015840(120591119896+ 120590)

120593 (119886119896119909 (120591119896))

le

119906 (120591119896+ 120590)

120583119886119896

(32)

Applying Lemma 1 it follows from (30) (31) and (32) that

119906 (119905) le 119906 (120591+

119895) prod

120591119895lt120591119896lt119905

119888119896

prod

120591119895lt120591119896+120590lt119905

1

120583119886119896

minus int

119905

120591119895

[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)

)]

sdot prod

119904lt120591119896lt119905

119888119896

prod

119904lt120591119896+120590lt119905

1

120583119886119896

d119904

le prod

120591119895lt120591119896lt119905

119888119896

prod

120591119895lt120591119896+120590lt119905

1

120583119886119896

times (119906 (120591+

119895) minus 120583int

119905

120591119895

[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)

)]

sdot prod

120591119895lt120591119896lt119904

1

119888119896

prod

120591119895lt120591119896+120590lt119904

119886119896d119904)

(33)

Abstract and Applied Analysis 5

which yields 119906(119905) lt 0 for all large 119905 This is contrary to 119906(119905) gt

0 and so case (i) in Lemma 2 is not possibleIf 119909(119905) satisfies the case (ii) in Lemma 2 that is 11990910158401015840(120591+

119896) gt

0 11990910158401015840(119905) gt 0 and 1199091015840(120591+

119896) lt 0 1199091015840(119905) lt 0 which proves that the

solution 119909(119905) is positive and decreasing Integrating (1) from119904 to 119905 (119905 ge 119904 ge 119879) we obtain

11990910158401015840

(119905) minus sum

119904lt120591119896lt119905

(119888119896minus 1) 119909

10158401015840(120591119896) minus 11990910158401015840

(119904)

+ int

119905

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0

(34)

Noting 119888119896le 1 and 119909

10158401015840(119905) gt 0 then it holds that

11990910158401015840

(119905) minus 11990910158401015840

(119904) + int

119905

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (35)

which leads to

minus11990910158401015840

(119904) + int

119905

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (36)

and hence

minus11990910158401015840

(119904) + int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (37)

Integrating the above inequality again from 119906 to 119905 (119905 ge 119906 ge 119879)one has

minus 1199091015840

(119905) + sum

119904lt120591119896lt119905

(119887119896minus 1) 119909

1015840(120591119896) + 1199091015840

(119906)

+ int

119905

119906

int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0

(38)

Using 1199091015840(119905) lt 0 and 119887

119896ge 1 we have

1199091015840

(119906) + int

infin

119906

int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0 (39)

Now we integrate the last inequality from 120578 to 119905 (119905 ge 120578 ge 119879)to obtain

119909 (119905) minus sum

120578lt120591119896lt119905

(119886119896minus 1) 119909 (120591

119896) minus 119909 (120578)

+ int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 d119906 le 0

(40)

Since 119886119896

le 1 (119896 isin N) and 119909(119905) is decreasing then for 119905 isin

(120591119896 120591119896+1

] 119909(120591119896+1

) le 119909(119905) le 119909(120591+

119896) = 119886119896119909(120591119896) le 119909(120591

119896) 119896 isin N

Thus we get

120583119909 (120578) int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) d120579 d119904 d119906 le minus119909 (119905) + 119909 (120578) le 119909 (120578)

(41)

and then

int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) d120579 d119904 d119906 le

1

120583

(42)

which contradicts the condition (19) The proof is complete

Replace the condition (19) with (15) we may obtain thefollowing asymptotic results

Theorem 7 Assume that (5) (6) and (15) hold and 119909(119905) is asolution of (1) If there exists a positive differentiable function 119903

such that (21) hold then 119909(119905) is either oscillatory or has a finitelimit

Proof By the proof of Theorem 6 we know the case (i) inLemma 2 is not possible too since the condition (19) is notrequired to prove it So it suffices to show if there is a solutionsatisfying case (ii) in Lemma 2 that is if

11990910158401015840(120591+

119896) gt 0 119909

10158401015840

(119905) gt 0 1199091015840(120591+

119896) lt 0 119909

1015840

(119905) lt 0 (43)

with 119905 isin (120591119896 120591119896+1

] and 120591119896ge 119879 then lim

119905rarrinfin119909(119905) exists This

is obtained by applying Lemma 5 which leads to lim119905rarrinfin

119909(119905)

exists The proof is complete

Corollary 8 In addition to the assumption of Theorem 7assume that

lim119905rarrinfin

int

119905

1199050

prod

1199050lt120591119896lt119904

1

119888119896

119901 (119904) d119904 = infin (44)

Then solution 119909(119905) of (1) either oscillates or satisfieslim119905rarrinfin

119909(119905) = 0

Proof By the proof ofTheorem 7 lim119905rarrinfin

119909(119905) exists and wedefine it by lim

119905rarrinfin119909(119905) = 120574 ge 0 We now show 120574 = 0 If not

then 120574 gt 0 So lim119905rarrinfin

120593(119909(119905 minus 120590)) = 120593(120574) = 120581 gt 0 Hencethere exists 120591

119895gt 119879 such that 120593(119909(119905minus120590)) gt 1205812 for 119905 gt 120591

119895Then

119909101584010158401015840

(119905) = minus119891 (119905 119909 (119905) 119909 (119905 minus 120590))

le minus119901 (119905) 120593 (119909 (119905 minus 120590)) le minus

120581

2

119901 (119905) 119905 ge 120591119895

(45)

and note that 11990910158401015840(120591+119896) le 11988811989611990910158401015840(120591119896) since 11990910158401015840(119905) gt 0 which imply

that

11990910158401015840

(119905) le 11990910158401015840(120591+

119895) prod

120591119895lt120591119896lt119905

119888119896minus

120581

2

int

119905

120591119895

prod

119904lt120591119896lt119905

119888119896119901 (119904) d119904

le prod

120591119895lt120591119896lt119905

119888119896[

[

11990910158401015840(120591+

119895) minus

120581

2

int

119905

120591119895

prod

120591119895lt120591119896lt119904

1

119888119896

119901 (119904) d119904]

]

(46)

Thus in virtue of (44) it holds that 11990910158401015840(119905) lt 0 and contradicts11990910158401015840(119905) gt 0 for 119905 large enough the proof is complete

Remark 9 Theorem 6 and Corollary 8 extend the results in[11 Theorem 31] and [9 Corollary 1] respectively In factwhen 119886

119896= 119887119896

= 119888119896

= 1 for 119896 isin N which implies that theimpulses in (1) disappear In such a case (5) and (6) holdnaturally and (21) and (44) are reduced to

lim119905rarrinfin

int

119905

1199050

[

[

119901 (119904) 119903 (119904) minus

[1199031015840(119904)]

2

4120583119903 (119904) (119904 minus 119879)

]

]

d119904 = infin

lim119905rarrinfin

int

119905

1199050

119901 (119904) d119904 = infin

(47)

6 Abstract and Applied Analysis

which are similar to those in [11 Theorem 31] and [9Corollary 1] respectively

Next we present some new oscillation results for (1) byusing an integral averaging condition of Kamenevrsquos type

Theorem 10 Assume (5) (6) and (19) hold Furthermore119886119896

le 1120583 le 1 119887119896

ge 1 and 119888119896

le 1 119896 isin N If there exists apositive differentiable function 119903 such that

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 = infin

(48)

where 119860(119904 120591119896(119904)

) is defined by (21) and 119898 ge 1 Then everysolution of (1) is oscillatory

Proof We choose 119879 large enough such that Lemma 2 holdsBy Lemma 2 there are two possible cases First if the case (i)holds proceeding as in the proof of Theorem 6 we will endup with (32) By (30) we have

119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)

) le minus1199061015840

(119905) 119905 ge 119879 119905 = 120591119896 (49)

If 119905 isin (120591119896+ 120590 120591119896+1

] sub (120591119896 120591119896+1

] for 120591119895ge 119879 we obtain

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904

le minusint

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904(50)

An integration by parts of the right-hand side leads to

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904

= (int

120591119895+120590

120591119895

+int

120591119895+1

120591119895+120590

+ sdot sdot sdot + int

120591119896+120590

120591119896

+int

119905

120591119896+120590

) (119905 minus 119904)1198981199061015840

(119904) d119904

= int

119905

120591119895

119906 (119904)

119898

(119905 minus 119904)119898minus1d119904

+

119896

sum

119894=119895

[119905 minus (120591119894+ 120590)]119898

[119906 (120591119894+ 120590) minus 119906 (120591

+

119894+ 120590)]

+

119896

sum

119894=119895+1

(119905 minus 120591119894)119898

[119906 (120591119894) minus 119906 (120591

+

119894)] minus (119905 minus 120591

119895)

119898

119906 (120591+

119895)

(51)

Take into account (31) (32) 119886119896le 1120583 and 119888

119896le 1 we have

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904

ge

119896

sum

119894=119895+1

(119905 minus 120591119894)119898

(1 minus 119888119894) 119906 (120591119894) minus (119905 minus 120591

119895)

119898

119906 (120591+

119895)

ge minus(119905 minus 120591119895)

119898

119906 (120591+

119895)

(52)

If 119905 isin (120591119896 120591119896+ 120590] similarly we also get

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904 ge minus(119905 minus 120591119895)

119898

119906 (120591+

119895) (53)

So it yields

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 le (119905 minus 120591

119895)

119898

119906 (120591+

119895)

(54)

which follows that1

119905119898

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904

le (

119905 minus 120591119895

119905

)

119898

119906 (120591+

119895)

(55)

Hence

lim sup119905rarrinfin

1

119905119898

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 le 119906 (120591

+

119895)

(56)

which is a contradiction of (48)If case (ii) holds then as a manner with case (ii) in

Theorem 6 it is not possible too The proof is complete

Corollary 11 Assume (19) holds and 119886119896

= 119887119896

= 119888119896

= 1 for119896 isin N If there exists a positive differential function 119903 such that

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898[

[

119901 (119904) 119903 (119904) minus

[1199031015840(119904)]

2

4120583119903 (119904) (119904 minus 119879)

]

]

d119904 = infin

(57)

where 119879 is large enough such that Lemma 2 holds Then everysolution of (1) is oscillatory

Remark 12 Corollary 11 is an extension of [11 Theorem 32]into impulsive case Especially let 119903(119905) equiv 1 in (48) it reducesto

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898119901 (119904) d119904 = infin (58)

naturally which can be considered as the extension ofKamenev-type oscillation criteria for third-order impulsivedifferential equations with delay (see [8 14 15])

4 Examples

Example 13 Consider the third-order impulsive differentialequation with delay

119909101584010158401015840

(119905) + (1 + 1205721199092

(119905)) 119909 (119905 minus 120590) = 0 119905 gt 1199050 119905 = 120591

119896

119909 (120591+

119896) = 119886119896119909 (120591119896) 119909

1015840(120591+

119896) = 1198871198961199091015840(120591119896)

11990910158401015840(120591+

119896) = 11988811989611990910158401015840(120591119896) 119896 isin N

(59)

where 120590 gt 0 120572 ge 0 are constants 120591119896minus 120591119896minus1

gt 120590 for any 119896 isin N

Abstract and Applied Analysis 7

When 119886119896

= 119887119896

= 119888119896

= 1 for any 119896 isin N the impulses in(59) disappear by [16 Theorem 4] (59) is nonoscillatory if120590 lt e3 and 120572 = 0 However we may change its oscillation byproper impulsive control In fact let 120590 lt e3 and 120572 = 0 and120591119896= 1199050+ 119896120590 (119896 isin N) in (59) choose 120593(119909) = 119909 119901(119905) equiv 1 and

119903(119905) = 1 a simple calculation leads to

int

120591119899

1199050

prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896119901 (119904) d119904

= [int

1205911

1199050

+(int

1205911+120590

1205911

+int

1205912

1205911+120590

) + + (int

120591119899minus1+120590

120591119899minus1

+int

120591119899

120591119899minus1+120590

)]

sdot prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896d119904

= (1205911minus 1199050) + [

120590

1198881

+

1198861

1198881

(1205912minus 1205911minus 120590)] + sdot sdot sdot +

11988611198862sdot sdot sdot 119886119899minus2

120590

11988811198882sdot sdot sdot 119888119899minus1

+

11988611198862sdot sdot sdot 119886119899minus1

11988811198882sdot sdot sdot 119888119899minus1

(120591119899minus 120591119899minus1

minus 120590)

= 120590(1 +

1

1198881

+

1

1198882

+ sdot sdot sdot +

1

119888119899minus1

)

(60)

