Research ArticleOscillation for Forced Second-Order Impulsive NonlinearDynamic Equations on Time Scales

Mugen Huang1 and KunwenWen 2

1School of Statistics and Mathematics Guangdong University of Finance and Economics Guangzhou Guangdong 514015 China2Department of Mathematics Jiaying University Meizhou Guangdong 514015 China

Correspondence should be addressed to Kunwen Wen wenkunwen126com

Received 27 October 2018 Revised 29 November 2018 Accepted 11 December 2018 Published 3 January 2019

Academic Editor Douglas R Anderson

Copyright copy 2019 Mugen Huang and Kunwen Wen This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

As the unification and development of impulsive differential equations and difference equations impulsive dynamic equations ontime scales are a powerful tool to simulate the natural and social phenomena In this paper we study the interval oscillation of a typeof forced second-order nonlinear impulsive dynamic equationswith changing signs coefficients By using the Riccati transformationtechnique we obtain some new interval oscillation criteria based only on information of a sequence of subintervals of positive axisIn addition we provide an example to illustrate the use of our oscillatory results

1 Introduction

To unify continuous and discrete dynamics in a syntheticaltheory Hilger introduced the theory of time scales in 1990 [1]which provides a powerful tool to study some traditionallyseparated fields in an uniform model [2 3] As an adequatemathematical tool to model the natural and social phenom-ena observed in physics chemistry biology economics neu-ral networks and social sciences the topic is in spotlight withsignificant implications and has been a hot area of researchfor several years [4ndash7] For example Anderson and Zafer[5] established interval oscillation criteria for second-ordersuper-half-linear functional dynamic equations with delayand advance arguments Next Agarwal et al [6] extendedthe delay dynamic equations in [5] tomixed nonlinearities Inaddition they give analogous results for related advance typeequations as well as extended delay and advance equations

Impulsive dynamic (or differential) equation with forcedterm is a powerful mathematical tool to simulate externalaction on a system for example an external force actedon a physical system or some new species introduced to abiological system A nonoscillatory spring can switch to beoscillatory by being appended a series of appropriate externalforces It is very interesting and meaningful to study theimpacts of impulse and forced term on the oscillation of

the system and its formation mechanism and impulse is animportant mechanism generates oscillation for differentialequations or dynamic equations [8ndash12] For instance Ozbek-ler and Zafer [10] derived new interval oscillation criteria forforced super-half-linear differential equations with impulseeffects Huang et al [13] studied a general second-ordernonlinear dynamic equations with impulses and proved intheory that suitable impulses would convert nonoscillatorysolutions to be oscillatory Based information only on asequence of intervals on the half interval [0infin) Wong[12] introduced interval oscillatory criteria to differentialequation with forced term and proved external force canproduce oscillation Liu extended Wongrsquos interval oscillationcriteria to a super-linear second-order impulsive forceddifferential equations Huang et al [14] extended Wongrsquosmethods to dynamic equation on time scales with forcedterm

To cover and develop the oscillatory criteria establishedby Wong and Liu it is naturally to ask if we can extend theinterval oscillation criteria to impulsive dynamic equationswith forced term In this paper we study the oscillation of aforced second-order nonlinear impulsive dynamic equationon time scales

To be clear and self-contained we introduce some fun-damental conceptions and symbols used in time scales [1ndash3]

HindawiDiscrete Dynamics in Nature and SocietyVolume 2019 Article ID 7524743 7 pageshttpsdoiorg10115520197524743

2 Discrete Dynamics in Nature and Society

Denote T a time scale with 0 isin T Define the forward andbackward jump operators by

120590 (119905) = inf 119904 isin T 119904 gt 119905 120588 (119905) = sup 119904 isin T 119904 lt 119905 (1)

where inf 120601 = sup T sup120601 = inf T and 120601 denotes the emptyset Anonmaximal element 119905 isin T is called right-dense if120590(119905) =119905 or right-scattered if 120590(119905) gt 119905 Similarly define a nonminimalelement 119905 isin T being le-dense or le-scattered by reversingthe inequality Denote 120583(119905) = 120590(119905) minus 119905 as the graininess

A function 119891 T 997888rarr R is called rd-continuous if itis continuous at right-dense points and its left-sided limitsexist at left-dense points in T Let 119862119903119889(T R) be the set of rd-continuous functions from T to R A mapping 119891 T 997888rarr X

is differentiable at 119905 isin T if there is 119887 isin X such that forany 120576 gt 0 there exists a neighborhood U of 119905 satisfying|[119891(120590(119905)) minus 119891(119904)] minus 119887[120590(119905) minus 119904]| le 120576|120590(119905) minus 119904| for all 119904 isin UIf 119891Δ(119905) exist for all 119905 isin T then 119891 is delta differentiable (or inshort differentiable) on T The delta-derivative and forwardjump operator 120590 are related by the formula

119891 (120590 (119905)) = 119891 (119905) + 120583 (119905) 119891Δ (119905) (2)

Consider the following forced second-order nonlinearimpulsive dynamic equations

119909ΔΔ (119905) + 119901 (119905) 119891 (119909120590 (119905)) = 119890 (119905) 119905 isin JT fl [0infin) cap T 119905 = 119905119896 119896 = 1 2

119909 (119905+119896 ) = 119886119896119909 (119905minus119896 ) 119909Δ (119905+119896 ) = 119887119896119909Δ (119905minus119896 )

119896 = 1 2 119909 (1199050) = 1199090119909Δ (1199050) = 119909Δ0

(3)

where 119886119896 ge 119887119896 gt 0 119896 = 1 2 sdot sdot sdot are constants 119890 119901 isin119862119903119889(T R) 119905119896 isin T and 0 le 1199050 lt 1199051 lt 1199052 lt sdot sdot sdot lt 119905119896 ltsdot sdot sdot lim119896997888rarrinfin119905119896 = infin

119909 (119905+119896 ) = limℎ997888rarr0+

119909 (119905119896 + ℎ) 119909Δ (119905+119896 ) = lim

ℎ997888rarr0+119909Δ (119905119896 + ℎ) (4)

which represent right limits of 119909(119905) and 119909Δ(119905) at 119905 = 119905119896 in thesense of time scales If 119905119896 is right-scattered then 119909(119905+119896 ) = 119909(119905119896)119909Δ(119905+119896 ) = 119909Δ(119905119896) Similarly we can define 119909(119905minus119896 ) and 119909Δ(119905minus119896 )Assume that all the impulsive points 119905119896 119896 = 1 2 sdot sdot sdot areright-dense To define the solutions of (3) we introduce thefollowing space119860119862119894 = 119909 JT 997888rarr R is 119894minustimes Δminusdifferentiable whose119894minusth delta-derivative 119909Δ(119894) is absolutely continuous 119875119862 = 119909 JT 997888rarr R is rd-continuous except at the points119905119896 119896 = 1 2 sdot sdot sdot for which 119909(119905minus119896 ) 119909(119905+119896 ) 119909Δ(119905minus119896 ) and 119909Δ(119905+119896 ) existwith 119909(119905minus119896 ) = 119909(119905119896) 119909Δ(119905minus119896 ) = 119909Δ(119905119896)

Definition 1 A function 119909 isin 119875119862 cap 1198601198622(JT 1199051 1199052 sdot sdot sdot R) issaid to be a solution of (3) if it satisfies119909ΔΔ (119905)+119901(119905)119891(119909120590(119905)) =119890(119905) ae on JT 119905119896 119896 = 1 2 sdot sdot sdot and for each 119896 = 1 2 119909satisfies the impulsive condition 119909(119905+119896 ) = 119886119896119909(119905minus119896 ) 119909Δ(119905+119896 ) =119887119896119909Δ(119905minus119896 ) and the initial conditions 119909(119905+0 ) = 1199090 119909Δ(119905+0 ) = 119909Δ0 Definition 2 A solution 119909 of (3) is said to be oscillatory if it isneither eventually positive nor eventually negative otherwiseit is called nonoscillatory Equation (3) is called oscillatory ifall solutions are oscillatory

By using generalized Riccati transformation techniquewe study the interval oscillation criteria of (3) with changingsign coefficients 119901(119905) and forced term 119890(119905) in Section 2 Theresults extend and improve those of the earlier publicationsWe provide an example to illustrate the use of our mainoscillatory results in Section 3

2 Main Results

In this section we develop some interval oscillation criteriaof (3) which base information on a couple of intervals [1198881 1198891]and [1198882 1198892] To model the real world more accurate weconsider the coefficients 119901(119905) and 119890(119905) of (3) being changingsigns Without loss of generality assume that 119901(119905) and 119890(119905)satisfy (C) 119888119894 119889119894 notin 119905119896 119894 = 1 2 with 1198881 lt 1198891 1198882 lt 1198892 and

119901 (119905) ge 0 119905 isin [1198881 1198891] cup [1198882 1198892] (5)

119890 (119905)le 0 119905 isin [1198881 1198891] ge 0 119905 isin [1198882 1198892] (6)

Denote ℎ(119904) = max119894 1199050 lt 119905119894 lt 119904 sumΔ 2Δ 1 = 0 if Δ 2 lt Δ 1 and119863(119888119894 119889119894)

= 119906 isin 1198621119903119889 [119888119894 119889119894] 119906 (119905) equiv0 119906 (119888119894) = 119906 (119889119894) = 0 119894 = 1 2

(7)

We formulate the well-known inequality for conveniencewhich will be used in the proof of our main results

Lemma 3 For any nonnegative 119860 and 119861 we have119860120582 minus 120582119860119861120582minus1 + (120582 minus 1) 119861120582 ge 0 120582 gt 1 (8)

and the equality holds if and only if 119860 = 119861Theorem 4 Let 119891(119909)119909 ge 120572 gt 0 for 119909 = 0 Suppose thatfor any 119879 ge 0 there exists constants 1198881 1198891 1198882 1198892 isin T with119879 le 1198881 lt 1198891 119879 le 1198882 lt 1198892 and condition (119862) holds If thereexists 119906 isin 119863(119888119894 119889119894) such that

int119889119894119888119894

120572119901 (119905) 1199062 (120590 (119905))

minus 120583 (119905) + 119905 minus 119905ℎ(119889119894)4 (119905 minus 119905ℎ(119889119894)) [119906 (119905) + 119906 (120590 (119905))119906 (120590 (119905)) 119906Δ (119905)]2Δ119905ge 119867(119906 119888119894 119889119894)

(9)

Discrete Dynamics in Nature and Society 3

where119867(119906 119888119894 119889119894) = 0 for ℎ(119888119894) = ℎ(119889119894) and119867(119906 119888119894 119889119894) = 119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1 (119905ℎ(119888119894)+1 minus 119888119894)119906

2 (119905ℎ(119888119894)+1)

+ ℎ(119889119894)sum119895=ℎ(119888119894)+2

119887119894 minus 119886119894119886119894 (119905119894 minus 119905119894minus1)1199062 (119905119894)

(10)

for ℎ(119888119894) lt ℎ(119889119894) 119894 = 1 2 then (3) is oscillatory

Proof We prove the oscillation of (3) by argument of con-tradiction Suppose that (3) has a nonoscillatory solution 119909which is eventually positive Say 119909(119905) gt 0 119909120590(119905) gt 0 for 119905 ge1199050 ge 0 Define the Riccati transformation 119908(119905) = minus119909Δ(119905)119909(119905)for 119905 ge 1199050 The derivative of 119908(119905) along with the solutions of(3) takes the form

119908Δ (119905) = minus119909ΔΔ (119905)119909120590 (119905) + 119909 (119905)119909120590 (119905) (119909Δ (119905)119909 (119905) )

2

= 11 + 120583 (119905) (119909Δ (119905) 119909 (119905))1199082 (119905)+ 119901 (119905) 119891 (119909120590 (119905))119909120590 (119905) minus 119890 (119905)119909120590 (119905)

(11)

with 119905 = 119905119896 119896 = 1 2 sdot sdot sdot By the assumption (119862) we can select1198881 1198891 isin T with 1198891 gt 1198881 ge 1199050 and 119901(119905) ge 0 119890(119905) le 0 119905 isin [1198881 1198891]such that

119909ΔΔ (119905) = 119890 (119905) minus 119901 (119905) 119891 (119909120590 (119905)) le 0119905 isin [1198881 119905ℎ(1198881)+1] cup (119905ℎ(1198881)+1 119905ℎ(1198881)+2] cup sdot sdot sdot cup (119905ℎ(1198891) 1198891]

(12)

which gives 119909Δ(119905) nonincreasing on 119905 isin [1198881 119905ℎ(1198881)+1] cup (119905ℎ(1198881)+1119905ℎ(1198881)+2] cup sdot sdot sdot cup (119905ℎ(1198891) 1198891]For 119905 isin [1198881 119905ℎ(1198881)+1] we have119909 (119905) ge 119909 (119905) minus 119909 (1198881) = int119905

1198881

119909Δ (119904) Δ119904 ge 119909Δ (119905) (119905 minus 1198881) (13)

Hence

119909Δ (119905)119909 (119905) le 1119905 minus 1198881 119905 isin (1198881 119905ℎ(1198881)+1] (14)

Making similar analysis on the intervals (119905119896 119905119896+1] and(119905ℎ(1198891) 1198891] 119894 = ℎ(1198881) + 1 ℎ(1198891) minus 1 we obtain119909Δ (119905)119909 (119905) le 1119905 minus 119905119896

119905 isin (119905119896 119905119896+1] 119896 = ℎ (1198881) + 1 ℎ (1198891) minus 1(15)

and

119909Δ (119905)119909 (119905) le 1119905 minus 119905ℎ(1198891) 119905 isin (119905ℎ(1198891) 1198891] (16)

By using (14) (15) (16) and the condition 119891(119909)119909 ge 120572 (11)yields

119908Δ (119905) ge 119905 minus 119905ℎ(1198891)120583 (119905) + 119905 minus 119905ℎ(1198891)1199082 (119905) + 120572119901 (119905)

= 1205821 (119905) 1199082 (119905) + 120572119901 (119905) 119905 = 119905119896 119896 = 1 2 (17)

where 1205821(119905) = (119905 minus 119905ℎ(1198891))(120583(119905) + 119905 minus 119905ℎ(1198891)) For the impulsivemoments 119905 = 119905119896 119896 = 1 2 sdot sdot sdot it follows from (3) that

119908 (119905+119896) = 119887119896119886119896119908 (119905119896) (18)

For the case ℎ(1198881) lt ℎ(1198891) all the impulsive moments in[1198881 1198891] are 119905ℎ(1198881)+1 119905ℎ(1198881)+2 119905ℎ(1198891) Let 119906(119905) isin 119863(1198881 1198891) bea function satisfying the assumption of the theory Multiplyboth sides of (17) by 1199062(120590(119905)) and integrate it from 1198881 to 1198891using the integration by parts formula

int119887119886119891Δ (119905) 119892 (119905) Δ119905 = 119891 (119905) 119892 (119905)1003816100381610038161003816119887119886 minus int119887

119886119891120590 (119905) 119892Δ (119905) Δ119905 (19)

for any constants 119886 119887 and differential functions 119891 119892 weobtain

ℎ(1198891)sum119896=ℎ(1198881)+1

1199062 (119905119894) [119908 (119905119896) minus 119908 (119905+119896 )] ge int11988911198881

120572119901 (119905)

sdot 1199062 (120590 (119905)) Δ119905 + int119905ℎ(1198881)+11198881

[1205821 (119905) 1199082 (119905) 1199062 (120590 (119905))+ (119906 (119905) + 119906120590 (119905)) 119906Δ (119905) 119908 (119905)] Δ119905+ ℎ(1198891)minus1sum119896=ℎ(1198881)+1

int119905119896+1119905119896

[1205821 (119905) 1199082 (119905) 1199062 (120590 (119905))+ (119906 (119905) + 119906120590 (119905)) 119906Δ (119905) 119908 (119905)] Δ119905+ int1198891119905ℎ(1198891)

[1205821 (119905) 1199082 (119905) 1199062 (120590 (119905)) + (119906 (119905) + 119906120590 (119905))

sdot 119906Δ (119905) 119908 (119905)] Δ119905 = int11988911198881

[120572119901 (119905) 1199062 (120590 (119905))

minus 141205821 (119905) (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905

+ int119905ℎ(1198881 )+11198881

[radic1205821 (119905)119908 (119905) 119906120590 (119905) + 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)

sdot 119906Δ (119905)]2 Δ119905

+ ℎ(1198891)minus1sum119896=ℎ(1198881)+1

int119905119896+1119905119896

[radic1205821 (119905)119908 (119905) 119906120590 (119905)

4 Discrete Dynamics in Nature and Society

+ 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)119906

Δ (119905)]2 Δ119905

+ int1198891119905ℎ(1198891)

[radic1205821 (119905)119908 (119905) 119906120590 (119905) + 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)

sdot 119906Δ (119905)]2 Δ119905 gt int11988911198881

[120572119901 (119905) 1199062 (120590 (119905))

minus 141205821 (119905) (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905

(20)

Using (18) we get

ℎ(1198891)sum119896=ℎ(1198881)+1

119886119896 minus 119887119896119886119896 1199062 (119905119896) 119908 (119905119896) gt int11988911198881

[120572119901 (119905) 1199062 (120590 (119905))

minus 141205821 (119905) (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905

(21)

It follows from (14)-(16) that

119908(119905ℎ(1198881)+1) = minus119909Δ (119905ℎ(1198881)+1)119909 (119905ℎ(1198881)+1) ge minus 1119905ℎ(1198881)+1 minus 1198881 (22)

and

119908 (119905119896) ge minus 1119905119896 minus 119905119896minus1 119896 = ℎ (1198881) + 2 ℎ (1198891) (23)

The conditions 119887119896 ge 119886119896 gt 0 119896 = 1 2 sdot sdot sdot (22) and (23) yield

ℎ(1198891)sum119896=ℎ(1198881)+1

119887119896 minus 119886119896119886119896 1199062 (119905119896) 119908 (119905119896)

ge minus 119887ℎ(1198881)+1 minus 119886ℎ(1198881)+1119886ℎ(1198881)+1 (119905ℎ(1198881)+1 minus 1198881)1199062 (119905ℎ(1198881)+1)

minus ℎ(1198891)sum119896=ℎ(1198881)+2

119887119896 minus 119886119896119886119896 (119905119896 minus 119905119896minus1)1199062 (119905119896)

(24)

Thus

ℎ(1198891)sum119896=ℎ(1198881)+1

119886119896 minus 119887119896119886119896 1199062 (119905119896) 119908 (119905119896)

le 119887ℎ(1198881)+1 minus 119886ℎ(1198881)+1119886ℎ(1198881)+1 (119905ℎ(1198881)+1 minus 1198881)1199062 (119905ℎ(1198881)+1)

+ ℎ(1198891)sum119896=ℎ(1198881)+2

119887119896 minus 119886119896119886119896 (119905119896 minus 119905119896minus1)1199062 (119905119896) = 119867 (119906 1198881 1198891)

(25)

According to (21) we have

int11988911198881

[120572119901 (119905) 1199062 (120590 (119905))

minus 141205821 (119905) (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905

lt 119867 (119906 1198881 1198891)

(26)

which is contradicted with (9)For the case ℎ(1198881) = ℎ(1198891)119867(119906 1198881 1198891) = 0 and there are

no impulsive points in [1198881 1198891] Similar to the proof of (21) weobtain

int11988911198881

[120572119901 (119905) 1199062 (120590 (119905))

minus 141205821 (119905) (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905 lt 0

(27)

which is contradicted with (9) againFor the case 119909(119905) lt 0 119905 ge 1199050 ge 0 by letting 119910 = minus119909 then119910 is positive solution of the dynamic equations

119909ΔΔ (119905) + 119901 (119905) 119891 (119909120590 (119905)) = minus119890 (119905) 119905 isin JT fl [0infin) cap T 119905 = 119905119896 119896 = 1 2

119909 (119905+119896 ) = 119886119896119909 (119905minus119896 ) 119909Δ (119905+119896 ) = 119887119896119909Δ (119905minus119896 )

119896 = 1 2 119909 (1199050) = 1199090119909Δ (1199050) = 119909Δ0

(28)

Repeating the above procedure on the interval [1198882 1198892] wewill obtain a contradiction again This completes the proofof Theorem 4

Theorem 5 Let 119909119891(119909) gt 0 for 119909 = 0 and |119891(119909)| ge |119909|119903for 119903 gt 1 Suppose that for any 119879 ge 0 there exists constants1198881 1198891 1198882 1198892 isin T with 119879 le 1198881 lt 1198891 119879 le 1198882 lt 1198892 and (119862)holds If there exists 119906 isin 119863(119888119894 119889119894) such that

int119889119894119888119894

119903 (119903 minus 1)(1minus119903)119903 1199011119903 (119905) |119890 (119905)|(119903minus1)119903 1199062 (120590 (119905))

minus 120583 (119905) + 119905 minus 119905ℎ(119889119894)4 (119905 minus 119905ℎ(119889119894)) [119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905)]2Δ119905ge 119867(119906 119888119894 119889119894)

(29)

where119867 is given by (10) then (3) is oscillatory

Proof Assume that 119909(119905) gt 0 119909120590(119905) gt 0 119905 ge 1199050 ge 0is a nonoscillatory solution of (3) Let 119860 = 1199011119903(119905)119909120590(119905)

Discrete Dynamics in Nature and Society 5

119861 = (minus119890(119905)(119903 minus 1))1119903 and choose 1198881 1198891 isin T with 1198891 gt 1198881 ge1199050 ge 0 and 119901(119905) ge 0 119890(119905) le 0 for 119905 isin [1198881 1198891] Hence 119860 gt 0119861 gt 0 for 119903 gt 1 Lemma 3 gives

119901 (119905) 119909119903 (120590 (119905)) minus 119890 (119905)ge 119903 (119903 minus 1)(1minus119903)119903 1199011119903 (119905) |119890 (119905)|(119903minus1)119903 119909 (120590 (119905))= 12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 119909 (120590 (119905))

(30)

where 1205822 = 119903(119903 minus 1)(1minus119903)119903 is a positive constant By (3) and(30) we obtain

119909ΔΔ (119905) + 12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 119909 (120590 (119905)) le 0 (31)

Let119908(119905) = 119909Δ(119905)119909(119905) By using (14)-(16) the formulas (2) and

(119891119892)Δ = 119891Δ119892 minus 119891119892Δ119892119892120590 (32)

we have

119908Δ (119905) = 119909ΔΔ (119905)119909120590 (119905) minus 119909 (119905)119909120590 (119905) [119909Δ (119905)119909 (119905) ]

2

le minus12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 minus 1205821 (119905) 1199082 (119905) (33)

where 119905 = 119905119896 119896 = 1 2 sdot sdot sdot and 1205821(119905) = (119905 minus 119905ℎ(1198891))(120583(119905) + 119905 minus119905ℎ(1198891))In the case ℎ(1198881) lt ℎ(1198891) all the impulsive moments in[1198881 1198891] are 119905ℎ(1198881)+1 119905ℎ(1198881)+2 119905ℎ(1198891) Let 119906(119905) isin 119863(1198881 1198891) be

given as in the theorem Multiplying both sides of (33) by1199062(120590(119905)) and integrating it from 1198881 to 1198891 giveℎ(1198891)sum119896=ℎ(1198881)+1

1199062 (119905119896) [119908 (119905119896) minus 119908 (119905+119896 )] le minusint11988911198881

12058221199011119903 (119905)

sdot |119890 (119905)|(119903minus1)119903 1199062 (120590 (119905)) Δ119905 minus int119905ℎ(1198881 )+11198881