Then let

119886119896= 119888119896=

119896

119896 + 1

119887119896= 1 119896 isin N (61)

Obviously (5) (6) and (19) hold and

int

120591119899

1199050

prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896119901 (119904) d119904

= 120590 (1 +

2

1

+

3

2

+ sdot sdot sdot +

119899

119899 minus 1

) 997888rarr infin (119899 997888rarr infin)

(62)

Thus (21) is also satisfied By Theorem 6 every solution of(59) is oscillatory

If we let

119886119896= 1 +

1

1198962 119887119896= 119888119896=

119896 + 1

119896

119896 isin N (63)

In this case it is easily to verify (5) (6) (15) (44) and(21) hold By Corollary 8 every solution of (59) is eitheroscillatory or tends to zero

Remark 14 It is easy to verify that in [7 Theorems 1 2 and3] cannot be applied to (59) On the other hand Theorem 7is not applicable for the condition (61) sincesuminfin

119899=1|119886119896minus1| does

not convergence

Example 15 Consider the third-order impulsive differentialequation with delay

119909101584010158401015840

(119905) + e119905 cosh (|119909 (119905)|120572minus1

119909 (119905)) 119909 (119905 minus 1) = 0

119905 ge 0 119905 = 2119896

119909 (120591+

119896) = 119886119896119909 (120591119896) 119909

1015840(120591+

119896) = 1198871198961199091015840(120591119896)

11990910158401015840(120591+

119896) = 11988811989611990910158401015840(120591119896) 120591

119896= 2119896 119896 isin N

(64)

where 120572 gt 0 119886119896= 1 119887119896= 119888119896= 119896(119896 + 1) 119896 isin N

Let 120593(119909) = 119909 119901(119905) = e119905 it is easy to verify that (5) (6)and (19) hold Choose 119903(119905) equiv 1 and119898 = 1 we have

1

119905

int

119905

119879

(119905 minus 119904) e119904d119904 =

e119905

119905

+ (

119879 minus 1

119905

minus 1) e119879 rarr infin (119905 rarr infin)

(65)

Acknowledgments

Theauthor is very grateful to ProfessorH Saker who presentsthe references [8ndash12 17] and gives many helpful suggestionswhich leads to an improvement of this paper The workis supported in part by the NSF of Guangdong province(S2012010010034)

References

[1] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificand Technical 1993

[2] G Ballinger and X Liu ldquoExistence uniqueness and bounded-ness results for impulsive delay differential equationsrdquo Applica-ble Analysis vol 74 no 1-2 pp 71ndash93 2000

[3] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing New Jersey NJ USA 1989

[4] A M Samoılenko and N A Perestyuk Impulsive DifferentialEquations vol 14 World Scientific Publishing New Jersey NJUSA 1995

[5] Y S Chen andW Z Feng ldquoOscillations of second order nonlin-ear ODE with impulsesrdquo Journal of Mathematical Analysis andApplications vol 210 no 1 pp 150ndash169 1997

[6] R P Agarwal F Karakoc and A Zafer ldquoA survey on oscilla-tion of impulsive ordinary differential equationsrdquo Advances inDifferential Equations vol 2010 pp 1ndash52 2010

[7] W-H Mao and A-HWan ldquoOscillatory and asymptotic behav-ior of solutions for nonlinear impulsive delay differential equa-tionsrdquo Acta Mathematicae Applicatae Sinica vol 22 EnglishSeries no 3 pp 387ndash396 2006

[8] L Erbe A Peterson and S H Saker ldquoAsymptotic behavior ofsolutions of a third-order nonlinear dynamic equation on timescalesrdquo Journal of Computational and Applied Mathematics vol181 no 1 pp 92ndash102 2005

[9] L Erbe A Peterson and S H Saker ldquoOscillation and asymp-totic behavior of a third-order nonlinear dynamic equationrdquoThe Canadian Applied Mathematics Quarterly vol 14 no 2 pp124ndash147 2006

8 Abstract and Applied Analysis

[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007

[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006

[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010

[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997

[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003

[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978

[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984

[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008

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Differential EquationsInternational Journal of

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Abstract and Applied Analysis 3

we get that 1199091015840(119905) gt 0 for any 119905 isin (120591119896 120591119896+1

] 119896 = 119895 + 1 119895 + 2 So the following impulsive differential inequalities hold

11990910158401015840

(119905) le minus120572 prod

120591119895+1lt120591119896lt119905

119888119896 119905 gt 120591

119895+1 119905 = 120591

119896

1199091015840(120591+

119896) le 119887119896119909 (120591119896) 119896 = 119895 + 1 119895 + 2

(13)

According to Lemma 1 we get

1199091015840

(119905) le 1199091015840(120591+

119895+1) prod

120591119895+1lt120591119896lt119905

119887119896minus 120572int

119905

120591119895+1

prod

119904lt120591119896lt119905

119887119896

prod

120591119895+1lt120591119896lt119904

119888119896d119904

= prod

120591119895+1lt120591119896lt119905

119887119896(1199091015840(120591+

119895+1) minus 120572int

119905

120591119895+1

prod

120591119895+1lt120591119896lt119904

119888119896

119887119896

d119904)

(14)

Hence the condition (6) implies that 1199091015840(119905) lt 0 when 119905 is

sufficiently large which contradicts to 1199091015840(119905) gt 0 for 119905 gt 120591

119895+1

again In terms of the above discussion we see that11990910158401015840(120591119896) gt 0

for any 120591119896gt 119879 with sufficiently large 119879 Consequently noting

that 119909101584010158401015840(119905) lt 0 for any 119905 isin (120591

119896 120591119896+1

] we have 11990910158401015840(119905) gt

11990910158401015840(120591119896+1

) gt 0Next if there exists a 120591

119895gt 119879 such that 1199091015840(120591

119895) ge 0 then

1199091015840(120591+

119895) = 119887119895(119909(120591+

119895) ge 0 1199091015840(120591

119895+1) gt 119909

1015840(120591+

119895) ge 0 Therefore by

induction we have 1199091015840(119905) gt 1199091015840(120591+

119894) gt 0 for 119905 isin (120591

119894 120591119894+1

] 119894 = 119895 +

1 119895 + 2 So case (i) is satisfied Otherwise if 1199091015840(120591119896) lt 0 for

all 120591119896ge 119879 then 119909

1015840(120591+

119896) = 119887119896119909(120591+

119896) lt 0 Thus for 119905 isin (120591

119896 120591119896+1

]using 119909

10158401015840(119905) gt 0 we have 1199091015840(119905) lt 119909

1015840(120591119896+1

) lt 0 hence case (ii)is satisfied This completes the proof

Remark 3 Suppose that 119909(119905) is an eventually negative solu-tion of (1) If (5) and (6) hold one can prove it holds that oneof the following two cases in a similar way as Lemma 2

(i) 11990910158401015840(120591+

119896) lt 0 11990910158401015840(119905) lt 0 and 119909

1015840(120591+

119896) lt 0 1199091015840(119905) lt 0

(ii) 11990910158401015840(120591+

119896) lt 0 11990910158401015840(119905) lt 0 and 119909

1015840(120591+

119896) gt 0 1199091015840(119905) gt 0 with

119905 isin (120591119896 120591119896+1

] and 120591119896ge 119879

Lemma 4 Let 119909(119905) be a piecewise continuous function oncup119896isinN(120591119896 120591119896+1] which is continuous at 119905 = 120591

119896and is left contin-

uous at 119905 = 120591119896 If

(1) 119909(119905) ge 0 (le 0) for 119905 ge 1199050

(2) 119909(119905) is monotone nonincreasing (monotone nonde-creasing) on (120591

119896 120591119896+1

] (120591119896ge 119879) for 119879 large enough

(3) suminfin

119896=1[119909(120591+

119896) minus 119909(120591

119896)] converges

then lim119905rarrinfin

119909(119905) = 119886 ge 0 (le 0)

The proof of Lemma 4 is similar to that of [13 Theorem5] and hence is omitted

Lemma 5 Assume that 119909(119905) is a solution of (1) which satisfiescase (ii) in Lemma 2 In addition if

infin

sum

119896=1

1003816100381610038161003816119886119896minus 1

1003816100381610038161003816lt infin

infin

prod

119896=1

119886119896119894119904 119887119900119906119899119889119890119889 (15)

then lim119905rarrinfin

119909(119905) exists (finite)

Proof First we claim thatsuminfin119896=1

[119909(120591+

119896)minus119909(120591

119896)] is convergence

In fact since 119909(119905) is decreasing on (120591119896 120591119896+1

] (120591119896ge 119879 and 119879 is

defined in Lemma 2) then

119909 (120591119896+1

) le 119909 (120591+

119896) 119909 (120591

+

119896+1) le 119886119896+1

119909 (120591119896+1

) le 119886119896+1

119909 (120591+

119896)

(16)

Obviously by induction we can get

119909 (120591119896+119899

) le 119886119896+119899minus1

sdot sdot sdot 119886119896+1

119909 (120591+

119896)

119909 (120591+

119896+119899) le 119886119896+119899

sdot sdot sdot 119886119896+1

119909 (120591+

119896)

(17)

Since prodinfin

119896=1119886119896

is bounded we conclude that 119909(120591119896) is

bounded which follows that there exists1198721gt 0 such that

1003816100381610038161003816119909 (120591+

119896) minus 119909 (120591

119896)1003816100381610038161003816=

1003816100381610038161003816119886119896minus 1

1003816100381610038161003816119909 (120591119896) le 119872

1

1003816100381610038161003816119886119896minus 1

1003816100381610038161003816 (18)

Hencesuminfin119896=1

[119909(120591+

119896)minus119909(120591

119896)] is convergence sincesuminfin

119896=1|119886119896minus1| is

convergence which follows from Lemma 4 that lim119905rarrinfin

119909(119905)

exists The proof is complete

3 Main Results

In this section we establish some sufficient conditions whichguarantee that every solution 119909(119905) of (1) either oscillates orhas a finite limit Occasionally we will make the additionalassumption

lim sup119905rarrinfin

int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) d120579 d119904 d119906 = infin (19)

where here it is understood that

int

infin

1199050

119901 (119905) d119905 lt infin int

infin

1199050

int

infin

119906

119901 (120579) d120579 d119906 lt infin (20)

Now we are ready to state and prove the main results inthis paper The results will be proved by making use of thetechnique in [11]

Theorem 6 Assume that (5) (6) and (19) hold and 119909(119905) is asolution of (1) Furthermore assume that 119886

119896le 1 119887

119896ge 1 and

119888119896le 1 for 119896 isin N If there exists a positive differentiable function

119903 such that

lim119905rarrinfin

int

119905

1199050

[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)

)] prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896d119904 = infin

(21)

where 119896(119904) = max119896 120591119896lt 119904 and

119860 (119904 120591119896(119904)

)

=

119888119896(119904)[1199031015840(119904)]2

4120583119903 (119904) (119904 minus 120591119896(119904))

119904 isin (120591119896 120591119896+ 120590]

[1199031015840(119904)]2

4120583119903 (119904) (119904 minus 120591119896(119904)minus ((119888119896(119904)minus 1) 119888

119896(119904)) 120590)

119904 isin (120591119896+ 120590 120591119896+1]

(22)

Then 119909(119905) is oscillatory

4 Abstract and Applied Analysis

Proof Let 119909(119905) be a nonoscillatory solution of (1) withoutloss of generality we may assume that 119909(119905) gt 0 eventually(119909(119905) lt 0 eventually can be achieved in the similar way)By Lemma 2 either case (i) or case (ii) in Lemma 2 holdsAssume that 119909(119905) satisfies case (i) then 119909

10158401015840(119905) gt 0 1199091015840(119905) gt 0

for 119905 isin (120591119895 120591119895+1

] 120591119895ge 119879 (119879 is defined in Lemma 2) Define the

Riccati transformation 119906 by

119906 (119905) =

119903 (119905) 11990910158401015840(119905)

120593 (119909 (119905 minus 120590))

119905 ge 120591119895 119905 = 120591

119896 (23)

Thus 119906(119905) gt 0 for 119905 isin (120591119896 120591119896+1

] 119896 = 119895 119895 + 1 and

1199061015840

(119905) =

119903 (119905)

120593 (119909 (119905 minus 120590))

119909101584010158401015840

(119905)

+

1199031015840(119905) 120593 (119909 (119905 minus 120590)) minus 119903 (119905) 120593

1015840(119909 (119905 minus 120590)) 119909

1015840(119905 minus 120590)