[1205821 (119905) 1199082 (119905)sdot 1199062 (120590 (119905)) + (119906 (119905) + 119906120590 (119905)) 119906Δ (119905) 119908 (119905)] Δ119905minus ℎ(1198891)minus1sum119896=ℎ(1198881)+1

int119905119896+1119905119894

[1205821 (119905) 1199082 (119905) 1199062 (120590 (119905))+ (119906 (119905) + 119906120590 (119905)) 119906Δ (119905) 119908 (119905)] Δ119905minus int1198891119905ℎ(1198891)

[1205821 (119905) 1199082 (119905) 1199062 (120590 (119905)) + (119906 (119905) + 119906120590 (119905))

sdot 119906Δ (119905) 119908 (119905)] Δ119905 = minusint11988911198881

[12058221199011119903 (119905)sdot |119890 (119905)|(119903minus1)119903 1199062 (120590 (119905))minus 141205821 (119905) (

119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905

minus int119905ℎ(1198881 )+11198881

[radic1205821 (119905)119908 (119905) 119906120590 (119905) + 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)

sdot 119906Δ (119905)]2 Δ119905

minus ℎ(1198891)minus1sum119896=ℎ(1198881)+1

int119905119896+1119905119896

[radic1205821 (119905)119908 (119905) 119906120590 (119905)

+ 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)119906

Δ (119905)]2 Δ119905

minus int1198891119905ℎ(1198891)

[radic1205821 (119905)119908 (119905) 119906120590 (119905) + 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)

sdot 119906Δ (119905)]2 Δ119905 lt minusint11988911198881

[12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903sdot 1199062 (120590 (119905))minus 141205821 (119905) (

119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905(34)

Similar to the proof of Theorem 4 by using (18) we have

ℎ(1198891)sum119896=ℎ(1198881)+1

1199062 (119905119896) [119908 (119905119896) minus 119908 (119905+119896 )]

= ℎ(1198891)sum119896=ℎ(1198881)+1

119886119896 minus 119887119896119886119896 1199062 (119905119896) 119908 (119905119896) ge minus119867(119906 1198881 1198891) (35)

Thus (34) yields

int119889119894119888119894

[12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 1199062 (120590 (119905))

minus 141205821 (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905 lt 119867 (119906 1198881

1198891)

(36)

which is contradicted with (29)In the case ℎ(1198881) = ℎ(1198891) we have 119867(119906 1198881 1198891) = 0 and

there are no impulsive moments in [1198881 1198891] Similar to theproof of Theorem 4 we get a contradiction again

If 119909(119905) lt 0 119905 ge 1199050 ge 0 we can derive a similar contradic-tion on [1198882 1198892] This completes the proof of Theorem 5

3 Example

Since the time scale P119886119887 = ⋃infin119899=0[119899(119886 + 119887) 119899(119886 + 119887) + 119886]is an important and useful tool used extensively to simulatebiological population electric circuit and so on we give anexample in P119886119887 to show the application of our oscillatorycriteria

6 Discrete Dynamics in Nature and Society

Example 6 Consider the following system

119909ΔΔ (119905) + 119898 sin 119905119909 (120590 (119905)) = cos t119905 isin P120587120587 = infin⋃

119899=0

[2119899120587 (2119899 + 1) 120587] 119905 = (2119896 + 12)120587 plusmn 38120587 119896 = 0 1 2

119909 (119905+119896 ) = 119886119896119909 (119905minus119896 ) 119909Δ (119905+119896 ) = 119887119896119909Δ (119905minus119896 )

119905119896 = (2119896 + 12)120587 plusmn 38120587 119896 = 0 1 2 119909 (1199050) = 1199090119909Δ (1199050) = 119909Δ0

(37)

with the transition condition

119909 (2119899120587) = 119909 ((2119899 minus 1) 120587) 119899 ge 1 (38)

where 119887119896 ge 119886119896 gt 0119898 gt 0 are constantsHence 119901(119905) = 119898 sin 119905 119890(119905) = cos 119905 and 119891(119909(120590(119905))) =119909(120590(119905)) For any 119879 ge 0 choose 1198881 = 2119899120587+31205874 1198891 = 2119899120587+1205871198882 = 2119899120587 1198892 = 2119899120587 + 1205874 (119888119894 119889119894 isin P120587120587 119894 = 1 2) such

that 119888119894 ge 119879 for sufficiently large 119899 119894 = 1 2 Hence 119901(119905) ge 0for 119905 isin [1198881 1198891] cup [1198882 1198892] and 119890(119905) le 0 for 119905 isin [1198881 1198891]and 119890(119905) ge 0 for 119905 isin [1198882 1198892] Let 119906(119905) = sin 2119905 cos 2119905 then119906(119905) isin 119863(119888119894 119889119894) 119894 = 1 2 Furthermore we have 120590(119905) = 119905120583(119905) = 0 for 119905 isin [119888119894 119889119894] 119894 = 1 2 By some simple calculationfor 119894 = 1 2 we have

int119889119894119888119894

119896119901 (119905) 1199062 (120590 (119905))

minus 120583 (119905) + 119905 minus 119905ℎ(119889119894)4 (119905 minus 119905ℎ(119889119894)) [119906 (119905) + 119906 (120590 (119905))119906 (120590 (119905))sdot 119906Δ (119905)]2Δ119905 = int119889119894

119888119894

[119901 (119905) 1199062 (119905)

minus (119906Δ (119905))2] 119889119905 = int119889119894119888119894

[119898 sin 119905 sin2 2119905 cos2 2119905minus 4 cos2 4119905] 119889119905

int119889119894119888119894

4 cos2 4119905119889119905 = 1205872

(39)

and

int119889119894119888119894

119898 sin 119905 sin2 2119905 cos2 2119905119889119905 = 863 (1 minusradic22 )119898 (40)

Therefore

int119889119894119888119894

119896119901 (119905) 1199062 (120590 (119905))

minus 120583 (119905) + 119905 minus 119905ℎ(119889119894)4 (119905 minus 119905ℎ(119889119894)) [119906 (119905) + 119906 (120590 (119905))119906 (120590 (119905))sdot 119906Δ (119905)]2Δ119905

(41)

= 863 (1 minusradic22 )119898 minus 1205872 (42)

The impulsive points in [1198881 1198891] = [2119899120587 + 31205874 2119899120587 + 120587]are 2119899120587 + (78)120587 and in [1198882 1198892] = [2119899120587 2119899120587 + 1205874] are 2119899120587 +(18)120587 Thus

119867(119906 119888119894 119889119894) = 119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1 (119905ℎ(119888119894)+1 minus 119888119894)1199062 (119905ℎ(119888119894)+1)

= 2120587119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1

(43)

ByTheorems 4 (3) is oscillatory if

863 (1 minusradic22 )119898 minus 1205872 ge 2120587

119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1 (44)

For example letting 119886119896 = 1 119887119896 = (119896 + 1)119896 yields863 (1 minus

radic22 )119898 ge 1205872 + 2(2119899 + 78) 1205872 (45)

for sufficiently large 119898 and so (38) holds It follows fromTheorem 5 that all the solutions of (37)-(38) are oscillatory

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the Natural Science Foundationof Guangdong Province (Grants nos 2017A030310597 and2018A030307024)

References

[1] S Hilger ldquoAnalysis on measure chains a unified approach tocontinuous and discrete calculusrdquo Results in Mathematics vol18 no 1-2 pp 18ndash56 1990

Discrete Dynamics in Nature and Society 7

[2] R P AgarwalM Bohner andDOrsquoRegan ldquoDynamic equationson time scales a surveyrdquo Journal of Computational and AppliedMathematics vol 141 no 1-2 pp 1ndash26 2002

[3] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhauser Boston MassUSA 2001

[4] R P Agarwal M Benchohra D OrsquoRegan and A OuahabldquoSecond order impulsive dynamic equations on time scalesrdquoFunctional Differential Equations vol 11 no 3-4 pp 223ndash2342004

[5] D R Anderson and A Zafer ldquoInterval criteria for second-order super-half-linear functional dynamic equations withdelay and advance argumentsrdquo Journal of Difference Equationsand Applications vol 16 no 8 pp 917ndash930 2010

[6] R P Agarwal D R Anderson and A k Zafer ldquoInterval oscil-lation criteria for second-order forced delay dynamic equationswith mixed nonlinearitiesrdquo Computers and Mathematics withApplications An International Journal vol 59 no 2 pp 977ndash993 2010

[7] M Benchohra S Hamani and J Henderson ldquoOscillation andnonoscillation for impulsive dynamic equations on certain timescalesrdquo Advances in Difference Equations pp 1ndash12 2006

[8] M A El-Sayed ldquoAn oscillation criterion for a forced secondorder linear differential equationrdquo Proceedings of the AmericanMathematical Society vol 118 no 3 pp 813ndash817 1993

[9] W-T Li and R P Agarwal ldquoInterval oscillation criteria for aforced second order nonlinear ordinary differential equationrdquoApplicable Analysis An International Journal vol 75 no 3-4pp 341ndash347 2000

[10] A Ozbekler and A Zafer ldquoInterval criteria for the forcedoscillation of super-half-linear differential equations underimpulse effectsrdquoMathematical and ComputerModelling vol 50no 1-2 pp 59ndash65 2009

[11] Y Sun ldquoA note on Nasrs and Wongs papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003

[12] J S Wong ldquoOscillation criteria for a forced second-order lineardifferential equationrdquo Journal of Mathematical Analysis andApplications vol 231 no 1 pp 235ndash240 1999

[13] M Huang and W Feng ldquoOscillation criteria for impulsivedynamic equations on time scalesrdquo Electronic Journal of Differ-ential Equations vol 2007 no 72 pp 1ndash9 2007

[14] M Huang and W Feng ldquoOscillation for forced second-ordernonlinear dynamic equations on time scalesrdquo Electronic Journalof Differential Equations vol 2006 pp 1ndash8 2006

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Discrete Dynamics in Nature and Society 3

where119867(119906 119888119894 119889119894) = 0 for ℎ(119888119894) = ℎ(119889119894) and119867(119906 119888119894 119889119894) = 119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1 (119905ℎ(119888119894)+1 minus 119888119894)119906

2 (119905ℎ(119888119894)+1)

+ ℎ(119889119894)sum119895=ℎ(119888119894)+2

119887119894 minus 119886119894119886119894 (119905119894 minus 119905119894minus1)1199062 (119905119894)

(10)

for ℎ(119888119894) lt ℎ(119889119894) 119894 = 1 2 then (3) is oscillatory

Proof We prove the oscillation of (3) by argument of con-tradiction Suppose that (3) has a nonoscillatory solution 119909which is eventually positive Say 119909(119905) gt 0 119909120590(119905) gt 0 for 119905 ge1199050 ge 0 Define the Riccati transformation 119908(119905) = minus119909Δ(119905)119909(119905)for 119905 ge 1199050 The derivative of 119908(119905) along with the solutions of(3) takes the form

119908Δ (119905) = minus119909ΔΔ (119905)119909120590 (119905) + 119909 (119905)119909120590 (119905) (119909Δ (119905)119909 (119905) )

2

= 11 + 120583 (119905) (119909Δ (119905) 119909 (119905))1199082 (119905)+ 119901 (119905) 119891 (119909120590 (119905))119909120590 (119905) minus 119890 (119905)119909120590 (119905)

(11)

with 119905 = 119905119896 119896 = 1 2 sdot sdot sdot By the assumption (119862) we can select1198881 1198891 isin T with 1198891 gt 1198881 ge 1199050 and 119901(119905) ge 0 119890(119905) le 0 119905 isin [1198881 1198891]such that

119909ΔΔ (119905) = 119890 (119905) minus 119901 (119905) 119891 (119909120590 (119905)) le 0119905 isin [1198881 119905ℎ(1198881)+1] cup (119905ℎ(1198881)+1 119905ℎ(1198881)+2] cup sdot sdot sdot cup (119905ℎ(1198891) 1198891]

(12)

which gives 119909Δ(119905) nonincreasing on 119905 isin [1198881 119905ℎ(1198881)+1] cup (119905ℎ(1198881)+1119905ℎ(1198881)+2] cup sdot sdot sdot cup (119905ℎ(1198891) 1198891]For 119905 isin [1198881 119905ℎ(1198881)+1] we have119909 (119905) ge 119909 (119905) minus 119909 (1198881) = int119905

1198881

119909Δ (119904) Δ119904 ge 119909Δ (119905) (119905 minus 1198881) (13)

Hence

119909Δ (119905)119909 (119905) le 1119905 minus 1198881 119905 isin (1198881 119905ℎ(1198881)+1] (14)

Making similar analysis on the intervals (119905119896 119905119896+1] and(119905ℎ(1198891) 1198891] 119894 = ℎ(1198881) + 1 ℎ(1198891) minus 1 we obtain119909Δ (119905)119909 (119905) le 1119905 minus 119905119896

119905 isin (119905119896 119905119896+1] 119896 = ℎ (1198881) + 1 ℎ (1198891) minus 1(15)

and

119909Δ (119905)119909 (119905) le 1119905 minus 119905ℎ(1198891) 119905 isin (119905ℎ(1198891) 1198891] (16)

By using (14) (15) (16) and the condition 119891(119909)119909 ge 120572 (11)yields

119908Δ (119905) ge 119905 minus 119905ℎ(1198891)120583 (119905) + 119905 minus 119905ℎ(1198891)1199082 (119905) + 120572119901 (119905)

= 1205821 (119905) 1199082 (119905) + 120572119901 (119905) 119905 = 119905119896 119896 = 1 2 (17)

where 1205821(119905) = (119905 minus 119905ℎ(1198891))(120583(119905) + 119905 minus 119905ℎ(1198891)) For the impulsivemoments 119905 = 119905119896 119896 = 1 2 sdot sdot sdot it follows from (3) that

119908 (119905+119896) = 119887119896119886119896119908 (119905119896) (18)

For the case ℎ(1198881) lt ℎ(1198891) all the impulsive moments in[1198881 1198891] are 119905ℎ(1198881)+1 119905ℎ(1198881)+2 119905ℎ(1198891) Let 119906(119905) isin 119863(1198881 1198891) bea function satisfying the assumption of the theory Multiplyboth sides of (17) by 1199062(120590(119905)) and integrate it from 1198881 to 1198891using the integration by parts formula

int119887119886119891Δ (119905) 119892 (119905) Δ119905 = 119891 (119905) 119892 (119905)1003816100381610038161003816119887119886 minus int119887

119886119891120590 (119905) 119892Δ (119905) Δ119905 (19)

for any constants 119886 119887 and differential functions 119891 119892 weobtain

ℎ(1198891)sum119896=ℎ(1198881)+1

1199062 (119905119894) [119908 (119905119896) minus 119908 (119905+119896 )] ge int11988911198881

120572119901 (119905)

sdot 1199062 (120590 (119905)) Δ119905 + int119905ℎ(1198881)+11198881

[1205821 (119905) 1199082 (119905) 1199062 (120590 (119905))+ (119906 (119905) + 119906120590 (119905)) 119906Δ (119905) 119908 (119905)] Δ119905+ ℎ(1198891)minus1sum119896=ℎ(1198881)+1

int119905119896+1119905119896

[1205821 (119905) 1199082 (119905) 1199062 (120590 (119905))+ (119906 (119905) + 119906120590 (119905)) 119906Δ (119905) 119908 (119905)] Δ119905+ int1198891119905ℎ(1198891)

[1205821 (119905) 1199082 (119905) 1199062 (120590 (119905)) + (119906 (119905) + 119906120590 (119905))

sdot 119906Δ (119905) 119908 (119905)] Δ119905 = int11988911198881

[120572119901 (119905) 1199062 (120590 (119905))

minus 141205821 (119905) (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905

+ int119905ℎ(1198881 )+11198881

[radic1205821 (119905)119908 (119905) 119906120590 (119905) + 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)

sdot 119906Δ (119905)]2 Δ119905

+ ℎ(1198891)minus1sum119896=ℎ(1198881)+1

int119905119896+1119905119896

[radic1205821 (119905)119908 (119905) 119906120590 (119905)

4 Discrete Dynamics in Nature and Society

+ 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)119906

Δ (119905)]2 Δ119905

+ int1198891119905ℎ(1198891)

[radic1205821 (119905)119908 (119905) 119906120590 (119905) + 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)

sdot 119906Δ (119905)]2 Δ119905 gt int11988911198881

[120572119901 (119905) 1199062 (120590 (119905))

minus 141205821 (119905) (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905

(20)

Using (18) we get

ℎ(1198891)sum119896=ℎ(1198881)+1

119886119896 minus 119887119896119886119896 1199062 (119905119896) 119908 (119905119896) gt int11988911198881

[120572119901 (119905) 1199062 (120590 (119905))

minus 141205821 (119905) (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905

(21)

It follows from (14)-(16) that

119908(119905ℎ(1198881)+1) = minus119909Δ (119905ℎ(1198881)+1)119909 (119905ℎ(1198881)+1) ge minus 1119905ℎ(1198881)+1 minus 1198881 (22)

and

119908 (119905119896) ge minus 1119905119896 minus 119905119896minus1 119896 = ℎ (1198881) + 2 ℎ (1198891) (23)

The conditions 119887119896 ge 119886119896 gt 0 119896 = 1 2 sdot sdot sdot (22) and (23) yield

ℎ(1198891)sum119896=ℎ(1198881)+1

119887119896 minus 119886119896119886119896 1199062 (119905119896) 119908 (119905119896)

ge minus 119887ℎ(1198881)+1 minus 119886ℎ(1198881)+1119886ℎ(1198881)+1 (119905ℎ(1198881)+1 minus 1198881)1199062 (119905ℎ(1198881)+1)

minus ℎ(1198891)sum119896=ℎ(1198881)+2

119887119896 minus 119886119896119886119896 (119905119896 minus 119905119896minus1)1199062 (119905119896)

(24)

Thus

ℎ(1198891)sum119896=ℎ(1198881)+1

119886119896 minus 119887119896119886119896 1199062 (119905119896) 119908 (119905119896)

le 119887ℎ(1198881)+1 minus 119886ℎ(1198881)+1119886ℎ(1198881)+1 (119905ℎ(1198881)+1 minus 1198881)1199062 (119905ℎ(1198881)+1)

+ ℎ(1198891)sum119896=ℎ(1198881)+2

119887119896 minus 119886119896119886119896 (119905119896 minus 119905119896minus1)1199062 (119905119896) = 119867 (119906 1198881 1198891)

(25)

According to (21) we have

int11988911198881

[120572119901 (119905) 1199062 (120590 (119905))

minus 141205821 (119905) (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905

lt 119867 (119906 1198881 1198891)

(26)

which is contradicted with (9)For the case ℎ(1198881) = ℎ(1198891)119867(119906 1198881 1198891) = 0 and there are

no impulsive points in [1198881 1198891] Similar to the proof of (21) weobtain

int11988911198881

[120572119901 (119905) 1199062 (120590 (119905))

minus 141205821 (119905) (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905 lt 0

(27)

which is contradicted with (9) againFor the case 119909(119905) lt 0 119905 ge 1199050 ge 0 by letting 119910 = minus119909 then119910 is positive solution of the dynamic equations

119909ΔΔ (119905) + 119901 (119905) 119891 (119909120590 (119905)) = minus119890 (119905) 119905 isin JT fl [0infin) cap T 119905 = 119905119896 119896 = 1 2

119909 (119905+119896 ) = 119886119896119909 (119905minus119896 ) 119909Δ (119905+119896 ) = 119887119896119909Δ (119905minus119896 )

119896 = 1 2 119909 (1199050) = 1199090119909Δ (1199050) = 119909Δ0

(28)

Repeating the above procedure on the interval [1198882 1198892] wewill obtain a contradiction again This completes the proofof Theorem 4

Theorem 5 Let 119909119891(119909) gt 0 for 119909 = 0 and |119891(119909)| ge |119909|119903for 119903 gt 1 Suppose that for any 119879 ge 0 there exists constants1198881 1198891 1198882 1198892 isin T with 119879 le 1198881 lt 1198891 119879 le 1198882 lt 1198892 and (119862)holds If there exists 119906 isin 119863(119888119894 119889119894) such that

int119889119894119888119894

119903 (119903 minus 1)(1minus119903)119903 1199011119903 (119905) |119890 (119905)|(119903minus1)119903 1199062 (120590 (119905))

minus 120583 (119905) + 119905 minus 119905ℎ(119889119894)4 (119905 minus 119905ℎ(119889119894)) [119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905)]2Δ119905ge 119867(119906 119888119894 119889119894)

(29)

where119867 is given by (10) then (3) is oscillatory

Proof Assume that 119909(119905) gt 0 119909120590(119905) gt 0 119905 ge 1199050 ge 0is a nonoscillatory solution of (3) Let 119860 = 1199011119903(119905)119909120590(119905)

Discrete Dynamics in Nature and Society 5

119861 = (minus119890(119905)(119903 minus 1))1119903 and choose 1198881 1198891 isin T with 1198891 gt 1198881 ge1199050 ge 0 and 119901(119905) ge 0 119890(119905) le 0 for 119905 isin [1198881 1198891] Hence 119860 gt 0119861 gt 0 for 119903 gt 1 Lemma 3 gives

119901 (119905) 119909119903 (120590 (119905)) minus 119890 (119905)ge 119903 (119903 minus 1)(1minus119903)119903 1199011119903 (119905) |119890 (119905)|(119903minus1)119903 119909 (120590 (119905))= 12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 119909 (120590 (119905))

(30)

where 1205822 = 119903(119903 minus 1)(1minus119903)119903 is a positive constant By (3) and(30) we obtain

119909ΔΔ (119905) + 12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 119909 (120590 (119905)) le 0 (31)

Let119908(119905) = 119909Δ(119905)119909(119905) By using (14)-(16) the formulas (2) and

(119891119892)Δ = 119891Δ119892 minus 119891119892Δ119892119892120590 (32)

we have

119908Δ (119905) = 119909ΔΔ (119905)119909120590 (119905) minus 119909 (119905)119909120590 (119905) [119909Δ (119905)119909 (119905) ]

2

le minus12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 minus 1205821 (119905) 1199082 (119905) (33)

where 119905 = 119905119896 119896 = 1 2 sdot sdot sdot and 1205821(119905) = (119905 minus 119905ℎ(1198891))(120583(119905) + 119905 minus119905ℎ(1198891))In the case ℎ(1198881) lt ℎ(1198891) all the impulsive moments in[1198881 1198891] are 119905ℎ(1198881)+1 119905ℎ(1198881)+2 119905ℎ(1198891) Let 119906(119905) isin 119863(1198881 1198891) be

given as in the theorem Multiplying both sides of (33) by1199062(120590(119905)) and integrating it from 1198881 to 1198891 giveℎ(1198891)sum119896=ℎ(1198881)+1

1199062 (119905119896) [119908 (119905119896) minus 119908 (119905+119896 )] le minusint11988911198881

12058221199011119903 (119905)

sdot |119890 (119905)|(119903minus1)119903 1199062 (120590 (119905)) Δ119905 minus int119905ℎ(1198881 )+11198881

[1205821 (119905) 1199082 (119905)sdot 1199062 (120590 (119905)) + (119906 (119905) + 119906120590 (119905)) 119906Δ (119905) 119908 (119905)] Δ119905minus ℎ(1198891)minus1sum119896=ℎ(1198881)+1

int119905119896+1119905119894

[1205821 (119905) 1199082 (119905) 1199062 (120590 (119905))+ (119906 (119905) + 119906120590 (119905)) 119906Δ (119905) 119908 (119905)] Δ119905minus int1198891119905ℎ(1198891)

[1205821 (119905) 1199082 (119905) 1199062 (120590 (119905)) + (119906 (119905) + 119906120590 (119905))

sdot 119906Δ (119905) 119908 (119905)] Δ119905 = minusint11988911198881

[12058221199011119903 (119905)sdot |119890 (119905)|(119903minus1)119903 1199062 (120590 (119905))minus 141205821 (119905) (