1205932(119909 (119905 minus 120590))

times 11990910158401015840

(119905)

le minus 119901 (119905) 119903 (119905) +

1199031015840(119905)

119903 (119905)

119906 (119905)

minus

119903 (119905) 1205831199091015840(119905 minus 120590)

1205932(119909 (119905 minus 120590))

11990910158401015840

(119905)

(24)

If 119905 isin (120591119896 120591119896+120590] sub (120591

119896 120591119896+1

] namely 119905 minus120590 le 120591119896lt 119905 11990910158401015840(119905)

is decreasing in (119905 minus 120590 120591119896] and (120591

119896 119905] respectively In view of

the following

11990910158401015840

(119905 minus 120590) gt 11990910158401015840(120591119896) ge

11990910158401015840(120591+

119896)

119888119896

gt

11990910158401015840(119905)

119888119896

(25)

we have

1199091015840

(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int

119905minus120590

120591119896minus120590

11990910158401015840

(119904) d119904

gt 11990910158401015840

(119905 minus 120590) (119905 minus 120591119896) gt

11990910158401015840(119905)

119888119896

(119905 minus 120591119896)

(26)

Thus

1199061015840

(119905) le minus 119901 (119905) 119903 (119905) +

1199031015840(119905)

119903 (119905)

119906 (119905) minus

120583119903 (119905) (119905 minus 120591119896)

1198881198961205932(119909 (119905 minus 120590))

[11990910158401015840

(119905)]

2

= minus 119901 (119905) 119903 (119905) +

1199031015840(119905)

119903 (119905)

119906 (119905) minus

120583 (119905 minus 120591119896)

119888119896119903 (119905)

[119906 (119905)]2

= minus (radic

120583 (119905 minus 120591119896)

119888119896119903 (119905)

119906 (119905) minus radic

119888119896

4120583119903 (119905) (119905 minus 120591119896)

1199031015840

(119905))

2

minus[

[

119901 (119905) 119903 (119905) minus

119888119896[1199031015840(119905)]

2

4120583119903 (119905) (119905 minus 120591119896)

]

]

le minus[

[

119901 (119905) 119903 (119905) minus

119888119896[1199031015840(119905)]

2

4120583119903 (119905) (119905 minus 120591119896)

]

]

(27)

If 119905 isin (120591119896+120590 120591119896+1

] sub (120591119896 120591119896+1

] that is 120591119896lt 119905minus120590 lt 119905 le 120591

119896+1

then

1199091015840

(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int

120591119896

120591119896minus120590

11990910158401015840

(119904) d119904 + int

119905minus120590

120591119896

11990910158401015840

(119904) d119904

ge 11990910158401015840(120591119896) 120590 + 119909

10158401015840

(119905 minus 120590) (119905 minus 120591119896minus 120590)

ge

11990910158401015840(119905)

119888119896

120590 + 11990910158401015840

(119905) (119905 minus 120591119896minus 120590)

ge 11990910158401015840

(119905) (119905 minus 120591119896minus

119888119896minus 1

119888119896

120590)

(28)

Similarly we have

1199061015840

(119905) le minus[

[

119901 (119905) 119903 (119905) minus

[1199031015840(119905)]

2

4120583119903 (119905) (119905 minus 120591119896minus ((119888119896minus 1) 119888

119896) 120590)

]

]

(29)

Thus we obtain

1199061015840

(119905) le minus [119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)

)] for 119905 isin (120591119896 120591119896+1

]

(30)

On the other hand

119906 (120591+

119896) =

119903 (120591119896) 11990910158401015840(120591+

119896)

120593 (119909 (120591119896minus 120590))

le 119888119896119906 (120591119896) (31)

Observing that 120593(119906) ge 120583119906 we have

119906 (120591+

119896+ 120590) =

119903 (120591119896+ 120590) 119909

10158401015840(120591119896+ 120590)

120593 (119909 (120591+

119896))

le

119903 (120591119896+ 120590) 119909

10158401015840(120591119896+ 120590)

120593 (119886119896119909 (120591119896))

le

119906 (120591119896+ 120590)

120583119886119896

(32)

Applying Lemma 1 it follows from (30) (31) and (32) that

119906 (119905) le 119906 (120591+

119895) prod

120591119895lt120591119896lt119905

119888119896

prod

120591119895lt120591119896+120590lt119905

1

120583119886119896

minus int

119905

120591119895

[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)

)]

sdot prod

119904lt120591119896lt119905

119888119896

prod

119904lt120591119896+120590lt119905

1

120583119886119896

d119904

le prod

120591119895lt120591119896lt119905

119888119896

prod

120591119895lt120591119896+120590lt119905

1

120583119886119896

times (119906 (120591+

119895) minus 120583int

119905

120591119895

[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)

)]

sdot prod

120591119895lt120591119896lt119904

1

119888119896

prod

120591119895lt120591119896+120590lt119904

119886119896d119904)

(33)

Abstract and Applied Analysis 5

which yields 119906(119905) lt 0 for all large 119905 This is contrary to 119906(119905) gt

0 and so case (i) in Lemma 2 is not possibleIf 119909(119905) satisfies the case (ii) in Lemma 2 that is 11990910158401015840(120591+

119896) gt

0 11990910158401015840(119905) gt 0 and 1199091015840(120591+

119896) lt 0 1199091015840(119905) lt 0 which proves that the

solution 119909(119905) is positive and decreasing Integrating (1) from119904 to 119905 (119905 ge 119904 ge 119879) we obtain

11990910158401015840

(119905) minus sum

119904lt120591119896lt119905

(119888119896minus 1) 119909

10158401015840(120591119896) minus 11990910158401015840

(119904)

+ int

119905

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0

(34)

Noting 119888119896le 1 and 119909

10158401015840(119905) gt 0 then it holds that

11990910158401015840

(119905) minus 11990910158401015840

(119904) + int

119905

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (35)

which leads to

minus11990910158401015840

(119904) + int

119905

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (36)

and hence

minus11990910158401015840

(119904) + int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (37)

Integrating the above inequality again from 119906 to 119905 (119905 ge 119906 ge 119879)one has

minus 1199091015840

(119905) + sum

119904lt120591119896lt119905

(119887119896minus 1) 119909

1015840(120591119896) + 1199091015840

(119906)

+ int

119905

119906

int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0

(38)

Using 1199091015840(119905) lt 0 and 119887

119896ge 1 we have

1199091015840

(119906) + int

infin

119906

int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0 (39)

Now we integrate the last inequality from 120578 to 119905 (119905 ge 120578 ge 119879)to obtain

119909 (119905) minus sum

120578lt120591119896lt119905

(119886119896minus 1) 119909 (120591

119896) minus 119909 (120578)

+ int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 d119906 le 0

(40)

Since 119886119896

le 1 (119896 isin N) and 119909(119905) is decreasing then for 119905 isin

(120591119896 120591119896+1

] 119909(120591119896+1

) le 119909(119905) le 119909(120591+

119896) = 119886119896119909(120591119896) le 119909(120591

119896) 119896 isin N

Thus we get

120583119909 (120578) int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) d120579 d119904 d119906 le minus119909 (119905) + 119909 (120578) le 119909 (120578)

(41)

and then

int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) d120579 d119904 d119906 le

1

120583

(42)

which contradicts the condition (19) The proof is complete

Replace the condition (19) with (15) we may obtain thefollowing asymptotic results

Theorem 7 Assume that (5) (6) and (15) hold and 119909(119905) is asolution of (1) If there exists a positive differentiable function 119903

such that (21) hold then 119909(119905) is either oscillatory or has a finitelimit

Proof By the proof of Theorem 6 we know the case (i) inLemma 2 is not possible too since the condition (19) is notrequired to prove it So it suffices to show if there is a solutionsatisfying case (ii) in Lemma 2 that is if

11990910158401015840(120591+

119896) gt 0 119909

10158401015840

(119905) gt 0 1199091015840(120591+

119896) lt 0 119909

1015840

(119905) lt 0 (43)

with 119905 isin (120591119896 120591119896+1

] and 120591119896ge 119879 then lim

119905rarrinfin119909(119905) exists This

is obtained by applying Lemma 5 which leads to lim119905rarrinfin

119909(119905)

exists The proof is complete

Corollary 8 In addition to the assumption of Theorem 7assume that

lim119905rarrinfin

int

119905

1199050

prod

1199050lt120591119896lt119904

1

119888119896

119901 (119904) d119904 = infin (44)

Then solution 119909(119905) of (1) either oscillates or satisfieslim119905rarrinfin

119909(119905) = 0

Proof By the proof ofTheorem 7 lim119905rarrinfin

119909(119905) exists and wedefine it by lim

119905rarrinfin119909(119905) = 120574 ge 0 We now show 120574 = 0 If not

then 120574 gt 0 So lim119905rarrinfin

120593(119909(119905 minus 120590)) = 120593(120574) = 120581 gt 0 Hencethere exists 120591

119895gt 119879 such that 120593(119909(119905minus120590)) gt 1205812 for 119905 gt 120591

119895Then

119909101584010158401015840

(119905) = minus119891 (119905 119909 (119905) 119909 (119905 minus 120590))

le minus119901 (119905) 120593 (119909 (119905 minus 120590)) le minus

120581

2

119901 (119905) 119905 ge 120591119895

(45)

and note that 11990910158401015840(120591+119896) le 11988811989611990910158401015840(120591119896) since 11990910158401015840(119905) gt 0 which imply

that

11990910158401015840

(119905) le 11990910158401015840(120591+

119895) prod

120591119895lt120591119896lt119905

119888119896minus

120581

2

int

119905

120591119895

prod

119904lt120591119896lt119905

119888119896119901 (119904) d119904

le prod

120591119895lt120591119896lt119905

119888119896[

[

11990910158401015840(120591+

119895) minus

120581

2

int

119905

120591119895

prod

120591119895lt120591119896lt119904

1

119888119896

119901 (119904) d119904]

]

(46)

Thus in virtue of (44) it holds that 11990910158401015840(119905) lt 0 and contradicts11990910158401015840(119905) gt 0 for 119905 large enough the proof is complete

Remark 9 Theorem 6 and Corollary 8 extend the results in[11 Theorem 31] and [9 Corollary 1] respectively In factwhen 119886

119896= 119887119896

= 119888119896

= 1 for 119896 isin N which implies that theimpulses in (1) disappear In such a case (5) and (6) holdnaturally and (21) and (44) are reduced to

lim119905rarrinfin

int

119905

1199050

[

[

119901 (119904) 119903 (119904) minus

[1199031015840(119904)]

2

4120583119903 (119904) (119904 minus 119879)

]

]

d119904 = infin

lim119905rarrinfin

int

119905

1199050

119901 (119904) d119904 = infin

(47)

6 Abstract and Applied Analysis

which are similar to those in [11 Theorem 31] and [9Corollary 1] respectively

Next we present some new oscillation results for (1) byusing an integral averaging condition of Kamenevrsquos type

Theorem 10 Assume (5) (6) and (19) hold Furthermore119886119896

le 1120583 le 1 119887119896

ge 1 and 119888119896

le 1 119896 isin N If there exists apositive differentiable function 119903 such that

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 = infin

(48)

where 119860(119904 120591119896(119904)

) is defined by (21) and 119898 ge 1 Then everysolution of (1) is oscillatory

Proof We choose 119879 large enough such that Lemma 2 holdsBy Lemma 2 there are two possible cases First if the case (i)holds proceeding as in the proof of Theorem 6 we will endup with (32) By (30) we have

119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)

) le minus1199061015840

(119905) 119905 ge 119879 119905 = 120591119896 (49)

If 119905 isin (120591119896+ 120590 120591119896+1

] sub (120591119896 120591119896+1

] for 120591119895ge 119879 we obtain

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904

le minusint

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904(50)

An integration by parts of the right-hand side leads to

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904

= (int

120591119895+120590

120591119895

+int

120591119895+1

120591119895+120590

+ sdot sdot sdot + int

120591119896+120590

120591119896

+int

119905

120591119896+120590

) (119905 minus 119904)1198981199061015840

(119904) d119904

= int

119905

120591119895

119906 (119904)

119898

(119905 minus 119904)119898minus1d119904

+

119896

sum

119894=119895

[119905 minus (120591119894+ 120590)]119898

[119906 (120591119894+ 120590) minus 119906 (120591

+

119894+ 120590)]

+

119896

sum

119894=119895+1

(119905 minus 120591119894)119898

[119906 (120591119894) minus 119906 (120591

+

119894)] minus (119905 minus 120591

119895)

119898

119906 (120591+

119895)

(51)