119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905

minus int119905ℎ(1198881 )+11198881

[radic1205821 (119905)119908 (119905) 119906120590 (119905) + 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)

sdot 119906Δ (119905)]2 Δ119905

minus ℎ(1198891)minus1sum119896=ℎ(1198881)+1

int119905119896+1119905119896

[radic1205821 (119905)119908 (119905) 119906120590 (119905)

+ 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)119906

Δ (119905)]2 Δ119905

minus int1198891119905ℎ(1198891)

[radic1205821 (119905)119908 (119905) 119906120590 (119905) + 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)

sdot 119906Δ (119905)]2 Δ119905 lt minusint11988911198881

[12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903sdot 1199062 (120590 (119905))minus 141205821 (119905) (

119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905(34)

Similar to the proof of Theorem 4 by using (18) we have

ℎ(1198891)sum119896=ℎ(1198881)+1

1199062 (119905119896) [119908 (119905119896) minus 119908 (119905+119896 )]

= ℎ(1198891)sum119896=ℎ(1198881)+1

119886119896 minus 119887119896119886119896 1199062 (119905119896) 119908 (119905119896) ge minus119867(119906 1198881 1198891) (35)

Thus (34) yields

int119889119894119888119894

[12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 1199062 (120590 (119905))

minus 141205821 (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905 lt 119867 (119906 1198881

1198891)

(36)

which is contradicted with (29)In the case ℎ(1198881) = ℎ(1198891) we have 119867(119906 1198881 1198891) = 0 and

there are no impulsive moments in [1198881 1198891] Similar to theproof of Theorem 4 we get a contradiction again

If 119909(119905) lt 0 119905 ge 1199050 ge 0 we can derive a similar contradic-tion on [1198882 1198892] This completes the proof of Theorem 5

3 Example

Since the time scale P119886119887 = ⋃infin119899=0[119899(119886 + 119887) 119899(119886 + 119887) + 119886]is an important and useful tool used extensively to simulatebiological population electric circuit and so on we give anexample in P119886119887 to show the application of our oscillatorycriteria

6 Discrete Dynamics in Nature and Society

Example 6 Consider the following system

119909ΔΔ (119905) + 119898 sin 119905119909 (120590 (119905)) = cos t119905 isin P120587120587 = infin⋃

119899=0

[2119899120587 (2119899 + 1) 120587] 119905 = (2119896 + 12)120587 plusmn 38120587 119896 = 0 1 2

119909 (119905+119896 ) = 119886119896119909 (119905minus119896 ) 119909Δ (119905+119896 ) = 119887119896119909Δ (119905minus119896 )

119905119896 = (2119896 + 12)120587 plusmn 38120587 119896 = 0 1 2 119909 (1199050) = 1199090119909Δ (1199050) = 119909Δ0

(37)

with the transition condition

119909 (2119899120587) = 119909 ((2119899 minus 1) 120587) 119899 ge 1 (38)

where 119887119896 ge 119886119896 gt 0119898 gt 0 are constantsHence 119901(119905) = 119898 sin 119905 119890(119905) = cos 119905 and 119891(119909(120590(119905))) =119909(120590(119905)) For any 119879 ge 0 choose 1198881 = 2119899120587+31205874 1198891 = 2119899120587+1205871198882 = 2119899120587 1198892 = 2119899120587 + 1205874 (119888119894 119889119894 isin P120587120587 119894 = 1 2) such

that 119888119894 ge 119879 for sufficiently large 119899 119894 = 1 2 Hence 119901(119905) ge 0for 119905 isin [1198881 1198891] cup [1198882 1198892] and 119890(119905) le 0 for 119905 isin [1198881 1198891]and 119890(119905) ge 0 for 119905 isin [1198882 1198892] Let 119906(119905) = sin 2119905 cos 2119905 then119906(119905) isin 119863(119888119894 119889119894) 119894 = 1 2 Furthermore we have 120590(119905) = 119905120583(119905) = 0 for 119905 isin [119888119894 119889119894] 119894 = 1 2 By some simple calculationfor 119894 = 1 2 we have

int119889119894119888119894

119896119901 (119905) 1199062 (120590 (119905))

minus 120583 (119905) + 119905 minus 119905ℎ(119889119894)4 (119905 minus 119905ℎ(119889119894)) [119906 (119905) + 119906 (120590 (119905))119906 (120590 (119905))sdot 119906Δ (119905)]2Δ119905 = int119889119894

119888119894

[119901 (119905) 1199062 (119905)

minus (119906Δ (119905))2] 119889119905 = int119889119894119888119894

[119898 sin 119905 sin2 2119905 cos2 2119905minus 4 cos2 4119905] 119889119905

int119889119894119888119894

4 cos2 4119905119889119905 = 1205872

(39)

and

int119889119894119888119894

119898 sin 119905 sin2 2119905 cos2 2119905119889119905 = 863 (1 minusradic22 )119898 (40)

Therefore

int119889119894119888119894

119896119901 (119905) 1199062 (120590 (119905))

minus 120583 (119905) + 119905 minus 119905ℎ(119889119894)4 (119905 minus 119905ℎ(119889119894)) [119906 (119905) + 119906 (120590 (119905))119906 (120590 (119905))sdot 119906Δ (119905)]2Δ119905

(41)

= 863 (1 minusradic22 )119898 minus 1205872 (42)

The impulsive points in [1198881 1198891] = [2119899120587 + 31205874 2119899120587 + 120587]are 2119899120587 + (78)120587 and in [1198882 1198892] = [2119899120587 2119899120587 + 1205874] are 2119899120587 +(18)120587 Thus

119867(119906 119888119894 119889119894) = 119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1 (119905ℎ(119888119894)+1 minus 119888119894)1199062 (119905ℎ(119888119894)+1)

= 2120587119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1

(43)

ByTheorems 4 (3) is oscillatory if

863 (1 minusradic22 )119898 minus 1205872 ge 2120587

119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1 (44)

For example letting 119886119896 = 1 119887119896 = (119896 + 1)119896 yields863 (1 minus

radic22 )119898 ge 1205872 + 2(2119899 + 78) 1205872 (45)

for sufficiently large 119898 and so (38) holds It follows fromTheorem 5 that all the solutions of (37)-(38) are oscillatory

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the Natural Science Foundationof Guangdong Province (Grants nos 2017A030310597 and2018A030307024)

References

[1] S Hilger ldquoAnalysis on measure chains a unified approach tocontinuous and discrete calculusrdquo Results in Mathematics vol18 no 1-2 pp 18ndash56 1990

Discrete Dynamics in Nature and Society 7

[2] R P AgarwalM Bohner andDOrsquoRegan ldquoDynamic equationson time scales a surveyrdquo Journal of Computational and AppliedMathematics vol 141 no 1-2 pp 1ndash26 2002

[3] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhauser Boston MassUSA 2001

[4] R P Agarwal M Benchohra D OrsquoRegan and A OuahabldquoSecond order impulsive dynamic equations on time scalesrdquoFunctional Differential Equations vol 11 no 3-4 pp 223ndash2342004

[5] D R Anderson and A Zafer ldquoInterval criteria for second-order super-half-linear functional dynamic equations withdelay and advance argumentsrdquo Journal of Difference Equationsand Applications vol 16 no 8 pp 917ndash930 2010

[6] R P Agarwal D R Anderson and A k Zafer ldquoInterval oscil-lation criteria for second-order forced delay dynamic equationswith mixed nonlinearitiesrdquo Computers and Mathematics withApplications An International Journal vol 59 no 2 pp 977ndash993 2010

[7] M Benchohra S Hamani and J Henderson ldquoOscillation andnonoscillation for impulsive dynamic equations on certain timescalesrdquo Advances in Difference Equations pp 1ndash12 2006

[8] M A El-Sayed ldquoAn oscillation criterion for a forced secondorder linear differential equationrdquo Proceedings of the AmericanMathematical Society vol 118 no 3 pp 813ndash817 1993

[9] W-T Li and R P Agarwal ldquoInterval oscillation criteria for aforced second order nonlinear ordinary differential equationrdquoApplicable Analysis An International Journal vol 75 no 3-4pp 341ndash347 2000

[10] A Ozbekler and A Zafer ldquoInterval criteria for the forcedoscillation of super-half-linear differential equations underimpulse effectsrdquoMathematical and ComputerModelling vol 50no 1-2 pp 59ndash65 2009

[11] Y Sun ldquoA note on Nasrs and Wongs papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003

[12] J S Wong ldquoOscillation criteria for a forced second-order lineardifferential equationrdquo Journal of Mathematical Analysis andApplications vol 231 no 1 pp 235ndash240 1999

[13] M Huang and W Feng ldquoOscillation criteria for impulsivedynamic equations on time scalesrdquo Electronic Journal of Differ-ential Equations vol 2007 no 72 pp 1ndash9 2007

[14] M Huang and W Feng ldquoOscillation for forced second-ordernonlinear dynamic equations on time scalesrdquo Electronic Journalof Differential Equations vol 2006 pp 1ndash8 2006

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4 Discrete Dynamics in Nature and Society

+ 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)119906

Δ (119905)]2 Δ119905

+ int1198891119905ℎ(1198891)

[radic1205821 (119905)119908 (119905) 119906120590 (119905) + 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)

sdot 119906Δ (119905)]2 Δ119905 gt int11988911198881

[120572119901 (119905) 1199062 (120590 (119905))

minus 141205821 (119905) (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905

(20)

Using (18) we get

ℎ(1198891)sum119896=ℎ(1198881)+1

119886119896 minus 119887119896119886119896 1199062 (119905119896) 119908 (119905119896) gt int11988911198881

[120572119901 (119905) 1199062 (120590 (119905))

minus 141205821 (119905) (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905

(21)

It follows from (14)-(16) that

119908(119905ℎ(1198881)+1) = minus119909Δ (119905ℎ(1198881)+1)119909 (119905ℎ(1198881)+1) ge minus 1119905ℎ(1198881)+1 minus 1198881 (22)

and

119908 (119905119896) ge minus 1119905119896 minus 119905119896minus1 119896 = ℎ (1198881) + 2 ℎ (1198891) (23)

The conditions 119887119896 ge 119886119896 gt 0 119896 = 1 2 sdot sdot sdot (22) and (23) yield

ℎ(1198891)sum119896=ℎ(1198881)+1

119887119896 minus 119886119896119886119896 1199062 (119905119896) 119908 (119905119896)

ge minus 119887ℎ(1198881)+1 minus 119886ℎ(1198881)+1119886ℎ(1198881)+1 (119905ℎ(1198881)+1 minus 1198881)1199062 (119905ℎ(1198881)+1)

minus ℎ(1198891)sum119896=ℎ(1198881)+2

119887119896 minus 119886119896119886119896 (119905119896 minus 119905119896minus1)1199062 (119905119896)

(24)

Thus

ℎ(1198891)sum119896=ℎ(1198881)+1

119886119896 minus 119887119896119886119896 1199062 (119905119896) 119908 (119905119896)

le 119887ℎ(1198881)+1 minus 119886ℎ(1198881)+1119886ℎ(1198881)+1 (119905ℎ(1198881)+1 minus 1198881)1199062 (119905ℎ(1198881)+1)

+ ℎ(1198891)sum119896=ℎ(1198881)+2

119887119896 minus 119886119896119886119896 (119905119896 minus 119905119896minus1)1199062 (119905119896) = 119867 (119906 1198881 1198891)

(25)

According to (21) we have

int11988911198881

[120572119901 (119905) 1199062 (120590 (119905))

minus 141205821 (119905) (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905

lt 119867 (119906 1198881 1198891)

(26)

which is contradicted with (9)For the case ℎ(1198881) = ℎ(1198891)119867(119906 1198881 1198891) = 0 and there are

no impulsive points in [1198881 1198891] Similar to the proof of (21) weobtain

int11988911198881

[120572119901 (119905) 1199062 (120590 (119905))

minus 141205821 (119905) (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905 lt 0

(27)

which is contradicted with (9) againFor the case 119909(119905) lt 0 119905 ge 1199050 ge 0 by letting 119910 = minus119909 then119910 is positive solution of the dynamic equations

119909ΔΔ (119905) + 119901 (119905) 119891 (119909120590 (119905)) = minus119890 (119905) 119905 isin JT fl [0infin) cap T 119905 = 119905119896 119896 = 1 2

119909 (119905+119896 ) = 119886119896119909 (119905minus119896 ) 119909Δ (119905+119896 ) = 119887119896119909Δ (119905minus119896 )

119896 = 1 2 119909 (1199050) = 1199090119909Δ (1199050) = 119909Δ0

(28)

Repeating the above procedure on the interval [1198882 1198892] wewill obtain a contradiction again This completes the proofof Theorem 4

Theorem 5 Let 119909119891(119909) gt 0 for 119909 = 0 and |119891(119909)| ge |119909|119903for 119903 gt 1 Suppose that for any 119879 ge 0 there exists constants1198881 1198891 1198882 1198892 isin T with 119879 le 1198881 lt 1198891 119879 le 1198882 lt 1198892 and (119862)holds If there exists 119906 isin 119863(119888119894 119889119894) such that

int119889119894119888119894

119903 (119903 minus 1)(1minus119903)119903 1199011119903 (119905) |119890 (119905)|(119903minus1)119903 1199062 (120590 (119905))

minus 120583 (119905) + 119905 minus 119905ℎ(119889119894)4 (119905 minus 119905ℎ(119889119894)) [119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905)]2Δ119905ge 119867(119906 119888119894 119889119894)

(29)

where119867 is given by (10) then (3) is oscillatory

Proof Assume that 119909(119905) gt 0 119909120590(119905) gt 0 119905 ge 1199050 ge 0is a nonoscillatory solution of (3) Let 119860 = 1199011119903(119905)119909120590(119905)

Discrete Dynamics in Nature and Society 5

119861 = (minus119890(119905)(119903 minus 1))1119903 and choose 1198881 1198891 isin T with 1198891 gt 1198881 ge1199050 ge 0 and 119901(119905) ge 0 119890(119905) le 0 for 119905 isin [1198881 1198891] Hence 119860 gt 0119861 gt 0 for 119903 gt 1 Lemma 3 gives

119901 (119905) 119909119903 (120590 (119905)) minus 119890 (119905)ge 119903 (119903 minus 1)(1minus119903)119903 1199011119903 (119905) |119890 (119905)|(119903minus1)119903 119909 (120590 (119905))= 12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 119909 (120590 (119905))

(30)

where 1205822 = 119903(119903 minus 1)(1minus119903)119903 is a positive constant By (3) and(30) we obtain

119909ΔΔ (119905) + 12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 119909 (120590 (119905)) le 0 (31)

Let119908(119905) = 119909Δ(119905)119909(119905) By using (14)-(16) the formulas (2) and

(119891119892)Δ = 119891Δ119892 minus 119891119892Δ119892119892120590 (32)

we have

119908Δ (119905) = 119909ΔΔ (119905)119909120590 (119905) minus 119909 (119905)119909120590 (119905) [119909Δ (119905)119909 (119905) ]

2

le minus12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 minus 1205821 (119905) 1199082 (119905) (33)

where 119905 = 119905119896 119896 = 1 2 sdot sdot sdot and 1205821(119905) = (119905 minus 119905ℎ(1198891))(120583(119905) + 119905 minus119905ℎ(1198891))In the case ℎ(1198881) lt ℎ(1198891) all the impulsive moments in[1198881 1198891] are 119905ℎ(1198881)+1 119905ℎ(1198881)+2 119905ℎ(1198891) Let 119906(119905) isin 119863(1198881 1198891) be

given as in the theorem Multiplying both sides of (33) by1199062(120590(119905)) and integrating it from 1198881 to 1198891 giveℎ(1198891)sum119896=ℎ(1198881)+1

1199062 (119905119896) [119908 (119905119896) minus 119908 (119905+119896 )] le minusint11988911198881

12058221199011119903 (119905)

sdot |119890 (119905)|(119903minus1)119903 1199062 (120590 (119905)) Δ119905 minus int119905ℎ(1198881 )+11198881

[1205821 (119905) 1199082 (119905)sdot 1199062 (120590 (119905)) + (119906 (119905) + 119906120590 (119905)) 119906Δ (119905) 119908 (119905)] Δ119905minus ℎ(1198891)minus1sum119896=ℎ(1198881)+1

int119905119896+1119905119894

[1205821 (119905) 1199082 (119905) 1199062 (120590 (119905))+ (119906 (119905) + 119906120590 (119905)) 119906Δ (119905) 119908 (119905)] Δ119905minus int1198891119905ℎ(1198891)

[1205821 (119905) 1199082 (119905) 1199062 (120590 (119905)) + (119906 (119905) + 119906120590 (119905))

sdot 119906Δ (119905) 119908 (119905)] Δ119905 = minusint11988911198881

[12058221199011119903 (119905)sdot |119890 (119905)|(119903minus1)119903 1199062 (120590 (119905))minus 141205821 (119905) (

119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905

minus int119905ℎ(1198881 )+11198881

[radic1205821 (119905)119908 (119905) 119906120590 (119905) + 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)

sdot 119906Δ (119905)]2 Δ119905

minus ℎ(1198891)minus1sum119896=ℎ(1198881)+1

int119905119896+1119905119896

[radic1205821 (119905)119908 (119905) 119906120590 (119905)

+ 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)119906

Δ (119905)]2 Δ119905

minus int1198891119905ℎ(1198891)

[radic1205821 (119905)119908 (119905) 119906120590 (119905) + 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)

sdot 119906Δ (119905)]2 Δ119905 lt minusint11988911198881

[12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903sdot 1199062 (120590 (119905))minus 141205821 (119905) (

119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905(34)

Similar to the proof of Theorem 4 by using (18) we have

ℎ(1198891)sum119896=ℎ(1198881)+1

1199062 (119905119896) [119908 (119905119896) minus 119908 (119905+119896 )]

= ℎ(1198891)sum119896=ℎ(1198881)+1

119886119896 minus 119887119896119886119896 1199062 (119905119896) 119908 (119905119896) ge minus119867(119906 1198881 1198891) (35)

Thus (34) yields

int119889119894119888119894

[12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 1199062 (120590 (119905))

minus 141205821 (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905 lt 119867 (119906 1198881

1198891)

(36)

which is contradicted with (29)In the case ℎ(1198881) = ℎ(1198891) we have 119867(119906 1198881 1198891) = 0 and

there are no impulsive moments in [1198881 1198891] Similar to theproof of Theorem 4 we get a contradiction again

If 119909(119905) lt 0 119905 ge 1199050 ge 0 we can derive a similar contradic-tion on [1198882 1198892] This completes the proof of Theorem 5

3 Example

Since the time scale P119886119887 = ⋃infin119899=0[119899(119886 + 119887) 119899(119886 + 119887) + 119886]is an important and useful tool used extensively to simulatebiological population electric circuit and so on we give anexample in P119886119887 to show the application of our oscillatorycriteria

6 Discrete Dynamics in Nature and Society

Example 6 Consider the following system

119909ΔΔ (119905) + 119898 sin 119905119909 (120590 (119905)) = cos t119905 isin P120587120587 = infin⋃

119899=0

[2119899120587 (2119899 + 1) 120587] 119905 = (2119896 + 12)120587 plusmn 38120587 119896 = 0 1 2

119909 (119905+119896 ) = 119886119896119909 (119905minus119896 ) 119909Δ (119905+119896 ) = 119887119896119909Δ (119905minus119896 )

119905119896 = (2119896 + 12)120587 plusmn 38120587 119896 = 0 1 2 119909 (1199050) = 1199090119909Δ (1199050) = 119909Δ0

(37)

with the transition condition

119909 (2119899120587) = 119909 ((2119899 minus 1) 120587) 119899 ge 1 (38)

where 119887119896 ge 119886119896 gt 0119898 gt 0 are constantsHence 119901(119905) = 119898 sin 119905 119890(119905) = cos 119905 and 119891(119909(120590(119905))) =119909(120590(119905)) For any 119879 ge 0 choose 1198881 = 2119899120587+31205874 1198891 = 2119899120587+1205871198882 = 2119899120587 1198892 = 2119899120587 + 1205874 (119888119894 119889119894 isin P120587120587 119894 = 1 2) such

that 119888119894 ge 119879 for sufficiently large 119899 119894 = 1 2 Hence 119901(119905) ge 0for 119905 isin [1198881 1198891] cup [1198882 1198892] and 119890(119905) le 0 for 119905 isin [1198881 1198891]and 119890(119905) ge 0 for 119905 isin [1198882 1198892] Let 119906(119905) = sin 2119905 cos 2119905 then119906(119905) isin 119863(119888119894 119889119894) 119894 = 1 2 Furthermore we have 120590(119905) = 119905120583(119905) = 0 for 119905 isin [119888119894 119889119894] 119894 = 1 2 By some simple calculationfor 119894 = 1 2 we have

int119889119894119888119894

119896119901 (119905) 1199062 (120590 (119905))

minus 120583 (119905) + 119905 minus 119905ℎ(119889119894)4 (119905 minus 119905ℎ(119889119894)) [119906 (119905) + 119906 (120590 (119905))119906 (120590 (119905))sdot 119906Δ (119905)]2Δ119905 = int119889119894

119888119894

[119901 (119905) 1199062 (119905)

minus (119906Δ (119905))2] 119889119905 = int119889119894119888119894

[119898 sin 119905 sin2 2119905 cos2 2119905minus 4 cos2 4119905] 119889119905

int119889119894119888119894

4 cos2 4119905119889119905 = 1205872

(39)

and

int119889119894119888119894

119898 sin 119905 sin2 2119905 cos2 2119905119889119905 = 863 (1 minusradic22 )119898 (40)

Therefore

int119889119894119888119894

119896119901 (119905) 1199062 (120590 (119905))

minus 120583 (119905) + 119905 minus 119905ℎ(119889119894)4 (119905 minus 119905ℎ(119889119894)) [119906 (119905) + 119906 (120590 (119905))119906 (120590 (119905))sdot 119906Δ (119905)]2Δ119905

(41)

= 863 (1 minusradic22 )119898 minus 1205872 (42)

The impulsive points in [1198881 1198891] = [2119899120587 + 31205874 2119899120587 + 120587]are 2119899120587 + (78)120587 and in [1198882 1198892] = [2119899120587 2119899120587 + 1205874] are 2119899120587 +(18)120587 Thus

119867(119906 119888119894 119889119894) = 119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1 (119905ℎ(119888119894)+1 minus 119888119894)1199062 (119905ℎ(119888119894)+1)

= 2120587119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1

(43)

ByTheorems 4 (3) is oscillatory if

863 (1 minusradic22 )119898 minus 1205872 ge 2120587

119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1 (44)

For example letting 119886119896 = 1 119887119896 = (119896 + 1)119896 yields863 (1 minus

radic22 )119898 ge 1205872 + 2(2119899 + 78) 1205872 (45)

for sufficiently large 119898 and so (38) holds It follows fromTheorem 5 that all the solutions of (37)-(38) are oscillatory

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the Natural Science Foundationof Guangdong Province (Grants nos 2017A030310597 and2018A030307024)

References

[1] S Hilger ldquoAnalysis on measure chains a unified approach tocontinuous and discrete calculusrdquo Results in Mathematics vol18 no 1-2 pp 18ndash56 1990

Discrete Dynamics in Nature and Society 7

[2] R P AgarwalM Bohner andDOrsquoRegan ldquoDynamic equationson time scales a surveyrdquo Journal of Computational and AppliedMathematics vol 141 no 1-2 pp 1ndash26 2002

[3] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhauser Boston MassUSA 2001

[4] R P Agarwal M Benchohra D OrsquoRegan and A OuahabldquoSecond order impulsive dynamic equations on time scalesrdquoFunctional Differential Equations vol 11 no 3-4 pp 223ndash2342004