Take into account (31) (32) 119886119896le 1120583 and 119888

119896le 1 we have

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904

ge

119896

sum

119894=119895+1

(119905 minus 120591119894)119898

(1 minus 119888119894) 119906 (120591119894) minus (119905 minus 120591

119895)

119898

119906 (120591+

119895)

ge minus(119905 minus 120591119895)

119898

119906 (120591+

119895)

(52)

If 119905 isin (120591119896 120591119896+ 120590] similarly we also get

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904 ge minus(119905 minus 120591119895)

119898

119906 (120591+

119895) (53)

So it yields

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 le (119905 minus 120591

119895)

119898

119906 (120591+

119895)

(54)

which follows that1

119905119898

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904

le (

119905 minus 120591119895

119905

)

119898

119906 (120591+

119895)

(55)

Hence

lim sup119905rarrinfin

1

119905119898

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 le 119906 (120591

+

119895)

(56)

which is a contradiction of (48)If case (ii) holds then as a manner with case (ii) in

Theorem 6 it is not possible too The proof is complete

Corollary 11 Assume (19) holds and 119886119896

= 119887119896

= 119888119896

= 1 for119896 isin N If there exists a positive differential function 119903 such that

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898[

[

119901 (119904) 119903 (119904) minus

[1199031015840(119904)]

2

4120583119903 (119904) (119904 minus 119879)

]

]

d119904 = infin

(57)

where 119879 is large enough such that Lemma 2 holds Then everysolution of (1) is oscillatory

Remark 12 Corollary 11 is an extension of [11 Theorem 32]into impulsive case Especially let 119903(119905) equiv 1 in (48) it reducesto

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898119901 (119904) d119904 = infin (58)

naturally which can be considered as the extension ofKamenev-type oscillation criteria for third-order impulsivedifferential equations with delay (see [8 14 15])

4 Examples

Example 13 Consider the third-order impulsive differentialequation with delay

119909101584010158401015840

(119905) + (1 + 1205721199092

(119905)) 119909 (119905 minus 120590) = 0 119905 gt 1199050 119905 = 120591

119896

119909 (120591+

119896) = 119886119896119909 (120591119896) 119909

1015840(120591+

119896) = 1198871198961199091015840(120591119896)

11990910158401015840(120591+

119896) = 11988811989611990910158401015840(120591119896) 119896 isin N

(59)

where 120590 gt 0 120572 ge 0 are constants 120591119896minus 120591119896minus1

gt 120590 for any 119896 isin N

Abstract and Applied Analysis 7

When 119886119896

= 119887119896

= 119888119896

= 1 for any 119896 isin N the impulses in(59) disappear by [16 Theorem 4] (59) is nonoscillatory if120590 lt e3 and 120572 = 0 However we may change its oscillation byproper impulsive control In fact let 120590 lt e3 and 120572 = 0 and120591119896= 1199050+ 119896120590 (119896 isin N) in (59) choose 120593(119909) = 119909 119901(119905) equiv 1 and

119903(119905) = 1 a simple calculation leads to

int

120591119899

1199050

prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896119901 (119904) d119904

= [int

1205911

1199050

+(int

1205911+120590

1205911

+int

1205912

1205911+120590

) + + (int

120591119899minus1+120590

120591119899minus1

+int

120591119899

120591119899minus1+120590

)]

sdot prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896d119904

= (1205911minus 1199050) + [

120590

1198881

+

1198861

1198881

(1205912minus 1205911minus 120590)] + sdot sdot sdot +

11988611198862sdot sdot sdot 119886119899minus2

120590

11988811198882sdot sdot sdot 119888119899minus1

+

11988611198862sdot sdot sdot 119886119899minus1

11988811198882sdot sdot sdot 119888119899minus1

(120591119899minus 120591119899minus1

minus 120590)

= 120590(1 +

1

1198881

+

1

1198882

+ sdot sdot sdot +

1

119888119899minus1

)

(60)

Then let

119886119896= 119888119896=

119896

119896 + 1

119887119896= 1 119896 isin N (61)

Obviously (5) (6) and (19) hold and

int

120591119899

1199050

prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896119901 (119904) d119904

= 120590 (1 +

2

1

+

3

2

+ sdot sdot sdot +

119899

119899 minus 1

) 997888rarr infin (119899 997888rarr infin)

(62)

Thus (21) is also satisfied By Theorem 6 every solution of(59) is oscillatory

If we let

119886119896= 1 +

1

1198962 119887119896= 119888119896=

119896 + 1

119896

119896 isin N (63)

In this case it is easily to verify (5) (6) (15) (44) and(21) hold By Corollary 8 every solution of (59) is eitheroscillatory or tends to zero

Remark 14 It is easy to verify that in [7 Theorems 1 2 and3] cannot be applied to (59) On the other hand Theorem 7is not applicable for the condition (61) sincesuminfin

119899=1|119886119896minus1| does

not convergence

Example 15 Consider the third-order impulsive differentialequation with delay

119909101584010158401015840

(119905) + e119905 cosh (|119909 (119905)|120572minus1

119909 (119905)) 119909 (119905 minus 1) = 0

119905 ge 0 119905 = 2119896

119909 (120591+

119896) = 119886119896119909 (120591119896) 119909

1015840(120591+

119896) = 1198871198961199091015840(120591119896)

11990910158401015840(120591+

119896) = 11988811989611990910158401015840(120591119896) 120591

119896= 2119896 119896 isin N

(64)

where 120572 gt 0 119886119896= 1 119887119896= 119888119896= 119896(119896 + 1) 119896 isin N

Let 120593(119909) = 119909 119901(119905) = e119905 it is easy to verify that (5) (6)and (19) hold Choose 119903(119905) equiv 1 and119898 = 1 we have

1

119905

int

119905

119879

(119905 minus 119904) e119904d119904 =

e119905

119905

+ (

119879 minus 1

119905

minus 1) e119879 rarr infin (119905 rarr infin)

(65)

Acknowledgments

Theauthor is very grateful to ProfessorH Saker who presentsthe references [8ndash12 17] and gives many helpful suggestionswhich leads to an improvement of this paper The workis supported in part by the NSF of Guangdong province(S2012010010034)

References

[1] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificand Technical 1993

[2] G Ballinger and X Liu ldquoExistence uniqueness and bounded-ness results for impulsive delay differential equationsrdquo Applica-ble Analysis vol 74 no 1-2 pp 71ndash93 2000

[3] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing New Jersey NJ USA 1989

[4] A M Samoılenko and N A Perestyuk Impulsive DifferentialEquations vol 14 World Scientific Publishing New Jersey NJUSA 1995

[5] Y S Chen andW Z Feng ldquoOscillations of second order nonlin-ear ODE with impulsesrdquo Journal of Mathematical Analysis andApplications vol 210 no 1 pp 150ndash169 1997

[6] R P Agarwal F Karakoc and A Zafer ldquoA survey on oscilla-tion of impulsive ordinary differential equationsrdquo Advances inDifferential Equations vol 2010 pp 1ndash52 2010

[7] W-H Mao and A-HWan ldquoOscillatory and asymptotic behav-ior of solutions for nonlinear impulsive delay differential equa-tionsrdquo Acta Mathematicae Applicatae Sinica vol 22 EnglishSeries no 3 pp 387ndash396 2006

[8] L Erbe A Peterson and S H Saker ldquoAsymptotic behavior ofsolutions of a third-order nonlinear dynamic equation on timescalesrdquo Journal of Computational and Applied Mathematics vol181 no 1 pp 92ndash102 2005

[9] L Erbe A Peterson and S H Saker ldquoOscillation and asymp-totic behavior of a third-order nonlinear dynamic equationrdquoThe Canadian Applied Mathematics Quarterly vol 14 no 2 pp124ndash147 2006

8 Abstract and Applied Analysis

[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007

[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006

[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010

[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997

[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003

[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978

[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984

[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

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Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Proof Let 119909(119905) be a nonoscillatory solution of (1) withoutloss of generality we may assume that 119909(119905) gt 0 eventually(119909(119905) lt 0 eventually can be achieved in the similar way)By Lemma 2 either case (i) or case (ii) in Lemma 2 holdsAssume that 119909(119905) satisfies case (i) then 119909

10158401015840(119905) gt 0 1199091015840(119905) gt 0

for 119905 isin (120591119895 120591119895+1

] 120591119895ge 119879 (119879 is defined in Lemma 2) Define the

Riccati transformation 119906 by

119906 (119905) =

119903 (119905) 11990910158401015840(119905)

120593 (119909 (119905 minus 120590))

119905 ge 120591119895 119905 = 120591

119896 (23)

Thus 119906(119905) gt 0 for 119905 isin (120591119896 120591119896+1

] 119896 = 119895 119895 + 1 and

1199061015840

(119905) =

119903 (119905)

120593 (119909 (119905 minus 120590))

119909101584010158401015840

(119905)

+

1199031015840(119905) 120593 (119909 (119905 minus 120590)) minus 119903 (119905) 120593

1015840(119909 (119905 minus 120590)) 119909

1015840(119905 minus 120590)

1205932(119909 (119905 minus 120590))

times 11990910158401015840

(119905)

le minus 119901 (119905) 119903 (119905) +

1199031015840(119905)

119903 (119905)

119906 (119905)

minus

119903 (119905) 1205831199091015840(119905 minus 120590)

1205932(119909 (119905 minus 120590))

11990910158401015840

(119905)

(24)

If 119905 isin (120591119896 120591119896+120590] sub (120591

119896 120591119896+1

] namely 119905 minus120590 le 120591119896lt 119905 11990910158401015840(119905)

is decreasing in (119905 minus 120590 120591119896] and (120591

119896 119905] respectively In view of

the following

11990910158401015840

(119905 minus 120590) gt 11990910158401015840(120591119896) ge

11990910158401015840(120591+

119896)

119888119896

gt

11990910158401015840(119905)

119888119896

(25)

we have

1199091015840

(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int

119905minus120590

120591119896minus120590

11990910158401015840

(119904) d119904

gt 11990910158401015840

(119905 minus 120590) (119905 minus 120591119896) gt

11990910158401015840(119905)

119888119896

(119905 minus 120591119896)

(26)

Thus

1199061015840

(119905) le minus 119901 (119905) 119903 (119905) +

1199031015840(119905)

119903 (119905)

119906 (119905) minus

120583119903 (119905) (119905 minus 120591119896)

1198881198961205932(119909 (119905 minus 120590))

[11990910158401015840

(119905)]

2

= minus 119901 (119905) 119903 (119905) +

1199031015840(119905)

119903 (119905)

119906 (119905) minus

120583 (119905 minus 120591119896)

119888119896119903 (119905)

[119906 (119905)]2

= minus (radic

120583 (119905 minus 120591119896)

119888119896119903 (119905)

119906 (119905) minus radic

119888119896

4120583119903 (119905) (119905 minus 120591119896)

1199031015840

(119905))

2

minus[

[

119901 (119905) 119903 (119905) minus

119888119896[1199031015840(119905)]

2

4120583119903 (119905) (119905 minus 120591119896)

]

]

le minus[

[

119901 (119905) 119903 (119905) minus

119888119896[1199031015840(119905)]

2

4120583119903 (119905) (119905 minus 120591119896)

]

]

(27)

If 119905 isin (120591119896+120590 120591119896+1

] sub (120591119896 120591119896+1

] that is 120591119896lt 119905minus120590 lt 119905 le 120591

119896+1

then

1199091015840

(119905 minus 120590) = 1199091015840(120591119896minus 120590) + int

120591119896

120591119896minus120590

11990910158401015840

(119904) d119904 + int

119905minus120590

120591119896

11990910158401015840

(119904) d119904

ge 11990910158401015840(120591119896) 120590 + 119909

10158401015840

(119905 minus 120590) (119905 minus 120591119896minus 120590)

ge

11990910158401015840(119905)

119888119896

120590 + 11990910158401015840

(119905) (119905 minus 120591119896minus 120590)

ge 11990910158401015840

(119905) (119905 minus 120591119896minus

119888119896minus 1

119888119896

120590)

(28)

Similarly we have

1199061015840

(119905) le minus[

[

119901 (119905) 119903 (119905) minus

[1199031015840(119905)]

2

4120583119903 (119905) (119905 minus 120591119896minus ((119888119896minus 1) 119888

119896) 120590)

]

]

(29)

Thus we obtain

1199061015840

(119905) le minus [119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)

)] for 119905 isin (120591119896 120591119896+1

]

(30)

On the other hand

119906 (120591+

119896) =

119903 (120591119896) 11990910158401015840(120591+

119896)

120593 (119909 (120591119896minus 120590))

le 119888119896119906 (120591119896) (31)

Observing that 120593(119906) ge 120583119906 we have

119906 (120591+

119896+ 120590) =

119903 (120591119896+ 120590) 119909

10158401015840(120591119896+ 120590)

120593 (119909 (120591+

119896))

le

119903 (120591119896+ 120590) 119909

10158401015840(120591119896+ 120590)

120593 (119886119896119909 (120591119896))

le

119906 (120591119896+ 120590)

120583119886119896

(32)

Applying Lemma 1 it follows from (30) (31) and (32) that

119906 (119905) le 119906 (120591+

119895) prod

120591119895lt120591119896lt119905

119888119896

prod

120591119895lt120591119896+120590lt119905

1

120583119886119896

minus int

119905

120591119895

[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)

)]

sdot prod

119904lt120591119896lt119905

119888119896

prod

119904lt120591119896+120590lt119905

1

120583119886119896

d119904

le prod

120591119895lt120591119896lt119905

119888119896

prod

120591119895lt120591119896+120590lt119905

1

120583119886119896

times (119906 (120591+

119895) minus 120583int

119905

120591119895

[119901 (119904) 119903 (119904) minus 119860 (119904 120591119896(119904)

)]

sdot prod

120591119895lt120591119896lt119904

1

119888119896

prod

120591119895lt120591119896+120590lt119904

119886119896d119904)

(33)

Abstract and Applied Analysis 5

which yields 119906(119905) lt 0 for all large 119905 This is contrary to 119906(119905) gt

0 and so case (i) in Lemma 2 is not possibleIf 119909(119905) satisfies the case (ii) in Lemma 2 that is 11990910158401015840(120591+

119896) gt

0 11990910158401015840(119905) gt 0 and 1199091015840(120591+

119896) lt 0 1199091015840(119905) lt 0 which proves that the

solution 119909(119905) is positive and decreasing Integrating (1) from119904 to 119905 (119905 ge 119904 ge 119879) we obtain

11990910158401015840

(119905) minus sum

119904lt120591119896lt119905

(119888119896minus 1) 119909

10158401015840(120591119896) minus 11990910158401015840

(119904)

+ int

119905

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0

(34)

Noting 119888119896le 1 and 119909

10158401015840(119905) gt 0 then it holds that

11990910158401015840

(119905) minus 11990910158401015840

(119904) + int

119905

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (35)

which leads to

minus11990910158401015840

(119904) + int

119905

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (36)

and hence

minus11990910158401015840

(119904) + int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (37)

Integrating the above inequality again from 119906 to 119905 (119905 ge 119906 ge 119879)one has

minus 1199091015840

(119905) + sum

119904lt120591119896lt119905

(119887119896minus 1) 119909

1015840(120591119896) + 1199091015840

(119906)

+ int

119905

119906

int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0

(38)

Using 1199091015840(119905) lt 0 and 119887

119896ge 1 we have

1199091015840

(119906) + int

infin

119906

int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0 (39)

Now we integrate the last inequality from 120578 to 119905 (119905 ge 120578 ge 119879)to obtain

119909 (119905) minus sum

120578lt120591119896lt119905

(119886119896minus 1) 119909 (120591

119896) minus 119909 (120578)

+ int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 d119906 le 0

(40)

Since 119886119896

le 1 (119896 isin N) and 119909(119905) is decreasing then for 119905 isin

(120591119896 120591119896+1

] 119909(120591119896+1

) le 119909(119905) le 119909(120591+

119896) = 119886119896119909(120591119896) le 119909(120591

119896) 119896 isin N

Thus we get

120583119909 (120578) int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) d120579 d119904 d119906 le minus119909 (119905) + 119909 (120578) le 119909 (120578)

(41)

and then

int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) d120579 d119904 d119906 le

1

120583

(42)

which contradicts the condition (19) The proof is complete

Replace the condition (19) with (15) we may obtain thefollowing asymptotic results

Theorem 7 Assume that (5) (6) and (15) hold and 119909(119905) is asolution of (1) If there exists a positive differentiable function 119903

such that (21) hold then 119909(119905) is either oscillatory or has a finitelimit

Proof By the proof of Theorem 6 we know the case (i) inLemma 2 is not possible too since the condition (19) is notrequired to prove it So it suffices to show if there is a solutionsatisfying case (ii) in Lemma 2 that is if

11990910158401015840(120591+

119896) gt 0 119909

10158401015840

(119905) gt 0 1199091015840(120591+

119896) lt 0 119909

1015840

(119905) lt 0 (43)

with 119905 isin (120591119896 120591119896+1

] and 120591119896ge 119879 then lim

119905rarrinfin119909(119905) exists This

is obtained by applying Lemma 5 which leads to lim119905rarrinfin

119909(119905)

exists The proof is complete

Corollary 8 In addition to the assumption of Theorem 7assume that

lim119905rarrinfin

int

119905

1199050

prod

1199050lt120591119896lt119904

1

119888119896

119901 (119904) d119904 = infin (44)

Then solution 119909(119905) of (1) either oscillates or satisfieslim119905rarrinfin

119909(119905) = 0

Proof By the proof ofTheorem 7 lim119905rarrinfin

119909(119905) exists and wedefine it by lim

119905rarrinfin119909(119905) = 120574 ge 0 We now show 120574 = 0 If not

then 120574 gt 0 So lim119905rarrinfin

120593(119909(119905 minus 120590)) = 120593(120574) = 120581 gt 0 Hencethere exists 120591

119895gt 119879 such that 120593(119909(119905minus120590)) gt 1205812 for 119905 gt 120591

119895Then

119909101584010158401015840

(119905) = minus119891 (119905 119909 (119905) 119909 (119905 minus 120590))

le minus119901 (119905) 120593 (119909 (119905 minus 120590)) le minus

120581

2

119901 (119905) 119905 ge 120591119895

(45)

and note that 11990910158401015840(120591+119896) le 11988811989611990910158401015840(120591119896) since 11990910158401015840(119905) gt 0 which imply

that

11990910158401015840

(119905) le 11990910158401015840(120591+

119895) prod

120591119895lt120591119896lt119905

119888119896minus

120581

2

int

119905

120591119895

prod

119904lt120591119896lt119905

119888119896119901 (119904) d119904

le prod

120591119895lt120591119896lt119905

119888119896[

[

11990910158401015840(120591+

119895) minus

120581

2

int

119905

120591119895

prod

120591119895lt120591119896lt119904

1

119888119896

119901 (119904) d119904]

]

(46)

Thus in virtue of (44) it holds that 11990910158401015840(119905) lt 0 and contradicts11990910158401015840(119905) gt 0 for 119905 large enough the proof is complete

Remark 9 Theorem 6 and Corollary 8 extend the results in[11 Theorem 31] and [9 Corollary 1] respectively In factwhen 119886

119896= 119887119896

= 119888119896

= 1 for 119896 isin N which implies that theimpulses in (1) disappear In such a case (5) and (6) holdnaturally and (21) and (44) are reduced to

lim119905rarrinfin

int

119905

1199050

[

[

119901 (119904) 119903 (119904) minus

[1199031015840(119904)]

2

4120583119903 (119904) (119904 minus 119879)

]

]

d119904 = infin

lim119905rarrinfin

int

119905

1199050

119901 (119904) d119904 = infin

(47)

6 Abstract and Applied Analysis

which are similar to those in [11 Theorem 31] and [9Corollary 1] respectively

Next we present some new oscillation results for (1) byusing an integral averaging condition of Kamenevrsquos type

Theorem 10 Assume (5) (6) and (19) hold Furthermore119886119896

le 1120583 le 1 119887119896

ge 1 and 119888119896

le 1 119896 isin N If there exists apositive differentiable function 119903 such that

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 = infin

(48)

where 119860(119904 120591119896(119904)

) is defined by (21) and 119898 ge 1 Then everysolution of (1) is oscillatory

Proof We choose 119879 large enough such that Lemma 2 holdsBy Lemma 2 there are two possible cases First if the case (i)holds proceeding as in the proof of Theorem 6 we will endup with (32) By (30) we have

119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)

) le minus1199061015840

(119905) 119905 ge 119879 119905 = 120591119896 (49)

If 119905 isin (120591119896+ 120590 120591119896+1

] sub (120591119896 120591119896+1

] for 120591119895ge 119879 we obtain

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904

le minusint

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904(50)

An integration by parts of the right-hand side leads to

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904

= (int

120591119895+120590

120591119895

+int

120591119895+1

120591119895+120590

+ sdot sdot sdot + int

120591119896+120590

120591119896

+int

119905

120591119896+120590

) (119905 minus 119904)1198981199061015840

(119904) d119904

= int

119905

120591119895

119906 (119904)

119898

(119905 minus 119904)119898minus1d119904

+

119896

sum

119894=119895

[119905 minus (120591119894+ 120590)]119898

[119906 (120591119894+ 120590) minus 119906 (120591

+

119894+ 120590)]

+

119896

sum

119894=119895+1

(119905 minus 120591119894)119898

[119906 (120591119894) minus 119906 (120591

+

119894)] minus (119905 minus 120591

119895)

119898

119906 (120591+

119895)

(51)

Take into account (31) (32) 119886119896le 1120583 and 119888

119896le 1 we have

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904

ge

119896

sum

119894=119895+1

(119905 minus 120591119894)119898

(1 minus 119888119894) 119906 (120591119894) minus (119905 minus 120591

119895)

119898

119906 (120591+

119895)

ge minus(119905 minus 120591119895)

119898

119906 (120591+

119895)

(52)

If 119905 isin (120591119896 120591119896+ 120590] similarly we also get

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904 ge minus(119905 minus 120591119895)

119898

119906 (120591+

119895) (53)

So it yields

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 le (119905 minus 120591

119895)

119898

119906 (120591+

119895)

(54)

which follows that1

119905119898

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904

le (

119905 minus 120591119895

119905

)

119898

119906 (120591+

119895)

(55)

Hence

lim sup119905rarrinfin

1

119905119898

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 le 119906 (120591

+

119895)

(56)

which is a contradiction of (48)If case (ii) holds then as a manner with case (ii) in

Theorem 6 it is not possible too The proof is complete

Corollary 11 Assume (19) holds and 119886119896

= 119887119896

= 119888119896

= 1 for119896 isin N If there exists a positive differential function 119903 such that

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898[

[

119901 (119904) 119903 (119904) minus

[1199031015840(119904)]

2

4120583119903 (119904) (119904 minus 119879)

]

]

d119904 = infin

(57)

where 119879 is large enough such that Lemma 2 holds Then everysolution of (1) is oscillatory

Remark 12 Corollary 11 is an extension of [11 Theorem 32]into impulsive case Especially let 119903(119905) equiv 1 in (48) it reducesto

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898119901 (119904) d119904 = infin (58)

naturally which can be considered as the extension ofKamenev-type oscillation criteria for third-order impulsivedifferential equations with delay (see [8 14 15])

4 Examples

Example 13 Consider the third-order impulsive differentialequation with delay

119909101584010158401015840

(119905) + (1 + 1205721199092

(119905)) 119909 (119905 minus 120590) = 0 119905 gt 1199050 119905 = 120591

119896

119909 (120591+

119896) = 119886119896119909 (120591119896) 119909

1015840(120591+

119896) = 1198871198961199091015840(120591119896)

11990910158401015840(120591+

119896) = 11988811989611990910158401015840(120591119896) 119896 isin N

(59)

where 120590 gt 0 120572 ge 0 are constants 120591119896minus 120591119896minus1

gt 120590 for any 119896 isin N

Abstract and Applied Analysis 7

When 119886119896

= 119887119896

= 119888119896

= 1 for any 119896 isin N the impulses in(59) disappear by [16 Theorem 4] (59) is nonoscillatory if120590 lt e3 and 120572 = 0 However we may change its oscillation byproper impulsive control In fact let 120590 lt e3 and 120572 = 0 and120591119896= 1199050+ 119896120590 (119896 isin N) in (59) choose 120593(119909) = 119909 119901(119905) equiv 1 and

119903(119905) = 1 a simple calculation leads to

int

120591119899

1199050

prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896119901 (119904) d119904

= [int

1205911

1199050

+(int

1205911+120590

1205911

+int

1205912

1205911+120590

) + + (int

120591119899minus1+120590

120591119899minus1

+int

120591119899

120591119899minus1+120590

)]

sdot prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896d119904

= (1205911minus 1199050) + [

120590

1198881

+

1198861

1198881

(1205912minus 1205911minus 120590)] + sdot sdot sdot +

11988611198862sdot sdot sdot 119886119899minus2

120590

11988811198882sdot sdot sdot 119888119899minus1

+

11988611198862sdot sdot sdot 119886119899minus1

11988811198882sdot sdot sdot 119888119899minus1

(120591119899minus 120591119899minus1

minus 120590)

= 120590(1 +

1

1198881

+

1

1198882

+ sdot sdot sdot +

1

119888119899minus1

)

(60)

Then let

119886119896= 119888119896=

119896

119896 + 1

119887119896= 1 119896 isin N (61)

Obviously (5) (6) and (19) hold and

int

120591119899

1199050

prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896119901 (119904) d119904

= 120590 (1 +

2

1

+

3

2

+ sdot sdot sdot +

119899

119899 minus 1

) 997888rarr infin (119899 997888rarr infin)

(62)

Thus (21) is also satisfied By Theorem 6 every solution of(59) is oscillatory

If we let

119886119896= 1 +

1

1198962 119887119896= 119888119896=

119896 + 1

119896

119896 isin N (63)

In this case it is easily to verify (5) (6) (15) (44) and(21) hold By Corollary 8 every solution of (59) is eitheroscillatory or tends to zero

Remark 14 It is easy to verify that in [7 Theorems 1 2 and3] cannot be applied to (59) On the other hand Theorem 7is not applicable for the condition (61) sincesuminfin

119899=1|119886119896minus1| does

not convergence

Example 15 Consider the third-order impulsive differentialequation with delay

119909101584010158401015840

(119905) + e119905 cosh (|119909 (119905)|120572minus1

119909 (119905)) 119909 (119905 minus 1) = 0

119905 ge 0 119905 = 2119896

119909 (120591+

119896) = 119886119896119909 (120591119896) 119909

1015840(120591+

119896) = 1198871198961199091015840(120591119896)

11990910158401015840(120591+

119896) = 11988811989611990910158401015840(120591119896) 120591

119896= 2119896 119896 isin N

(64)

where 120572 gt 0 119886119896= 1 119887119896= 119888119896= 119896(119896 + 1) 119896 isin N

Let 120593(119909) = 119909 119901(119905) = e119905 it is easy to verify that (5) (6)and (19) hold Choose 119903(119905) equiv 1 and119898 = 1 we have

1

119905

int

119905

119879

(119905 minus 119904) e119904d119904 =

e119905

119905

+ (

119879 minus 1

119905

minus 1) e119879 rarr infin (119905 rarr infin)

(65)

Acknowledgments

Theauthor is very grateful to ProfessorH Saker who presentsthe references [8ndash12 17] and gives many helpful suggestionswhich leads to an improvement of this paper The workis supported in part by the NSF of Guangdong province(S2012010010034)

References

[1] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificand Technical 1993

[2] G Ballinger and X Liu ldquoExistence uniqueness and bounded-ness results for impulsive delay differential equationsrdquo Applica-ble Analysis vol 74 no 1-2 pp 71ndash93 2000

[3] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing New Jersey NJ USA 1989

[4] A M Samoılenko and N A Perestyuk Impulsive DifferentialEquations vol 14 World Scientific Publishing New Jersey NJUSA 1995

[5] Y S Chen andW Z Feng ldquoOscillations of second order nonlin-ear ODE with impulsesrdquo Journal of Mathematical Analysis andApplications vol 210 no 1 pp 150ndash169 1997

[6] R P Agarwal F Karakoc and A Zafer ldquoA survey on oscilla-tion of impulsive ordinary differential equationsrdquo Advances inDifferential Equations vol 2010 pp 1ndash52 2010

[7] W-H Mao and A-HWan ldquoOscillatory and asymptotic behav-ior of solutions for nonlinear impulsive delay differential equa-tionsrdquo Acta Mathematicae Applicatae Sinica vol 22 EnglishSeries no 3 pp 387ndash396 2006

[8] L Erbe A Peterson and S H Saker ldquoAsymptotic behavior ofsolutions of a third-order nonlinear dynamic equation on timescalesrdquo Journal of Computational and Applied Mathematics vol181 no 1 pp 92ndash102 2005

[9] L Erbe A Peterson and S H Saker ldquoOscillation and asymp-totic behavior of a third-order nonlinear dynamic equationrdquoThe Canadian Applied Mathematics Quarterly vol 14 no 2 pp124ndash147 2006

8 Abstract and Applied Analysis

[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007

[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006

[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010

[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997

[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003

[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978

[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984

[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008

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Volume 2014

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Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

which yields 119906(119905) lt 0 for all large 119905 This is contrary to 119906(119905) gt

0 and so case (i) in Lemma 2 is not possibleIf 119909(119905) satisfies the case (ii) in Lemma 2 that is 11990910158401015840(120591+

119896) gt

0 11990910158401015840(119905) gt 0 and 1199091015840(120591+

119896) lt 0 1199091015840(119905) lt 0 which proves that the

solution 119909(119905) is positive and decreasing Integrating (1) from119904 to 119905 (119905 ge 119904 ge 119879) we obtain

11990910158401015840

(119905) minus sum

119904lt120591119896lt119905

(119888119896minus 1) 119909

10158401015840(120591119896) minus 11990910158401015840

(119904)

+ int

119905

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0

(34)

Noting 119888119896le 1 and 119909

10158401015840(119905) gt 0 then it holds that

11990910158401015840

(119905) minus 11990910158401015840

(119904) + int

119905

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (35)

which leads to

minus11990910158401015840

(119904) + int

119905

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (36)

and hence

minus11990910158401015840

(119904) + int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 le 0 (37)

Integrating the above inequality again from 119906 to 119905 (119905 ge 119906 ge 119879)one has

minus 1199091015840

(119905) + sum

119904lt120591119896lt119905

(119887119896minus 1) 119909

1015840(120591119896) + 1199091015840

(119906)

+ int

119905

119906

int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0

(38)

Using 1199091015840(119905) lt 0 and 119887

119896ge 1 we have

1199091015840

(119906) + int

infin

119906

int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 le 0 (39)

Now we integrate the last inequality from 120578 to 119905 (119905 ge 120578 ge 119879)to obtain

119909 (119905) minus sum

120578lt120591119896lt119905

(119886119896minus 1) 119909 (120591

119896) minus 119909 (120578)

+ int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) 120593 (119909 (120579 minus 120590)) d120579 d119904 d119906 le 0

(40)

Since 119886119896

le 1 (119896 isin N) and 119909(119905) is decreasing then for 119905 isin

(120591119896 120591119896+1

] 119909(120591119896+1

) le 119909(119905) le 119909(120591+

119896) = 119886119896119909(120591119896) le 119909(120591

119896) 119896 isin N

Thus we get

120583119909 (120578) int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) d120579 d119904 d119906 le minus119909 (119905) + 119909 (120578) le 119909 (120578)

(41)

and then

int

119905

120578

int

infin

119906

int

infin

119904

119901 (120579) d120579 d119904 d119906 le

1

120583

(42)

which contradicts the condition (19) The proof is complete

Replace the condition (19) with (15) we may obtain thefollowing asymptotic results

Theorem 7 Assume that (5) (6) and (15) hold and 119909(119905) is asolution of (1) If there exists a positive differentiable function 119903

such that (21) hold then 119909(119905) is either oscillatory or has a finitelimit

Proof By the proof of Theorem 6 we know the case (i) inLemma 2 is not possible too since the condition (19) is notrequired to prove it So it suffices to show if there is a solutionsatisfying case (ii) in Lemma 2 that is if

11990910158401015840(120591+

119896) gt 0 119909

10158401015840

(119905) gt 0 1199091015840(120591+

119896) lt 0 119909

1015840

(119905) lt 0 (43)

with 119905 isin (120591119896 120591119896+1

] and 120591119896ge 119879 then lim

119905rarrinfin119909(119905) exists This

is obtained by applying Lemma 5 which leads to lim119905rarrinfin

119909(119905)

exists The proof is complete

Corollary 8 In addition to the assumption of Theorem 7assume that

lim119905rarrinfin

int

119905

1199050

prod

1199050lt120591119896lt119904

1

119888119896

119901 (119904) d119904 = infin (44)

Then solution 119909(119905) of (1) either oscillates or satisfieslim119905rarrinfin

119909(119905) = 0

Proof By the proof ofTheorem 7 lim119905rarrinfin

119909(119905) exists and wedefine it by lim

119905rarrinfin119909(119905) = 120574 ge 0 We now show 120574 = 0 If not

then 120574 gt 0 So lim119905rarrinfin

120593(119909(119905 minus 120590)) = 120593(120574) = 120581 gt 0 Hencethere exists 120591

119895gt 119879 such that 120593(119909(119905minus120590)) gt 1205812 for 119905 gt 120591

119895Then

119909101584010158401015840

(119905) = minus119891 (119905 119909 (119905) 119909 (119905 minus 120590))

le minus119901 (119905) 120593 (119909 (119905 minus 120590)) le minus

120581

2

119901 (119905) 119905 ge 120591119895

(45)

and note that 11990910158401015840(120591+119896) le 11988811989611990910158401015840(120591119896) since 11990910158401015840(119905) gt 0 which imply

that

11990910158401015840

(119905) le 11990910158401015840(120591+

119895) prod

120591119895lt120591119896lt119905

119888119896minus

120581

2

int

119905

120591119895

prod

119904lt120591119896lt119905

119888119896119901 (119904) d119904

le prod

120591119895lt120591119896lt119905

119888119896[

[

11990910158401015840(120591+

119895) minus

120581

2

int

119905

120591119895

prod

120591119895lt120591119896lt119904

1

119888119896

119901 (119904) d119904]

]

(46)

Thus in virtue of (44) it holds that 11990910158401015840(119905) lt 0 and contradicts11990910158401015840(119905) gt 0 for 119905 large enough the proof is complete

Remark 9 Theorem 6 and Corollary 8 extend the results in[11 Theorem 31] and [9 Corollary 1] respectively In factwhen 119886

119896= 119887119896

= 119888119896

= 1 for 119896 isin N which implies that theimpulses in (1) disappear In such a case (5) and (6) holdnaturally and (21) and (44) are reduced to

lim119905rarrinfin

int

119905

1199050

[

[

119901 (119904) 119903 (119904) minus

[1199031015840(119904)]

2

4120583119903 (119904) (119904 minus 119879)

]

]

d119904 = infin

lim119905rarrinfin

int

119905

1199050

119901 (119904) d119904 = infin

(47)

6 Abstract and Applied Analysis

which are similar to those in [11 Theorem 31] and [9Corollary 1] respectively

Next we present some new oscillation results for (1) byusing an integral averaging condition of Kamenevrsquos type

Theorem 10 Assume (5) (6) and (19) hold Furthermore119886119896

le 1120583 le 1 119887119896

ge 1 and 119888119896

le 1 119896 isin N If there exists apositive differentiable function 119903 such that

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 = infin

(48)

where 119860(119904 120591119896(119904)

) is defined by (21) and 119898 ge 1 Then everysolution of (1) is oscillatory

Proof We choose 119879 large enough such that Lemma 2 holdsBy Lemma 2 there are two possible cases First if the case (i)holds proceeding as in the proof of Theorem 6 we will endup with (32) By (30) we have

119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)

) le minus1199061015840

(119905) 119905 ge 119879 119905 = 120591119896 (49)

If 119905 isin (120591119896+ 120590 120591119896+1

] sub (120591119896 120591119896+1

] for 120591119895ge 119879 we obtain

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904

le minusint

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904(50)

An integration by parts of the right-hand side leads to

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904

= (int

120591119895+120590

120591119895

+int

120591119895+1

120591119895+120590

+ sdot sdot sdot + int

120591119896+120590

120591119896

+int

119905

120591119896+120590

) (119905 minus 119904)1198981199061015840

(119904) d119904

= int

119905

120591119895

119906 (119904)

119898

(119905 minus 119904)119898minus1d119904

+

119896

sum

119894=119895

[119905 minus (120591119894+ 120590)]119898

[119906 (120591119894+ 120590) minus 119906 (120591

+

119894+ 120590)]

+

119896

sum

119894=119895+1

(119905 minus 120591119894)119898

[119906 (120591119894) minus 119906 (120591

+

119894)] minus (119905 minus 120591

119895)

119898

119906 (120591+

119895)

(51)

Take into account (31) (32) 119886119896le 1120583 and 119888

119896le 1 we have

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904

ge

119896

sum

119894=119895+1

(119905 minus 120591119894)119898

(1 minus 119888119894) 119906 (120591119894) minus (119905 minus 120591

119895)

119898

119906 (120591+

119895)

ge minus(119905 minus 120591119895)

119898

119906 (120591+

119895)

(52)

If 119905 isin (120591119896 120591119896+ 120590] similarly we also get

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904 ge minus(119905 minus 120591119895)

119898

119906 (120591+

119895) (53)

So it yields

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 le (119905 minus 120591

119895)

119898

119906 (120591+

119895)

(54)

which follows that1

119905119898

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904

le (

119905 minus 120591119895

119905

)

119898

119906 (120591+

119895)

(55)

Hence

lim sup119905rarrinfin

1

119905119898

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 le 119906 (120591

+

119895)

(56)

which is a contradiction of (48)If case (ii) holds then as a manner with case (ii) in

Theorem 6 it is not possible too The proof is complete

Corollary 11 Assume (19) holds and 119886119896

= 119887119896

= 119888119896

= 1 for119896 isin N If there exists a positive differential function 119903 such that

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898[

[

119901 (119904) 119903 (119904) minus

[1199031015840(119904)]

2

4120583119903 (119904) (119904 minus 119879)

]

]

d119904 = infin

(57)

where 119879 is large enough such that Lemma 2 holds Then everysolution of (1) is oscillatory

Remark 12 Corollary 11 is an extension of [11 Theorem 32]into impulsive case Especially let 119903(119905) equiv 1 in (48) it reducesto

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898119901 (119904) d119904 = infin (58)

naturally which can be considered as the extension ofKamenev-type oscillation criteria for third-order impulsivedifferential equations with delay (see [8 14 15])

4 Examples

Example 13 Consider the third-order impulsive differentialequation with delay

119909101584010158401015840

(119905) + (1 + 1205721199092

(119905)) 119909 (119905 minus 120590) = 0 119905 gt 1199050 119905 = 120591

119896

119909 (120591+

119896) = 119886119896119909 (120591119896) 119909

1015840(120591+

119896) = 1198871198961199091015840(120591119896)

11990910158401015840(120591+

119896) = 11988811989611990910158401015840(120591119896) 119896 isin N

(59)

where 120590 gt 0 120572 ge 0 are constants 120591119896minus 120591119896minus1

gt 120590 for any 119896 isin N

Abstract and Applied Analysis 7

When 119886119896

= 119887119896

= 119888119896

= 1 for any 119896 isin N the impulses in(59) disappear by [16 Theorem 4] (59) is nonoscillatory if120590 lt e3 and 120572 = 0 However we may change its oscillation byproper impulsive control In fact let 120590 lt e3 and 120572 = 0 and120591119896= 1199050+ 119896120590 (119896 isin N) in (59) choose 120593(119909) = 119909 119901(119905) equiv 1 and

119903(119905) = 1 a simple calculation leads to

int

120591119899

1199050

prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896119901 (119904) d119904

= [int

1205911

1199050

+(int

1205911+120590

1205911

+int

1205912

1205911+120590

) + + (int

120591119899minus1+120590

120591119899minus1

+int

120591119899

120591119899minus1+120590

)]

sdot prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896d119904

= (1205911minus 1199050) + [

120590

1198881

+

1198861

1198881

(1205912minus 1205911minus 120590)] + sdot sdot sdot +

11988611198862sdot sdot sdot 119886119899minus2

120590

11988811198882sdot sdot sdot 119888119899minus1

+

11988611198862sdot sdot sdot 119886119899minus1

11988811198882sdot sdot sdot 119888119899minus1

(120591119899minus 120591119899minus1

minus 120590)

= 120590(1 +

1

1198881

+

1

1198882

+ sdot sdot sdot +

1

119888119899minus1

)

(60)

Then let

119886119896= 119888119896=

119896

119896 + 1

119887119896= 1 119896 isin N (61)

Obviously (5) (6) and (19) hold and

int

120591119899

1199050

prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896119901 (119904) d119904

= 120590 (1 +

2

1

+

3

2

+ sdot sdot sdot +

119899

119899 minus 1

) 997888rarr infin (119899 997888rarr infin)

(62)

Thus (21) is also satisfied By Theorem 6 every solution of(59) is oscillatory

If we let

119886119896= 1 +

1

1198962 119887119896= 119888119896=

119896 + 1

119896

119896 isin N (63)

In this case it is easily to verify (5) (6) (15) (44) and(21) hold By Corollary 8 every solution of (59) is eitheroscillatory or tends to zero

Remark 14 It is easy to verify that in [7 Theorems 1 2 and3] cannot be applied to (59) On the other hand Theorem 7is not applicable for the condition (61) sincesuminfin

119899=1|119886119896minus1| does

not convergence

Example 15 Consider the third-order impulsive differentialequation with delay

119909101584010158401015840

(119905) + e119905 cosh (|119909 (119905)|120572minus1

119909 (119905)) 119909 (119905 minus 1) = 0

119905 ge 0 119905 = 2119896

119909 (120591+

119896) = 119886119896119909 (120591119896) 119909

1015840(120591+

119896) = 1198871198961199091015840(120591119896)

11990910158401015840(120591+

119896) = 11988811989611990910158401015840(120591119896) 120591

119896= 2119896 119896 isin N

(64)

where 120572 gt 0 119886119896= 1 119887119896= 119888119896= 119896(119896 + 1) 119896 isin N

Let 120593(119909) = 119909 119901(119905) = e119905 it is easy to verify that (5) (6)and (19) hold Choose 119903(119905) equiv 1 and119898 = 1 we have

1

119905

int

119905

119879

(119905 minus 119904) e119904d119904 =

e119905

119905

+ (

119879 minus 1

119905

minus 1) e119879 rarr infin (119905 rarr infin)

(65)

Acknowledgments

Theauthor is very grateful to ProfessorH Saker who presentsthe references [8ndash12 17] and gives many helpful suggestionswhich leads to an improvement of this paper The workis supported in part by the NSF of Guangdong province(S2012010010034)

References

[1] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificand Technical 1993

[2] G Ballinger and X Liu ldquoExistence uniqueness and bounded-ness results for impulsive delay differential equationsrdquo Applica-ble Analysis vol 74 no 1-2 pp 71ndash93 2000

[3] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing New Jersey NJ USA 1989

[4] A M Samoılenko and N A Perestyuk Impulsive DifferentialEquations vol 14 World Scientific Publishing New Jersey NJUSA 1995

[5] Y S Chen andW Z Feng ldquoOscillations of second order nonlin-ear ODE with impulsesrdquo Journal of Mathematical Analysis andApplications vol 210 no 1 pp 150ndash169 1997

[6] R P Agarwal F Karakoc and A Zafer ldquoA survey on oscilla-tion of impulsive ordinary differential equationsrdquo Advances inDifferential Equations vol 2010 pp 1ndash52 2010

[7] W-H Mao and A-HWan ldquoOscillatory and asymptotic behav-ior of solutions for nonlinear impulsive delay differential equa-tionsrdquo Acta Mathematicae Applicatae Sinica vol 22 EnglishSeries no 3 pp 387ndash396 2006

[8] L Erbe A Peterson and S H Saker ldquoAsymptotic behavior ofsolutions of a third-order nonlinear dynamic equation on timescalesrdquo Journal of Computational and Applied Mathematics vol181 no 1 pp 92ndash102 2005

[9] L Erbe A Peterson and S H Saker ldquoOscillation and asymp-totic behavior of a third-order nonlinear dynamic equationrdquoThe Canadian Applied Mathematics Quarterly vol 14 no 2 pp124ndash147 2006

8 Abstract and Applied Analysis

[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007

[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006

[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010

[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997

[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003

[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978

[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984

[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008

Submit your manuscripts athttpwwwhindawicom

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Function Spaces

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

which are similar to those in [11 Theorem 31] and [9Corollary 1] respectively

Next we present some new oscillation results for (1) byusing an integral averaging condition of Kamenevrsquos type

Theorem 10 Assume (5) (6) and (19) hold Furthermore119886119896

le 1120583 le 1 119887119896

ge 1 and 119888119896

le 1 119896 isin N If there exists apositive differentiable function 119903 such that

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 = infin

(48)

where 119860(119904 120591119896(119904)

) is defined by (21) and 119898 ge 1 Then everysolution of (1) is oscillatory

Proof We choose 119879 large enough such that Lemma 2 holdsBy Lemma 2 there are two possible cases First if the case (i)holds proceeding as in the proof of Theorem 6 we will endup with (32) By (30) we have

119901 (119905) 119903 (119905) minus 119860 (119905 120591119896(119905)

) le minus1199061015840

(119905) 119905 ge 119879 119905 = 120591119896 (49)

If 119905 isin (120591119896+ 120590 120591119896+1

] sub (120591119896 120591119896+1

] for 120591119895ge 119879 we obtain

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904

le minusint

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904(50)

An integration by parts of the right-hand side leads to

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904

= (int

120591119895+120590

120591119895

+int

120591119895+1

120591119895+120590

+ sdot sdot sdot + int

120591119896+120590

120591119896

+int

119905

120591119896+120590

) (119905 minus 119904)1198981199061015840

(119904) d119904

= int

119905

120591119895

119906 (119904)

119898

(119905 minus 119904)119898minus1d119904

+

119896

sum

119894=119895

[119905 minus (120591119894+ 120590)]119898

[119906 (120591119894+ 120590) minus 119906 (120591

+

119894+ 120590)]

+

119896

sum

119894=119895+1

(119905 minus 120591119894)119898

[119906 (120591119894) minus 119906 (120591

+

119894)] minus (119905 minus 120591

119895)

119898

119906 (120591+

119895)

(51)

Take into account (31) (32) 119886119896le 1120583 and 119888

119896le 1 we have

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904

ge

119896

sum

119894=119895+1

(119905 minus 120591119894)119898

(1 minus 119888119894) 119906 (120591119894) minus (119905 minus 120591

119895)

119898

119906 (120591+

119895)

ge minus(119905 minus 120591119895)

119898

119906 (120591+

119895)

(52)

If 119905 isin (120591119896 120591119896+ 120590] similarly we also get

int

119905

120591119895

(119905 minus 119904)1198981199061015840

(119904) d119904 ge minus(119905 minus 120591119895)

119898

119906 (120591+

119895) (53)

So it yields

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 le (119905 minus 120591

119895)

119898

119906 (120591+

119895)

(54)

which follows that1

119905119898

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904

le (

119905 minus 120591119895

119905

)

119898

119906 (120591+

119895)

(55)

Hence

lim sup119905rarrinfin

1

119905119898

int

119905

120591119895

(119905 minus 119904)119898[119901 (119904) 119903 (119904) minus 119860 (119904 120591

119896(119904))] d119904 le 119906 (120591

+

119895)

(56)

which is a contradiction of (48)If case (ii) holds then as a manner with case (ii) in

Theorem 6 it is not possible too The proof is complete

Corollary 11 Assume (19) holds and 119886119896

= 119887119896

= 119888119896

= 1 for119896 isin N If there exists a positive differential function 119903 such that

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898[

[

119901 (119904) 119903 (119904) minus

[1199031015840(119904)]

2

4120583119903 (119904) (119904 minus 119879)

]

]

d119904 = infin

(57)

where 119879 is large enough such that Lemma 2 holds Then everysolution of (1) is oscillatory

Remark 12 Corollary 11 is an extension of [11 Theorem 32]into impulsive case Especially let 119903(119905) equiv 1 in (48) it reducesto

lim sup119905rarrinfin

1

119905119898

int

119905

119879

(119905 minus 119904)119898119901 (119904) d119904 = infin (58)

naturally which can be considered as the extension ofKamenev-type oscillation criteria for third-order impulsivedifferential equations with delay (see [8 14 15])

4 Examples

Example 13 Consider the third-order impulsive differentialequation with delay

119909101584010158401015840

(119905) + (1 + 1205721199092

(119905)) 119909 (119905 minus 120590) = 0 119905 gt 1199050 119905 = 120591

119896

119909 (120591+

119896) = 119886119896119909 (120591119896) 119909

1015840(120591+

119896) = 1198871198961199091015840(120591119896)

11990910158401015840(120591+

119896) = 11988811989611990910158401015840(120591119896) 119896 isin N

(59)

where 120590 gt 0 120572 ge 0 are constants 120591119896minus 120591119896minus1

gt 120590 for any 119896 isin N

Abstract and Applied Analysis 7

When 119886119896

= 119887119896

= 119888119896

= 1 for any 119896 isin N the impulses in(59) disappear by [16 Theorem 4] (59) is nonoscillatory if120590 lt e3 and 120572 = 0 However we may change its oscillation byproper impulsive control In fact let 120590 lt e3 and 120572 = 0 and120591119896= 1199050+ 119896120590 (119896 isin N) in (59) choose 120593(119909) = 119909 119901(119905) equiv 1 and

119903(119905) = 1 a simple calculation leads to

int

120591119899

1199050

prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896119901 (119904) d119904

= [int

1205911

1199050

+(int

1205911+120590

1205911

+int

1205912

1205911+120590

) + + (int

120591119899minus1+120590

120591119899minus1

+int

120591119899

120591119899minus1+120590

)]

sdot prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896d119904

= (1205911minus 1199050) + [

120590

1198881

+

1198861

1198881

(1205912minus 1205911minus 120590)] + sdot sdot sdot +

11988611198862sdot sdot sdot 119886119899minus2

120590

11988811198882sdot sdot sdot 119888119899minus1

+

11988611198862sdot sdot sdot 119886119899minus1

11988811198882sdot sdot sdot 119888119899minus1

(120591119899minus 120591119899minus1

minus 120590)

= 120590(1 +

1

1198881

+

1

1198882

+ sdot sdot sdot +

1

119888119899minus1

)

(60)

Then let

119886119896= 119888119896=

119896

119896 + 1

119887119896= 1 119896 isin N (61)

Obviously (5) (6) and (19) hold and

int

120591119899

1199050

prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896119901 (119904) d119904

= 120590 (1 +

2

1

+

3

2

+ sdot sdot sdot +

119899

119899 minus 1

) 997888rarr infin (119899 997888rarr infin)

(62)

Thus (21) is also satisfied By Theorem 6 every solution of(59) is oscillatory

If we let

119886119896= 1 +

1

1198962 119887119896= 119888119896=

119896 + 1

119896

119896 isin N (63)

In this case it is easily to verify (5) (6) (15) (44) and(21) hold By Corollary 8 every solution of (59) is eitheroscillatory or tends to zero

Remark 14 It is easy to verify that in [7 Theorems 1 2 and3] cannot be applied to (59) On the other hand Theorem 7is not applicable for the condition (61) sincesuminfin

119899=1|119886119896minus1| does

not convergence

Example 15 Consider the third-order impulsive differentialequation with delay

119909101584010158401015840

(119905) + e119905 cosh (|119909 (119905)|120572minus1

119909 (119905)) 119909 (119905 minus 1) = 0

119905 ge 0 119905 = 2119896

119909 (120591+

119896) = 119886119896119909 (120591119896) 119909

1015840(120591+

119896) = 1198871198961199091015840(120591119896)

11990910158401015840(120591+

119896) = 11988811989611990910158401015840(120591119896) 120591

119896= 2119896 119896 isin N

(64)

where 120572 gt 0 119886119896= 1 119887119896= 119888119896= 119896(119896 + 1) 119896 isin N

Let 120593(119909) = 119909 119901(119905) = e119905 it is easy to verify that (5) (6)and (19) hold Choose 119903(119905) equiv 1 and119898 = 1 we have

1

119905

int

119905

119879

(119905 minus 119904) e119904d119904 =

e119905

119905

+ (

119879 minus 1

119905

minus 1) e119879 rarr infin (119905 rarr infin)

(65)

Acknowledgments

Theauthor is very grateful to ProfessorH Saker who presentsthe references [8ndash12 17] and gives many helpful suggestionswhich leads to an improvement of this paper The workis supported in part by the NSF of Guangdong province(S2012010010034)

References

[1] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificand Technical 1993

[2] G Ballinger and X Liu ldquoExistence uniqueness and bounded-ness results for impulsive delay differential equationsrdquo Applica-ble Analysis vol 74 no 1-2 pp 71ndash93 2000

[3] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing New Jersey NJ USA 1989

[4] A M Samoılenko and N A Perestyuk Impulsive DifferentialEquations vol 14 World Scientific Publishing New Jersey NJUSA 1995

[5] Y S Chen andW Z Feng ldquoOscillations of second order nonlin-ear ODE with impulsesrdquo Journal of Mathematical Analysis andApplications vol 210 no 1 pp 150ndash169 1997

[6] R P Agarwal F Karakoc and A Zafer ldquoA survey on oscilla-tion of impulsive ordinary differential equationsrdquo Advances inDifferential Equations vol 2010 pp 1ndash52 2010

[7] W-H Mao and A-HWan ldquoOscillatory and asymptotic behav-ior of solutions for nonlinear impulsive delay differential equa-tionsrdquo Acta Mathematicae Applicatae Sinica vol 22 EnglishSeries no 3 pp 387ndash396 2006

[8] L Erbe A Peterson and S H Saker ldquoAsymptotic behavior ofsolutions of a third-order nonlinear dynamic equation on timescalesrdquo Journal of Computational and Applied Mathematics vol181 no 1 pp 92ndash102 2005

[9] L Erbe A Peterson and S H Saker ldquoOscillation and asymp-totic behavior of a third-order nonlinear dynamic equationrdquoThe Canadian Applied Mathematics Quarterly vol 14 no 2 pp124ndash147 2006

8 Abstract and Applied Analysis

[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007

[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006

[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010

[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997

[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003

[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978

[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984

[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

When 119886119896

= 119887119896

= 119888119896

= 1 for any 119896 isin N the impulses in(59) disappear by [16 Theorem 4] (59) is nonoscillatory if120590 lt e3 and 120572 = 0 However we may change its oscillation byproper impulsive control In fact let 120590 lt e3 and 120572 = 0 and120591119896= 1199050+ 119896120590 (119896 isin N) in (59) choose 120593(119909) = 119909 119901(119905) equiv 1 and

119903(119905) = 1 a simple calculation leads to

int

120591119899

1199050

prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896119901 (119904) d119904

= [int

1205911

1199050

+(int

1205911+120590

1205911

+int

1205912

1205911+120590

) + + (int

120591119899minus1+120590

120591119899minus1

+int

120591119899

120591119899minus1+120590

)]

sdot prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896d119904

= (1205911minus 1199050) + [

120590

1198881

+

1198861

1198881

(1205912minus 1205911minus 120590)] + sdot sdot sdot +

11988611198862sdot sdot sdot 119886119899minus2

120590

11988811198882sdot sdot sdot 119888119899minus1

+

11988611198862sdot sdot sdot 119886119899minus1

11988811198882sdot sdot sdot 119888119899minus1

(120591119899minus 120591119899minus1

minus 120590)

= 120590(1 +

1

1198881

+

1

1198882

+ sdot sdot sdot +

1

119888119899minus1

)

(60)

Then let

119886119896= 119888119896=

119896

119896 + 1

119887119896= 1 119896 isin N (61)

Obviously (5) (6) and (19) hold and

int

120591119899

1199050

prod

1199050lt120591119896lt119904

1

119888119896

prod

1199050lt120591119896+120590lt119904

119886119896119901 (119904) d119904

= 120590 (1 +

2

1

+

3

2

+ sdot sdot sdot +

119899

119899 minus 1

) 997888rarr infin (119899 997888rarr infin)

(62)

Thus (21) is also satisfied By Theorem 6 every solution of(59) is oscillatory

If we let

119886119896= 1 +

1

1198962 119887119896= 119888119896=

119896 + 1

119896

119896 isin N (63)

In this case it is easily to verify (5) (6) (15) (44) and(21) hold By Corollary 8 every solution of (59) is eitheroscillatory or tends to zero

Remark 14 It is easy to verify that in [7 Theorems 1 2 and3] cannot be applied to (59) On the other hand Theorem 7is not applicable for the condition (61) sincesuminfin

119899=1|119886119896minus1| does

not convergence

Example 15 Consider the third-order impulsive differentialequation with delay

119909101584010158401015840

(119905) + e119905 cosh (|119909 (119905)|120572minus1

119909 (119905)) 119909 (119905 minus 1) = 0

119905 ge 0 119905 = 2119896

119909 (120591+

119896) = 119886119896119909 (120591119896) 119909

1015840(120591+

119896) = 1198871198961199091015840(120591119896)

11990910158401015840(120591+

119896) = 11988811989611990910158401015840(120591119896) 120591

119896= 2119896 119896 isin N

(64)

where 120572 gt 0 119886119896= 1 119887119896= 119888119896= 119896(119896 + 1) 119896 isin N

Let 120593(119909) = 119909 119901(119905) = e119905 it is easy to verify that (5) (6)and (19) hold Choose 119903(119905) equiv 1 and119898 = 1 we have

1

119905

int

119905

119879

(119905 minus 119904) e119904d119904 =

e119905

119905

+ (

119879 minus 1

119905

minus 1) e119879 rarr infin (119905 rarr infin)

(65)

Acknowledgments

Theauthor is very grateful to ProfessorH Saker who presentsthe references [8ndash12 17] and gives many helpful suggestionswhich leads to an improvement of this paper The workis supported in part by the NSF of Guangdong province(S2012010010034)

References

[1] D D Bainov and P S Simeonov Impulsive Differential Equa-tions Periodic Solutions and Applications Longman Scientificand Technical 1993

[2] G Ballinger and X Liu ldquoExistence uniqueness and bounded-ness results for impulsive delay differential equationsrdquo Applica-ble Analysis vol 74 no 1-2 pp 71ndash93 2000

[3] V Lakshmikantham D D Baınov and P S Simeonov Theoryof Impulsive Differential Equations vol 6 World ScientificPublishing New Jersey NJ USA 1989

[4] A M Samoılenko and N A Perestyuk Impulsive DifferentialEquations vol 14 World Scientific Publishing New Jersey NJUSA 1995

[5] Y S Chen andW Z Feng ldquoOscillations of second order nonlin-ear ODE with impulsesrdquo Journal of Mathematical Analysis andApplications vol 210 no 1 pp 150ndash169 1997

[6] R P Agarwal F Karakoc and A Zafer ldquoA survey on oscilla-tion of impulsive ordinary differential equationsrdquo Advances inDifferential Equations vol 2010 pp 1ndash52 2010

[7] W-H Mao and A-HWan ldquoOscillatory and asymptotic behav-ior of solutions for nonlinear impulsive delay differential equa-tionsrdquo Acta Mathematicae Applicatae Sinica vol 22 EnglishSeries no 3 pp 387ndash396 2006

[8] L Erbe A Peterson and S H Saker ldquoAsymptotic behavior ofsolutions of a third-order nonlinear dynamic equation on timescalesrdquo Journal of Computational and Applied Mathematics vol181 no 1 pp 92ndash102 2005

[9] L Erbe A Peterson and S H Saker ldquoOscillation and asymp-totic behavior of a third-order nonlinear dynamic equationrdquoThe Canadian Applied Mathematics Quarterly vol 14 no 2 pp124ndash147 2006

8 Abstract and Applied Analysis

[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007

[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006

[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010

[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997

[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003

[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978

[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984

[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

[10] L Erbe A Peterson and S H Saker ldquoHille and Nehari typecriteria for third-order dynamic equationsrdquo Journal of Mathe-matical Analysis and Applications vol 329 no 1 pp 112ndash1312007

[11] S H Saker ldquoOscillation criteria of third-order nonlinear delaydifferential equationsrdquo Mathematica Slovaca vol 56 no 4 pp433ndash450 2006

[12] S H Saker and J Dzurina ldquoOn the oscillation of certain classof third-order nonlinear delay differential equationsrdquo Mathe-matica Bohemica vol 135 no 3 pp 225ndash237 2010

[13] J Yu and J Yan ldquoPositive solutions and asymptotic behavior ofdelay differential equations with nonlinear impulsesrdquo Journal ofMathematical Analysis and Applications vol 207 no 2 pp 388ndash396 1997

[14] T Candan and R S Dahiya ldquoOscillation of third order func-tional differential equations with delayrdquo in Proceedings of the5th Mississippi State Conference on Differential Equations andComputational Simulations vol 10 p 7988 2003

[15] I V Kamenev ldquoAn integral criterion for oscillation of linear dif-ferential equations of second orderrdquo Matematicheskie Zametkivol 23 pp 136ndash138 1978

[16] G Ladas Y G Sficas and I P Stavroulakis ldquoNecessary and suf-ficient conditions for oscillations of higher order delay differ-ential equationsrdquo Transactions of the American MathematicalSociety vol 285 no 1 pp 81ndash90 1984

[17] B Baculıkova E M Elabbasy S H Saker and J DzurinaldquoOscillation criteria for third-order nonlinear differential equa-tionsrdquoMathematica Slovaca vol 58 no 2 pp 201ndash220 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of