[5] D R Anderson and A Zafer ldquoInterval criteria for second-order super-half-linear functional dynamic equations withdelay and advance argumentsrdquo Journal of Difference Equationsand Applications vol 16 no 8 pp 917ndash930 2010

[6] R P Agarwal D R Anderson and A k Zafer ldquoInterval oscil-lation criteria for second-order forced delay dynamic equationswith mixed nonlinearitiesrdquo Computers and Mathematics withApplications An International Journal vol 59 no 2 pp 977ndash993 2010

[7] M Benchohra S Hamani and J Henderson ldquoOscillation andnonoscillation for impulsive dynamic equations on certain timescalesrdquo Advances in Difference Equations pp 1ndash12 2006

[8] M A El-Sayed ldquoAn oscillation criterion for a forced secondorder linear differential equationrdquo Proceedings of the AmericanMathematical Society vol 118 no 3 pp 813ndash817 1993

[9] W-T Li and R P Agarwal ldquoInterval oscillation criteria for aforced second order nonlinear ordinary differential equationrdquoApplicable Analysis An International Journal vol 75 no 3-4pp 341ndash347 2000

[10] A Ozbekler and A Zafer ldquoInterval criteria for the forcedoscillation of super-half-linear differential equations underimpulse effectsrdquoMathematical and ComputerModelling vol 50no 1-2 pp 59ndash65 2009

[11] Y Sun ldquoA note on Nasrs and Wongs papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003

[12] J S Wong ldquoOscillation criteria for a forced second-order lineardifferential equationrdquo Journal of Mathematical Analysis andApplications vol 231 no 1 pp 235ndash240 1999

[13] M Huang and W Feng ldquoOscillation criteria for impulsivedynamic equations on time scalesrdquo Electronic Journal of Differ-ential Equations vol 2007 no 72 pp 1ndash9 2007

[14] M Huang and W Feng ldquoOscillation for forced second-ordernonlinear dynamic equations on time scalesrdquo Electronic Journalof Differential Equations vol 2006 pp 1ndash8 2006

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Discrete Dynamics in Nature and Society 5

119861 = (minus119890(119905)(119903 minus 1))1119903 and choose 1198881 1198891 isin T with 1198891 gt 1198881 ge1199050 ge 0 and 119901(119905) ge 0 119890(119905) le 0 for 119905 isin [1198881 1198891] Hence 119860 gt 0119861 gt 0 for 119903 gt 1 Lemma 3 gives

119901 (119905) 119909119903 (120590 (119905)) minus 119890 (119905)ge 119903 (119903 minus 1)(1minus119903)119903 1199011119903 (119905) |119890 (119905)|(119903minus1)119903 119909 (120590 (119905))= 12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 119909 (120590 (119905))

(30)

where 1205822 = 119903(119903 minus 1)(1minus119903)119903 is a positive constant By (3) and(30) we obtain

119909ΔΔ (119905) + 12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 119909 (120590 (119905)) le 0 (31)

Let119908(119905) = 119909Δ(119905)119909(119905) By using (14)-(16) the formulas (2) and

(119891119892)Δ = 119891Δ119892 minus 119891119892Δ119892119892120590 (32)

we have

119908Δ (119905) = 119909ΔΔ (119905)119909120590 (119905) minus 119909 (119905)119909120590 (119905) [119909Δ (119905)119909 (119905) ]

2

le minus12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 minus 1205821 (119905) 1199082 (119905) (33)

where 119905 = 119905119896 119896 = 1 2 sdot sdot sdot and 1205821(119905) = (119905 minus 119905ℎ(1198891))(120583(119905) + 119905 minus119905ℎ(1198891))In the case ℎ(1198881) lt ℎ(1198891) all the impulsive moments in[1198881 1198891] are 119905ℎ(1198881)+1 119905ℎ(1198881)+2 119905ℎ(1198891) Let 119906(119905) isin 119863(1198881 1198891) be

given as in the theorem Multiplying both sides of (33) by1199062(120590(119905)) and integrating it from 1198881 to 1198891 giveℎ(1198891)sum119896=ℎ(1198881)+1

1199062 (119905119896) [119908 (119905119896) minus 119908 (119905+119896 )] le minusint11988911198881

12058221199011119903 (119905)

sdot |119890 (119905)|(119903minus1)119903 1199062 (120590 (119905)) Δ119905 minus int119905ℎ(1198881 )+11198881

[1205821 (119905) 1199082 (119905)sdot 1199062 (120590 (119905)) + (119906 (119905) + 119906120590 (119905)) 119906Δ (119905) 119908 (119905)] Δ119905minus ℎ(1198891)minus1sum119896=ℎ(1198881)+1

int119905119896+1119905119894

[1205821 (119905) 1199082 (119905) 1199062 (120590 (119905))+ (119906 (119905) + 119906120590 (119905)) 119906Δ (119905) 119908 (119905)] Δ119905minus int1198891119905ℎ(1198891)

[1205821 (119905) 1199082 (119905) 1199062 (120590 (119905)) + (119906 (119905) + 119906120590 (119905))

sdot 119906Δ (119905) 119908 (119905)] Δ119905 = minusint11988911198881

[12058221199011119903 (119905)sdot |119890 (119905)|(119903minus1)119903 1199062 (120590 (119905))minus 141205821 (119905) (

119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905

minus int119905ℎ(1198881 )+11198881

[radic1205821 (119905)119908 (119905) 119906120590 (119905) + 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)

sdot 119906Δ (119905)]2 Δ119905

minus ℎ(1198891)minus1sum119896=ℎ(1198881)+1

int119905119896+1119905119896

[radic1205821 (119905)119908 (119905) 119906120590 (119905)

+ 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)119906

Δ (119905)]2 Δ119905

minus int1198891119905ℎ(1198891)

[radic1205821 (119905)119908 (119905) 119906120590 (119905) + 119906 (119905) + 119906120590 (119905)2radic1205821 (119905)119906120590 (119905)

sdot 119906Δ (119905)]2 Δ119905 lt minusint11988911198881

[12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903sdot 1199062 (120590 (119905))minus 141205821 (119905) (

119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905(34)

Similar to the proof of Theorem 4 by using (18) we have

ℎ(1198891)sum119896=ℎ(1198881)+1

1199062 (119905119896) [119908 (119905119896) minus 119908 (119905+119896 )]

= ℎ(1198891)sum119896=ℎ(1198881)+1

119886119896 minus 119887119896119886119896 1199062 (119905119896) 119908 (119905119896) ge minus119867(119906 1198881 1198891) (35)

Thus (34) yields

int119889119894119888119894

[12058221199011119903 (119905) |119890 (119905)|(119903minus1)119903 1199062 (120590 (119905))

minus 141205821 (119906 (119905) + 119906120590 (119905)119906120590 (119905) 119906Δ (119905))2]Δ119905 lt 119867 (119906 1198881

1198891)

(36)

which is contradicted with (29)In the case ℎ(1198881) = ℎ(1198891) we have 119867(119906 1198881 1198891) = 0 and

there are no impulsive moments in [1198881 1198891] Similar to theproof of Theorem 4 we get a contradiction again

If 119909(119905) lt 0 119905 ge 1199050 ge 0 we can derive a similar contradic-tion on [1198882 1198892] This completes the proof of Theorem 5

3 Example

Since the time scale P119886119887 = ⋃infin119899=0[119899(119886 + 119887) 119899(119886 + 119887) + 119886]is an important and useful tool used extensively to simulatebiological population electric circuit and so on we give anexample in P119886119887 to show the application of our oscillatorycriteria

6 Discrete Dynamics in Nature and Society

Example 6 Consider the following system

119909ΔΔ (119905) + 119898 sin 119905119909 (120590 (119905)) = cos t119905 isin P120587120587 = infin⋃

119899=0

[2119899120587 (2119899 + 1) 120587] 119905 = (2119896 + 12)120587 plusmn 38120587 119896 = 0 1 2

119909 (119905+119896 ) = 119886119896119909 (119905minus119896 ) 119909Δ (119905+119896 ) = 119887119896119909Δ (119905minus119896 )

119905119896 = (2119896 + 12)120587 plusmn 38120587 119896 = 0 1 2 119909 (1199050) = 1199090119909Δ (1199050) = 119909Δ0

(37)

with the transition condition

119909 (2119899120587) = 119909 ((2119899 minus 1) 120587) 119899 ge 1 (38)

where 119887119896 ge 119886119896 gt 0119898 gt 0 are constantsHence 119901(119905) = 119898 sin 119905 119890(119905) = cos 119905 and 119891(119909(120590(119905))) =119909(120590(119905)) For any 119879 ge 0 choose 1198881 = 2119899120587+31205874 1198891 = 2119899120587+1205871198882 = 2119899120587 1198892 = 2119899120587 + 1205874 (119888119894 119889119894 isin P120587120587 119894 = 1 2) such

that 119888119894 ge 119879 for sufficiently large 119899 119894 = 1 2 Hence 119901(119905) ge 0for 119905 isin [1198881 1198891] cup [1198882 1198892] and 119890(119905) le 0 for 119905 isin [1198881 1198891]and 119890(119905) ge 0 for 119905 isin [1198882 1198892] Let 119906(119905) = sin 2119905 cos 2119905 then119906(119905) isin 119863(119888119894 119889119894) 119894 = 1 2 Furthermore we have 120590(119905) = 119905120583(119905) = 0 for 119905 isin [119888119894 119889119894] 119894 = 1 2 By some simple calculationfor 119894 = 1 2 we have

int119889119894119888119894

119896119901 (119905) 1199062 (120590 (119905))

minus 120583 (119905) + 119905 minus 119905ℎ(119889119894)4 (119905 minus 119905ℎ(119889119894)) [119906 (119905) + 119906 (120590 (119905))119906 (120590 (119905))sdot 119906Δ (119905)]2Δ119905 = int119889119894

119888119894

[119901 (119905) 1199062 (119905)

minus (119906Δ (119905))2] 119889119905 = int119889119894119888119894

[119898 sin 119905 sin2 2119905 cos2 2119905minus 4 cos2 4119905] 119889119905

int119889119894119888119894

4 cos2 4119905119889119905 = 1205872

(39)

and

int119889119894119888119894

119898 sin 119905 sin2 2119905 cos2 2119905119889119905 = 863 (1 minusradic22 )119898 (40)

Therefore

int119889119894119888119894

119896119901 (119905) 1199062 (120590 (119905))

minus 120583 (119905) + 119905 minus 119905ℎ(119889119894)4 (119905 minus 119905ℎ(119889119894)) [119906 (119905) + 119906 (120590 (119905))119906 (120590 (119905))sdot 119906Δ (119905)]2Δ119905

(41)

= 863 (1 minusradic22 )119898 minus 1205872 (42)

The impulsive points in [1198881 1198891] = [2119899120587 + 31205874 2119899120587 + 120587]are 2119899120587 + (78)120587 and in [1198882 1198892] = [2119899120587 2119899120587 + 1205874] are 2119899120587 +(18)120587 Thus

119867(119906 119888119894 119889119894) = 119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1 (119905ℎ(119888119894)+1 minus 119888119894)1199062 (119905ℎ(119888119894)+1)

= 2120587119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1

(43)

ByTheorems 4 (3) is oscillatory if

863 (1 minusradic22 )119898 minus 1205872 ge 2120587

119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1 (44)

For example letting 119886119896 = 1 119887119896 = (119896 + 1)119896 yields863 (1 minus

radic22 )119898 ge 1205872 + 2(2119899 + 78) 1205872 (45)

for sufficiently large 119898 and so (38) holds It follows fromTheorem 5 that all the solutions of (37)-(38) are oscillatory

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the Natural Science Foundationof Guangdong Province (Grants nos 2017A030310597 and2018A030307024)

References

[1] S Hilger ldquoAnalysis on measure chains a unified approach tocontinuous and discrete calculusrdquo Results in Mathematics vol18 no 1-2 pp 18ndash56 1990

Discrete Dynamics in Nature and Society 7

[2] R P AgarwalM Bohner andDOrsquoRegan ldquoDynamic equationson time scales a surveyrdquo Journal of Computational and AppliedMathematics vol 141 no 1-2 pp 1ndash26 2002

[3] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhauser Boston MassUSA 2001

[4] R P Agarwal M Benchohra D OrsquoRegan and A OuahabldquoSecond order impulsive dynamic equations on time scalesrdquoFunctional Differential Equations vol 11 no 3-4 pp 223ndash2342004

[5] D R Anderson and A Zafer ldquoInterval criteria for second-order super-half-linear functional dynamic equations withdelay and advance argumentsrdquo Journal of Difference Equationsand Applications vol 16 no 8 pp 917ndash930 2010

[6] R P Agarwal D R Anderson and A k Zafer ldquoInterval oscil-lation criteria for second-order forced delay dynamic equationswith mixed nonlinearitiesrdquo Computers and Mathematics withApplications An International Journal vol 59 no 2 pp 977ndash993 2010

[7] M Benchohra S Hamani and J Henderson ldquoOscillation andnonoscillation for impulsive dynamic equations on certain timescalesrdquo Advances in Difference Equations pp 1ndash12 2006

[8] M A El-Sayed ldquoAn oscillation criterion for a forced secondorder linear differential equationrdquo Proceedings of the AmericanMathematical Society vol 118 no 3 pp 813ndash817 1993

[9] W-T Li and R P Agarwal ldquoInterval oscillation criteria for aforced second order nonlinear ordinary differential equationrdquoApplicable Analysis An International Journal vol 75 no 3-4pp 341ndash347 2000

[10] A Ozbekler and A Zafer ldquoInterval criteria for the forcedoscillation of super-half-linear differential equations underimpulse effectsrdquoMathematical and ComputerModelling vol 50no 1-2 pp 59ndash65 2009

[11] Y Sun ldquoA note on Nasrs and Wongs papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003

[12] J S Wong ldquoOscillation criteria for a forced second-order lineardifferential equationrdquo Journal of Mathematical Analysis andApplications vol 231 no 1 pp 235ndash240 1999

[13] M Huang and W Feng ldquoOscillation criteria for impulsivedynamic equations on time scalesrdquo Electronic Journal of Differ-ential Equations vol 2007 no 72 pp 1ndash9 2007

[14] M Huang and W Feng ldquoOscillation for forced second-ordernonlinear dynamic equations on time scalesrdquo Electronic Journalof Differential Equations vol 2006 pp 1ndash8 2006

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Discrete Dynamics in Nature and Society

Example 6 Consider the following system

119909ΔΔ (119905) + 119898 sin 119905119909 (120590 (119905)) = cos t119905 isin P120587120587 = infin⋃

119899=0

[2119899120587 (2119899 + 1) 120587] 119905 = (2119896 + 12)120587 plusmn 38120587 119896 = 0 1 2

119909 (119905+119896 ) = 119886119896119909 (119905minus119896 ) 119909Δ (119905+119896 ) = 119887119896119909Δ (119905minus119896 )

119905119896 = (2119896 + 12)120587 plusmn 38120587 119896 = 0 1 2 119909 (1199050) = 1199090119909Δ (1199050) = 119909Δ0

(37)

with the transition condition

119909 (2119899120587) = 119909 ((2119899 minus 1) 120587) 119899 ge 1 (38)

where 119887119896 ge 119886119896 gt 0119898 gt 0 are constantsHence 119901(119905) = 119898 sin 119905 119890(119905) = cos 119905 and 119891(119909(120590(119905))) =119909(120590(119905)) For any 119879 ge 0 choose 1198881 = 2119899120587+31205874 1198891 = 2119899120587+1205871198882 = 2119899120587 1198892 = 2119899120587 + 1205874 (119888119894 119889119894 isin P120587120587 119894 = 1 2) such

that 119888119894 ge 119879 for sufficiently large 119899 119894 = 1 2 Hence 119901(119905) ge 0for 119905 isin [1198881 1198891] cup [1198882 1198892] and 119890(119905) le 0 for 119905 isin [1198881 1198891]and 119890(119905) ge 0 for 119905 isin [1198882 1198892] Let 119906(119905) = sin 2119905 cos 2119905 then119906(119905) isin 119863(119888119894 119889119894) 119894 = 1 2 Furthermore we have 120590(119905) = 119905120583(119905) = 0 for 119905 isin [119888119894 119889119894] 119894 = 1 2 By some simple calculationfor 119894 = 1 2 we have

int119889119894119888119894

119896119901 (119905) 1199062 (120590 (119905))

minus 120583 (119905) + 119905 minus 119905ℎ(119889119894)4 (119905 minus 119905ℎ(119889119894)) [119906 (119905) + 119906 (120590 (119905))119906 (120590 (119905))sdot 119906Δ (119905)]2Δ119905 = int119889119894

119888119894

[119901 (119905) 1199062 (119905)

minus (119906Δ (119905))2] 119889119905 = int119889119894119888119894

[119898 sin 119905 sin2 2119905 cos2 2119905minus 4 cos2 4119905] 119889119905

int119889119894119888119894

4 cos2 4119905119889119905 = 1205872

(39)

and

int119889119894119888119894

119898 sin 119905 sin2 2119905 cos2 2119905119889119905 = 863 (1 minusradic22 )119898 (40)

Therefore

int119889119894119888119894

119896119901 (119905) 1199062 (120590 (119905))

minus 120583 (119905) + 119905 minus 119905ℎ(119889119894)4 (119905 minus 119905ℎ(119889119894)) [119906 (119905) + 119906 (120590 (119905))119906 (120590 (119905))sdot 119906Δ (119905)]2Δ119905

(41)

= 863 (1 minusradic22 )119898 minus 1205872 (42)

The impulsive points in [1198881 1198891] = [2119899120587 + 31205874 2119899120587 + 120587]are 2119899120587 + (78)120587 and in [1198882 1198892] = [2119899120587 2119899120587 + 1205874] are 2119899120587 +(18)120587 Thus

119867(119906 119888119894 119889119894) = 119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1 (119905ℎ(119888119894)+1 minus 119888119894)1199062 (119905ℎ(119888119894)+1)

= 2120587119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1

(43)

ByTheorems 4 (3) is oscillatory if

863 (1 minusradic22 )119898 minus 1205872 ge 2120587

119887ℎ(119888119894)+1 minus 119886ℎ(119888119894)+1119886ℎ(119888119894)+1 (44)

For example letting 119886119896 = 1 119887119896 = (119896 + 1)119896 yields863 (1 minus

radic22 )119898 ge 1205872 + 2(2119899 + 78) 1205872 (45)

for sufficiently large 119898 and so (38) holds It follows fromTheorem 5 that all the solutions of (37)-(38) are oscillatory

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the Natural Science Foundationof Guangdong Province (Grants nos 2017A030310597 and2018A030307024)

References

[1] S Hilger ldquoAnalysis on measure chains a unified approach tocontinuous and discrete calculusrdquo Results in Mathematics vol18 no 1-2 pp 18ndash56 1990

Discrete Dynamics in Nature and Society 7

[2] R P AgarwalM Bohner andDOrsquoRegan ldquoDynamic equationson time scales a surveyrdquo Journal of Computational and AppliedMathematics vol 141 no 1-2 pp 1ndash26 2002

[3] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhauser Boston MassUSA 2001

[4] R P Agarwal M Benchohra D OrsquoRegan and A OuahabldquoSecond order impulsive dynamic equations on time scalesrdquoFunctional Differential Equations vol 11 no 3-4 pp 223ndash2342004

[5] D R Anderson and A Zafer ldquoInterval criteria for second-order super-half-linear functional dynamic equations withdelay and advance argumentsrdquo Journal of Difference Equationsand Applications vol 16 no 8 pp 917ndash930 2010

[6] R P Agarwal D R Anderson and A k Zafer ldquoInterval oscil-lation criteria for second-order forced delay dynamic equationswith mixed nonlinearitiesrdquo Computers and Mathematics withApplications An International Journal vol 59 no 2 pp 977ndash993 2010

[7] M Benchohra S Hamani and J Henderson ldquoOscillation andnonoscillation for impulsive dynamic equations on certain timescalesrdquo Advances in Difference Equations pp 1ndash12 2006

[8] M A El-Sayed ldquoAn oscillation criterion for a forced secondorder linear differential equationrdquo Proceedings of the AmericanMathematical Society vol 118 no 3 pp 813ndash817 1993

[9] W-T Li and R P Agarwal ldquoInterval oscillation criteria for aforced second order nonlinear ordinary differential equationrdquoApplicable Analysis An International Journal vol 75 no 3-4pp 341ndash347 2000

[10] A Ozbekler and A Zafer ldquoInterval criteria for the forcedoscillation of super-half-linear differential equations underimpulse effectsrdquoMathematical and ComputerModelling vol 50no 1-2 pp 59ndash65 2009

[11] Y Sun ldquoA note on Nasrs and Wongs papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003

[12] J S Wong ldquoOscillation criteria for a forced second-order lineardifferential equationrdquo Journal of Mathematical Analysis andApplications vol 231 no 1 pp 235ndash240 1999

[13] M Huang and W Feng ldquoOscillation criteria for impulsivedynamic equations on time scalesrdquo Electronic Journal of Differ-ential Equations vol 2007 no 72 pp 1ndash9 2007

[14] M Huang and W Feng ldquoOscillation for forced second-ordernonlinear dynamic equations on time scalesrdquo Electronic Journalof Differential Equations vol 2006 pp 1ndash8 2006

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 7

[2] R P AgarwalM Bohner andDOrsquoRegan ldquoDynamic equationson time scales a surveyrdquo Journal of Computational and AppliedMathematics vol 141 no 1-2 pp 1ndash26 2002

[3] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhauser Boston MassUSA 2001

[4] R P Agarwal M Benchohra D OrsquoRegan and A OuahabldquoSecond order impulsive dynamic equations on time scalesrdquoFunctional Differential Equations vol 11 no 3-4 pp 223ndash2342004

[5] D R Anderson and A Zafer ldquoInterval criteria for second-order super-half-linear functional dynamic equations withdelay and advance argumentsrdquo Journal of Difference Equationsand Applications vol 16 no 8 pp 917ndash930 2010

[6] R P Agarwal D R Anderson and A k Zafer ldquoInterval oscil-lation criteria for second-order forced delay dynamic equationswith mixed nonlinearitiesrdquo Computers and Mathematics withApplications An International Journal vol 59 no 2 pp 977ndash993 2010

[7] M Benchohra S Hamani and J Henderson ldquoOscillation andnonoscillation for impulsive dynamic equations on certain timescalesrdquo Advances in Difference Equations pp 1ndash12 2006

[8] M A El-Sayed ldquoAn oscillation criterion for a forced secondorder linear differential equationrdquo Proceedings of the AmericanMathematical Society vol 118 no 3 pp 813ndash817 1993

[9] W-T Li and R P Agarwal ldquoInterval oscillation criteria for aforced second order nonlinear ordinary differential equationrdquoApplicable Analysis An International Journal vol 75 no 3-4pp 341ndash347 2000

[10] A Ozbekler and A Zafer ldquoInterval criteria for the forcedoscillation of super-half-linear differential equations underimpulse effectsrdquoMathematical and ComputerModelling vol 50no 1-2 pp 59ndash65 2009

[11] Y Sun ldquoA note on Nasrs and Wongs papersrdquo Journal ofMathematical Analysis and Applications vol 286 no 1 pp 363ndash367 2003

[12] J S Wong ldquoOscillation criteria for a forced second-order lineardifferential equationrdquo Journal of Mathematical Analysis andApplications vol 231 no 1 pp 235ndash240 1999

[13] M Huang and W Feng ldquoOscillation criteria for impulsivedynamic equations on time scalesrdquo Electronic Journal of Differ-ential Equations vol 2007 no 72 pp 1ndash9 2007

[14] M Huang and W Feng ldquoOscillation for forced second-ordernonlinear dynamic equations on time scalesrdquo Electronic Journalof Differential Equations vol 2006 pp 1ndash8 2006

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom