Oscillation Criteria for Delay and Advanced Differential...

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Research Article Oscillation Criteria for Delay and Advanced Differential Equations with Nonmonotone Arguments George E. Chatzarakis 1 and Tongxing Li 2,3 1 Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education (ASPETE), N. Heraklio, 14121 Athens, Greece 2 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China 3 School of Information Science and Engineering, Linyi University, Linyi, Shandong 276005, China Correspondence should be addressed to Tongxing Li; [email protected] Received 16 April 2017; Revised 27 June 2017; Accepted 6 March 2018; Published 18 April 2018 Academic Editor: Shanmugam Lakshmanan Copyright © 2018 George E. Chatzarakis and Tongxing Li. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the oscillatory behavior of differential equations with nonmonotone deviating arguments and nonnegative coefficients. New oscillation criteria, involving limsup and liminf, are obtained based on an iterative method. Examples, numerically solved in MATLAB, are given to illustrate the applicability and strength of the obtained conditions over known ones. 1. Introduction In mathematics, delay differential equations (DDEs) are that type of differential equations where the derivative of the unknown function, at a certain time, is given in terms of the values of the function, at previous times. DDEs are also referred in the literature as time-delay systems, systems with aſtereffect or dead-time, hereditary systems, or equations with delay arguments. Mathematical modelling involving DDEs is widely used for analysis and predictions in various areas of the life sciences, for example, population dynamics, epidemiology, immunology, physiology, neural networks. See, for example, [1–10] and the references cited therein. e time delays add to these models memory effects, taking into account the dependence of the model’s present state on its past history [9]. e delay can be related to the duration of certain hidden processes, like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period, and so on. In analogy, advanced differential equations (ADEs) are used in many applied problems where the evolution rate depends not only on the present, but also on the future. While delays in DDEs represent the retrospective memory of the past, advances in ADEs represent the prospective memory of the future, accounting for the influence on the system of potential future actions, which are available, at the present time. For instance, population dynamics, economics problems, or mechanical control engineering are typical fields where such phenomena are thought to occur (see [11, 12] for details). e earliest delay model in mathematical biology is Hutchinson’s equation, in 1948 [6]. Hutchinson modified the classical logistic equation, with a delay term to incorporate hatching and maturation periods into the model and account for oscillations, in the population of Daphnia, () = () (1 − ( − ) ), (1) where () denotes the size of the population, in the present time , () describes the change of this size, at time , (−) is the size, in some past time −, > 0 is the delay, representing the time for new eggs to hatch, and is the reproduction rate of the population, while is the carrying capacity, for the population. Hindawi Complexity Volume 2018, Article ID 8237634, 18 pages https://doi.org/10.1155/2018/8237634

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Research ArticleOscillation Criteria for Delay and Advanced DifferentialEquations with Nonmonotone Arguments

George E Chatzarakis1 and Tongxing Li 23

1Department of Electrical and Electronic Engineering Educators School of Pedagogical and Technological Education (ASPETE)N Heraklio 14121 Athens Greece2School of Mathematical Sciences Qufu Normal University Qufu Shandong 273165 China3School of Information Science and Engineering Linyi University Linyi Shandong 276005 China

Correspondence should be addressed to Tongxing Li litongx2007163com

Received 16 April 2017 Revised 27 June 2017 Accepted 6 March 2018 Published 18 April 2018

Academic Editor Shanmugam Lakshmanan

Copyright copy 2018 George E Chatzarakis and Tongxing Li 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

We study the oscillatory behavior of differential equations with nonmonotone deviating arguments and nonnegative coefficientsNew oscillation criteria involving limsup and liminf are obtained based on an iterative method Examples numerically solved inMATLAB are given to illustrate the applicability and strength of the obtained conditions over known ones

1 Introduction

In mathematics delay differential equations (DDEs) are thattype of differential equations where the derivative of theunknown function at a certain time is given in terms ofthe values of the function at previous times DDEs are alsoreferred in the literature as time-delay systems systems withaftereffect or dead-time hereditary systems or equationswith delay arguments

Mathematical modelling involving DDEs is widely usedfor analysis and predictions in various areas of the lifesciences for example population dynamics epidemiologyimmunology physiology neural networks See for example[1ndash10] and the references cited therein The time delays addto these models memory effects taking into account thedependence of the modelrsquos present state on its past history[9]The delay can be related to the duration of certain hiddenprocesses like the stages of the life cycle the time betweeninfection of a cell and the production of new viruses theduration of the infectious period the immune period and soon

In analogy advanced differential equations (ADEs) areused in many applied problems where the evolution ratedepends not only on the present but also on the future

While delays in DDEs represent the retrospective memoryof the past advances in ADEs represent the prospectivememory of the future accounting for the influence on thesystem of potential future actions which are available at thepresent time For instance population dynamics economicsproblems ormechanical control engineering are typical fieldswhere such phenomena are thought to occur (see [11 12] fordetails)

The earliest delay model in mathematical biology isHutchinsonrsquos equation in 1948 [6] Hutchinson modified theclassical logistic equation with a delay term to incorporatehatching andmaturation periods into the model and accountfor oscillations in the population of Daphnia

1199101015840 (119905) = 119903119910 (119905) (1 minus 119910 (119905 minus 120591)119870 ) (1)

where 119910(119905) denotes the size of the population in the presenttime 119905 1199101015840(119905) describes the change of this size at time 119905 119910(119905minus120591)is the size in some past time 119905 minus 120591 120591 gt 0 is the delayrepresenting the time for new eggs to hatch and 119903 is thereproduction rate of the population while 119870 is the carryingcapacity for the population

HindawiComplexityVolume 2018 Article ID 8237634 18 pageshttpsdoiorg10115520188237634

2 Complexity

Many physiological processes including the concentra-tion of red blood cells the concentration of CO2 in the bloodcausing the observed periodic oscillations in the breathingfrequency and the production of new blood cells in the bonemarrow exhibit oscillations and several DDE models havebeen proposed to model these processes

Below we present two applications indicating the rel-evance of the DDEs we study in this paper to real worldproblems The two examples are taken from the areas ofphysiology and population dynamics

Application 1 (blood cells production [9]) The production ofred and white blood cells in the bonemarrow is regulated bythe level of oxygen in the bloodA reduction in the number ofcells in the blood as a result of the loss of cells causes the levelof oxygen in the blood to decrease When the level of oxygenin the blood decreases a substance is released that in turnleads to the release of blood elements from the bonemarrowThus the concentration 119888(119905)of cells in the blood stream at anytime 119905 changes according to the loss of cells and the releaseof new cells from the bone marrow But the bone marrowresponds to a reduction in the number of blood cells and thedecrease in the level of oxygen with a delay that is in theorder of 6 days That means the release of new cells into theblood stream at time 119905 depends on the cell concentration atan earlier time namely 119905 minus 120591 where 120591 is the delay with whichthe bonemarrow responds to a reduced level of oxygen in thebloodThe simplest model of the concentration of the cells inthe blood stream can be described by the DDE

1198881015840 (119905) = 120582119888 (119905 minus 120591) minus 120574119888 (119905) (2)

where 120582 represents the flux of cells into the blood stream 120574is the death rate and 120591 is the delay All of them are positiveconstantsThe solutions of the above equation exhibit similaroscillations to the actual oscillatory pattern observed in theconcentration of cells in the blood stream

Application 2 Imagine a biological population composed ofadult and juvenile individuals Let 119873(119905) denote the densityof adults at time 119905 Assume that the length of the juvenileperiod is exactly ℎ units of time for each individual Assumethat adults produce offspring at a per capita rate 120572 and thattheir probability per unit of time of dying is 120583 Assume that anewborn survives the juvenile period with probability 120588 andput 119905 = 120572120588 Then the dynamics of119873 can be described by thedifferential equation

1198731015840 (119905) = minus120583119873 (119905) + 119903119873 (119905 minus ℎ) (3)

which involves a nonlocal term 119903119873(119905 minus ℎ) meaning thatnewborns become adults with some delay So the timevariation of the population density119873 involves the current aswell as the past values of119873

The use of DDEs from the initial application in popula-tion dynamics has spread to every area of the life sciencesimmunology physiology epidemiology and cell growthTheoriginal delay logistic equation has led to several new DDEforms likeVolterrarsquos integrodifferential equations and neutral

DDEs [9] and several newmodels from the delayedHopfieldmodel in neural networks to the SIRmodel in epidemiology[7]More recently the idea of state dependent delays has beenintroduced involving ldquoa delay that itself is governed by adifferential equation that represents adaptation to the systemrsquosstaterdquo [9]

From the above review of DDEs in the biological sci-ences it is apparent that if DDEs are so extensively used inthis area this is because the dynamics of those equationsnamely the stability and oscillatory properties of the solu-tions of those equations replicate the stability and oscillatorypatterns we actually observe in processes in those areasThus the study of the stability and oscillatory behavior of thesolutions of DDEs has become the principal subject of theresearch on those equations For more advanced treatises onoscillation theory the reader is referred to [13ndash33]

In the paper we consider a differential equation withdelay argument of the form

1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) = 0 119905 ge 1199050 (E)where 119901 is a function of nonnegative real numbers and 120591 is afunction of positive real numbers such that

120591 (119905) lt 119905 119905 ge 1199050lim119905rarrinfin

120591 (119905) = infin (4)

By a solution of (E) we understand a continuously differ-entiable function defined on [120591(1198790)infin) for some 1198790 ge 1199050and such that (E) is satisfied for 119905 ge 1198790 Such a solution iscalled oscillatory if it has arbitrarily large zeros and otherwiseit is called nonoscillatory An equation is oscillatory if all itssolutions oscillate

A parallel problem to that of establishing oscillationcriteria for the solutions of equation (E) is the one concerningthe solutions of the advanced differential equation (ADE)

1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) = 0 119905 ge 1199050 (E1015840)where 119902 is a function of nonnegative real numbers and 120590 is afunction of positive real numbers such that

120590 (119905) gt 119905 119905 ge 1199050 (5)

The objective of this paper is to consider the oscillatorydynamics of both delay and advanced differential equationsfrom the perspective of the qualitative analysis of thoseequations In that framework (i) we formulate new iterativeoscillation conditions for testing whether all solutions of aDDE of the form of (E) or an ADE of the form of (E1015840) areoscillatory (ii) we show that these tests significantly improveon all the previous iterative and noniterative oscillationcriteria which briefly are reviewed in the Historical andChronological Review in Section 2 requiring fewer iterationsto determine whether an equation of the considered form isoscillatory and (iii) these criteria apply to amore general classof equations having nonmonotone arguments 120591(119905) or 120590(119905) incontrast to the large majority of the other studies where thecriteria apply to equations with nondecreasing arguments

Complexity 3

From this point onward we will use the notation

120572 fl lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904

120573 fl lim inf119905rarrinfin

int120590(119905)119905119902 (119904) 119889119904

119863 (120596) fl 0 if 120596 gt 1119890 1 minus 120596 minus radic1 minus 2120596 minus 12059622 if 120596 isin [0 1119890 ]

LD fl lim sup119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904where 120591 (119905) is nondecreasing

LA fl lim sup119905rarrinfin

int120590(119905)119905119902 (119904) 119889119904where 120590 (119905) is nondecreasing

(6)

2 Historical and Chronological Review

21 DDEs The first systematic study for the oscillation of allsolutions of equation (E) was made by Myskis in 1950 [31]when he proved that every solution of (E) oscillates if

lim sup119905rarrinfin

[119905 minus 120591 (119905)] lt infinlim inf119905rarrinfin

[119905 minus 120591 (119905)] lim inf119905rarrinfin

119901 (119905) gt 1119890 (7)

In 1972 Ladas et al [27] proved that if

LD gt 1 (8)

then all solutions of (E) are oscillatoryIn 1982 Koplatadze and Chanturiya [24] improved (7) to

120572 gt 1119890 (9)

Regarding the constant 1119890 in (9) it should be remarked thatif the inequality

int119905120591(119905)119901 (119904) 119889119904 le 1119890 (10)

holds eventually then according to [24] (E) has a nonoscil-latory solution

It is apparent that there is a gap between conditions (8)and (9) when

lim119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 (11)

does not exist How to fill this gap is an interesting problemwhich has been investigated by several authors For example

in 2000 Jaros and Stavroulakis [23] proved that if 1205820 is thesmaller root of the equation 120582 = 119890120572120582 and

LD gt 1 + ln 12058201205820 minus 119863 (120572) (12)

then all solutions of (E) oscillateNowwe come to the general case where the argument 120591(119905)

is nonmonotone Set

ℎ (119905) fl sup119904le119905120591 (119904) 119905 ge 1199050 (13)

Clearly the function ℎ(119905) is nondecreasing and 120591(119905) le ℎ(119905) lt119905 for all 119905 ge 1199050In 1994 Koplatadze and Kvinikadze [25] proved that if

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

ℎ(119904)119901 (119906) 120595119895 (119906) 119889119906)119889119904

gt 1 minus 119863 (120572) (14)

where

1205951 (119905) = 0120595119895 (119905) = exp(int119905

120591(119905)119901 (119906) 120595119895minus1 (119906) 119889119906) 119895 ge 2 (15)

then all solutions of (E) oscillateIn 2011 Braverman and Karpuz [14] proved that if

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904 gt 1 (16)

then all solutions of (E) oscillate while in 2014 Stavroulakis[32] improved (16) to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

gt 1 minus 119863 (120572) (17)

In 2016 El-Morshedy and Attia [30] proved that if

lim sup119905rarrinfin

[[int119905

119892(119905)119901119899 (119904) 119889119904

+ 119863 (120572) exp(int119905119892(119905)

119899minus1sum119895=0

119901119895 (119904) 119889119904)]] gt 1(18)

where 1199010(119905) = 119901(119905) and119901119899 (119905)= 119901119899minus1 (119905) int119905

119892(119905)119901119899minus1 (119904) exp(int119905

119892(119904)119901119899minus1 (119906) 119889119906)119889119904

119899 ge 1(19)

4 Complexity

then all solutions of (E) are oscillatory Here 119892(119905) is a nonde-creasing continuous function such that 120591(119905) le 119892(119905) le 119905 119905 ge1199051 for some 1199051 ge 1199050 Clearly 119892(119905) is more general than ℎ(119905)defined by (13)

Recently Chatzarakis [15 16] proved that if for some 119895 isinN

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901119895 (119906) 119889119906)119889119904 gt 1 (20)

or

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901119895 (119906) 119889119906)119889119904

gt 1 minus 119863 (120572) (21)

where

119901119895 (119905)= 119901 (119905) [1 + int119905

120591(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901119895minus1 (119906) 119889119906)119889119904] (22)

with 1199010(119905) = 119901(119905) then all solutions of (E) are oscillatoryLately Chatzarakis [17] studied a more general form of(E) namely

1199091015840 (119905) + 119898sum119894=1

119901119894 (119905) 119909 (120591119894 (119905)) = 0 119905 ge 1199050 (23)

and established sufficient oscillation conditions Those con-ditions can lead to (20) and (21) when119898 = 122 ADEs ByTheorem 243 [29] if

LA gt 1 (24)

then all solutions of (E1015840) are oscillatoryIn 1984 Fukagai and Kusano [21] proved that if

120573 gt 1119890 (25)

then all solutions of (E1015840) are oscillatory while ifint120590(119905)119905119902 (119904) 119889119904 le 1119890 for all sufficiently large 119905 (26)

then (E1015840) has a nonoscillatory solutionAssume that the argument 120590(119905) is not necessarily mono-

tone Set

120588 (119905) = inf119904ge119905120590 (119904) 119905 ge 1199050 (27)

Clearly the function 120588(119905) is nondecreasing and 120590(119905) ge 120588(119905) gt119905 for all 119905 ge 1199050In 2015 Chatzarakis and Ocalan [18] proved that if

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904 gt 1 (28)

or

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904 gt 1119890 (29)

then all solutions of (E1015840) are oscillatoryRecently Chatzarakis [15 16] proved that if for some 119895 isin

N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902119895 (119906) 119889119906)119889119904 gt 1 (30)

or

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902119895 (119906) 119889119906)119889119904

gt 1 minus 119863 (120573) (31)

where119902119895 (119905)= 119902 (119905) [1 + int120590(119905)

119905119902 (119904) exp(int120590(119904)

120588(119905)119902119895minus1 (119906) 119889119906)119889119904]

119895 ge 1(32)

with 1199020(119905) = 119902(119905) then all solutions of (E1015840) oscillateLately Chatzarakis [17] studied a more general form of(E1015840) namely

1199091015840 (119905) minus 119898sum119894=1

119902119894 (119905) 119909 (120590119894 (119905)) = 0 119905 ge 1199050 (33)

and established sufficient oscillation conditions Those con-ditions can lead to (30) and (31) when119898 = 13 Main Results

31 DDEs In our main results we state theorems establish-ing new sufficient oscillation conditions For the proofs ofthose theorems we use the following lemmas

Lemma 3 (see [19 Lemma 211]) Assume that ℎ(119905) is definedby (13) Then

120572 fl lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 = lim inf

119905rarrinfinint119905ℎ(119905)119901 (119904) 119889119904 (34)

Lemma 4 (see [19 Lemma 213]) Assume that ℎ(119905) is definedby (13) 120572 isin (0 1119890] and 119909(119905) is an eventually positive solutionof (E) Then

lim inf119905rarrinfin

119909 (119905)119909 (ℎ (119905)) ge 119863 (120572) (35)

Lemma 5 (see [26]) Assume that ℎ(119905) is defined by (13) 120572 isin(0 1119890] and 119909(119905) is an eventually positive solution of (E) Then

lim inf119905rarrinfin

119909 (ℎ (119905))119909 (119905) ge 1205820 (36)

where 1205820 is the smaller root of the equation 120582 = 119890120572120582

Complexity 5

Theorem 6 Let ℎ(119905) be defined by (13) and for some 119895 isin Nlim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1(37)

where

119875119895 (119905) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895minus1 (120585) 119889120585) 119889119906)119889119904]

(38)

with 1198750(119905) = 1205820119901(119905) and let 1205820 be the smaller root of theequation 120582 = 119890120572120582 Then all solutions of (E) oscillateProof Assume for the sake of contradiction that there existsa nonoscillatory solution 119909(119905) of (E) Since minus119909(119905) is also asolution of (E) we can confine our discussion only to thecase where the solution 119909(119905) is eventually positiveThen thereexists a real number 1199051 gt 1199050 such that 119909(119905) 119909(120591(119905)) gt 0 for all119905 ge 1199051 Thus from (E) we have

1199091015840 (119905) = minus119901 (119905) 119909 (120591 (119905)) le 0 forall119905 ge 1199051 (39)

which means that 119909(119905) is an eventually nonincreasing func-tion of positive numbers Taking into account the fact that120591(119905) le ℎ(119905) (E) implies that

1199091015840 (119905) + 119901 (119905) 119909 (ℎ (119905)) le 0 119905 ge 1199051 (40)

Observe that (36) implies that for each 120598 gt 0 there exists areal number 119905120598 such that

119909 (ℎ (119905))119909 (119905) gt 1205820 minus 120598 forall119905 ge 119905120598 ge 1199051 (41)

Combining inequalities (40) and (41) we obtain

1199091015840 (119905) + 119901 (119905) (1205820 minus 120598) 119909 (119905) le 0 119905 ge 119905120598 (42)

or

1199091015840 (119905) + 1198750 (119905 120598) 119909 (119905) le 0 119905 ge 119905120598 (43)

where

1198750 (119905 120598) = 119901 (119905) (1205820 minus 120598) (44)

Applying the Gronwall inequality in (43) we conclude that

119909 (119904) ge 119909 (119905) exp(int1199051199041198750 (120585 120598) 119889120585) 119905 ge 119904 ge 119905120598 (45)

Now we divide (E) by 119909(119905) gt 0 and integrate on [119904 119905] sominusint119905119904

1199091015840 (119906)119909 (119906) 119889119906 = int119905

119904119901 (119906) 119909 (120591 (119906))119909 (119906) 119889119906 (46)

or

ln 119909 (119904)119909 (119905) = int119905

119904119901 (119906) 119909 (120591 (119906))119909 (119906) 119889119906 119905 ge 119904 ge 119905120598 (47)

Since 120591(119906) lt 119906 equality (47) givesln 119909 (119904)119909 (119905) = int

119905

119904119901 (119906) 119909 (120591 (119906))119909 (119906) 119889119906

ge int119905119904119901 (119906) 119909 (119906)119909 (119906) exp(int

119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906

= int119905119904119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906

(48)

or

119909 (119904)ge 119909 (119905) exp(int119905

119904119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906) (49)

Substituting 120591(119904) for 119904 in (49) we get

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906) (50)

Integrating (E) from 120591(119905) to 119905 we have119909 (119905) minus 119909 (120591 (119905)) + int119905

120591(119905)119901 (119904) 119909 (120591 (119904)) 119889119904 = 0 (51)

Combining (50) and (51) we obtain

119909 (119905) minus 119909 (120591 (119905)) + 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904

le 0(52)

Multiplying inequality (52) by 119901(119905) we find119901 (119905) 119909 (119905) minus 119901 (119905) 119909 (120591 (119905)) + 119901 (119905) 119909 (119905) int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904

le 0(53)

which in view of (E) becomes

1199091015840 (119905) + 119901 (119905) 119909 (119905) + 119901 (119905) 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904

le 0(54)

6 Complexity

Hence for sufficiently large 1199051199091015840 (119905) + 119901 (119905) [1 + int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904]

sdot 119909 (119905) le 0(55)

or

1199091015840 (119905) + 1198751 (119905 120598) 119909 (119905) le 0 (56)

where

1198751 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904]

(57)

Clearly (56) resembles (43) if we replace 1198750 by 1198751 Thusintegrating (56) on [119904 119905] yields

119909 (119904) ge 119909 (119905) exp(int1199051199041198751 (120585 120598) 119889120585) (58)

Repeating steps (45) through (50) we can see that 119909 satisfiesthe inequality

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906) (59)

Combining now (51) and (59) we obtain

119909 (119905) minus 119909 (120591 (119905)) + 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904

le 0(60)

Multiplying inequality (60) by 119901(119905) as before we find1199091015840 (119905) + 119901 (119905) [1 + int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904]

sdot 119909 (119905) le 0(61)

Therefore for sufficiently large 119905 we have1199091015840 (119905) + 1198752 (119905 120598) 119909 (119905) le 0 (62)

where

1198752 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904]

(63)

It becomes apparent now that by repeating the above stepswe can build inequalities on 1199091015840(119905) with progressively higherindices 119875119895(119905 120598) 119895 isin N In general for sufficiently large 119905 thepositive solution 119909(119905) satisfies the inequality

1199091015840 (119905) + 119875119895 (119905 120598) 119909 (119905) le 0 119895 isin N (64)

where

119875119895 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895minus1 (120585 120598) 119889120585) 119889119906)119889119904]

(65)

Proceeding to final step we recall that ℎ(119905) defined by (13) isa nondecreasing function Since 120591(119904) le ℎ(119904) le ℎ(119905) we have119909 (120591 (119904)) ge 119909 (ℎ (119905))sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (66)

Hence

119909 (119905) minus 119909 (ℎ (119905)) + 119909 (ℎ (119905)) int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(67)

or

119909 (ℎ (119905)) [int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1] lt 0(68)

Thus

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1 lt 0(69)

Taking the limit as 119905 rarr infin we have

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1(70)

Complexity 7

Since 120598may be taken arbitrarily small this inequality contra-dicts (37)

This completes the proof of the theorem

Theorem 7 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1 minus 119863 (120572) (71)

where 119875119895 is defined by (38) then all solutions of (E) oscillateProof Assume 119909 is an eventually positive solution of (E)Clearly (67) is satisfied for sufficiently large 119905 Thus

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus 119909 (119905)119909 (ℎ (119905)) (72)

which implies that

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus lim inf119905rarrinfin

119909 (119905)119909 (ℎ (119905)) (73)

Using Lemmas 3 and 4 it is evident that inequality (35) issatisfied Thus (73) leads to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus 119863 (120572) (74)

Since 120598may be taken arbitrarily small this inequality contra-dicts (71)

This completes the proof of the theorem

Theorem 8 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119863 (120572) (75)

where 119875119895 is defined by (38) then all solutions of (E) oscillateProof Assume 119909 is an eventually positive solution of (E)Then as in the proof of Theorem 6 for sufficiently large 119905we conclude that

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (76)

Integrating (E) from ℎ(119905) to 119905 and using (76) we obtain

119909 (119905) minus 119909 (ℎ (119905)) + int119905ℎ(119905)119901 (119904) 119909 (119905)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(77)

or

minus 119909 (ℎ (119905)) + int119905ℎ(119905)119901 (119904) 119909 (119905)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 0(78)

Hence

119909 (ℎ (119905)) [ 119909 (119905)119909 (ℎ (119905)) int119905

ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1] lt 0(79)

which yields for all sufficiently large 119905int119905ℎ(119905)119901 (119904)sdot exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 119909 (ℎ (119905))119909 (119905)

(80)

8 Complexity

and consequently

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le lim sup119905rarrinfin

119909 (ℎ (119905))119909 (119905) (81)

Taking into account the fact that (35) is satisfied inequality(81) leads to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1119863 (120572) (82)

which contradicts (75) when 120598 rarr 0This completes the proof of the theorem

Theorem 9 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120572) (83)

where 119875119895 is defined by (38) and 1205820 is the smaller root of theequation 120582 = 119890120572120582 then all solutions of (E) oscillateProof Let 119909 be an eventually positive solution of (E) As inthe proof ofTheorem 8 we can show that (76) holds namely

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (84)

Since 120591(119904) le ℎ(119904) inequality (84) gives119909 (120591 (119904)) ge 119909 (ℎ (119904))sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (85)

By Lemma 5 for each 120598 gt 0 there exists a real number 119905120598 suchthat

119909 (ℎ (119905))119909 (119905) gt 1205820 minus 120598 forall119905 ge 119905120598 ge 1199051 (86)

Note that by the nondecreasing nature of the function119909(ℎ(119905))119909(119904) in 119904 it holds1 = 119909 (ℎ (119905))119909 (ℎ (119905)) le 119909 (ℎ (119905))119909 (119904) le 119909 (ℎ (119905))119909 (119905)

119905120598 le ℎ (119905) le 119904 le 119905(87)

In particular for 120598 isin (0 1205820 minus 1) by continuity we concludethat there exists a real number 119905lowast isin (ℎ(119905) 119905] satisfying

1 lt 1205820 minus 120598 = 119909 (ℎ (119905))119909 (119905lowast) (88)

Integrating (E) from 119905lowast to 119905 and using (85) we obtain

119909 (119905) minus 119909 (119905lowast) + 119909 (ℎ (119905)) int119905119905lowast119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(89)

or

int119905119905lowast119901 (119904) exp(intℎ(119904)

120591(119904)119901 (119906)

sdot exp(int119906120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 119909 (119905lowast)119909 (ℎ (119905)) minus 119909 (119905)119909 (ℎ (119905)) (90)

Using (88) andLemma4we deduce that for the 120598 consideredthere exists a real number 1199051015840120598 ge 119905120598 such that

int119905119905lowast119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 11205820 minus 120598 minus 119863 (120572) + 120598(91)

for 119905 ge 1199051015840120598Dividing (E) by119909(119905) integrating from ℎ(119905) to 119905lowast and using

(85) we deduce that

int119905lowastℎ(119905)119901 (119904)sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(92)

Complexity 9

Clearly by means of (36) 119909(ℎ(119904))119909(119904) gt 1205820minus120598 for 119904 ge ℎ(119905) ge1199051015840120598 Hence for all sufficiently large 119905 we conclude that(1205820 minus 120598)int119905

lowast

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(93)

or

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minus 11205820 minus 120598 int119905lowast

ℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904 = 11205820 minus 120598 ln119909 (ℎ (119905))119909 (119905lowast)

= ln (1205820 minus 120598)1205820 minus 120598

(94)

that is

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt ln (1205820 minus 120598)1205820 minus 120598 (95)

Using (91) along with (95) we get

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 + ln (1205820 minus 120598)1205820 minus 120598 minus 119863 (120572) + 120598(96)

which contradicts (83) when 120598 rarr 0This completes the proof of the theorem

Theorem 10 Let ℎ(119905) be defined by (13) If for some 119895 isin Nlim inf119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119890 (97)

where 119875119895 is defined by (38) then all solutions of (E) oscillate

Proof For the sake of contradiction let 119909 be a nonincreasingeventually positive solution and 1199051 gt 1199050 be such that 119909(119905) gt 0and 119909(120591(119905)) gt 0 for all 119905 ge 1199051 We note that wemay obtain (85)as in the proof of Theorem 9

Dividing (E) by 119909(119905) and integrating from ℎ(119905) to 119905 wehave

ln(119909 (ℎ (119905))119909 (119905) ) = int119905

ℎ(119905)119901 (119904) 119909 (120591 (119904))119909 (119904) 119889119904 forall119905 ge 1199052 ge 1199051 (98)

from which in view of 120591(119904) le ℎ(119904) and (85) we get

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(99)

Since 119909 is nonincreasing and ℎ(119904) lt 119904 inequality (99)becomes

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(100)

From (97) it is clear that there exists a constant 119888 gt 0 suchthat

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

ge 119888 gt 1119890 (101)

Choose 1198881015840 such that 119888 gt 1198881015840 gt 1119890 For every 120598 gt 0 such that119888 minus 120598 gt 1198881015840 we haveint119905ℎ(t)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 119888 minus 120598 gt 1198881015840 gt 1119890 (102)

Combining inequalities (100) and (102) we obtain

ln(119909 (ℎ (119905))119909 (119905) ) gt 1198881015840 (103)

or119909 (ℎ (119905))119909 (119905) gt 1198901198881015840 gt 1198901198881015840 gt 1 (104)

which yields

119909 (ℎ (119905)) gt (1198901198881015840) 119909 (119905) (105)

10 Complexity

Following the above steps we can inductively show that forany positive integer 119896

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 for sufficiently large 119905 (106)

Since 1198901198881015840 gt 1 there is a natural number 119896 isin N satisfying119896 gt 2[ln 2 minus ln 1198881015840](1 + ln 1198881015840) such that for 119905 sufficiently large

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 gt ( 21198881015840 )2 (107)

Further (cf [13 24]) for sufficiently large 119905 there exists a realnumber 119905119898 isin (ℎ(119905) 119905) such that

int119905119898ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402 int119905119905119898

119901 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402

(108)

Integrating (E) from ℎ(119905) to 119905119898 using (85) and the fact that119909(119905) gt 0 we obtain119909 (ℎ (119905)) gt 119909 (ℎ (119905119898)) int119905119898

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(109)

which in view of the first inequality in (108) implies that

119909 (ℎ (119905)) gt 11988810158402 119909 (ℎ (119905119898)) (110)

Similarly integrating (E) from 119905119898 to 119905 using (85) and the factthat 119909(119905) gt 0 we have119909 (119905119898) gt 119909 (ℎ (119905)) int119905

119905119898

119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(111)

which in view of the second inequality in (108) yields

119909 (119905119898) gt 11988810158402 119909 (ℎ (119905)) (112)

Combining inequalities (110) and (112) we deduce that

119909 (ℎ (119905119898)) lt 21198881015840 119909 (ℎ (119905)) lt ( 21198881015840 )2 119909 (119905119898) (113)

which contradicts (107)The proof of the theorem is complete

32 ADEs Analogous oscillation conditions to thoseobtained for the delay equation (E) can be derived for the(dual) advanced differential equation (E1015840) by followingsimilar arguments with the ones employed for obtainingTheorems 6minus10Theorem 11 Let 120588(119905) be defined by (27) and for some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1

(114)

where

119876119895 (119905) = 119902 (119905) [1 + int120590(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895minus1 (120585) 119889120585)119889119906)119889119904]

(115)

with 1198760(119905) = 1205820119902(119905) and let 1205820 be the smaller root of theequation 120582 = 119890120573120582 Then all solutions of (E1015840) oscillateTheorem 12 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 minus 119863 (120573)

(116)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillateTheorem 13 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119863 (120573) (117)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate

Complexity 11

Theorem 14 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120573) (118)

where 119876119895 is defined by (115) and 1205820 is the smaller root of theequation 120582 = 119890120573120582 then all solutions of (E1015840) oscillateTheorem 15 Let 120588(119905) be defined by (27) If for some 119895 isin N

lim inf119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119890 (119)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate33 Differential Inequalities A slight modification in theproofs of Theorems 6minus15 leads to the following results aboutdifferential inequalities

Theorem 16 Assume that all the conditions ofTheorem 6 [11] 7 [12] 8 [13] 9 [14] or 10 [15] hold Then(119894) the delay [advanced] differential inequality

1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) le 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) ge 0]

119905 ge 1199050(120)

has no eventually positive solutions(119894119894) the delay [advanced] differential inequality1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) ge 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) le 0]

119905 ge 1199050(121)

has no eventually negative solutions

Remark 17 The oscillation criteria established in this paperall depend on 1205820 (see eg (37) and (71)) in contrast to theconditions obtained in [15 16] and in [17 for m = 1] In factthe left-hand side of conditions (37) and (71) depends on 1205820which is not the casewith the left-hand side of conditions (20)and (21) Since 1205820 gt 1 when 120572 isin (0 1119890] it is obvious that

1198750 (119905) = 1205820119901 (119905) gt 119901 (119905) = 1199010 (119905) (122)

Consequently the left-hand side of conditions (37) and(71) is greater than the corresponding parts of (20) and(21) respectively This is the reason why the conditions inthis paper improve on all known conditions mentioned inSection 2

4 Examples and Comments

The oscillation tests we have proposed and established in themain results involve an iterative procedure We iterativelycompute limsup and liminf on the terms 119875119895(119905) and119876119895(119905) 119895 isinN of a recurrent relation defined on the coefficients and thedeviating argument of an equation of the form (E) or (E1015840)to determine whether that equation is oscillatory But thiscomputation cannot be performed on paper but by meansof a program numerically computing limsup and liminfThe examples below illustrate the significance of our resultsand indicate the high level of improvement in the oscillationcriteria The calculations were performed using MATLABcode

Example 1 Consider the delay differential equation

1199091015840 (119905) + 325119909 (120591 (119905)) = 0 119905 ge 0 (123)

with (see Figure 1(a))

120591 (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]minus4119905 + 40119896 + 9 if 119905 isin [8119896 + 2 8119896 + 3]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 3 8119896 + 4]minus4119905 + 40119896 + 18 if 119905 isin [8119896 + 4 8119896 + 5]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 5 8119896 + 6]minus2119905 + 24119896 + 15 if 119905 isin [8119896 + 6 8119896 + 7]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 7 8119896 + 8]

(124)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (13) we see (Figure 1(b)) that

ℎ (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]8119896 + 1 if 119905 isin [8119896 + 2 8119896 + 195 ]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 195 8119896 + 4]8119896 + 2 if 119905 isin [8119896 + 4 8119896 + 295 ]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 295 8119896 + 6]8119896 + 3 if 119905 isin [8119896 + 6 8119896 + 446 ]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 446 8119896 + 8]

(125)

It is obvious that

120572 = lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 = lim inf

119905rarrinfinint8119896+28119896+1

325119889119904= 012

(126)

12 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

minus1minus2

651 2 3 4 8 9 10 11 12 13 15 16

1415

minus3

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

minus1minus2

t

195 295

1415

7 14651 2 3 4 8 9 10 11 12 13 15 16

446

y = tℎ(t)

(b)

Figure 1 The graphs of 120591(119905) and ℎ(119905)

and therefore the smaller root of 119890012120582 = 120582 is 1205820 = 114765Observe that the function 119865119895 R0 rarr R+ defined as

119865119895 (119905) = int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

(127)

attains its maximum at 119905 = 8119896+446 119896 isin N0 for every 119895 isin NSpecifically

1198651 (119905 = 8119896 + 446 ) = int8119896+446

8119896+3119901 (119904)

sdot exp(int8119896+3120591(119904)

119901 (119906) exp(int119906120591(119906)1198751 (120585) 119889120585) 119889119906)119889119904

(128)

with

1198751 (120585) = 119901 (120585) [1 + int120585120591(120585)119901 (V)

sdot exp(int120585120591(V)119901 (119908) exp(int119908

120591(119908)1205820119901 (119911) 119889119911) 119889119908)119889V]

(129)

Using MATLAB we obtain

1198651 (119905 = 8119896 + 446 ) ≃ 10417 (130)

and therefore

lim sup119905rarrinfin

1198651 (119905) ≃ 10417 gt 1 (131)

Hence condition (37) of Theorem 6 is satisfied for 119895 = 1Consequently all solutions of (123) are oscillatory

Observe however that

LD = lim sup119896rarrinfin

int8119896+4468119896+3

335119889119904 = 052 lt 1120572 = 012 lt 1119890

052 lt 1 + ln 12058201205820 minus 119863 (120572) ≃ 09831(132)

Note that the functionΦ119895 defined by

Φ119895 (119905) = int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

ℎ(119904)119901 (119906) 120595119895 (119906) 119889119906)119889119904

119895 ge 2(133)

attains its maximum at 119905 = 8119896 + 446 119896 isin N0 for every 119895 ge 2Specifically

Φ2 (8119896 + 446 )= int8119896+4468119896+3

119901 (119904) exp(int8119896+3ℎ(119904)

119901 (119904) 1205952 (119906) 119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 exp(int119906

120591(119906)

325 sdot 0 119889119908)119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 sdot 1 119889119906)119889119904= 325 sdot [int

8119896+195

8119896+3exp( 325 int

8119896+3

8119896+1119889119906)119889119904

Complexity 13

+ int8119896+48119896+195

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+2958119896+4

exp( 325 int8119896+3

8119896+2119889119906)119889119904

+ int8119896+68119896+295

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+4468119896+6

exp( 325 int8119896+3

8119896+3119889119906)119889119904] ≃ 057983

(134)

Thus

lim sup119905rarrinfin

Φ2 (119905) ≃ 057983 lt 1 minus 119863 (120572) ≃ 099174 (135)

Also

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int8119896+4468119896+3

325 exp(int8119896+3

120591(119904)

325119889119906)119889119904= 325 sdot lim sup

119905rarrinfin[int8119896+48119896+3

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+58119896+4

exp( 325 int8119896+3

minus4119904+40119896+18119889119906)119889119904

+ int8119896+68119896+5

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+78119896+6

exp( 325 int8119896+3

minus2119904+24119896+15119889119906)119889119904

+ int8119896+4468119896+7

exp( 325 int8119896+3

6119904minus40119896minus41119889119906)119889119904] ≃ 07043

lt 107043 lt 1 minus 119863 (120572) ≃ 099174

(136)

In addition

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)1199011 (119906) 119889119906)119889119904 ≃ 08052

lt 108052 lt 1 minus 119863 (120572) ≃ 099174

(137)

That is none of conditions (8) (9) (12) (14) (for 119895 = 2) (16)(17) (20) (for 119895 = 1) and (21) (for 119895 = 1) is satisfiedComment The improvement of condition (37) over thecorresponding condition (8) is significant approximately10033 We get this measure by comparing the values inthe left-hand side of those conditions Also the improvementover conditions (14) (16) and (20) is very satisfactory

around 7966 479 and 2937 respectively In additioncondition (37) is satisfied from the first iteration whileconditions (14) (20) and (21) need more than one iteration

Example 2 (taken and adapted from [17]) Consider theadvanced differential equation

1199091015840 (119905) minus 3332500119909 (120590 (119905)) = 0 119905 ge 0 (138)

with (see Figure 2(a))

120590 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 2]minus3119905 + 20119896 + 13 if 119905 isin [5119896 + 2 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 4]minus119905 + 10119896 + 13 if 119905 isin [5119896 + 4 5119896 + 5]

(139)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (27) we see (Figure 2(b)) that

120588 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 125]5119896 + 4 if 119905 isin [5119896 + 125 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 38]5119896 + 8 if 119905 isin [5119896 + 38 5119896 + 5]

(140)

It is obvious that

120573 = lim inf119905rarrinfin

int5119896+45119896+3

3332500119889119904 = 01332 (141)

and therefore the smaller root of 11989001332120582 = 120582 is 1205820 = 116839Observe that the function 119866119895 R0 rarr R+ defined as

119866119895 (119905) = int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

(142)

attains its maximum at 119905 = 5119896 + 38 119896 isin N0 for every 119895 isin NSpecifically

1198661 (119905 = 5119896 + 38) = int5119896+85119896+38

119902 (119904)sdot exp(int120590(119904)

5119896+8119902 (119906) exp(int120590(119906)

1199061198761 (120585) 119889120585)119889119906)119889119904

(143)

with

1198761 (120585) = 119902 (120585) [1 + int120590(120585)120585119902 (V)

sdot exp(int120590(V)120585119902 (119908) exp(int120590(119908)

1199081205820119902 (119911) 119889119911)119889119908)119889V]

(144)

14 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

651 2 3 4 8 9 10 11 12 13

1415

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

7

t

14651

2 3 4 8 9 10 11 12 13

38

1415

125

y = t

(t)

(b)

Figure 2 The graphs of 120590(119905) and 120588(119905)

Using MATLAB we obtain

1198661 (119905 = 5119896 + 38) ≃ 09915 (145)

Thereforelim sup119905rarrinfin

1198661 (119905) ≃ 09915 gt 1 minus 119863 (120573) ≃ 09896 (146)

Hence condition (116) of Theorem 12 is satisfied for 119895 = 1Consequently all solutions of (138) oscillate

Observe however that

LA = lim sup119896rarrinfin

int5119896+85119896+38

3332500119889119904 = 055944 lt 1120573 = 01332 lt 1119890 lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int5119896+85119896+38

119902 (119904) exp(int120590(119904)5119896+8

119902 (119906) 119889119906)119889119904= lim sup119896rarrinfin

[int5119896+45119896+38

119902 (119904)sdot exp(int5119904minus20119896minus11

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+5

5119896+4119902 (119904)

sdot exp(intminus119904+10119896+135119896+8

119902 (119906) 119889119906)119889119904 + int5119896+65119896+5

119902 (119904)sdot exp(int5119896+8

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+7

5119896+6119902 (119904)

sdot exp(int4119904minus15119896minus165119896+8

119902 (119906) 119889119906)119889119904 + int5119896+85119896+7

119902 (119904)sdot exp(intminus3119904+20119896+33

5119896+8119902 (119906) 119889119906)119889119904] ≃ 06672 lt 1

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim inf119905rarrinfin

int5119896+45119896+3

119902 (119904) exp(int120590(119904)5119896+4

119902 (119906) 119889119906)119889119904≃ 01893 lt 1119890

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)1199021 (119906) 119889119906)119889119904 ≃ 07196

lt 107196 lt 1 minus 119863 (120573) ≃ 09896

(147)

That is none of conditions (24) (25) (28) (29) (30) (for 119895 =1) and (31) (for 119895 = 1) is satisfiedComment The improvement of condition (116) over thecorresponding condition (24) is significant approximately7723 We get this measure by comparing the values in theleft-hand side of those conditions Also the improvementover conditions (28) and (30) is very satisfactory around4861 and 3778 respectively In addition condition (116)is satisfied from the first iteration while conditions (30) and(31) need more than one iteration

Complexity 15

Remark 3 Similarly one can provide examples illustratingthe other main results

5 Concluding Remarks

In the present paper we have considered the oscillatorydynamics of differential equations having nonmonotonedeviating arguments and nonnegative coefficients New suf-ficient conditions have been established for the oscillation ofall solutions of (E) and (E1015840) These conditions include (37)(71) (75) (83) and (97) and (114) (116) (117) (118) and(119) for (E) and (E1015840) respectively Applying these conditionsinvolves a procedure that checks for oscillations by iterativelycomputing limsup and liminf on terms recursively definedon the equationrsquos coefficients and deviating argument

The main advantage of these conditions is that theyachieve a major improvement over all the related oscillationconditions for (E) [(E1015840)] in the literature For example con-dition (37) [(114)] improves upon the noniterative conditionsthat are reviewed in the introduction namely conditions (8)[(24)] (12) (16)equiv(20) (for 119895 = 1) [(28)equiv(30) (for 119895 = 1)]and (17)equiv(21) (for 119895 = 1) [(31) (for 119895 = 1)] That immediatelybecomes evident by inspecting the left-hand side of (37)[(114)] and the left-hand side of each of the above conditions

The improvement of (37) [(114)] over the other iterativeconditions namely (14) (for j gt 2) (20) (for 119895 gt 1) [(30)(for 119895 gt 1)] and (21) (for 119895 gt 1) [(31) (for 119895 gt 1)] is that itrequires far fewer iterations to establish oscillation than theother conditions

This advantage easily can be verified computationallyby running the MATLAB programs (see Appendix) forcomputing limsup and liminf and comparing the number ofiterations required by each condition to establish oscillationThen we see that we achieve a significant improvement overall known oscillation criteria

Another advantage and a significant departure from thelarge majority of the other studies is that the criteria in thispaper apply to a more general class of equations havingnonmonotone arguments 120591(119905) or 120590(119905) in contrast to mostof the other oscillation criteria that apply to equations withnondecreasing arguments

Appendix

In this appendix for completeness we give the algorithmon MATLAB software used in Example 1 for calculation oflim sup119905rarrinfin1198651(119905) ≃ 10417 For Example 2 the algorithm isomitted since it is similar to the one in Example 1

Algorithm for Example 1

clear clcc = 12119899 = 501205820 = 114765a5 = 19b5 = 703h5 = (b5 minus a5)119899

for i5 = 1 1 119899 + 1x5 = a5 + (i5 minus 1) lowast h5a4 = TFunction(x5)b4 = 19h4 = (b4 minus a4)119899for i4 = 1 1 119899 + 1x4 = a4 + (i4 minus 1) lowast h4a3 = TFunction(x4)b3 = x4h3 = (b3 minus a3)119899for i3 = 1 1 119899 + 1x3 = a3 + (i3 minus 1) lowast h3a2 = TFunction(x3)b2 = x3h2 = (b2 minus a2)119899for i2 = 1 1 119899 + 1x2 = a2 + (i2 minus 1) lowast h2a1 = TFunction(x2)b1 = x3h1 = (b1 minus a1)119899for i1 = 1 1 119899 + 1x1 = a1 + (i1 minus 1) lowast h1f1(i1) = c lowast exp(1205820 lowast c lowast (x1 minus TFunction(x1)))endI1 = f1(1) + f1(119899 + 1)for i1 = 2 2 119899I1 = I1 + f1(i1) lowast 4endfor i1 = 3 2 119899 minus 1I1 = I1 + f1(i1) lowast 2endI1 = I1 lowast h13f2(i2) = c lowast exp(I1)endI2 = f2(1) + f2(119899 + 1)for i2 = 2 2 119899I2 = I2 + f2(i2) lowast 4endfor i2 = 3 2 119899 minus 1I2 = I2 + f2(i2) lowast 2endI2 = I2 lowast h23f3(i3) = c lowast (1 + I2)end

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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Page 2: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

2 Complexity

Many physiological processes including the concentra-tion of red blood cells the concentration of CO2 in the bloodcausing the observed periodic oscillations in the breathingfrequency and the production of new blood cells in the bonemarrow exhibit oscillations and several DDE models havebeen proposed to model these processes

Below we present two applications indicating the rel-evance of the DDEs we study in this paper to real worldproblems The two examples are taken from the areas ofphysiology and population dynamics

Application 1 (blood cells production [9]) The production ofred and white blood cells in the bonemarrow is regulated bythe level of oxygen in the bloodA reduction in the number ofcells in the blood as a result of the loss of cells causes the levelof oxygen in the blood to decrease When the level of oxygenin the blood decreases a substance is released that in turnleads to the release of blood elements from the bonemarrowThus the concentration 119888(119905)of cells in the blood stream at anytime 119905 changes according to the loss of cells and the releaseof new cells from the bone marrow But the bone marrowresponds to a reduction in the number of blood cells and thedecrease in the level of oxygen with a delay that is in theorder of 6 days That means the release of new cells into theblood stream at time 119905 depends on the cell concentration atan earlier time namely 119905 minus 120591 where 120591 is the delay with whichthe bonemarrow responds to a reduced level of oxygen in thebloodThe simplest model of the concentration of the cells inthe blood stream can be described by the DDE

1198881015840 (119905) = 120582119888 (119905 minus 120591) minus 120574119888 (119905) (2)

where 120582 represents the flux of cells into the blood stream 120574is the death rate and 120591 is the delay All of them are positiveconstantsThe solutions of the above equation exhibit similaroscillations to the actual oscillatory pattern observed in theconcentration of cells in the blood stream

Application 2 Imagine a biological population composed ofadult and juvenile individuals Let 119873(119905) denote the densityof adults at time 119905 Assume that the length of the juvenileperiod is exactly ℎ units of time for each individual Assumethat adults produce offspring at a per capita rate 120572 and thattheir probability per unit of time of dying is 120583 Assume that anewborn survives the juvenile period with probability 120588 andput 119905 = 120572120588 Then the dynamics of119873 can be described by thedifferential equation

1198731015840 (119905) = minus120583119873 (119905) + 119903119873 (119905 minus ℎ) (3)

which involves a nonlocal term 119903119873(119905 minus ℎ) meaning thatnewborns become adults with some delay So the timevariation of the population density119873 involves the current aswell as the past values of119873

The use of DDEs from the initial application in popula-tion dynamics has spread to every area of the life sciencesimmunology physiology epidemiology and cell growthTheoriginal delay logistic equation has led to several new DDEforms likeVolterrarsquos integrodifferential equations and neutral

DDEs [9] and several newmodels from the delayedHopfieldmodel in neural networks to the SIRmodel in epidemiology[7]More recently the idea of state dependent delays has beenintroduced involving ldquoa delay that itself is governed by adifferential equation that represents adaptation to the systemrsquosstaterdquo [9]

From the above review of DDEs in the biological sci-ences it is apparent that if DDEs are so extensively used inthis area this is because the dynamics of those equationsnamely the stability and oscillatory properties of the solu-tions of those equations replicate the stability and oscillatorypatterns we actually observe in processes in those areasThus the study of the stability and oscillatory behavior of thesolutions of DDEs has become the principal subject of theresearch on those equations For more advanced treatises onoscillation theory the reader is referred to [13ndash33]

In the paper we consider a differential equation withdelay argument of the form

1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) = 0 119905 ge 1199050 (E)where 119901 is a function of nonnegative real numbers and 120591 is afunction of positive real numbers such that

120591 (119905) lt 119905 119905 ge 1199050lim119905rarrinfin

120591 (119905) = infin (4)

By a solution of (E) we understand a continuously differ-entiable function defined on [120591(1198790)infin) for some 1198790 ge 1199050and such that (E) is satisfied for 119905 ge 1198790 Such a solution iscalled oscillatory if it has arbitrarily large zeros and otherwiseit is called nonoscillatory An equation is oscillatory if all itssolutions oscillate

A parallel problem to that of establishing oscillationcriteria for the solutions of equation (E) is the one concerningthe solutions of the advanced differential equation (ADE)

1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) = 0 119905 ge 1199050 (E1015840)where 119902 is a function of nonnegative real numbers and 120590 is afunction of positive real numbers such that

120590 (119905) gt 119905 119905 ge 1199050 (5)

The objective of this paper is to consider the oscillatorydynamics of both delay and advanced differential equationsfrom the perspective of the qualitative analysis of thoseequations In that framework (i) we formulate new iterativeoscillation conditions for testing whether all solutions of aDDE of the form of (E) or an ADE of the form of (E1015840) areoscillatory (ii) we show that these tests significantly improveon all the previous iterative and noniterative oscillationcriteria which briefly are reviewed in the Historical andChronological Review in Section 2 requiring fewer iterationsto determine whether an equation of the considered form isoscillatory and (iii) these criteria apply to amore general classof equations having nonmonotone arguments 120591(119905) or 120590(119905) incontrast to the large majority of the other studies where thecriteria apply to equations with nondecreasing arguments

Complexity 3

From this point onward we will use the notation

120572 fl lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904

120573 fl lim inf119905rarrinfin

int120590(119905)119905119902 (119904) 119889119904

119863 (120596) fl 0 if 120596 gt 1119890 1 minus 120596 minus radic1 minus 2120596 minus 12059622 if 120596 isin [0 1119890 ]

LD fl lim sup119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904where 120591 (119905) is nondecreasing

LA fl lim sup119905rarrinfin

int120590(119905)119905119902 (119904) 119889119904where 120590 (119905) is nondecreasing

(6)

2 Historical and Chronological Review

21 DDEs The first systematic study for the oscillation of allsolutions of equation (E) was made by Myskis in 1950 [31]when he proved that every solution of (E) oscillates if

lim sup119905rarrinfin

[119905 minus 120591 (119905)] lt infinlim inf119905rarrinfin

[119905 minus 120591 (119905)] lim inf119905rarrinfin

119901 (119905) gt 1119890 (7)

In 1972 Ladas et al [27] proved that if

LD gt 1 (8)

then all solutions of (E) are oscillatoryIn 1982 Koplatadze and Chanturiya [24] improved (7) to

120572 gt 1119890 (9)

Regarding the constant 1119890 in (9) it should be remarked thatif the inequality

int119905120591(119905)119901 (119904) 119889119904 le 1119890 (10)

holds eventually then according to [24] (E) has a nonoscil-latory solution

It is apparent that there is a gap between conditions (8)and (9) when

lim119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 (11)

does not exist How to fill this gap is an interesting problemwhich has been investigated by several authors For example

in 2000 Jaros and Stavroulakis [23] proved that if 1205820 is thesmaller root of the equation 120582 = 119890120572120582 and

LD gt 1 + ln 12058201205820 minus 119863 (120572) (12)

then all solutions of (E) oscillateNowwe come to the general case where the argument 120591(119905)

is nonmonotone Set

ℎ (119905) fl sup119904le119905120591 (119904) 119905 ge 1199050 (13)

Clearly the function ℎ(119905) is nondecreasing and 120591(119905) le ℎ(119905) lt119905 for all 119905 ge 1199050In 1994 Koplatadze and Kvinikadze [25] proved that if

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

ℎ(119904)119901 (119906) 120595119895 (119906) 119889119906)119889119904

gt 1 minus 119863 (120572) (14)

where

1205951 (119905) = 0120595119895 (119905) = exp(int119905

120591(119905)119901 (119906) 120595119895minus1 (119906) 119889119906) 119895 ge 2 (15)

then all solutions of (E) oscillateIn 2011 Braverman and Karpuz [14] proved that if

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904 gt 1 (16)

then all solutions of (E) oscillate while in 2014 Stavroulakis[32] improved (16) to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

gt 1 minus 119863 (120572) (17)

In 2016 El-Morshedy and Attia [30] proved that if

lim sup119905rarrinfin

[[int119905

119892(119905)119901119899 (119904) 119889119904

+ 119863 (120572) exp(int119905119892(119905)

119899minus1sum119895=0

119901119895 (119904) 119889119904)]] gt 1(18)

where 1199010(119905) = 119901(119905) and119901119899 (119905)= 119901119899minus1 (119905) int119905

119892(119905)119901119899minus1 (119904) exp(int119905

119892(119904)119901119899minus1 (119906) 119889119906)119889119904

119899 ge 1(19)

4 Complexity

then all solutions of (E) are oscillatory Here 119892(119905) is a nonde-creasing continuous function such that 120591(119905) le 119892(119905) le 119905 119905 ge1199051 for some 1199051 ge 1199050 Clearly 119892(119905) is more general than ℎ(119905)defined by (13)

Recently Chatzarakis [15 16] proved that if for some 119895 isinN

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901119895 (119906) 119889119906)119889119904 gt 1 (20)

or

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901119895 (119906) 119889119906)119889119904

gt 1 minus 119863 (120572) (21)

where

119901119895 (119905)= 119901 (119905) [1 + int119905

120591(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901119895minus1 (119906) 119889119906)119889119904] (22)

with 1199010(119905) = 119901(119905) then all solutions of (E) are oscillatoryLately Chatzarakis [17] studied a more general form of(E) namely

1199091015840 (119905) + 119898sum119894=1

119901119894 (119905) 119909 (120591119894 (119905)) = 0 119905 ge 1199050 (23)

and established sufficient oscillation conditions Those con-ditions can lead to (20) and (21) when119898 = 122 ADEs ByTheorem 243 [29] if

LA gt 1 (24)

then all solutions of (E1015840) are oscillatoryIn 1984 Fukagai and Kusano [21] proved that if

120573 gt 1119890 (25)

then all solutions of (E1015840) are oscillatory while ifint120590(119905)119905119902 (119904) 119889119904 le 1119890 for all sufficiently large 119905 (26)

then (E1015840) has a nonoscillatory solutionAssume that the argument 120590(119905) is not necessarily mono-

tone Set

120588 (119905) = inf119904ge119905120590 (119904) 119905 ge 1199050 (27)

Clearly the function 120588(119905) is nondecreasing and 120590(119905) ge 120588(119905) gt119905 for all 119905 ge 1199050In 2015 Chatzarakis and Ocalan [18] proved that if

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904 gt 1 (28)

or

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904 gt 1119890 (29)

then all solutions of (E1015840) are oscillatoryRecently Chatzarakis [15 16] proved that if for some 119895 isin

N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902119895 (119906) 119889119906)119889119904 gt 1 (30)

or

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902119895 (119906) 119889119906)119889119904

gt 1 minus 119863 (120573) (31)

where119902119895 (119905)= 119902 (119905) [1 + int120590(119905)

119905119902 (119904) exp(int120590(119904)

120588(119905)119902119895minus1 (119906) 119889119906)119889119904]

119895 ge 1(32)

with 1199020(119905) = 119902(119905) then all solutions of (E1015840) oscillateLately Chatzarakis [17] studied a more general form of(E1015840) namely

1199091015840 (119905) minus 119898sum119894=1

119902119894 (119905) 119909 (120590119894 (119905)) = 0 119905 ge 1199050 (33)

and established sufficient oscillation conditions Those con-ditions can lead to (30) and (31) when119898 = 13 Main Results

31 DDEs In our main results we state theorems establish-ing new sufficient oscillation conditions For the proofs ofthose theorems we use the following lemmas

Lemma 3 (see [19 Lemma 211]) Assume that ℎ(119905) is definedby (13) Then

120572 fl lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 = lim inf

119905rarrinfinint119905ℎ(119905)119901 (119904) 119889119904 (34)

Lemma 4 (see [19 Lemma 213]) Assume that ℎ(119905) is definedby (13) 120572 isin (0 1119890] and 119909(119905) is an eventually positive solutionof (E) Then

lim inf119905rarrinfin

119909 (119905)119909 (ℎ (119905)) ge 119863 (120572) (35)

Lemma 5 (see [26]) Assume that ℎ(119905) is defined by (13) 120572 isin(0 1119890] and 119909(119905) is an eventually positive solution of (E) Then

lim inf119905rarrinfin

119909 (ℎ (119905))119909 (119905) ge 1205820 (36)

where 1205820 is the smaller root of the equation 120582 = 119890120572120582

Complexity 5

Theorem 6 Let ℎ(119905) be defined by (13) and for some 119895 isin Nlim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1(37)

where

119875119895 (119905) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895minus1 (120585) 119889120585) 119889119906)119889119904]

(38)

with 1198750(119905) = 1205820119901(119905) and let 1205820 be the smaller root of theequation 120582 = 119890120572120582 Then all solutions of (E) oscillateProof Assume for the sake of contradiction that there existsa nonoscillatory solution 119909(119905) of (E) Since minus119909(119905) is also asolution of (E) we can confine our discussion only to thecase where the solution 119909(119905) is eventually positiveThen thereexists a real number 1199051 gt 1199050 such that 119909(119905) 119909(120591(119905)) gt 0 for all119905 ge 1199051 Thus from (E) we have

1199091015840 (119905) = minus119901 (119905) 119909 (120591 (119905)) le 0 forall119905 ge 1199051 (39)

which means that 119909(119905) is an eventually nonincreasing func-tion of positive numbers Taking into account the fact that120591(119905) le ℎ(119905) (E) implies that

1199091015840 (119905) + 119901 (119905) 119909 (ℎ (119905)) le 0 119905 ge 1199051 (40)

Observe that (36) implies that for each 120598 gt 0 there exists areal number 119905120598 such that

119909 (ℎ (119905))119909 (119905) gt 1205820 minus 120598 forall119905 ge 119905120598 ge 1199051 (41)

Combining inequalities (40) and (41) we obtain

1199091015840 (119905) + 119901 (119905) (1205820 minus 120598) 119909 (119905) le 0 119905 ge 119905120598 (42)

or

1199091015840 (119905) + 1198750 (119905 120598) 119909 (119905) le 0 119905 ge 119905120598 (43)

where

1198750 (119905 120598) = 119901 (119905) (1205820 minus 120598) (44)

Applying the Gronwall inequality in (43) we conclude that

119909 (119904) ge 119909 (119905) exp(int1199051199041198750 (120585 120598) 119889120585) 119905 ge 119904 ge 119905120598 (45)

Now we divide (E) by 119909(119905) gt 0 and integrate on [119904 119905] sominusint119905119904

1199091015840 (119906)119909 (119906) 119889119906 = int119905

119904119901 (119906) 119909 (120591 (119906))119909 (119906) 119889119906 (46)

or

ln 119909 (119904)119909 (119905) = int119905

119904119901 (119906) 119909 (120591 (119906))119909 (119906) 119889119906 119905 ge 119904 ge 119905120598 (47)

Since 120591(119906) lt 119906 equality (47) givesln 119909 (119904)119909 (119905) = int

119905

119904119901 (119906) 119909 (120591 (119906))119909 (119906) 119889119906

ge int119905119904119901 (119906) 119909 (119906)119909 (119906) exp(int

119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906

= int119905119904119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906

(48)

or

119909 (119904)ge 119909 (119905) exp(int119905

119904119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906) (49)

Substituting 120591(119904) for 119904 in (49) we get

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906) (50)

Integrating (E) from 120591(119905) to 119905 we have119909 (119905) minus 119909 (120591 (119905)) + int119905

120591(119905)119901 (119904) 119909 (120591 (119904)) 119889119904 = 0 (51)

Combining (50) and (51) we obtain

119909 (119905) minus 119909 (120591 (119905)) + 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904

le 0(52)

Multiplying inequality (52) by 119901(119905) we find119901 (119905) 119909 (119905) minus 119901 (119905) 119909 (120591 (119905)) + 119901 (119905) 119909 (119905) int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904

le 0(53)

which in view of (E) becomes

1199091015840 (119905) + 119901 (119905) 119909 (119905) + 119901 (119905) 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904

le 0(54)

6 Complexity

Hence for sufficiently large 1199051199091015840 (119905) + 119901 (119905) [1 + int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904]

sdot 119909 (119905) le 0(55)

or

1199091015840 (119905) + 1198751 (119905 120598) 119909 (119905) le 0 (56)

where

1198751 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904]

(57)

Clearly (56) resembles (43) if we replace 1198750 by 1198751 Thusintegrating (56) on [119904 119905] yields

119909 (119904) ge 119909 (119905) exp(int1199051199041198751 (120585 120598) 119889120585) (58)

Repeating steps (45) through (50) we can see that 119909 satisfiesthe inequality

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906) (59)

Combining now (51) and (59) we obtain

119909 (119905) minus 119909 (120591 (119905)) + 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904

le 0(60)

Multiplying inequality (60) by 119901(119905) as before we find1199091015840 (119905) + 119901 (119905) [1 + int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904]

sdot 119909 (119905) le 0(61)

Therefore for sufficiently large 119905 we have1199091015840 (119905) + 1198752 (119905 120598) 119909 (119905) le 0 (62)

where

1198752 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904]

(63)

It becomes apparent now that by repeating the above stepswe can build inequalities on 1199091015840(119905) with progressively higherindices 119875119895(119905 120598) 119895 isin N In general for sufficiently large 119905 thepositive solution 119909(119905) satisfies the inequality

1199091015840 (119905) + 119875119895 (119905 120598) 119909 (119905) le 0 119895 isin N (64)

where

119875119895 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895minus1 (120585 120598) 119889120585) 119889119906)119889119904]

(65)

Proceeding to final step we recall that ℎ(119905) defined by (13) isa nondecreasing function Since 120591(119904) le ℎ(119904) le ℎ(119905) we have119909 (120591 (119904)) ge 119909 (ℎ (119905))sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (66)

Hence

119909 (119905) minus 119909 (ℎ (119905)) + 119909 (ℎ (119905)) int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(67)

or

119909 (ℎ (119905)) [int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1] lt 0(68)

Thus

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1 lt 0(69)

Taking the limit as 119905 rarr infin we have

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1(70)

Complexity 7

Since 120598may be taken arbitrarily small this inequality contra-dicts (37)

This completes the proof of the theorem

Theorem 7 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1 minus 119863 (120572) (71)

where 119875119895 is defined by (38) then all solutions of (E) oscillateProof Assume 119909 is an eventually positive solution of (E)Clearly (67) is satisfied for sufficiently large 119905 Thus

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus 119909 (119905)119909 (ℎ (119905)) (72)

which implies that

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus lim inf119905rarrinfin

119909 (119905)119909 (ℎ (119905)) (73)

Using Lemmas 3 and 4 it is evident that inequality (35) issatisfied Thus (73) leads to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus 119863 (120572) (74)

Since 120598may be taken arbitrarily small this inequality contra-dicts (71)

This completes the proof of the theorem

Theorem 8 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119863 (120572) (75)

where 119875119895 is defined by (38) then all solutions of (E) oscillateProof Assume 119909 is an eventually positive solution of (E)Then as in the proof of Theorem 6 for sufficiently large 119905we conclude that

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (76)

Integrating (E) from ℎ(119905) to 119905 and using (76) we obtain

119909 (119905) minus 119909 (ℎ (119905)) + int119905ℎ(119905)119901 (119904) 119909 (119905)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(77)

or

minus 119909 (ℎ (119905)) + int119905ℎ(119905)119901 (119904) 119909 (119905)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 0(78)

Hence

119909 (ℎ (119905)) [ 119909 (119905)119909 (ℎ (119905)) int119905

ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1] lt 0(79)

which yields for all sufficiently large 119905int119905ℎ(119905)119901 (119904)sdot exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 119909 (ℎ (119905))119909 (119905)

(80)

8 Complexity

and consequently

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le lim sup119905rarrinfin

119909 (ℎ (119905))119909 (119905) (81)

Taking into account the fact that (35) is satisfied inequality(81) leads to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1119863 (120572) (82)

which contradicts (75) when 120598 rarr 0This completes the proof of the theorem

Theorem 9 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120572) (83)

where 119875119895 is defined by (38) and 1205820 is the smaller root of theequation 120582 = 119890120572120582 then all solutions of (E) oscillateProof Let 119909 be an eventually positive solution of (E) As inthe proof ofTheorem 8 we can show that (76) holds namely

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (84)

Since 120591(119904) le ℎ(119904) inequality (84) gives119909 (120591 (119904)) ge 119909 (ℎ (119904))sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (85)

By Lemma 5 for each 120598 gt 0 there exists a real number 119905120598 suchthat

119909 (ℎ (119905))119909 (119905) gt 1205820 minus 120598 forall119905 ge 119905120598 ge 1199051 (86)

Note that by the nondecreasing nature of the function119909(ℎ(119905))119909(119904) in 119904 it holds1 = 119909 (ℎ (119905))119909 (ℎ (119905)) le 119909 (ℎ (119905))119909 (119904) le 119909 (ℎ (119905))119909 (119905)

119905120598 le ℎ (119905) le 119904 le 119905(87)

In particular for 120598 isin (0 1205820 minus 1) by continuity we concludethat there exists a real number 119905lowast isin (ℎ(119905) 119905] satisfying

1 lt 1205820 minus 120598 = 119909 (ℎ (119905))119909 (119905lowast) (88)

Integrating (E) from 119905lowast to 119905 and using (85) we obtain

119909 (119905) minus 119909 (119905lowast) + 119909 (ℎ (119905)) int119905119905lowast119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(89)

or

int119905119905lowast119901 (119904) exp(intℎ(119904)

120591(119904)119901 (119906)

sdot exp(int119906120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 119909 (119905lowast)119909 (ℎ (119905)) minus 119909 (119905)119909 (ℎ (119905)) (90)

Using (88) andLemma4we deduce that for the 120598 consideredthere exists a real number 1199051015840120598 ge 119905120598 such that

int119905119905lowast119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 11205820 minus 120598 minus 119863 (120572) + 120598(91)

for 119905 ge 1199051015840120598Dividing (E) by119909(119905) integrating from ℎ(119905) to 119905lowast and using

(85) we deduce that

int119905lowastℎ(119905)119901 (119904)sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(92)

Complexity 9

Clearly by means of (36) 119909(ℎ(119904))119909(119904) gt 1205820minus120598 for 119904 ge ℎ(119905) ge1199051015840120598 Hence for all sufficiently large 119905 we conclude that(1205820 minus 120598)int119905

lowast

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(93)

or

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minus 11205820 minus 120598 int119905lowast

ℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904 = 11205820 minus 120598 ln119909 (ℎ (119905))119909 (119905lowast)

= ln (1205820 minus 120598)1205820 minus 120598

(94)

that is

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt ln (1205820 minus 120598)1205820 minus 120598 (95)

Using (91) along with (95) we get

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 + ln (1205820 minus 120598)1205820 minus 120598 minus 119863 (120572) + 120598(96)

which contradicts (83) when 120598 rarr 0This completes the proof of the theorem

Theorem 10 Let ℎ(119905) be defined by (13) If for some 119895 isin Nlim inf119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119890 (97)

where 119875119895 is defined by (38) then all solutions of (E) oscillate

Proof For the sake of contradiction let 119909 be a nonincreasingeventually positive solution and 1199051 gt 1199050 be such that 119909(119905) gt 0and 119909(120591(119905)) gt 0 for all 119905 ge 1199051 We note that wemay obtain (85)as in the proof of Theorem 9

Dividing (E) by 119909(119905) and integrating from ℎ(119905) to 119905 wehave

ln(119909 (ℎ (119905))119909 (119905) ) = int119905

ℎ(119905)119901 (119904) 119909 (120591 (119904))119909 (119904) 119889119904 forall119905 ge 1199052 ge 1199051 (98)

from which in view of 120591(119904) le ℎ(119904) and (85) we get

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(99)

Since 119909 is nonincreasing and ℎ(119904) lt 119904 inequality (99)becomes

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(100)

From (97) it is clear that there exists a constant 119888 gt 0 suchthat

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

ge 119888 gt 1119890 (101)

Choose 1198881015840 such that 119888 gt 1198881015840 gt 1119890 For every 120598 gt 0 such that119888 minus 120598 gt 1198881015840 we haveint119905ℎ(t)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 119888 minus 120598 gt 1198881015840 gt 1119890 (102)

Combining inequalities (100) and (102) we obtain

ln(119909 (ℎ (119905))119909 (119905) ) gt 1198881015840 (103)

or119909 (ℎ (119905))119909 (119905) gt 1198901198881015840 gt 1198901198881015840 gt 1 (104)

which yields

119909 (ℎ (119905)) gt (1198901198881015840) 119909 (119905) (105)

10 Complexity

Following the above steps we can inductively show that forany positive integer 119896

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 for sufficiently large 119905 (106)

Since 1198901198881015840 gt 1 there is a natural number 119896 isin N satisfying119896 gt 2[ln 2 minus ln 1198881015840](1 + ln 1198881015840) such that for 119905 sufficiently large

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 gt ( 21198881015840 )2 (107)

Further (cf [13 24]) for sufficiently large 119905 there exists a realnumber 119905119898 isin (ℎ(119905) 119905) such that

int119905119898ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402 int119905119905119898

119901 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402

(108)

Integrating (E) from ℎ(119905) to 119905119898 using (85) and the fact that119909(119905) gt 0 we obtain119909 (ℎ (119905)) gt 119909 (ℎ (119905119898)) int119905119898

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(109)

which in view of the first inequality in (108) implies that

119909 (ℎ (119905)) gt 11988810158402 119909 (ℎ (119905119898)) (110)

Similarly integrating (E) from 119905119898 to 119905 using (85) and the factthat 119909(119905) gt 0 we have119909 (119905119898) gt 119909 (ℎ (119905)) int119905

119905119898

119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(111)

which in view of the second inequality in (108) yields

119909 (119905119898) gt 11988810158402 119909 (ℎ (119905)) (112)

Combining inequalities (110) and (112) we deduce that

119909 (ℎ (119905119898)) lt 21198881015840 119909 (ℎ (119905)) lt ( 21198881015840 )2 119909 (119905119898) (113)

which contradicts (107)The proof of the theorem is complete

32 ADEs Analogous oscillation conditions to thoseobtained for the delay equation (E) can be derived for the(dual) advanced differential equation (E1015840) by followingsimilar arguments with the ones employed for obtainingTheorems 6minus10Theorem 11 Let 120588(119905) be defined by (27) and for some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1

(114)

where

119876119895 (119905) = 119902 (119905) [1 + int120590(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895minus1 (120585) 119889120585)119889119906)119889119904]

(115)

with 1198760(119905) = 1205820119902(119905) and let 1205820 be the smaller root of theequation 120582 = 119890120573120582 Then all solutions of (E1015840) oscillateTheorem 12 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 minus 119863 (120573)

(116)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillateTheorem 13 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119863 (120573) (117)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate

Complexity 11

Theorem 14 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120573) (118)

where 119876119895 is defined by (115) and 1205820 is the smaller root of theequation 120582 = 119890120573120582 then all solutions of (E1015840) oscillateTheorem 15 Let 120588(119905) be defined by (27) If for some 119895 isin N

lim inf119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119890 (119)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate33 Differential Inequalities A slight modification in theproofs of Theorems 6minus15 leads to the following results aboutdifferential inequalities

Theorem 16 Assume that all the conditions ofTheorem 6 [11] 7 [12] 8 [13] 9 [14] or 10 [15] hold Then(119894) the delay [advanced] differential inequality

1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) le 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) ge 0]

119905 ge 1199050(120)

has no eventually positive solutions(119894119894) the delay [advanced] differential inequality1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) ge 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) le 0]

119905 ge 1199050(121)

has no eventually negative solutions

Remark 17 The oscillation criteria established in this paperall depend on 1205820 (see eg (37) and (71)) in contrast to theconditions obtained in [15 16] and in [17 for m = 1] In factthe left-hand side of conditions (37) and (71) depends on 1205820which is not the casewith the left-hand side of conditions (20)and (21) Since 1205820 gt 1 when 120572 isin (0 1119890] it is obvious that

1198750 (119905) = 1205820119901 (119905) gt 119901 (119905) = 1199010 (119905) (122)

Consequently the left-hand side of conditions (37) and(71) is greater than the corresponding parts of (20) and(21) respectively This is the reason why the conditions inthis paper improve on all known conditions mentioned inSection 2

4 Examples and Comments

The oscillation tests we have proposed and established in themain results involve an iterative procedure We iterativelycompute limsup and liminf on the terms 119875119895(119905) and119876119895(119905) 119895 isinN of a recurrent relation defined on the coefficients and thedeviating argument of an equation of the form (E) or (E1015840)to determine whether that equation is oscillatory But thiscomputation cannot be performed on paper but by meansof a program numerically computing limsup and liminfThe examples below illustrate the significance of our resultsand indicate the high level of improvement in the oscillationcriteria The calculations were performed using MATLABcode

Example 1 Consider the delay differential equation

1199091015840 (119905) + 325119909 (120591 (119905)) = 0 119905 ge 0 (123)

with (see Figure 1(a))

120591 (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]minus4119905 + 40119896 + 9 if 119905 isin [8119896 + 2 8119896 + 3]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 3 8119896 + 4]minus4119905 + 40119896 + 18 if 119905 isin [8119896 + 4 8119896 + 5]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 5 8119896 + 6]minus2119905 + 24119896 + 15 if 119905 isin [8119896 + 6 8119896 + 7]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 7 8119896 + 8]

(124)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (13) we see (Figure 1(b)) that

ℎ (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]8119896 + 1 if 119905 isin [8119896 + 2 8119896 + 195 ]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 195 8119896 + 4]8119896 + 2 if 119905 isin [8119896 + 4 8119896 + 295 ]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 295 8119896 + 6]8119896 + 3 if 119905 isin [8119896 + 6 8119896 + 446 ]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 446 8119896 + 8]

(125)

It is obvious that

120572 = lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 = lim inf

119905rarrinfinint8119896+28119896+1

325119889119904= 012

(126)

12 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

minus1minus2

651 2 3 4 8 9 10 11 12 13 15 16

1415

minus3

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

minus1minus2

t

195 295

1415

7 14651 2 3 4 8 9 10 11 12 13 15 16

446

y = tℎ(t)

(b)

Figure 1 The graphs of 120591(119905) and ℎ(119905)

and therefore the smaller root of 119890012120582 = 120582 is 1205820 = 114765Observe that the function 119865119895 R0 rarr R+ defined as

119865119895 (119905) = int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

(127)

attains its maximum at 119905 = 8119896+446 119896 isin N0 for every 119895 isin NSpecifically

1198651 (119905 = 8119896 + 446 ) = int8119896+446

8119896+3119901 (119904)

sdot exp(int8119896+3120591(119904)

119901 (119906) exp(int119906120591(119906)1198751 (120585) 119889120585) 119889119906)119889119904

(128)

with

1198751 (120585) = 119901 (120585) [1 + int120585120591(120585)119901 (V)

sdot exp(int120585120591(V)119901 (119908) exp(int119908

120591(119908)1205820119901 (119911) 119889119911) 119889119908)119889V]

(129)

Using MATLAB we obtain

1198651 (119905 = 8119896 + 446 ) ≃ 10417 (130)

and therefore

lim sup119905rarrinfin

1198651 (119905) ≃ 10417 gt 1 (131)

Hence condition (37) of Theorem 6 is satisfied for 119895 = 1Consequently all solutions of (123) are oscillatory

Observe however that

LD = lim sup119896rarrinfin

int8119896+4468119896+3

335119889119904 = 052 lt 1120572 = 012 lt 1119890

052 lt 1 + ln 12058201205820 minus 119863 (120572) ≃ 09831(132)

Note that the functionΦ119895 defined by

Φ119895 (119905) = int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

ℎ(119904)119901 (119906) 120595119895 (119906) 119889119906)119889119904

119895 ge 2(133)

attains its maximum at 119905 = 8119896 + 446 119896 isin N0 for every 119895 ge 2Specifically

Φ2 (8119896 + 446 )= int8119896+4468119896+3

119901 (119904) exp(int8119896+3ℎ(119904)

119901 (119904) 1205952 (119906) 119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 exp(int119906

120591(119906)

325 sdot 0 119889119908)119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 sdot 1 119889119906)119889119904= 325 sdot [int

8119896+195

8119896+3exp( 325 int

8119896+3

8119896+1119889119906)119889119904

Complexity 13

+ int8119896+48119896+195

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+2958119896+4

exp( 325 int8119896+3

8119896+2119889119906)119889119904

+ int8119896+68119896+295

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+4468119896+6

exp( 325 int8119896+3

8119896+3119889119906)119889119904] ≃ 057983

(134)

Thus

lim sup119905rarrinfin

Φ2 (119905) ≃ 057983 lt 1 minus 119863 (120572) ≃ 099174 (135)

Also

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int8119896+4468119896+3

325 exp(int8119896+3

120591(119904)

325119889119906)119889119904= 325 sdot lim sup

119905rarrinfin[int8119896+48119896+3

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+58119896+4

exp( 325 int8119896+3

minus4119904+40119896+18119889119906)119889119904

+ int8119896+68119896+5

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+78119896+6

exp( 325 int8119896+3

minus2119904+24119896+15119889119906)119889119904

+ int8119896+4468119896+7

exp( 325 int8119896+3

6119904minus40119896minus41119889119906)119889119904] ≃ 07043

lt 107043 lt 1 minus 119863 (120572) ≃ 099174

(136)

In addition

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)1199011 (119906) 119889119906)119889119904 ≃ 08052

lt 108052 lt 1 minus 119863 (120572) ≃ 099174

(137)

That is none of conditions (8) (9) (12) (14) (for 119895 = 2) (16)(17) (20) (for 119895 = 1) and (21) (for 119895 = 1) is satisfiedComment The improvement of condition (37) over thecorresponding condition (8) is significant approximately10033 We get this measure by comparing the values inthe left-hand side of those conditions Also the improvementover conditions (14) (16) and (20) is very satisfactory

around 7966 479 and 2937 respectively In additioncondition (37) is satisfied from the first iteration whileconditions (14) (20) and (21) need more than one iteration

Example 2 (taken and adapted from [17]) Consider theadvanced differential equation

1199091015840 (119905) minus 3332500119909 (120590 (119905)) = 0 119905 ge 0 (138)

with (see Figure 2(a))

120590 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 2]minus3119905 + 20119896 + 13 if 119905 isin [5119896 + 2 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 4]minus119905 + 10119896 + 13 if 119905 isin [5119896 + 4 5119896 + 5]

(139)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (27) we see (Figure 2(b)) that

120588 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 125]5119896 + 4 if 119905 isin [5119896 + 125 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 38]5119896 + 8 if 119905 isin [5119896 + 38 5119896 + 5]

(140)

It is obvious that

120573 = lim inf119905rarrinfin

int5119896+45119896+3

3332500119889119904 = 01332 (141)

and therefore the smaller root of 11989001332120582 = 120582 is 1205820 = 116839Observe that the function 119866119895 R0 rarr R+ defined as

119866119895 (119905) = int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

(142)

attains its maximum at 119905 = 5119896 + 38 119896 isin N0 for every 119895 isin NSpecifically

1198661 (119905 = 5119896 + 38) = int5119896+85119896+38

119902 (119904)sdot exp(int120590(119904)

5119896+8119902 (119906) exp(int120590(119906)

1199061198761 (120585) 119889120585)119889119906)119889119904

(143)

with

1198761 (120585) = 119902 (120585) [1 + int120590(120585)120585119902 (V)

sdot exp(int120590(V)120585119902 (119908) exp(int120590(119908)

1199081205820119902 (119911) 119889119911)119889119908)119889V]

(144)

14 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

651 2 3 4 8 9 10 11 12 13

1415

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

7

t

14651

2 3 4 8 9 10 11 12 13

38

1415

125

y = t

(t)

(b)

Figure 2 The graphs of 120590(119905) and 120588(119905)

Using MATLAB we obtain

1198661 (119905 = 5119896 + 38) ≃ 09915 (145)

Thereforelim sup119905rarrinfin

1198661 (119905) ≃ 09915 gt 1 minus 119863 (120573) ≃ 09896 (146)

Hence condition (116) of Theorem 12 is satisfied for 119895 = 1Consequently all solutions of (138) oscillate

Observe however that

LA = lim sup119896rarrinfin

int5119896+85119896+38

3332500119889119904 = 055944 lt 1120573 = 01332 lt 1119890 lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int5119896+85119896+38

119902 (119904) exp(int120590(119904)5119896+8

119902 (119906) 119889119906)119889119904= lim sup119896rarrinfin

[int5119896+45119896+38

119902 (119904)sdot exp(int5119904minus20119896minus11

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+5

5119896+4119902 (119904)

sdot exp(intminus119904+10119896+135119896+8

119902 (119906) 119889119906)119889119904 + int5119896+65119896+5

119902 (119904)sdot exp(int5119896+8

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+7

5119896+6119902 (119904)

sdot exp(int4119904minus15119896minus165119896+8

119902 (119906) 119889119906)119889119904 + int5119896+85119896+7

119902 (119904)sdot exp(intminus3119904+20119896+33

5119896+8119902 (119906) 119889119906)119889119904] ≃ 06672 lt 1

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim inf119905rarrinfin

int5119896+45119896+3

119902 (119904) exp(int120590(119904)5119896+4

119902 (119906) 119889119906)119889119904≃ 01893 lt 1119890

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)1199021 (119906) 119889119906)119889119904 ≃ 07196

lt 107196 lt 1 minus 119863 (120573) ≃ 09896

(147)

That is none of conditions (24) (25) (28) (29) (30) (for 119895 =1) and (31) (for 119895 = 1) is satisfiedComment The improvement of condition (116) over thecorresponding condition (24) is significant approximately7723 We get this measure by comparing the values in theleft-hand side of those conditions Also the improvementover conditions (28) and (30) is very satisfactory around4861 and 3778 respectively In addition condition (116)is satisfied from the first iteration while conditions (30) and(31) need more than one iteration

Complexity 15

Remark 3 Similarly one can provide examples illustratingthe other main results

5 Concluding Remarks

In the present paper we have considered the oscillatorydynamics of differential equations having nonmonotonedeviating arguments and nonnegative coefficients New suf-ficient conditions have been established for the oscillation ofall solutions of (E) and (E1015840) These conditions include (37)(71) (75) (83) and (97) and (114) (116) (117) (118) and(119) for (E) and (E1015840) respectively Applying these conditionsinvolves a procedure that checks for oscillations by iterativelycomputing limsup and liminf on terms recursively definedon the equationrsquos coefficients and deviating argument

The main advantage of these conditions is that theyachieve a major improvement over all the related oscillationconditions for (E) [(E1015840)] in the literature For example con-dition (37) [(114)] improves upon the noniterative conditionsthat are reviewed in the introduction namely conditions (8)[(24)] (12) (16)equiv(20) (for 119895 = 1) [(28)equiv(30) (for 119895 = 1)]and (17)equiv(21) (for 119895 = 1) [(31) (for 119895 = 1)] That immediatelybecomes evident by inspecting the left-hand side of (37)[(114)] and the left-hand side of each of the above conditions

The improvement of (37) [(114)] over the other iterativeconditions namely (14) (for j gt 2) (20) (for 119895 gt 1) [(30)(for 119895 gt 1)] and (21) (for 119895 gt 1) [(31) (for 119895 gt 1)] is that itrequires far fewer iterations to establish oscillation than theother conditions

This advantage easily can be verified computationallyby running the MATLAB programs (see Appendix) forcomputing limsup and liminf and comparing the number ofiterations required by each condition to establish oscillationThen we see that we achieve a significant improvement overall known oscillation criteria

Another advantage and a significant departure from thelarge majority of the other studies is that the criteria in thispaper apply to a more general class of equations havingnonmonotone arguments 120591(119905) or 120590(119905) in contrast to mostof the other oscillation criteria that apply to equations withnondecreasing arguments

Appendix

In this appendix for completeness we give the algorithmon MATLAB software used in Example 1 for calculation oflim sup119905rarrinfin1198651(119905) ≃ 10417 For Example 2 the algorithm isomitted since it is similar to the one in Example 1

Algorithm for Example 1

clear clcc = 12119899 = 501205820 = 114765a5 = 19b5 = 703h5 = (b5 minus a5)119899

for i5 = 1 1 119899 + 1x5 = a5 + (i5 minus 1) lowast h5a4 = TFunction(x5)b4 = 19h4 = (b4 minus a4)119899for i4 = 1 1 119899 + 1x4 = a4 + (i4 minus 1) lowast h4a3 = TFunction(x4)b3 = x4h3 = (b3 minus a3)119899for i3 = 1 1 119899 + 1x3 = a3 + (i3 minus 1) lowast h3a2 = TFunction(x3)b2 = x3h2 = (b2 minus a2)119899for i2 = 1 1 119899 + 1x2 = a2 + (i2 minus 1) lowast h2a1 = TFunction(x2)b1 = x3h1 = (b1 minus a1)119899for i1 = 1 1 119899 + 1x1 = a1 + (i1 minus 1) lowast h1f1(i1) = c lowast exp(1205820 lowast c lowast (x1 minus TFunction(x1)))endI1 = f1(1) + f1(119899 + 1)for i1 = 2 2 119899I1 = I1 + f1(i1) lowast 4endfor i1 = 3 2 119899 minus 1I1 = I1 + f1(i1) lowast 2endI1 = I1 lowast h13f2(i2) = c lowast exp(I1)endI2 = f2(1) + f2(119899 + 1)for i2 = 2 2 119899I2 = I2 + f2(i2) lowast 4endfor i2 = 3 2 119899 minus 1I2 = I2 + f2(i2) lowast 2endI2 = I2 lowast h23f3(i3) = c lowast (1 + I2)end

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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Page 3: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

Complexity 3

From this point onward we will use the notation

120572 fl lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904

120573 fl lim inf119905rarrinfin

int120590(119905)119905119902 (119904) 119889119904

119863 (120596) fl 0 if 120596 gt 1119890 1 minus 120596 minus radic1 minus 2120596 minus 12059622 if 120596 isin [0 1119890 ]

LD fl lim sup119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904where 120591 (119905) is nondecreasing

LA fl lim sup119905rarrinfin

int120590(119905)119905119902 (119904) 119889119904where 120590 (119905) is nondecreasing

(6)

2 Historical and Chronological Review

21 DDEs The first systematic study for the oscillation of allsolutions of equation (E) was made by Myskis in 1950 [31]when he proved that every solution of (E) oscillates if

lim sup119905rarrinfin

[119905 minus 120591 (119905)] lt infinlim inf119905rarrinfin

[119905 minus 120591 (119905)] lim inf119905rarrinfin

119901 (119905) gt 1119890 (7)

In 1972 Ladas et al [27] proved that if

LD gt 1 (8)

then all solutions of (E) are oscillatoryIn 1982 Koplatadze and Chanturiya [24] improved (7) to

120572 gt 1119890 (9)

Regarding the constant 1119890 in (9) it should be remarked thatif the inequality

int119905120591(119905)119901 (119904) 119889119904 le 1119890 (10)

holds eventually then according to [24] (E) has a nonoscil-latory solution

It is apparent that there is a gap between conditions (8)and (9) when

lim119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 (11)

does not exist How to fill this gap is an interesting problemwhich has been investigated by several authors For example

in 2000 Jaros and Stavroulakis [23] proved that if 1205820 is thesmaller root of the equation 120582 = 119890120572120582 and

LD gt 1 + ln 12058201205820 minus 119863 (120572) (12)

then all solutions of (E) oscillateNowwe come to the general case where the argument 120591(119905)

is nonmonotone Set

ℎ (119905) fl sup119904le119905120591 (119904) 119905 ge 1199050 (13)

Clearly the function ℎ(119905) is nondecreasing and 120591(119905) le ℎ(119905) lt119905 for all 119905 ge 1199050In 1994 Koplatadze and Kvinikadze [25] proved that if

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

ℎ(119904)119901 (119906) 120595119895 (119906) 119889119906)119889119904

gt 1 minus 119863 (120572) (14)

where

1205951 (119905) = 0120595119895 (119905) = exp(int119905

120591(119905)119901 (119906) 120595119895minus1 (119906) 119889119906) 119895 ge 2 (15)

then all solutions of (E) oscillateIn 2011 Braverman and Karpuz [14] proved that if

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904 gt 1 (16)

then all solutions of (E) oscillate while in 2014 Stavroulakis[32] improved (16) to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

gt 1 minus 119863 (120572) (17)

In 2016 El-Morshedy and Attia [30] proved that if

lim sup119905rarrinfin

[[int119905

119892(119905)119901119899 (119904) 119889119904

+ 119863 (120572) exp(int119905119892(119905)

119899minus1sum119895=0

119901119895 (119904) 119889119904)]] gt 1(18)

where 1199010(119905) = 119901(119905) and119901119899 (119905)= 119901119899minus1 (119905) int119905

119892(119905)119901119899minus1 (119904) exp(int119905

119892(119904)119901119899minus1 (119906) 119889119906)119889119904

119899 ge 1(19)

4 Complexity

then all solutions of (E) are oscillatory Here 119892(119905) is a nonde-creasing continuous function such that 120591(119905) le 119892(119905) le 119905 119905 ge1199051 for some 1199051 ge 1199050 Clearly 119892(119905) is more general than ℎ(119905)defined by (13)

Recently Chatzarakis [15 16] proved that if for some 119895 isinN

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901119895 (119906) 119889119906)119889119904 gt 1 (20)

or

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901119895 (119906) 119889119906)119889119904

gt 1 minus 119863 (120572) (21)

where

119901119895 (119905)= 119901 (119905) [1 + int119905

120591(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901119895minus1 (119906) 119889119906)119889119904] (22)

with 1199010(119905) = 119901(119905) then all solutions of (E) are oscillatoryLately Chatzarakis [17] studied a more general form of(E) namely

1199091015840 (119905) + 119898sum119894=1

119901119894 (119905) 119909 (120591119894 (119905)) = 0 119905 ge 1199050 (23)

and established sufficient oscillation conditions Those con-ditions can lead to (20) and (21) when119898 = 122 ADEs ByTheorem 243 [29] if

LA gt 1 (24)

then all solutions of (E1015840) are oscillatoryIn 1984 Fukagai and Kusano [21] proved that if

120573 gt 1119890 (25)

then all solutions of (E1015840) are oscillatory while ifint120590(119905)119905119902 (119904) 119889119904 le 1119890 for all sufficiently large 119905 (26)

then (E1015840) has a nonoscillatory solutionAssume that the argument 120590(119905) is not necessarily mono-

tone Set

120588 (119905) = inf119904ge119905120590 (119904) 119905 ge 1199050 (27)

Clearly the function 120588(119905) is nondecreasing and 120590(119905) ge 120588(119905) gt119905 for all 119905 ge 1199050In 2015 Chatzarakis and Ocalan [18] proved that if

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904 gt 1 (28)

or

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904 gt 1119890 (29)

then all solutions of (E1015840) are oscillatoryRecently Chatzarakis [15 16] proved that if for some 119895 isin

N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902119895 (119906) 119889119906)119889119904 gt 1 (30)

or

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902119895 (119906) 119889119906)119889119904

gt 1 minus 119863 (120573) (31)

where119902119895 (119905)= 119902 (119905) [1 + int120590(119905)

119905119902 (119904) exp(int120590(119904)

120588(119905)119902119895minus1 (119906) 119889119906)119889119904]

119895 ge 1(32)

with 1199020(119905) = 119902(119905) then all solutions of (E1015840) oscillateLately Chatzarakis [17] studied a more general form of(E1015840) namely

1199091015840 (119905) minus 119898sum119894=1

119902119894 (119905) 119909 (120590119894 (119905)) = 0 119905 ge 1199050 (33)

and established sufficient oscillation conditions Those con-ditions can lead to (30) and (31) when119898 = 13 Main Results

31 DDEs In our main results we state theorems establish-ing new sufficient oscillation conditions For the proofs ofthose theorems we use the following lemmas

Lemma 3 (see [19 Lemma 211]) Assume that ℎ(119905) is definedby (13) Then

120572 fl lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 = lim inf

119905rarrinfinint119905ℎ(119905)119901 (119904) 119889119904 (34)

Lemma 4 (see [19 Lemma 213]) Assume that ℎ(119905) is definedby (13) 120572 isin (0 1119890] and 119909(119905) is an eventually positive solutionof (E) Then

lim inf119905rarrinfin

119909 (119905)119909 (ℎ (119905)) ge 119863 (120572) (35)

Lemma 5 (see [26]) Assume that ℎ(119905) is defined by (13) 120572 isin(0 1119890] and 119909(119905) is an eventually positive solution of (E) Then

lim inf119905rarrinfin

119909 (ℎ (119905))119909 (119905) ge 1205820 (36)

where 1205820 is the smaller root of the equation 120582 = 119890120572120582

Complexity 5

Theorem 6 Let ℎ(119905) be defined by (13) and for some 119895 isin Nlim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1(37)

where

119875119895 (119905) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895minus1 (120585) 119889120585) 119889119906)119889119904]

(38)

with 1198750(119905) = 1205820119901(119905) and let 1205820 be the smaller root of theequation 120582 = 119890120572120582 Then all solutions of (E) oscillateProof Assume for the sake of contradiction that there existsa nonoscillatory solution 119909(119905) of (E) Since minus119909(119905) is also asolution of (E) we can confine our discussion only to thecase where the solution 119909(119905) is eventually positiveThen thereexists a real number 1199051 gt 1199050 such that 119909(119905) 119909(120591(119905)) gt 0 for all119905 ge 1199051 Thus from (E) we have

1199091015840 (119905) = minus119901 (119905) 119909 (120591 (119905)) le 0 forall119905 ge 1199051 (39)

which means that 119909(119905) is an eventually nonincreasing func-tion of positive numbers Taking into account the fact that120591(119905) le ℎ(119905) (E) implies that

1199091015840 (119905) + 119901 (119905) 119909 (ℎ (119905)) le 0 119905 ge 1199051 (40)

Observe that (36) implies that for each 120598 gt 0 there exists areal number 119905120598 such that

119909 (ℎ (119905))119909 (119905) gt 1205820 minus 120598 forall119905 ge 119905120598 ge 1199051 (41)

Combining inequalities (40) and (41) we obtain

1199091015840 (119905) + 119901 (119905) (1205820 minus 120598) 119909 (119905) le 0 119905 ge 119905120598 (42)

or

1199091015840 (119905) + 1198750 (119905 120598) 119909 (119905) le 0 119905 ge 119905120598 (43)

where

1198750 (119905 120598) = 119901 (119905) (1205820 minus 120598) (44)

Applying the Gronwall inequality in (43) we conclude that

119909 (119904) ge 119909 (119905) exp(int1199051199041198750 (120585 120598) 119889120585) 119905 ge 119904 ge 119905120598 (45)

Now we divide (E) by 119909(119905) gt 0 and integrate on [119904 119905] sominusint119905119904

1199091015840 (119906)119909 (119906) 119889119906 = int119905

119904119901 (119906) 119909 (120591 (119906))119909 (119906) 119889119906 (46)

or

ln 119909 (119904)119909 (119905) = int119905

119904119901 (119906) 119909 (120591 (119906))119909 (119906) 119889119906 119905 ge 119904 ge 119905120598 (47)

Since 120591(119906) lt 119906 equality (47) givesln 119909 (119904)119909 (119905) = int

119905

119904119901 (119906) 119909 (120591 (119906))119909 (119906) 119889119906

ge int119905119904119901 (119906) 119909 (119906)119909 (119906) exp(int

119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906

= int119905119904119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906

(48)

or

119909 (119904)ge 119909 (119905) exp(int119905

119904119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906) (49)

Substituting 120591(119904) for 119904 in (49) we get

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906) (50)

Integrating (E) from 120591(119905) to 119905 we have119909 (119905) minus 119909 (120591 (119905)) + int119905

120591(119905)119901 (119904) 119909 (120591 (119904)) 119889119904 = 0 (51)

Combining (50) and (51) we obtain

119909 (119905) minus 119909 (120591 (119905)) + 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904

le 0(52)

Multiplying inequality (52) by 119901(119905) we find119901 (119905) 119909 (119905) minus 119901 (119905) 119909 (120591 (119905)) + 119901 (119905) 119909 (119905) int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904

le 0(53)

which in view of (E) becomes

1199091015840 (119905) + 119901 (119905) 119909 (119905) + 119901 (119905) 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904

le 0(54)

6 Complexity

Hence for sufficiently large 1199051199091015840 (119905) + 119901 (119905) [1 + int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904]

sdot 119909 (119905) le 0(55)

or

1199091015840 (119905) + 1198751 (119905 120598) 119909 (119905) le 0 (56)

where

1198751 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904]

(57)

Clearly (56) resembles (43) if we replace 1198750 by 1198751 Thusintegrating (56) on [119904 119905] yields

119909 (119904) ge 119909 (119905) exp(int1199051199041198751 (120585 120598) 119889120585) (58)

Repeating steps (45) through (50) we can see that 119909 satisfiesthe inequality

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906) (59)

Combining now (51) and (59) we obtain

119909 (119905) minus 119909 (120591 (119905)) + 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904

le 0(60)

Multiplying inequality (60) by 119901(119905) as before we find1199091015840 (119905) + 119901 (119905) [1 + int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904]

sdot 119909 (119905) le 0(61)

Therefore for sufficiently large 119905 we have1199091015840 (119905) + 1198752 (119905 120598) 119909 (119905) le 0 (62)

where

1198752 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904]

(63)

It becomes apparent now that by repeating the above stepswe can build inequalities on 1199091015840(119905) with progressively higherindices 119875119895(119905 120598) 119895 isin N In general for sufficiently large 119905 thepositive solution 119909(119905) satisfies the inequality

1199091015840 (119905) + 119875119895 (119905 120598) 119909 (119905) le 0 119895 isin N (64)

where

119875119895 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895minus1 (120585 120598) 119889120585) 119889119906)119889119904]

(65)

Proceeding to final step we recall that ℎ(119905) defined by (13) isa nondecreasing function Since 120591(119904) le ℎ(119904) le ℎ(119905) we have119909 (120591 (119904)) ge 119909 (ℎ (119905))sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (66)

Hence

119909 (119905) minus 119909 (ℎ (119905)) + 119909 (ℎ (119905)) int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(67)

or

119909 (ℎ (119905)) [int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1] lt 0(68)

Thus

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1 lt 0(69)

Taking the limit as 119905 rarr infin we have

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1(70)

Complexity 7

Since 120598may be taken arbitrarily small this inequality contra-dicts (37)

This completes the proof of the theorem

Theorem 7 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1 minus 119863 (120572) (71)

where 119875119895 is defined by (38) then all solutions of (E) oscillateProof Assume 119909 is an eventually positive solution of (E)Clearly (67) is satisfied for sufficiently large 119905 Thus

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus 119909 (119905)119909 (ℎ (119905)) (72)

which implies that

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus lim inf119905rarrinfin

119909 (119905)119909 (ℎ (119905)) (73)

Using Lemmas 3 and 4 it is evident that inequality (35) issatisfied Thus (73) leads to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus 119863 (120572) (74)

Since 120598may be taken arbitrarily small this inequality contra-dicts (71)

This completes the proof of the theorem

Theorem 8 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119863 (120572) (75)

where 119875119895 is defined by (38) then all solutions of (E) oscillateProof Assume 119909 is an eventually positive solution of (E)Then as in the proof of Theorem 6 for sufficiently large 119905we conclude that

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (76)

Integrating (E) from ℎ(119905) to 119905 and using (76) we obtain

119909 (119905) minus 119909 (ℎ (119905)) + int119905ℎ(119905)119901 (119904) 119909 (119905)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(77)

or

minus 119909 (ℎ (119905)) + int119905ℎ(119905)119901 (119904) 119909 (119905)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 0(78)

Hence

119909 (ℎ (119905)) [ 119909 (119905)119909 (ℎ (119905)) int119905

ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1] lt 0(79)

which yields for all sufficiently large 119905int119905ℎ(119905)119901 (119904)sdot exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 119909 (ℎ (119905))119909 (119905)

(80)

8 Complexity

and consequently

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le lim sup119905rarrinfin

119909 (ℎ (119905))119909 (119905) (81)

Taking into account the fact that (35) is satisfied inequality(81) leads to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1119863 (120572) (82)

which contradicts (75) when 120598 rarr 0This completes the proof of the theorem

Theorem 9 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120572) (83)

where 119875119895 is defined by (38) and 1205820 is the smaller root of theequation 120582 = 119890120572120582 then all solutions of (E) oscillateProof Let 119909 be an eventually positive solution of (E) As inthe proof ofTheorem 8 we can show that (76) holds namely

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (84)

Since 120591(119904) le ℎ(119904) inequality (84) gives119909 (120591 (119904)) ge 119909 (ℎ (119904))sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (85)

By Lemma 5 for each 120598 gt 0 there exists a real number 119905120598 suchthat

119909 (ℎ (119905))119909 (119905) gt 1205820 minus 120598 forall119905 ge 119905120598 ge 1199051 (86)

Note that by the nondecreasing nature of the function119909(ℎ(119905))119909(119904) in 119904 it holds1 = 119909 (ℎ (119905))119909 (ℎ (119905)) le 119909 (ℎ (119905))119909 (119904) le 119909 (ℎ (119905))119909 (119905)

119905120598 le ℎ (119905) le 119904 le 119905(87)

In particular for 120598 isin (0 1205820 minus 1) by continuity we concludethat there exists a real number 119905lowast isin (ℎ(119905) 119905] satisfying

1 lt 1205820 minus 120598 = 119909 (ℎ (119905))119909 (119905lowast) (88)

Integrating (E) from 119905lowast to 119905 and using (85) we obtain

119909 (119905) minus 119909 (119905lowast) + 119909 (ℎ (119905)) int119905119905lowast119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(89)

or

int119905119905lowast119901 (119904) exp(intℎ(119904)

120591(119904)119901 (119906)

sdot exp(int119906120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 119909 (119905lowast)119909 (ℎ (119905)) minus 119909 (119905)119909 (ℎ (119905)) (90)

Using (88) andLemma4we deduce that for the 120598 consideredthere exists a real number 1199051015840120598 ge 119905120598 such that

int119905119905lowast119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 11205820 minus 120598 minus 119863 (120572) + 120598(91)

for 119905 ge 1199051015840120598Dividing (E) by119909(119905) integrating from ℎ(119905) to 119905lowast and using

(85) we deduce that

int119905lowastℎ(119905)119901 (119904)sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(92)

Complexity 9

Clearly by means of (36) 119909(ℎ(119904))119909(119904) gt 1205820minus120598 for 119904 ge ℎ(119905) ge1199051015840120598 Hence for all sufficiently large 119905 we conclude that(1205820 minus 120598)int119905

lowast

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(93)

or

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minus 11205820 minus 120598 int119905lowast

ℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904 = 11205820 minus 120598 ln119909 (ℎ (119905))119909 (119905lowast)

= ln (1205820 minus 120598)1205820 minus 120598

(94)

that is

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt ln (1205820 minus 120598)1205820 minus 120598 (95)

Using (91) along with (95) we get

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 + ln (1205820 minus 120598)1205820 minus 120598 minus 119863 (120572) + 120598(96)

which contradicts (83) when 120598 rarr 0This completes the proof of the theorem

Theorem 10 Let ℎ(119905) be defined by (13) If for some 119895 isin Nlim inf119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119890 (97)

where 119875119895 is defined by (38) then all solutions of (E) oscillate

Proof For the sake of contradiction let 119909 be a nonincreasingeventually positive solution and 1199051 gt 1199050 be such that 119909(119905) gt 0and 119909(120591(119905)) gt 0 for all 119905 ge 1199051 We note that wemay obtain (85)as in the proof of Theorem 9

Dividing (E) by 119909(119905) and integrating from ℎ(119905) to 119905 wehave

ln(119909 (ℎ (119905))119909 (119905) ) = int119905

ℎ(119905)119901 (119904) 119909 (120591 (119904))119909 (119904) 119889119904 forall119905 ge 1199052 ge 1199051 (98)

from which in view of 120591(119904) le ℎ(119904) and (85) we get

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(99)

Since 119909 is nonincreasing and ℎ(119904) lt 119904 inequality (99)becomes

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(100)

From (97) it is clear that there exists a constant 119888 gt 0 suchthat

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

ge 119888 gt 1119890 (101)

Choose 1198881015840 such that 119888 gt 1198881015840 gt 1119890 For every 120598 gt 0 such that119888 minus 120598 gt 1198881015840 we haveint119905ℎ(t)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 119888 minus 120598 gt 1198881015840 gt 1119890 (102)

Combining inequalities (100) and (102) we obtain

ln(119909 (ℎ (119905))119909 (119905) ) gt 1198881015840 (103)

or119909 (ℎ (119905))119909 (119905) gt 1198901198881015840 gt 1198901198881015840 gt 1 (104)

which yields

119909 (ℎ (119905)) gt (1198901198881015840) 119909 (119905) (105)

10 Complexity

Following the above steps we can inductively show that forany positive integer 119896

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 for sufficiently large 119905 (106)

Since 1198901198881015840 gt 1 there is a natural number 119896 isin N satisfying119896 gt 2[ln 2 minus ln 1198881015840](1 + ln 1198881015840) such that for 119905 sufficiently large

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 gt ( 21198881015840 )2 (107)

Further (cf [13 24]) for sufficiently large 119905 there exists a realnumber 119905119898 isin (ℎ(119905) 119905) such that

int119905119898ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402 int119905119905119898

119901 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402

(108)

Integrating (E) from ℎ(119905) to 119905119898 using (85) and the fact that119909(119905) gt 0 we obtain119909 (ℎ (119905)) gt 119909 (ℎ (119905119898)) int119905119898

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(109)

which in view of the first inequality in (108) implies that

119909 (ℎ (119905)) gt 11988810158402 119909 (ℎ (119905119898)) (110)

Similarly integrating (E) from 119905119898 to 119905 using (85) and the factthat 119909(119905) gt 0 we have119909 (119905119898) gt 119909 (ℎ (119905)) int119905

119905119898

119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(111)

which in view of the second inequality in (108) yields

119909 (119905119898) gt 11988810158402 119909 (ℎ (119905)) (112)

Combining inequalities (110) and (112) we deduce that

119909 (ℎ (119905119898)) lt 21198881015840 119909 (ℎ (119905)) lt ( 21198881015840 )2 119909 (119905119898) (113)

which contradicts (107)The proof of the theorem is complete

32 ADEs Analogous oscillation conditions to thoseobtained for the delay equation (E) can be derived for the(dual) advanced differential equation (E1015840) by followingsimilar arguments with the ones employed for obtainingTheorems 6minus10Theorem 11 Let 120588(119905) be defined by (27) and for some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1

(114)

where

119876119895 (119905) = 119902 (119905) [1 + int120590(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895minus1 (120585) 119889120585)119889119906)119889119904]

(115)

with 1198760(119905) = 1205820119902(119905) and let 1205820 be the smaller root of theequation 120582 = 119890120573120582 Then all solutions of (E1015840) oscillateTheorem 12 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 minus 119863 (120573)

(116)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillateTheorem 13 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119863 (120573) (117)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate

Complexity 11

Theorem 14 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120573) (118)

where 119876119895 is defined by (115) and 1205820 is the smaller root of theequation 120582 = 119890120573120582 then all solutions of (E1015840) oscillateTheorem 15 Let 120588(119905) be defined by (27) If for some 119895 isin N

lim inf119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119890 (119)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate33 Differential Inequalities A slight modification in theproofs of Theorems 6minus15 leads to the following results aboutdifferential inequalities

Theorem 16 Assume that all the conditions ofTheorem 6 [11] 7 [12] 8 [13] 9 [14] or 10 [15] hold Then(119894) the delay [advanced] differential inequality

1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) le 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) ge 0]

119905 ge 1199050(120)

has no eventually positive solutions(119894119894) the delay [advanced] differential inequality1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) ge 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) le 0]

119905 ge 1199050(121)

has no eventually negative solutions

Remark 17 The oscillation criteria established in this paperall depend on 1205820 (see eg (37) and (71)) in contrast to theconditions obtained in [15 16] and in [17 for m = 1] In factthe left-hand side of conditions (37) and (71) depends on 1205820which is not the casewith the left-hand side of conditions (20)and (21) Since 1205820 gt 1 when 120572 isin (0 1119890] it is obvious that

1198750 (119905) = 1205820119901 (119905) gt 119901 (119905) = 1199010 (119905) (122)

Consequently the left-hand side of conditions (37) and(71) is greater than the corresponding parts of (20) and(21) respectively This is the reason why the conditions inthis paper improve on all known conditions mentioned inSection 2

4 Examples and Comments

The oscillation tests we have proposed and established in themain results involve an iterative procedure We iterativelycompute limsup and liminf on the terms 119875119895(119905) and119876119895(119905) 119895 isinN of a recurrent relation defined on the coefficients and thedeviating argument of an equation of the form (E) or (E1015840)to determine whether that equation is oscillatory But thiscomputation cannot be performed on paper but by meansof a program numerically computing limsup and liminfThe examples below illustrate the significance of our resultsand indicate the high level of improvement in the oscillationcriteria The calculations were performed using MATLABcode

Example 1 Consider the delay differential equation

1199091015840 (119905) + 325119909 (120591 (119905)) = 0 119905 ge 0 (123)

with (see Figure 1(a))

120591 (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]minus4119905 + 40119896 + 9 if 119905 isin [8119896 + 2 8119896 + 3]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 3 8119896 + 4]minus4119905 + 40119896 + 18 if 119905 isin [8119896 + 4 8119896 + 5]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 5 8119896 + 6]minus2119905 + 24119896 + 15 if 119905 isin [8119896 + 6 8119896 + 7]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 7 8119896 + 8]

(124)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (13) we see (Figure 1(b)) that

ℎ (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]8119896 + 1 if 119905 isin [8119896 + 2 8119896 + 195 ]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 195 8119896 + 4]8119896 + 2 if 119905 isin [8119896 + 4 8119896 + 295 ]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 295 8119896 + 6]8119896 + 3 if 119905 isin [8119896 + 6 8119896 + 446 ]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 446 8119896 + 8]

(125)

It is obvious that

120572 = lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 = lim inf

119905rarrinfinint8119896+28119896+1

325119889119904= 012

(126)

12 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

minus1minus2

651 2 3 4 8 9 10 11 12 13 15 16

1415

minus3

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

minus1minus2

t

195 295

1415

7 14651 2 3 4 8 9 10 11 12 13 15 16

446

y = tℎ(t)

(b)

Figure 1 The graphs of 120591(119905) and ℎ(119905)

and therefore the smaller root of 119890012120582 = 120582 is 1205820 = 114765Observe that the function 119865119895 R0 rarr R+ defined as

119865119895 (119905) = int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

(127)

attains its maximum at 119905 = 8119896+446 119896 isin N0 for every 119895 isin NSpecifically

1198651 (119905 = 8119896 + 446 ) = int8119896+446

8119896+3119901 (119904)

sdot exp(int8119896+3120591(119904)

119901 (119906) exp(int119906120591(119906)1198751 (120585) 119889120585) 119889119906)119889119904

(128)

with

1198751 (120585) = 119901 (120585) [1 + int120585120591(120585)119901 (V)

sdot exp(int120585120591(V)119901 (119908) exp(int119908

120591(119908)1205820119901 (119911) 119889119911) 119889119908)119889V]

(129)

Using MATLAB we obtain

1198651 (119905 = 8119896 + 446 ) ≃ 10417 (130)

and therefore

lim sup119905rarrinfin

1198651 (119905) ≃ 10417 gt 1 (131)

Hence condition (37) of Theorem 6 is satisfied for 119895 = 1Consequently all solutions of (123) are oscillatory

Observe however that

LD = lim sup119896rarrinfin

int8119896+4468119896+3

335119889119904 = 052 lt 1120572 = 012 lt 1119890

052 lt 1 + ln 12058201205820 minus 119863 (120572) ≃ 09831(132)

Note that the functionΦ119895 defined by

Φ119895 (119905) = int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

ℎ(119904)119901 (119906) 120595119895 (119906) 119889119906)119889119904

119895 ge 2(133)

attains its maximum at 119905 = 8119896 + 446 119896 isin N0 for every 119895 ge 2Specifically

Φ2 (8119896 + 446 )= int8119896+4468119896+3

119901 (119904) exp(int8119896+3ℎ(119904)

119901 (119904) 1205952 (119906) 119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 exp(int119906

120591(119906)

325 sdot 0 119889119908)119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 sdot 1 119889119906)119889119904= 325 sdot [int

8119896+195

8119896+3exp( 325 int

8119896+3

8119896+1119889119906)119889119904

Complexity 13

+ int8119896+48119896+195

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+2958119896+4

exp( 325 int8119896+3

8119896+2119889119906)119889119904

+ int8119896+68119896+295

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+4468119896+6

exp( 325 int8119896+3

8119896+3119889119906)119889119904] ≃ 057983

(134)

Thus

lim sup119905rarrinfin

Φ2 (119905) ≃ 057983 lt 1 minus 119863 (120572) ≃ 099174 (135)

Also

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int8119896+4468119896+3

325 exp(int8119896+3

120591(119904)

325119889119906)119889119904= 325 sdot lim sup

119905rarrinfin[int8119896+48119896+3

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+58119896+4

exp( 325 int8119896+3

minus4119904+40119896+18119889119906)119889119904

+ int8119896+68119896+5

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+78119896+6

exp( 325 int8119896+3

minus2119904+24119896+15119889119906)119889119904

+ int8119896+4468119896+7

exp( 325 int8119896+3

6119904minus40119896minus41119889119906)119889119904] ≃ 07043

lt 107043 lt 1 minus 119863 (120572) ≃ 099174

(136)

In addition

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)1199011 (119906) 119889119906)119889119904 ≃ 08052

lt 108052 lt 1 minus 119863 (120572) ≃ 099174

(137)

That is none of conditions (8) (9) (12) (14) (for 119895 = 2) (16)(17) (20) (for 119895 = 1) and (21) (for 119895 = 1) is satisfiedComment The improvement of condition (37) over thecorresponding condition (8) is significant approximately10033 We get this measure by comparing the values inthe left-hand side of those conditions Also the improvementover conditions (14) (16) and (20) is very satisfactory

around 7966 479 and 2937 respectively In additioncondition (37) is satisfied from the first iteration whileconditions (14) (20) and (21) need more than one iteration

Example 2 (taken and adapted from [17]) Consider theadvanced differential equation

1199091015840 (119905) minus 3332500119909 (120590 (119905)) = 0 119905 ge 0 (138)

with (see Figure 2(a))

120590 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 2]minus3119905 + 20119896 + 13 if 119905 isin [5119896 + 2 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 4]minus119905 + 10119896 + 13 if 119905 isin [5119896 + 4 5119896 + 5]

(139)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (27) we see (Figure 2(b)) that

120588 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 125]5119896 + 4 if 119905 isin [5119896 + 125 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 38]5119896 + 8 if 119905 isin [5119896 + 38 5119896 + 5]

(140)

It is obvious that

120573 = lim inf119905rarrinfin

int5119896+45119896+3

3332500119889119904 = 01332 (141)

and therefore the smaller root of 11989001332120582 = 120582 is 1205820 = 116839Observe that the function 119866119895 R0 rarr R+ defined as

119866119895 (119905) = int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

(142)

attains its maximum at 119905 = 5119896 + 38 119896 isin N0 for every 119895 isin NSpecifically

1198661 (119905 = 5119896 + 38) = int5119896+85119896+38

119902 (119904)sdot exp(int120590(119904)

5119896+8119902 (119906) exp(int120590(119906)

1199061198761 (120585) 119889120585)119889119906)119889119904

(143)

with

1198761 (120585) = 119902 (120585) [1 + int120590(120585)120585119902 (V)

sdot exp(int120590(V)120585119902 (119908) exp(int120590(119908)

1199081205820119902 (119911) 119889119911)119889119908)119889V]

(144)

14 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

651 2 3 4 8 9 10 11 12 13

1415

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

7

t

14651

2 3 4 8 9 10 11 12 13

38

1415

125

y = t

(t)

(b)

Figure 2 The graphs of 120590(119905) and 120588(119905)

Using MATLAB we obtain

1198661 (119905 = 5119896 + 38) ≃ 09915 (145)

Thereforelim sup119905rarrinfin

1198661 (119905) ≃ 09915 gt 1 minus 119863 (120573) ≃ 09896 (146)

Hence condition (116) of Theorem 12 is satisfied for 119895 = 1Consequently all solutions of (138) oscillate

Observe however that

LA = lim sup119896rarrinfin

int5119896+85119896+38

3332500119889119904 = 055944 lt 1120573 = 01332 lt 1119890 lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int5119896+85119896+38

119902 (119904) exp(int120590(119904)5119896+8

119902 (119906) 119889119906)119889119904= lim sup119896rarrinfin

[int5119896+45119896+38

119902 (119904)sdot exp(int5119904minus20119896minus11

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+5

5119896+4119902 (119904)

sdot exp(intminus119904+10119896+135119896+8

119902 (119906) 119889119906)119889119904 + int5119896+65119896+5

119902 (119904)sdot exp(int5119896+8

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+7

5119896+6119902 (119904)

sdot exp(int4119904minus15119896minus165119896+8

119902 (119906) 119889119906)119889119904 + int5119896+85119896+7

119902 (119904)sdot exp(intminus3119904+20119896+33

5119896+8119902 (119906) 119889119906)119889119904] ≃ 06672 lt 1

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim inf119905rarrinfin

int5119896+45119896+3

119902 (119904) exp(int120590(119904)5119896+4

119902 (119906) 119889119906)119889119904≃ 01893 lt 1119890

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)1199021 (119906) 119889119906)119889119904 ≃ 07196

lt 107196 lt 1 minus 119863 (120573) ≃ 09896

(147)

That is none of conditions (24) (25) (28) (29) (30) (for 119895 =1) and (31) (for 119895 = 1) is satisfiedComment The improvement of condition (116) over thecorresponding condition (24) is significant approximately7723 We get this measure by comparing the values in theleft-hand side of those conditions Also the improvementover conditions (28) and (30) is very satisfactory around4861 and 3778 respectively In addition condition (116)is satisfied from the first iteration while conditions (30) and(31) need more than one iteration

Complexity 15

Remark 3 Similarly one can provide examples illustratingthe other main results

5 Concluding Remarks

In the present paper we have considered the oscillatorydynamics of differential equations having nonmonotonedeviating arguments and nonnegative coefficients New suf-ficient conditions have been established for the oscillation ofall solutions of (E) and (E1015840) These conditions include (37)(71) (75) (83) and (97) and (114) (116) (117) (118) and(119) for (E) and (E1015840) respectively Applying these conditionsinvolves a procedure that checks for oscillations by iterativelycomputing limsup and liminf on terms recursively definedon the equationrsquos coefficients and deviating argument

The main advantage of these conditions is that theyachieve a major improvement over all the related oscillationconditions for (E) [(E1015840)] in the literature For example con-dition (37) [(114)] improves upon the noniterative conditionsthat are reviewed in the introduction namely conditions (8)[(24)] (12) (16)equiv(20) (for 119895 = 1) [(28)equiv(30) (for 119895 = 1)]and (17)equiv(21) (for 119895 = 1) [(31) (for 119895 = 1)] That immediatelybecomes evident by inspecting the left-hand side of (37)[(114)] and the left-hand side of each of the above conditions

The improvement of (37) [(114)] over the other iterativeconditions namely (14) (for j gt 2) (20) (for 119895 gt 1) [(30)(for 119895 gt 1)] and (21) (for 119895 gt 1) [(31) (for 119895 gt 1)] is that itrequires far fewer iterations to establish oscillation than theother conditions

This advantage easily can be verified computationallyby running the MATLAB programs (see Appendix) forcomputing limsup and liminf and comparing the number ofiterations required by each condition to establish oscillationThen we see that we achieve a significant improvement overall known oscillation criteria

Another advantage and a significant departure from thelarge majority of the other studies is that the criteria in thispaper apply to a more general class of equations havingnonmonotone arguments 120591(119905) or 120590(119905) in contrast to mostof the other oscillation criteria that apply to equations withnondecreasing arguments

Appendix

In this appendix for completeness we give the algorithmon MATLAB software used in Example 1 for calculation oflim sup119905rarrinfin1198651(119905) ≃ 10417 For Example 2 the algorithm isomitted since it is similar to the one in Example 1

Algorithm for Example 1

clear clcc = 12119899 = 501205820 = 114765a5 = 19b5 = 703h5 = (b5 minus a5)119899

for i5 = 1 1 119899 + 1x5 = a5 + (i5 minus 1) lowast h5a4 = TFunction(x5)b4 = 19h4 = (b4 minus a4)119899for i4 = 1 1 119899 + 1x4 = a4 + (i4 minus 1) lowast h4a3 = TFunction(x4)b3 = x4h3 = (b3 minus a3)119899for i3 = 1 1 119899 + 1x3 = a3 + (i3 minus 1) lowast h3a2 = TFunction(x3)b2 = x3h2 = (b2 minus a2)119899for i2 = 1 1 119899 + 1x2 = a2 + (i2 minus 1) lowast h2a1 = TFunction(x2)b1 = x3h1 = (b1 minus a1)119899for i1 = 1 1 119899 + 1x1 = a1 + (i1 minus 1) lowast h1f1(i1) = c lowast exp(1205820 lowast c lowast (x1 minus TFunction(x1)))endI1 = f1(1) + f1(119899 + 1)for i1 = 2 2 119899I1 = I1 + f1(i1) lowast 4endfor i1 = 3 2 119899 minus 1I1 = I1 + f1(i1) lowast 2endI1 = I1 lowast h13f2(i2) = c lowast exp(I1)endI2 = f2(1) + f2(119899 + 1)for i2 = 2 2 119899I2 = I2 + f2(i2) lowast 4endfor i2 = 3 2 119899 minus 1I2 = I2 + f2(i2) lowast 2endI2 = I2 lowast h23f3(i3) = c lowast (1 + I2)end

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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Page 4: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

4 Complexity

then all solutions of (E) are oscillatory Here 119892(119905) is a nonde-creasing continuous function such that 120591(119905) le 119892(119905) le 119905 119905 ge1199051 for some 1199051 ge 1199050 Clearly 119892(119905) is more general than ℎ(119905)defined by (13)

Recently Chatzarakis [15 16] proved that if for some 119895 isinN

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901119895 (119906) 119889119906)119889119904 gt 1 (20)

or

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901119895 (119906) 119889119906)119889119904

gt 1 minus 119863 (120572) (21)

where

119901119895 (119905)= 119901 (119905) [1 + int119905

120591(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901119895minus1 (119906) 119889119906)119889119904] (22)

with 1199010(119905) = 119901(119905) then all solutions of (E) are oscillatoryLately Chatzarakis [17] studied a more general form of(E) namely

1199091015840 (119905) + 119898sum119894=1

119901119894 (119905) 119909 (120591119894 (119905)) = 0 119905 ge 1199050 (23)

and established sufficient oscillation conditions Those con-ditions can lead to (20) and (21) when119898 = 122 ADEs ByTheorem 243 [29] if

LA gt 1 (24)

then all solutions of (E1015840) are oscillatoryIn 1984 Fukagai and Kusano [21] proved that if

120573 gt 1119890 (25)

then all solutions of (E1015840) are oscillatory while ifint120590(119905)119905119902 (119904) 119889119904 le 1119890 for all sufficiently large 119905 (26)

then (E1015840) has a nonoscillatory solutionAssume that the argument 120590(119905) is not necessarily mono-

tone Set

120588 (119905) = inf119904ge119905120590 (119904) 119905 ge 1199050 (27)

Clearly the function 120588(119905) is nondecreasing and 120590(119905) ge 120588(119905) gt119905 for all 119905 ge 1199050In 2015 Chatzarakis and Ocalan [18] proved that if

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904 gt 1 (28)

or

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904 gt 1119890 (29)

then all solutions of (E1015840) are oscillatoryRecently Chatzarakis [15 16] proved that if for some 119895 isin

N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902119895 (119906) 119889119906)119889119904 gt 1 (30)

or

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902119895 (119906) 119889119906)119889119904

gt 1 minus 119863 (120573) (31)

where119902119895 (119905)= 119902 (119905) [1 + int120590(119905)

119905119902 (119904) exp(int120590(119904)

120588(119905)119902119895minus1 (119906) 119889119906)119889119904]

119895 ge 1(32)

with 1199020(119905) = 119902(119905) then all solutions of (E1015840) oscillateLately Chatzarakis [17] studied a more general form of(E1015840) namely

1199091015840 (119905) minus 119898sum119894=1

119902119894 (119905) 119909 (120590119894 (119905)) = 0 119905 ge 1199050 (33)

and established sufficient oscillation conditions Those con-ditions can lead to (30) and (31) when119898 = 13 Main Results

31 DDEs In our main results we state theorems establish-ing new sufficient oscillation conditions For the proofs ofthose theorems we use the following lemmas

Lemma 3 (see [19 Lemma 211]) Assume that ℎ(119905) is definedby (13) Then

120572 fl lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 = lim inf

119905rarrinfinint119905ℎ(119905)119901 (119904) 119889119904 (34)

Lemma 4 (see [19 Lemma 213]) Assume that ℎ(119905) is definedby (13) 120572 isin (0 1119890] and 119909(119905) is an eventually positive solutionof (E) Then

lim inf119905rarrinfin

119909 (119905)119909 (ℎ (119905)) ge 119863 (120572) (35)

Lemma 5 (see [26]) Assume that ℎ(119905) is defined by (13) 120572 isin(0 1119890] and 119909(119905) is an eventually positive solution of (E) Then

lim inf119905rarrinfin

119909 (ℎ (119905))119909 (119905) ge 1205820 (36)

where 1205820 is the smaller root of the equation 120582 = 119890120572120582

Complexity 5

Theorem 6 Let ℎ(119905) be defined by (13) and for some 119895 isin Nlim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1(37)

where

119875119895 (119905) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895minus1 (120585) 119889120585) 119889119906)119889119904]

(38)

with 1198750(119905) = 1205820119901(119905) and let 1205820 be the smaller root of theequation 120582 = 119890120572120582 Then all solutions of (E) oscillateProof Assume for the sake of contradiction that there existsa nonoscillatory solution 119909(119905) of (E) Since minus119909(119905) is also asolution of (E) we can confine our discussion only to thecase where the solution 119909(119905) is eventually positiveThen thereexists a real number 1199051 gt 1199050 such that 119909(119905) 119909(120591(119905)) gt 0 for all119905 ge 1199051 Thus from (E) we have

1199091015840 (119905) = minus119901 (119905) 119909 (120591 (119905)) le 0 forall119905 ge 1199051 (39)

which means that 119909(119905) is an eventually nonincreasing func-tion of positive numbers Taking into account the fact that120591(119905) le ℎ(119905) (E) implies that

1199091015840 (119905) + 119901 (119905) 119909 (ℎ (119905)) le 0 119905 ge 1199051 (40)

Observe that (36) implies that for each 120598 gt 0 there exists areal number 119905120598 such that

119909 (ℎ (119905))119909 (119905) gt 1205820 minus 120598 forall119905 ge 119905120598 ge 1199051 (41)

Combining inequalities (40) and (41) we obtain

1199091015840 (119905) + 119901 (119905) (1205820 minus 120598) 119909 (119905) le 0 119905 ge 119905120598 (42)

or

1199091015840 (119905) + 1198750 (119905 120598) 119909 (119905) le 0 119905 ge 119905120598 (43)

where

1198750 (119905 120598) = 119901 (119905) (1205820 minus 120598) (44)

Applying the Gronwall inequality in (43) we conclude that

119909 (119904) ge 119909 (119905) exp(int1199051199041198750 (120585 120598) 119889120585) 119905 ge 119904 ge 119905120598 (45)

Now we divide (E) by 119909(119905) gt 0 and integrate on [119904 119905] sominusint119905119904

1199091015840 (119906)119909 (119906) 119889119906 = int119905

119904119901 (119906) 119909 (120591 (119906))119909 (119906) 119889119906 (46)

or

ln 119909 (119904)119909 (119905) = int119905

119904119901 (119906) 119909 (120591 (119906))119909 (119906) 119889119906 119905 ge 119904 ge 119905120598 (47)

Since 120591(119906) lt 119906 equality (47) givesln 119909 (119904)119909 (119905) = int

119905

119904119901 (119906) 119909 (120591 (119906))119909 (119906) 119889119906

ge int119905119904119901 (119906) 119909 (119906)119909 (119906) exp(int

119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906

= int119905119904119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906

(48)

or

119909 (119904)ge 119909 (119905) exp(int119905

119904119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906) (49)

Substituting 120591(119904) for 119904 in (49) we get

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906) (50)

Integrating (E) from 120591(119905) to 119905 we have119909 (119905) minus 119909 (120591 (119905)) + int119905

120591(119905)119901 (119904) 119909 (120591 (119904)) 119889119904 = 0 (51)

Combining (50) and (51) we obtain

119909 (119905) minus 119909 (120591 (119905)) + 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904

le 0(52)

Multiplying inequality (52) by 119901(119905) we find119901 (119905) 119909 (119905) minus 119901 (119905) 119909 (120591 (119905)) + 119901 (119905) 119909 (119905) int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904

le 0(53)

which in view of (E) becomes

1199091015840 (119905) + 119901 (119905) 119909 (119905) + 119901 (119905) 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904

le 0(54)

6 Complexity

Hence for sufficiently large 1199051199091015840 (119905) + 119901 (119905) [1 + int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904]

sdot 119909 (119905) le 0(55)

or

1199091015840 (119905) + 1198751 (119905 120598) 119909 (119905) le 0 (56)

where

1198751 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904]

(57)

Clearly (56) resembles (43) if we replace 1198750 by 1198751 Thusintegrating (56) on [119904 119905] yields

119909 (119904) ge 119909 (119905) exp(int1199051199041198751 (120585 120598) 119889120585) (58)

Repeating steps (45) through (50) we can see that 119909 satisfiesthe inequality

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906) (59)

Combining now (51) and (59) we obtain

119909 (119905) minus 119909 (120591 (119905)) + 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904

le 0(60)

Multiplying inequality (60) by 119901(119905) as before we find1199091015840 (119905) + 119901 (119905) [1 + int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904]

sdot 119909 (119905) le 0(61)

Therefore for sufficiently large 119905 we have1199091015840 (119905) + 1198752 (119905 120598) 119909 (119905) le 0 (62)

where

1198752 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904]

(63)

It becomes apparent now that by repeating the above stepswe can build inequalities on 1199091015840(119905) with progressively higherindices 119875119895(119905 120598) 119895 isin N In general for sufficiently large 119905 thepositive solution 119909(119905) satisfies the inequality

1199091015840 (119905) + 119875119895 (119905 120598) 119909 (119905) le 0 119895 isin N (64)

where

119875119895 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895minus1 (120585 120598) 119889120585) 119889119906)119889119904]

(65)

Proceeding to final step we recall that ℎ(119905) defined by (13) isa nondecreasing function Since 120591(119904) le ℎ(119904) le ℎ(119905) we have119909 (120591 (119904)) ge 119909 (ℎ (119905))sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (66)

Hence

119909 (119905) minus 119909 (ℎ (119905)) + 119909 (ℎ (119905)) int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(67)

or

119909 (ℎ (119905)) [int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1] lt 0(68)

Thus

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1 lt 0(69)

Taking the limit as 119905 rarr infin we have

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1(70)

Complexity 7

Since 120598may be taken arbitrarily small this inequality contra-dicts (37)

This completes the proof of the theorem

Theorem 7 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1 minus 119863 (120572) (71)

where 119875119895 is defined by (38) then all solutions of (E) oscillateProof Assume 119909 is an eventually positive solution of (E)Clearly (67) is satisfied for sufficiently large 119905 Thus

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus 119909 (119905)119909 (ℎ (119905)) (72)

which implies that

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus lim inf119905rarrinfin

119909 (119905)119909 (ℎ (119905)) (73)

Using Lemmas 3 and 4 it is evident that inequality (35) issatisfied Thus (73) leads to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus 119863 (120572) (74)

Since 120598may be taken arbitrarily small this inequality contra-dicts (71)

This completes the proof of the theorem

Theorem 8 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119863 (120572) (75)

where 119875119895 is defined by (38) then all solutions of (E) oscillateProof Assume 119909 is an eventually positive solution of (E)Then as in the proof of Theorem 6 for sufficiently large 119905we conclude that

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (76)

Integrating (E) from ℎ(119905) to 119905 and using (76) we obtain

119909 (119905) minus 119909 (ℎ (119905)) + int119905ℎ(119905)119901 (119904) 119909 (119905)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(77)

or

minus 119909 (ℎ (119905)) + int119905ℎ(119905)119901 (119904) 119909 (119905)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 0(78)

Hence

119909 (ℎ (119905)) [ 119909 (119905)119909 (ℎ (119905)) int119905

ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1] lt 0(79)

which yields for all sufficiently large 119905int119905ℎ(119905)119901 (119904)sdot exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 119909 (ℎ (119905))119909 (119905)

(80)

8 Complexity

and consequently

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le lim sup119905rarrinfin

119909 (ℎ (119905))119909 (119905) (81)

Taking into account the fact that (35) is satisfied inequality(81) leads to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1119863 (120572) (82)

which contradicts (75) when 120598 rarr 0This completes the proof of the theorem

Theorem 9 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120572) (83)

where 119875119895 is defined by (38) and 1205820 is the smaller root of theequation 120582 = 119890120572120582 then all solutions of (E) oscillateProof Let 119909 be an eventually positive solution of (E) As inthe proof ofTheorem 8 we can show that (76) holds namely

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (84)

Since 120591(119904) le ℎ(119904) inequality (84) gives119909 (120591 (119904)) ge 119909 (ℎ (119904))sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (85)

By Lemma 5 for each 120598 gt 0 there exists a real number 119905120598 suchthat

119909 (ℎ (119905))119909 (119905) gt 1205820 minus 120598 forall119905 ge 119905120598 ge 1199051 (86)

Note that by the nondecreasing nature of the function119909(ℎ(119905))119909(119904) in 119904 it holds1 = 119909 (ℎ (119905))119909 (ℎ (119905)) le 119909 (ℎ (119905))119909 (119904) le 119909 (ℎ (119905))119909 (119905)

119905120598 le ℎ (119905) le 119904 le 119905(87)

In particular for 120598 isin (0 1205820 minus 1) by continuity we concludethat there exists a real number 119905lowast isin (ℎ(119905) 119905] satisfying

1 lt 1205820 minus 120598 = 119909 (ℎ (119905))119909 (119905lowast) (88)

Integrating (E) from 119905lowast to 119905 and using (85) we obtain

119909 (119905) minus 119909 (119905lowast) + 119909 (ℎ (119905)) int119905119905lowast119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(89)

or

int119905119905lowast119901 (119904) exp(intℎ(119904)

120591(119904)119901 (119906)

sdot exp(int119906120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 119909 (119905lowast)119909 (ℎ (119905)) minus 119909 (119905)119909 (ℎ (119905)) (90)

Using (88) andLemma4we deduce that for the 120598 consideredthere exists a real number 1199051015840120598 ge 119905120598 such that

int119905119905lowast119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 11205820 minus 120598 minus 119863 (120572) + 120598(91)

for 119905 ge 1199051015840120598Dividing (E) by119909(119905) integrating from ℎ(119905) to 119905lowast and using

(85) we deduce that

int119905lowastℎ(119905)119901 (119904)sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(92)

Complexity 9

Clearly by means of (36) 119909(ℎ(119904))119909(119904) gt 1205820minus120598 for 119904 ge ℎ(119905) ge1199051015840120598 Hence for all sufficiently large 119905 we conclude that(1205820 minus 120598)int119905

lowast

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(93)

or

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minus 11205820 minus 120598 int119905lowast

ℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904 = 11205820 minus 120598 ln119909 (ℎ (119905))119909 (119905lowast)

= ln (1205820 minus 120598)1205820 minus 120598

(94)

that is

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt ln (1205820 minus 120598)1205820 minus 120598 (95)

Using (91) along with (95) we get

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 + ln (1205820 minus 120598)1205820 minus 120598 minus 119863 (120572) + 120598(96)

which contradicts (83) when 120598 rarr 0This completes the proof of the theorem

Theorem 10 Let ℎ(119905) be defined by (13) If for some 119895 isin Nlim inf119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119890 (97)

where 119875119895 is defined by (38) then all solutions of (E) oscillate

Proof For the sake of contradiction let 119909 be a nonincreasingeventually positive solution and 1199051 gt 1199050 be such that 119909(119905) gt 0and 119909(120591(119905)) gt 0 for all 119905 ge 1199051 We note that wemay obtain (85)as in the proof of Theorem 9

Dividing (E) by 119909(119905) and integrating from ℎ(119905) to 119905 wehave

ln(119909 (ℎ (119905))119909 (119905) ) = int119905

ℎ(119905)119901 (119904) 119909 (120591 (119904))119909 (119904) 119889119904 forall119905 ge 1199052 ge 1199051 (98)

from which in view of 120591(119904) le ℎ(119904) and (85) we get

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(99)

Since 119909 is nonincreasing and ℎ(119904) lt 119904 inequality (99)becomes

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(100)

From (97) it is clear that there exists a constant 119888 gt 0 suchthat

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

ge 119888 gt 1119890 (101)

Choose 1198881015840 such that 119888 gt 1198881015840 gt 1119890 For every 120598 gt 0 such that119888 minus 120598 gt 1198881015840 we haveint119905ℎ(t)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 119888 minus 120598 gt 1198881015840 gt 1119890 (102)

Combining inequalities (100) and (102) we obtain

ln(119909 (ℎ (119905))119909 (119905) ) gt 1198881015840 (103)

or119909 (ℎ (119905))119909 (119905) gt 1198901198881015840 gt 1198901198881015840 gt 1 (104)

which yields

119909 (ℎ (119905)) gt (1198901198881015840) 119909 (119905) (105)

10 Complexity

Following the above steps we can inductively show that forany positive integer 119896

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 for sufficiently large 119905 (106)

Since 1198901198881015840 gt 1 there is a natural number 119896 isin N satisfying119896 gt 2[ln 2 minus ln 1198881015840](1 + ln 1198881015840) such that for 119905 sufficiently large

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 gt ( 21198881015840 )2 (107)

Further (cf [13 24]) for sufficiently large 119905 there exists a realnumber 119905119898 isin (ℎ(119905) 119905) such that

int119905119898ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402 int119905119905119898

119901 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402

(108)

Integrating (E) from ℎ(119905) to 119905119898 using (85) and the fact that119909(119905) gt 0 we obtain119909 (ℎ (119905)) gt 119909 (ℎ (119905119898)) int119905119898

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(109)

which in view of the first inequality in (108) implies that

119909 (ℎ (119905)) gt 11988810158402 119909 (ℎ (119905119898)) (110)

Similarly integrating (E) from 119905119898 to 119905 using (85) and the factthat 119909(119905) gt 0 we have119909 (119905119898) gt 119909 (ℎ (119905)) int119905

119905119898

119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(111)

which in view of the second inequality in (108) yields

119909 (119905119898) gt 11988810158402 119909 (ℎ (119905)) (112)

Combining inequalities (110) and (112) we deduce that

119909 (ℎ (119905119898)) lt 21198881015840 119909 (ℎ (119905)) lt ( 21198881015840 )2 119909 (119905119898) (113)

which contradicts (107)The proof of the theorem is complete

32 ADEs Analogous oscillation conditions to thoseobtained for the delay equation (E) can be derived for the(dual) advanced differential equation (E1015840) by followingsimilar arguments with the ones employed for obtainingTheorems 6minus10Theorem 11 Let 120588(119905) be defined by (27) and for some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1

(114)

where

119876119895 (119905) = 119902 (119905) [1 + int120590(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895minus1 (120585) 119889120585)119889119906)119889119904]

(115)

with 1198760(119905) = 1205820119902(119905) and let 1205820 be the smaller root of theequation 120582 = 119890120573120582 Then all solutions of (E1015840) oscillateTheorem 12 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 minus 119863 (120573)

(116)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillateTheorem 13 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119863 (120573) (117)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate

Complexity 11

Theorem 14 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120573) (118)

where 119876119895 is defined by (115) and 1205820 is the smaller root of theequation 120582 = 119890120573120582 then all solutions of (E1015840) oscillateTheorem 15 Let 120588(119905) be defined by (27) If for some 119895 isin N

lim inf119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119890 (119)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate33 Differential Inequalities A slight modification in theproofs of Theorems 6minus15 leads to the following results aboutdifferential inequalities

Theorem 16 Assume that all the conditions ofTheorem 6 [11] 7 [12] 8 [13] 9 [14] or 10 [15] hold Then(119894) the delay [advanced] differential inequality

1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) le 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) ge 0]

119905 ge 1199050(120)

has no eventually positive solutions(119894119894) the delay [advanced] differential inequality1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) ge 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) le 0]

119905 ge 1199050(121)

has no eventually negative solutions

Remark 17 The oscillation criteria established in this paperall depend on 1205820 (see eg (37) and (71)) in contrast to theconditions obtained in [15 16] and in [17 for m = 1] In factthe left-hand side of conditions (37) and (71) depends on 1205820which is not the casewith the left-hand side of conditions (20)and (21) Since 1205820 gt 1 when 120572 isin (0 1119890] it is obvious that

1198750 (119905) = 1205820119901 (119905) gt 119901 (119905) = 1199010 (119905) (122)

Consequently the left-hand side of conditions (37) and(71) is greater than the corresponding parts of (20) and(21) respectively This is the reason why the conditions inthis paper improve on all known conditions mentioned inSection 2

4 Examples and Comments

The oscillation tests we have proposed and established in themain results involve an iterative procedure We iterativelycompute limsup and liminf on the terms 119875119895(119905) and119876119895(119905) 119895 isinN of a recurrent relation defined on the coefficients and thedeviating argument of an equation of the form (E) or (E1015840)to determine whether that equation is oscillatory But thiscomputation cannot be performed on paper but by meansof a program numerically computing limsup and liminfThe examples below illustrate the significance of our resultsand indicate the high level of improvement in the oscillationcriteria The calculations were performed using MATLABcode

Example 1 Consider the delay differential equation

1199091015840 (119905) + 325119909 (120591 (119905)) = 0 119905 ge 0 (123)

with (see Figure 1(a))

120591 (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]minus4119905 + 40119896 + 9 if 119905 isin [8119896 + 2 8119896 + 3]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 3 8119896 + 4]minus4119905 + 40119896 + 18 if 119905 isin [8119896 + 4 8119896 + 5]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 5 8119896 + 6]minus2119905 + 24119896 + 15 if 119905 isin [8119896 + 6 8119896 + 7]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 7 8119896 + 8]

(124)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (13) we see (Figure 1(b)) that

ℎ (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]8119896 + 1 if 119905 isin [8119896 + 2 8119896 + 195 ]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 195 8119896 + 4]8119896 + 2 if 119905 isin [8119896 + 4 8119896 + 295 ]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 295 8119896 + 6]8119896 + 3 if 119905 isin [8119896 + 6 8119896 + 446 ]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 446 8119896 + 8]

(125)

It is obvious that

120572 = lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 = lim inf

119905rarrinfinint8119896+28119896+1

325119889119904= 012

(126)

12 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

minus1minus2

651 2 3 4 8 9 10 11 12 13 15 16

1415

minus3

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

minus1minus2

t

195 295

1415

7 14651 2 3 4 8 9 10 11 12 13 15 16

446

y = tℎ(t)

(b)

Figure 1 The graphs of 120591(119905) and ℎ(119905)

and therefore the smaller root of 119890012120582 = 120582 is 1205820 = 114765Observe that the function 119865119895 R0 rarr R+ defined as

119865119895 (119905) = int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

(127)

attains its maximum at 119905 = 8119896+446 119896 isin N0 for every 119895 isin NSpecifically

1198651 (119905 = 8119896 + 446 ) = int8119896+446

8119896+3119901 (119904)

sdot exp(int8119896+3120591(119904)

119901 (119906) exp(int119906120591(119906)1198751 (120585) 119889120585) 119889119906)119889119904

(128)

with

1198751 (120585) = 119901 (120585) [1 + int120585120591(120585)119901 (V)

sdot exp(int120585120591(V)119901 (119908) exp(int119908

120591(119908)1205820119901 (119911) 119889119911) 119889119908)119889V]

(129)

Using MATLAB we obtain

1198651 (119905 = 8119896 + 446 ) ≃ 10417 (130)

and therefore

lim sup119905rarrinfin

1198651 (119905) ≃ 10417 gt 1 (131)

Hence condition (37) of Theorem 6 is satisfied for 119895 = 1Consequently all solutions of (123) are oscillatory

Observe however that

LD = lim sup119896rarrinfin

int8119896+4468119896+3

335119889119904 = 052 lt 1120572 = 012 lt 1119890

052 lt 1 + ln 12058201205820 minus 119863 (120572) ≃ 09831(132)

Note that the functionΦ119895 defined by

Φ119895 (119905) = int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

ℎ(119904)119901 (119906) 120595119895 (119906) 119889119906)119889119904

119895 ge 2(133)

attains its maximum at 119905 = 8119896 + 446 119896 isin N0 for every 119895 ge 2Specifically

Φ2 (8119896 + 446 )= int8119896+4468119896+3

119901 (119904) exp(int8119896+3ℎ(119904)

119901 (119904) 1205952 (119906) 119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 exp(int119906

120591(119906)

325 sdot 0 119889119908)119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 sdot 1 119889119906)119889119904= 325 sdot [int

8119896+195

8119896+3exp( 325 int

8119896+3

8119896+1119889119906)119889119904

Complexity 13

+ int8119896+48119896+195

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+2958119896+4

exp( 325 int8119896+3

8119896+2119889119906)119889119904

+ int8119896+68119896+295

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+4468119896+6

exp( 325 int8119896+3

8119896+3119889119906)119889119904] ≃ 057983

(134)

Thus

lim sup119905rarrinfin

Φ2 (119905) ≃ 057983 lt 1 minus 119863 (120572) ≃ 099174 (135)

Also

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int8119896+4468119896+3

325 exp(int8119896+3

120591(119904)

325119889119906)119889119904= 325 sdot lim sup

119905rarrinfin[int8119896+48119896+3

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+58119896+4

exp( 325 int8119896+3

minus4119904+40119896+18119889119906)119889119904

+ int8119896+68119896+5

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+78119896+6

exp( 325 int8119896+3

minus2119904+24119896+15119889119906)119889119904

+ int8119896+4468119896+7

exp( 325 int8119896+3

6119904minus40119896minus41119889119906)119889119904] ≃ 07043

lt 107043 lt 1 minus 119863 (120572) ≃ 099174

(136)

In addition

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)1199011 (119906) 119889119906)119889119904 ≃ 08052

lt 108052 lt 1 minus 119863 (120572) ≃ 099174

(137)

That is none of conditions (8) (9) (12) (14) (for 119895 = 2) (16)(17) (20) (for 119895 = 1) and (21) (for 119895 = 1) is satisfiedComment The improvement of condition (37) over thecorresponding condition (8) is significant approximately10033 We get this measure by comparing the values inthe left-hand side of those conditions Also the improvementover conditions (14) (16) and (20) is very satisfactory

around 7966 479 and 2937 respectively In additioncondition (37) is satisfied from the first iteration whileconditions (14) (20) and (21) need more than one iteration

Example 2 (taken and adapted from [17]) Consider theadvanced differential equation

1199091015840 (119905) minus 3332500119909 (120590 (119905)) = 0 119905 ge 0 (138)

with (see Figure 2(a))

120590 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 2]minus3119905 + 20119896 + 13 if 119905 isin [5119896 + 2 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 4]minus119905 + 10119896 + 13 if 119905 isin [5119896 + 4 5119896 + 5]

(139)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (27) we see (Figure 2(b)) that

120588 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 125]5119896 + 4 if 119905 isin [5119896 + 125 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 38]5119896 + 8 if 119905 isin [5119896 + 38 5119896 + 5]

(140)

It is obvious that

120573 = lim inf119905rarrinfin

int5119896+45119896+3

3332500119889119904 = 01332 (141)

and therefore the smaller root of 11989001332120582 = 120582 is 1205820 = 116839Observe that the function 119866119895 R0 rarr R+ defined as

119866119895 (119905) = int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

(142)

attains its maximum at 119905 = 5119896 + 38 119896 isin N0 for every 119895 isin NSpecifically

1198661 (119905 = 5119896 + 38) = int5119896+85119896+38

119902 (119904)sdot exp(int120590(119904)

5119896+8119902 (119906) exp(int120590(119906)

1199061198761 (120585) 119889120585)119889119906)119889119904

(143)

with

1198761 (120585) = 119902 (120585) [1 + int120590(120585)120585119902 (V)

sdot exp(int120590(V)120585119902 (119908) exp(int120590(119908)

1199081205820119902 (119911) 119889119911)119889119908)119889V]

(144)

14 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

651 2 3 4 8 9 10 11 12 13

1415

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

7

t

14651

2 3 4 8 9 10 11 12 13

38

1415

125

y = t

(t)

(b)

Figure 2 The graphs of 120590(119905) and 120588(119905)

Using MATLAB we obtain

1198661 (119905 = 5119896 + 38) ≃ 09915 (145)

Thereforelim sup119905rarrinfin

1198661 (119905) ≃ 09915 gt 1 minus 119863 (120573) ≃ 09896 (146)

Hence condition (116) of Theorem 12 is satisfied for 119895 = 1Consequently all solutions of (138) oscillate

Observe however that

LA = lim sup119896rarrinfin

int5119896+85119896+38

3332500119889119904 = 055944 lt 1120573 = 01332 lt 1119890 lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int5119896+85119896+38

119902 (119904) exp(int120590(119904)5119896+8

119902 (119906) 119889119906)119889119904= lim sup119896rarrinfin

[int5119896+45119896+38

119902 (119904)sdot exp(int5119904minus20119896minus11

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+5

5119896+4119902 (119904)

sdot exp(intminus119904+10119896+135119896+8

119902 (119906) 119889119906)119889119904 + int5119896+65119896+5

119902 (119904)sdot exp(int5119896+8

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+7

5119896+6119902 (119904)

sdot exp(int4119904minus15119896minus165119896+8

119902 (119906) 119889119906)119889119904 + int5119896+85119896+7

119902 (119904)sdot exp(intminus3119904+20119896+33

5119896+8119902 (119906) 119889119906)119889119904] ≃ 06672 lt 1

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim inf119905rarrinfin

int5119896+45119896+3

119902 (119904) exp(int120590(119904)5119896+4

119902 (119906) 119889119906)119889119904≃ 01893 lt 1119890

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)1199021 (119906) 119889119906)119889119904 ≃ 07196

lt 107196 lt 1 minus 119863 (120573) ≃ 09896

(147)

That is none of conditions (24) (25) (28) (29) (30) (for 119895 =1) and (31) (for 119895 = 1) is satisfiedComment The improvement of condition (116) over thecorresponding condition (24) is significant approximately7723 We get this measure by comparing the values in theleft-hand side of those conditions Also the improvementover conditions (28) and (30) is very satisfactory around4861 and 3778 respectively In addition condition (116)is satisfied from the first iteration while conditions (30) and(31) need more than one iteration

Complexity 15

Remark 3 Similarly one can provide examples illustratingthe other main results

5 Concluding Remarks

In the present paper we have considered the oscillatorydynamics of differential equations having nonmonotonedeviating arguments and nonnegative coefficients New suf-ficient conditions have been established for the oscillation ofall solutions of (E) and (E1015840) These conditions include (37)(71) (75) (83) and (97) and (114) (116) (117) (118) and(119) for (E) and (E1015840) respectively Applying these conditionsinvolves a procedure that checks for oscillations by iterativelycomputing limsup and liminf on terms recursively definedon the equationrsquos coefficients and deviating argument

The main advantage of these conditions is that theyachieve a major improvement over all the related oscillationconditions for (E) [(E1015840)] in the literature For example con-dition (37) [(114)] improves upon the noniterative conditionsthat are reviewed in the introduction namely conditions (8)[(24)] (12) (16)equiv(20) (for 119895 = 1) [(28)equiv(30) (for 119895 = 1)]and (17)equiv(21) (for 119895 = 1) [(31) (for 119895 = 1)] That immediatelybecomes evident by inspecting the left-hand side of (37)[(114)] and the left-hand side of each of the above conditions

The improvement of (37) [(114)] over the other iterativeconditions namely (14) (for j gt 2) (20) (for 119895 gt 1) [(30)(for 119895 gt 1)] and (21) (for 119895 gt 1) [(31) (for 119895 gt 1)] is that itrequires far fewer iterations to establish oscillation than theother conditions

This advantage easily can be verified computationallyby running the MATLAB programs (see Appendix) forcomputing limsup and liminf and comparing the number ofiterations required by each condition to establish oscillationThen we see that we achieve a significant improvement overall known oscillation criteria

Another advantage and a significant departure from thelarge majority of the other studies is that the criteria in thispaper apply to a more general class of equations havingnonmonotone arguments 120591(119905) or 120590(119905) in contrast to mostof the other oscillation criteria that apply to equations withnondecreasing arguments

Appendix

In this appendix for completeness we give the algorithmon MATLAB software used in Example 1 for calculation oflim sup119905rarrinfin1198651(119905) ≃ 10417 For Example 2 the algorithm isomitted since it is similar to the one in Example 1

Algorithm for Example 1

clear clcc = 12119899 = 501205820 = 114765a5 = 19b5 = 703h5 = (b5 minus a5)119899

for i5 = 1 1 119899 + 1x5 = a5 + (i5 minus 1) lowast h5a4 = TFunction(x5)b4 = 19h4 = (b4 minus a4)119899for i4 = 1 1 119899 + 1x4 = a4 + (i4 minus 1) lowast h4a3 = TFunction(x4)b3 = x4h3 = (b3 minus a3)119899for i3 = 1 1 119899 + 1x3 = a3 + (i3 minus 1) lowast h3a2 = TFunction(x3)b2 = x3h2 = (b2 minus a2)119899for i2 = 1 1 119899 + 1x2 = a2 + (i2 minus 1) lowast h2a1 = TFunction(x2)b1 = x3h1 = (b1 minus a1)119899for i1 = 1 1 119899 + 1x1 = a1 + (i1 minus 1) lowast h1f1(i1) = c lowast exp(1205820 lowast c lowast (x1 minus TFunction(x1)))endI1 = f1(1) + f1(119899 + 1)for i1 = 2 2 119899I1 = I1 + f1(i1) lowast 4endfor i1 = 3 2 119899 minus 1I1 = I1 + f1(i1) lowast 2endI1 = I1 lowast h13f2(i2) = c lowast exp(I1)endI2 = f2(1) + f2(119899 + 1)for i2 = 2 2 119899I2 = I2 + f2(i2) lowast 4endfor i2 = 3 2 119899 minus 1I2 = I2 + f2(i2) lowast 2endI2 = I2 lowast h23f3(i3) = c lowast (1 + I2)end

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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Page 5: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

Complexity 5

Theorem 6 Let ℎ(119905) be defined by (13) and for some 119895 isin Nlim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1(37)

where

119875119895 (119905) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895minus1 (120585) 119889120585) 119889119906)119889119904]

(38)

with 1198750(119905) = 1205820119901(119905) and let 1205820 be the smaller root of theequation 120582 = 119890120572120582 Then all solutions of (E) oscillateProof Assume for the sake of contradiction that there existsa nonoscillatory solution 119909(119905) of (E) Since minus119909(119905) is also asolution of (E) we can confine our discussion only to thecase where the solution 119909(119905) is eventually positiveThen thereexists a real number 1199051 gt 1199050 such that 119909(119905) 119909(120591(119905)) gt 0 for all119905 ge 1199051 Thus from (E) we have

1199091015840 (119905) = minus119901 (119905) 119909 (120591 (119905)) le 0 forall119905 ge 1199051 (39)

which means that 119909(119905) is an eventually nonincreasing func-tion of positive numbers Taking into account the fact that120591(119905) le ℎ(119905) (E) implies that

1199091015840 (119905) + 119901 (119905) 119909 (ℎ (119905)) le 0 119905 ge 1199051 (40)

Observe that (36) implies that for each 120598 gt 0 there exists areal number 119905120598 such that

119909 (ℎ (119905))119909 (119905) gt 1205820 minus 120598 forall119905 ge 119905120598 ge 1199051 (41)

Combining inequalities (40) and (41) we obtain

1199091015840 (119905) + 119901 (119905) (1205820 minus 120598) 119909 (119905) le 0 119905 ge 119905120598 (42)

or

1199091015840 (119905) + 1198750 (119905 120598) 119909 (119905) le 0 119905 ge 119905120598 (43)

where

1198750 (119905 120598) = 119901 (119905) (1205820 minus 120598) (44)

Applying the Gronwall inequality in (43) we conclude that

119909 (119904) ge 119909 (119905) exp(int1199051199041198750 (120585 120598) 119889120585) 119905 ge 119904 ge 119905120598 (45)

Now we divide (E) by 119909(119905) gt 0 and integrate on [119904 119905] sominusint119905119904

1199091015840 (119906)119909 (119906) 119889119906 = int119905

119904119901 (119906) 119909 (120591 (119906))119909 (119906) 119889119906 (46)

or

ln 119909 (119904)119909 (119905) = int119905

119904119901 (119906) 119909 (120591 (119906))119909 (119906) 119889119906 119905 ge 119904 ge 119905120598 (47)

Since 120591(119906) lt 119906 equality (47) givesln 119909 (119904)119909 (119905) = int

119905

119904119901 (119906) 119909 (120591 (119906))119909 (119906) 119889119906

ge int119905119904119901 (119906) 119909 (119906)119909 (119906) exp(int

119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906

= int119905119904119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906

(48)

or

119909 (119904)ge 119909 (119905) exp(int119905

119904119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906) (49)

Substituting 120591(119904) for 119904 in (49) we get

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906) (50)

Integrating (E) from 120591(119905) to 119905 we have119909 (119905) minus 119909 (120591 (119905)) + int119905

120591(119905)119901 (119904) 119909 (120591 (119904)) 119889119904 = 0 (51)

Combining (50) and (51) we obtain

119909 (119905) minus 119909 (120591 (119905)) + 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904

le 0(52)

Multiplying inequality (52) by 119901(119905) we find119901 (119905) 119909 (119905) minus 119901 (119905) 119909 (120591 (119905)) + 119901 (119905) 119909 (119905) int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904

le 0(53)

which in view of (E) becomes

1199091015840 (119905) + 119901 (119905) 119909 (119905) + 119901 (119905) 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904

le 0(54)

6 Complexity

Hence for sufficiently large 1199051199091015840 (119905) + 119901 (119905) [1 + int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904]

sdot 119909 (119905) le 0(55)

or

1199091015840 (119905) + 1198751 (119905 120598) 119909 (119905) le 0 (56)

where

1198751 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904]

(57)

Clearly (56) resembles (43) if we replace 1198750 by 1198751 Thusintegrating (56) on [119904 119905] yields

119909 (119904) ge 119909 (119905) exp(int1199051199041198751 (120585 120598) 119889120585) (58)

Repeating steps (45) through (50) we can see that 119909 satisfiesthe inequality

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906) (59)

Combining now (51) and (59) we obtain

119909 (119905) minus 119909 (120591 (119905)) + 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904

le 0(60)

Multiplying inequality (60) by 119901(119905) as before we find1199091015840 (119905) + 119901 (119905) [1 + int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904]

sdot 119909 (119905) le 0(61)

Therefore for sufficiently large 119905 we have1199091015840 (119905) + 1198752 (119905 120598) 119909 (119905) le 0 (62)

where

1198752 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904]

(63)

It becomes apparent now that by repeating the above stepswe can build inequalities on 1199091015840(119905) with progressively higherindices 119875119895(119905 120598) 119895 isin N In general for sufficiently large 119905 thepositive solution 119909(119905) satisfies the inequality

1199091015840 (119905) + 119875119895 (119905 120598) 119909 (119905) le 0 119895 isin N (64)

where

119875119895 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895minus1 (120585 120598) 119889120585) 119889119906)119889119904]

(65)

Proceeding to final step we recall that ℎ(119905) defined by (13) isa nondecreasing function Since 120591(119904) le ℎ(119904) le ℎ(119905) we have119909 (120591 (119904)) ge 119909 (ℎ (119905))sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (66)

Hence

119909 (119905) minus 119909 (ℎ (119905)) + 119909 (ℎ (119905)) int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(67)

or

119909 (ℎ (119905)) [int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1] lt 0(68)

Thus

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1 lt 0(69)

Taking the limit as 119905 rarr infin we have

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1(70)

Complexity 7

Since 120598may be taken arbitrarily small this inequality contra-dicts (37)

This completes the proof of the theorem

Theorem 7 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1 minus 119863 (120572) (71)

where 119875119895 is defined by (38) then all solutions of (E) oscillateProof Assume 119909 is an eventually positive solution of (E)Clearly (67) is satisfied for sufficiently large 119905 Thus

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus 119909 (119905)119909 (ℎ (119905)) (72)

which implies that

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus lim inf119905rarrinfin

119909 (119905)119909 (ℎ (119905)) (73)

Using Lemmas 3 and 4 it is evident that inequality (35) issatisfied Thus (73) leads to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus 119863 (120572) (74)

Since 120598may be taken arbitrarily small this inequality contra-dicts (71)

This completes the proof of the theorem

Theorem 8 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119863 (120572) (75)

where 119875119895 is defined by (38) then all solutions of (E) oscillateProof Assume 119909 is an eventually positive solution of (E)Then as in the proof of Theorem 6 for sufficiently large 119905we conclude that

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (76)

Integrating (E) from ℎ(119905) to 119905 and using (76) we obtain

119909 (119905) minus 119909 (ℎ (119905)) + int119905ℎ(119905)119901 (119904) 119909 (119905)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(77)

or

minus 119909 (ℎ (119905)) + int119905ℎ(119905)119901 (119904) 119909 (119905)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 0(78)

Hence

119909 (ℎ (119905)) [ 119909 (119905)119909 (ℎ (119905)) int119905

ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1] lt 0(79)

which yields for all sufficiently large 119905int119905ℎ(119905)119901 (119904)sdot exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 119909 (ℎ (119905))119909 (119905)

(80)

8 Complexity

and consequently

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le lim sup119905rarrinfin

119909 (ℎ (119905))119909 (119905) (81)

Taking into account the fact that (35) is satisfied inequality(81) leads to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1119863 (120572) (82)

which contradicts (75) when 120598 rarr 0This completes the proof of the theorem

Theorem 9 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120572) (83)

where 119875119895 is defined by (38) and 1205820 is the smaller root of theequation 120582 = 119890120572120582 then all solutions of (E) oscillateProof Let 119909 be an eventually positive solution of (E) As inthe proof ofTheorem 8 we can show that (76) holds namely

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (84)

Since 120591(119904) le ℎ(119904) inequality (84) gives119909 (120591 (119904)) ge 119909 (ℎ (119904))sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (85)

By Lemma 5 for each 120598 gt 0 there exists a real number 119905120598 suchthat

119909 (ℎ (119905))119909 (119905) gt 1205820 minus 120598 forall119905 ge 119905120598 ge 1199051 (86)

Note that by the nondecreasing nature of the function119909(ℎ(119905))119909(119904) in 119904 it holds1 = 119909 (ℎ (119905))119909 (ℎ (119905)) le 119909 (ℎ (119905))119909 (119904) le 119909 (ℎ (119905))119909 (119905)

119905120598 le ℎ (119905) le 119904 le 119905(87)

In particular for 120598 isin (0 1205820 minus 1) by continuity we concludethat there exists a real number 119905lowast isin (ℎ(119905) 119905] satisfying

1 lt 1205820 minus 120598 = 119909 (ℎ (119905))119909 (119905lowast) (88)

Integrating (E) from 119905lowast to 119905 and using (85) we obtain

119909 (119905) minus 119909 (119905lowast) + 119909 (ℎ (119905)) int119905119905lowast119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(89)

or

int119905119905lowast119901 (119904) exp(intℎ(119904)

120591(119904)119901 (119906)

sdot exp(int119906120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 119909 (119905lowast)119909 (ℎ (119905)) minus 119909 (119905)119909 (ℎ (119905)) (90)

Using (88) andLemma4we deduce that for the 120598 consideredthere exists a real number 1199051015840120598 ge 119905120598 such that

int119905119905lowast119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 11205820 minus 120598 minus 119863 (120572) + 120598(91)

for 119905 ge 1199051015840120598Dividing (E) by119909(119905) integrating from ℎ(119905) to 119905lowast and using

(85) we deduce that

int119905lowastℎ(119905)119901 (119904)sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(92)

Complexity 9

Clearly by means of (36) 119909(ℎ(119904))119909(119904) gt 1205820minus120598 for 119904 ge ℎ(119905) ge1199051015840120598 Hence for all sufficiently large 119905 we conclude that(1205820 minus 120598)int119905

lowast

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(93)

or

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minus 11205820 minus 120598 int119905lowast

ℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904 = 11205820 minus 120598 ln119909 (ℎ (119905))119909 (119905lowast)

= ln (1205820 minus 120598)1205820 minus 120598

(94)

that is

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt ln (1205820 minus 120598)1205820 minus 120598 (95)

Using (91) along with (95) we get

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 + ln (1205820 minus 120598)1205820 minus 120598 minus 119863 (120572) + 120598(96)

which contradicts (83) when 120598 rarr 0This completes the proof of the theorem

Theorem 10 Let ℎ(119905) be defined by (13) If for some 119895 isin Nlim inf119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119890 (97)

where 119875119895 is defined by (38) then all solutions of (E) oscillate

Proof For the sake of contradiction let 119909 be a nonincreasingeventually positive solution and 1199051 gt 1199050 be such that 119909(119905) gt 0and 119909(120591(119905)) gt 0 for all 119905 ge 1199051 We note that wemay obtain (85)as in the proof of Theorem 9

Dividing (E) by 119909(119905) and integrating from ℎ(119905) to 119905 wehave

ln(119909 (ℎ (119905))119909 (119905) ) = int119905

ℎ(119905)119901 (119904) 119909 (120591 (119904))119909 (119904) 119889119904 forall119905 ge 1199052 ge 1199051 (98)

from which in view of 120591(119904) le ℎ(119904) and (85) we get

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(99)

Since 119909 is nonincreasing and ℎ(119904) lt 119904 inequality (99)becomes

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(100)

From (97) it is clear that there exists a constant 119888 gt 0 suchthat

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

ge 119888 gt 1119890 (101)

Choose 1198881015840 such that 119888 gt 1198881015840 gt 1119890 For every 120598 gt 0 such that119888 minus 120598 gt 1198881015840 we haveint119905ℎ(t)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 119888 minus 120598 gt 1198881015840 gt 1119890 (102)

Combining inequalities (100) and (102) we obtain

ln(119909 (ℎ (119905))119909 (119905) ) gt 1198881015840 (103)

or119909 (ℎ (119905))119909 (119905) gt 1198901198881015840 gt 1198901198881015840 gt 1 (104)

which yields

119909 (ℎ (119905)) gt (1198901198881015840) 119909 (119905) (105)

10 Complexity

Following the above steps we can inductively show that forany positive integer 119896

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 for sufficiently large 119905 (106)

Since 1198901198881015840 gt 1 there is a natural number 119896 isin N satisfying119896 gt 2[ln 2 minus ln 1198881015840](1 + ln 1198881015840) such that for 119905 sufficiently large

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 gt ( 21198881015840 )2 (107)

Further (cf [13 24]) for sufficiently large 119905 there exists a realnumber 119905119898 isin (ℎ(119905) 119905) such that

int119905119898ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402 int119905119905119898

119901 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402

(108)

Integrating (E) from ℎ(119905) to 119905119898 using (85) and the fact that119909(119905) gt 0 we obtain119909 (ℎ (119905)) gt 119909 (ℎ (119905119898)) int119905119898

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(109)

which in view of the first inequality in (108) implies that

119909 (ℎ (119905)) gt 11988810158402 119909 (ℎ (119905119898)) (110)

Similarly integrating (E) from 119905119898 to 119905 using (85) and the factthat 119909(119905) gt 0 we have119909 (119905119898) gt 119909 (ℎ (119905)) int119905

119905119898

119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(111)

which in view of the second inequality in (108) yields

119909 (119905119898) gt 11988810158402 119909 (ℎ (119905)) (112)

Combining inequalities (110) and (112) we deduce that

119909 (ℎ (119905119898)) lt 21198881015840 119909 (ℎ (119905)) lt ( 21198881015840 )2 119909 (119905119898) (113)

which contradicts (107)The proof of the theorem is complete

32 ADEs Analogous oscillation conditions to thoseobtained for the delay equation (E) can be derived for the(dual) advanced differential equation (E1015840) by followingsimilar arguments with the ones employed for obtainingTheorems 6minus10Theorem 11 Let 120588(119905) be defined by (27) and for some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1

(114)

where

119876119895 (119905) = 119902 (119905) [1 + int120590(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895minus1 (120585) 119889120585)119889119906)119889119904]

(115)

with 1198760(119905) = 1205820119902(119905) and let 1205820 be the smaller root of theequation 120582 = 119890120573120582 Then all solutions of (E1015840) oscillateTheorem 12 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 minus 119863 (120573)

(116)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillateTheorem 13 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119863 (120573) (117)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate

Complexity 11

Theorem 14 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120573) (118)

where 119876119895 is defined by (115) and 1205820 is the smaller root of theequation 120582 = 119890120573120582 then all solutions of (E1015840) oscillateTheorem 15 Let 120588(119905) be defined by (27) If for some 119895 isin N

lim inf119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119890 (119)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate33 Differential Inequalities A slight modification in theproofs of Theorems 6minus15 leads to the following results aboutdifferential inequalities

Theorem 16 Assume that all the conditions ofTheorem 6 [11] 7 [12] 8 [13] 9 [14] or 10 [15] hold Then(119894) the delay [advanced] differential inequality

1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) le 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) ge 0]

119905 ge 1199050(120)

has no eventually positive solutions(119894119894) the delay [advanced] differential inequality1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) ge 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) le 0]

119905 ge 1199050(121)

has no eventually negative solutions

Remark 17 The oscillation criteria established in this paperall depend on 1205820 (see eg (37) and (71)) in contrast to theconditions obtained in [15 16] and in [17 for m = 1] In factthe left-hand side of conditions (37) and (71) depends on 1205820which is not the casewith the left-hand side of conditions (20)and (21) Since 1205820 gt 1 when 120572 isin (0 1119890] it is obvious that

1198750 (119905) = 1205820119901 (119905) gt 119901 (119905) = 1199010 (119905) (122)

Consequently the left-hand side of conditions (37) and(71) is greater than the corresponding parts of (20) and(21) respectively This is the reason why the conditions inthis paper improve on all known conditions mentioned inSection 2

4 Examples and Comments

The oscillation tests we have proposed and established in themain results involve an iterative procedure We iterativelycompute limsup and liminf on the terms 119875119895(119905) and119876119895(119905) 119895 isinN of a recurrent relation defined on the coefficients and thedeviating argument of an equation of the form (E) or (E1015840)to determine whether that equation is oscillatory But thiscomputation cannot be performed on paper but by meansof a program numerically computing limsup and liminfThe examples below illustrate the significance of our resultsand indicate the high level of improvement in the oscillationcriteria The calculations were performed using MATLABcode

Example 1 Consider the delay differential equation

1199091015840 (119905) + 325119909 (120591 (119905)) = 0 119905 ge 0 (123)

with (see Figure 1(a))

120591 (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]minus4119905 + 40119896 + 9 if 119905 isin [8119896 + 2 8119896 + 3]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 3 8119896 + 4]minus4119905 + 40119896 + 18 if 119905 isin [8119896 + 4 8119896 + 5]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 5 8119896 + 6]minus2119905 + 24119896 + 15 if 119905 isin [8119896 + 6 8119896 + 7]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 7 8119896 + 8]

(124)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (13) we see (Figure 1(b)) that

ℎ (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]8119896 + 1 if 119905 isin [8119896 + 2 8119896 + 195 ]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 195 8119896 + 4]8119896 + 2 if 119905 isin [8119896 + 4 8119896 + 295 ]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 295 8119896 + 6]8119896 + 3 if 119905 isin [8119896 + 6 8119896 + 446 ]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 446 8119896 + 8]

(125)

It is obvious that

120572 = lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 = lim inf

119905rarrinfinint8119896+28119896+1

325119889119904= 012

(126)

12 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

minus1minus2

651 2 3 4 8 9 10 11 12 13 15 16

1415

minus3

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

minus1minus2

t

195 295

1415

7 14651 2 3 4 8 9 10 11 12 13 15 16

446

y = tℎ(t)

(b)

Figure 1 The graphs of 120591(119905) and ℎ(119905)

and therefore the smaller root of 119890012120582 = 120582 is 1205820 = 114765Observe that the function 119865119895 R0 rarr R+ defined as

119865119895 (119905) = int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

(127)

attains its maximum at 119905 = 8119896+446 119896 isin N0 for every 119895 isin NSpecifically

1198651 (119905 = 8119896 + 446 ) = int8119896+446

8119896+3119901 (119904)

sdot exp(int8119896+3120591(119904)

119901 (119906) exp(int119906120591(119906)1198751 (120585) 119889120585) 119889119906)119889119904

(128)

with

1198751 (120585) = 119901 (120585) [1 + int120585120591(120585)119901 (V)

sdot exp(int120585120591(V)119901 (119908) exp(int119908

120591(119908)1205820119901 (119911) 119889119911) 119889119908)119889V]

(129)

Using MATLAB we obtain

1198651 (119905 = 8119896 + 446 ) ≃ 10417 (130)

and therefore

lim sup119905rarrinfin

1198651 (119905) ≃ 10417 gt 1 (131)

Hence condition (37) of Theorem 6 is satisfied for 119895 = 1Consequently all solutions of (123) are oscillatory

Observe however that

LD = lim sup119896rarrinfin

int8119896+4468119896+3

335119889119904 = 052 lt 1120572 = 012 lt 1119890

052 lt 1 + ln 12058201205820 minus 119863 (120572) ≃ 09831(132)

Note that the functionΦ119895 defined by

Φ119895 (119905) = int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

ℎ(119904)119901 (119906) 120595119895 (119906) 119889119906)119889119904

119895 ge 2(133)

attains its maximum at 119905 = 8119896 + 446 119896 isin N0 for every 119895 ge 2Specifically

Φ2 (8119896 + 446 )= int8119896+4468119896+3

119901 (119904) exp(int8119896+3ℎ(119904)

119901 (119904) 1205952 (119906) 119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 exp(int119906

120591(119906)

325 sdot 0 119889119908)119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 sdot 1 119889119906)119889119904= 325 sdot [int

8119896+195

8119896+3exp( 325 int

8119896+3

8119896+1119889119906)119889119904

Complexity 13

+ int8119896+48119896+195

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+2958119896+4

exp( 325 int8119896+3

8119896+2119889119906)119889119904

+ int8119896+68119896+295

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+4468119896+6

exp( 325 int8119896+3

8119896+3119889119906)119889119904] ≃ 057983

(134)

Thus

lim sup119905rarrinfin

Φ2 (119905) ≃ 057983 lt 1 minus 119863 (120572) ≃ 099174 (135)

Also

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int8119896+4468119896+3

325 exp(int8119896+3

120591(119904)

325119889119906)119889119904= 325 sdot lim sup

119905rarrinfin[int8119896+48119896+3

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+58119896+4

exp( 325 int8119896+3

minus4119904+40119896+18119889119906)119889119904

+ int8119896+68119896+5

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+78119896+6

exp( 325 int8119896+3

minus2119904+24119896+15119889119906)119889119904

+ int8119896+4468119896+7

exp( 325 int8119896+3

6119904minus40119896minus41119889119906)119889119904] ≃ 07043

lt 107043 lt 1 minus 119863 (120572) ≃ 099174

(136)

In addition

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)1199011 (119906) 119889119906)119889119904 ≃ 08052

lt 108052 lt 1 minus 119863 (120572) ≃ 099174

(137)

That is none of conditions (8) (9) (12) (14) (for 119895 = 2) (16)(17) (20) (for 119895 = 1) and (21) (for 119895 = 1) is satisfiedComment The improvement of condition (37) over thecorresponding condition (8) is significant approximately10033 We get this measure by comparing the values inthe left-hand side of those conditions Also the improvementover conditions (14) (16) and (20) is very satisfactory

around 7966 479 and 2937 respectively In additioncondition (37) is satisfied from the first iteration whileconditions (14) (20) and (21) need more than one iteration

Example 2 (taken and adapted from [17]) Consider theadvanced differential equation

1199091015840 (119905) minus 3332500119909 (120590 (119905)) = 0 119905 ge 0 (138)

with (see Figure 2(a))

120590 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 2]minus3119905 + 20119896 + 13 if 119905 isin [5119896 + 2 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 4]minus119905 + 10119896 + 13 if 119905 isin [5119896 + 4 5119896 + 5]

(139)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (27) we see (Figure 2(b)) that

120588 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 125]5119896 + 4 if 119905 isin [5119896 + 125 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 38]5119896 + 8 if 119905 isin [5119896 + 38 5119896 + 5]

(140)

It is obvious that

120573 = lim inf119905rarrinfin

int5119896+45119896+3

3332500119889119904 = 01332 (141)

and therefore the smaller root of 11989001332120582 = 120582 is 1205820 = 116839Observe that the function 119866119895 R0 rarr R+ defined as

119866119895 (119905) = int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

(142)

attains its maximum at 119905 = 5119896 + 38 119896 isin N0 for every 119895 isin NSpecifically

1198661 (119905 = 5119896 + 38) = int5119896+85119896+38

119902 (119904)sdot exp(int120590(119904)

5119896+8119902 (119906) exp(int120590(119906)

1199061198761 (120585) 119889120585)119889119906)119889119904

(143)

with

1198761 (120585) = 119902 (120585) [1 + int120590(120585)120585119902 (V)

sdot exp(int120590(V)120585119902 (119908) exp(int120590(119908)

1199081205820119902 (119911) 119889119911)119889119908)119889V]

(144)

14 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

651 2 3 4 8 9 10 11 12 13

1415

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

7

t

14651

2 3 4 8 9 10 11 12 13

38

1415

125

y = t

(t)

(b)

Figure 2 The graphs of 120590(119905) and 120588(119905)

Using MATLAB we obtain

1198661 (119905 = 5119896 + 38) ≃ 09915 (145)

Thereforelim sup119905rarrinfin

1198661 (119905) ≃ 09915 gt 1 minus 119863 (120573) ≃ 09896 (146)

Hence condition (116) of Theorem 12 is satisfied for 119895 = 1Consequently all solutions of (138) oscillate

Observe however that

LA = lim sup119896rarrinfin

int5119896+85119896+38

3332500119889119904 = 055944 lt 1120573 = 01332 lt 1119890 lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int5119896+85119896+38

119902 (119904) exp(int120590(119904)5119896+8

119902 (119906) 119889119906)119889119904= lim sup119896rarrinfin

[int5119896+45119896+38

119902 (119904)sdot exp(int5119904minus20119896minus11

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+5

5119896+4119902 (119904)

sdot exp(intminus119904+10119896+135119896+8

119902 (119906) 119889119906)119889119904 + int5119896+65119896+5

119902 (119904)sdot exp(int5119896+8

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+7

5119896+6119902 (119904)

sdot exp(int4119904minus15119896minus165119896+8

119902 (119906) 119889119906)119889119904 + int5119896+85119896+7

119902 (119904)sdot exp(intminus3119904+20119896+33

5119896+8119902 (119906) 119889119906)119889119904] ≃ 06672 lt 1

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim inf119905rarrinfin

int5119896+45119896+3

119902 (119904) exp(int120590(119904)5119896+4

119902 (119906) 119889119906)119889119904≃ 01893 lt 1119890

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)1199021 (119906) 119889119906)119889119904 ≃ 07196

lt 107196 lt 1 minus 119863 (120573) ≃ 09896

(147)

That is none of conditions (24) (25) (28) (29) (30) (for 119895 =1) and (31) (for 119895 = 1) is satisfiedComment The improvement of condition (116) over thecorresponding condition (24) is significant approximately7723 We get this measure by comparing the values in theleft-hand side of those conditions Also the improvementover conditions (28) and (30) is very satisfactory around4861 and 3778 respectively In addition condition (116)is satisfied from the first iteration while conditions (30) and(31) need more than one iteration

Complexity 15

Remark 3 Similarly one can provide examples illustratingthe other main results

5 Concluding Remarks

In the present paper we have considered the oscillatorydynamics of differential equations having nonmonotonedeviating arguments and nonnegative coefficients New suf-ficient conditions have been established for the oscillation ofall solutions of (E) and (E1015840) These conditions include (37)(71) (75) (83) and (97) and (114) (116) (117) (118) and(119) for (E) and (E1015840) respectively Applying these conditionsinvolves a procedure that checks for oscillations by iterativelycomputing limsup and liminf on terms recursively definedon the equationrsquos coefficients and deviating argument

The main advantage of these conditions is that theyachieve a major improvement over all the related oscillationconditions for (E) [(E1015840)] in the literature For example con-dition (37) [(114)] improves upon the noniterative conditionsthat are reviewed in the introduction namely conditions (8)[(24)] (12) (16)equiv(20) (for 119895 = 1) [(28)equiv(30) (for 119895 = 1)]and (17)equiv(21) (for 119895 = 1) [(31) (for 119895 = 1)] That immediatelybecomes evident by inspecting the left-hand side of (37)[(114)] and the left-hand side of each of the above conditions

The improvement of (37) [(114)] over the other iterativeconditions namely (14) (for j gt 2) (20) (for 119895 gt 1) [(30)(for 119895 gt 1)] and (21) (for 119895 gt 1) [(31) (for 119895 gt 1)] is that itrequires far fewer iterations to establish oscillation than theother conditions

This advantage easily can be verified computationallyby running the MATLAB programs (see Appendix) forcomputing limsup and liminf and comparing the number ofiterations required by each condition to establish oscillationThen we see that we achieve a significant improvement overall known oscillation criteria

Another advantage and a significant departure from thelarge majority of the other studies is that the criteria in thispaper apply to a more general class of equations havingnonmonotone arguments 120591(119905) or 120590(119905) in contrast to mostof the other oscillation criteria that apply to equations withnondecreasing arguments

Appendix

In this appendix for completeness we give the algorithmon MATLAB software used in Example 1 for calculation oflim sup119905rarrinfin1198651(119905) ≃ 10417 For Example 2 the algorithm isomitted since it is similar to the one in Example 1

Algorithm for Example 1

clear clcc = 12119899 = 501205820 = 114765a5 = 19b5 = 703h5 = (b5 minus a5)119899

for i5 = 1 1 119899 + 1x5 = a5 + (i5 minus 1) lowast h5a4 = TFunction(x5)b4 = 19h4 = (b4 minus a4)119899for i4 = 1 1 119899 + 1x4 = a4 + (i4 minus 1) lowast h4a3 = TFunction(x4)b3 = x4h3 = (b3 minus a3)119899for i3 = 1 1 119899 + 1x3 = a3 + (i3 minus 1) lowast h3a2 = TFunction(x3)b2 = x3h2 = (b2 minus a2)119899for i2 = 1 1 119899 + 1x2 = a2 + (i2 minus 1) lowast h2a1 = TFunction(x2)b1 = x3h1 = (b1 minus a1)119899for i1 = 1 1 119899 + 1x1 = a1 + (i1 minus 1) lowast h1f1(i1) = c lowast exp(1205820 lowast c lowast (x1 minus TFunction(x1)))endI1 = f1(1) + f1(119899 + 1)for i1 = 2 2 119899I1 = I1 + f1(i1) lowast 4endfor i1 = 3 2 119899 minus 1I1 = I1 + f1(i1) lowast 2endI1 = I1 lowast h13f2(i2) = c lowast exp(I1)endI2 = f2(1) + f2(119899 + 1)for i2 = 2 2 119899I2 = I2 + f2(i2) lowast 4endfor i2 = 3 2 119899 minus 1I2 = I2 + f2(i2) lowast 2endI2 = I2 lowast h23f3(i3) = c lowast (1 + I2)end

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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Page 6: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

6 Complexity

Hence for sufficiently large 1199051199091015840 (119905) + 119901 (119905) [1 + int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904]

sdot 119909 (119905) le 0(55)

or

1199091015840 (119905) + 1198751 (119905 120598) 119909 (119905) le 0 (56)

where

1198751 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198750 (120585 120598) 119889120585) 119889119906)119889119904]

(57)

Clearly (56) resembles (43) if we replace 1198750 by 1198751 Thusintegrating (56) on [119904 119905] yields

119909 (119904) ge 119909 (119905) exp(int1199051199041198751 (120585 120598) 119889120585) (58)

Repeating steps (45) through (50) we can see that 119909 satisfiesthe inequality

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906) (59)

Combining now (51) and (59) we obtain

119909 (119905) minus 119909 (120591 (119905)) + 119909 (119905) int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904

le 0(60)

Multiplying inequality (60) by 119901(119905) as before we find1199091015840 (119905) + 119901 (119905) [1 + int119905

120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904]

sdot 119909 (119905) le 0(61)

Therefore for sufficiently large 119905 we have1199091015840 (119905) + 1198752 (119905 120598) 119909 (119905) le 0 (62)

where

1198752 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)1198751 (120585 120598) 119889120585) 119889119906)119889119904]

(63)

It becomes apparent now that by repeating the above stepswe can build inequalities on 1199091015840(119905) with progressively higherindices 119875119895(119905 120598) 119895 isin N In general for sufficiently large 119905 thepositive solution 119909(119905) satisfies the inequality

1199091015840 (119905) + 119875119895 (119905 120598) 119909 (119905) le 0 119895 isin N (64)

where

119875119895 (119905 120598) = 119901 (119905) [1 + int119905120591(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895minus1 (120585 120598) 119889120585) 119889119906)119889119904]

(65)

Proceeding to final step we recall that ℎ(119905) defined by (13) isa nondecreasing function Since 120591(119904) le ℎ(119904) le ℎ(119905) we have119909 (120591 (119904)) ge 119909 (ℎ (119905))sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (66)

Hence

119909 (119905) minus 119909 (ℎ (119905)) + 119909 (ℎ (119905)) int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(67)

or

119909 (ℎ (119905)) [int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1] lt 0(68)

Thus

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1 lt 0(69)

Taking the limit as 119905 rarr infin we have

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1(70)

Complexity 7

Since 120598may be taken arbitrarily small this inequality contra-dicts (37)

This completes the proof of the theorem

Theorem 7 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1 minus 119863 (120572) (71)

where 119875119895 is defined by (38) then all solutions of (E) oscillateProof Assume 119909 is an eventually positive solution of (E)Clearly (67) is satisfied for sufficiently large 119905 Thus

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus 119909 (119905)119909 (ℎ (119905)) (72)

which implies that

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus lim inf119905rarrinfin

119909 (119905)119909 (ℎ (119905)) (73)

Using Lemmas 3 and 4 it is evident that inequality (35) issatisfied Thus (73) leads to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus 119863 (120572) (74)

Since 120598may be taken arbitrarily small this inequality contra-dicts (71)

This completes the proof of the theorem

Theorem 8 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119863 (120572) (75)

where 119875119895 is defined by (38) then all solutions of (E) oscillateProof Assume 119909 is an eventually positive solution of (E)Then as in the proof of Theorem 6 for sufficiently large 119905we conclude that

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (76)

Integrating (E) from ℎ(119905) to 119905 and using (76) we obtain

119909 (119905) minus 119909 (ℎ (119905)) + int119905ℎ(119905)119901 (119904) 119909 (119905)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(77)

or

minus 119909 (ℎ (119905)) + int119905ℎ(119905)119901 (119904) 119909 (119905)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 0(78)

Hence

119909 (ℎ (119905)) [ 119909 (119905)119909 (ℎ (119905)) int119905

ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1] lt 0(79)

which yields for all sufficiently large 119905int119905ℎ(119905)119901 (119904)sdot exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 119909 (ℎ (119905))119909 (119905)

(80)

8 Complexity

and consequently

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le lim sup119905rarrinfin

119909 (ℎ (119905))119909 (119905) (81)

Taking into account the fact that (35) is satisfied inequality(81) leads to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1119863 (120572) (82)

which contradicts (75) when 120598 rarr 0This completes the proof of the theorem

Theorem 9 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120572) (83)

where 119875119895 is defined by (38) and 1205820 is the smaller root of theequation 120582 = 119890120572120582 then all solutions of (E) oscillateProof Let 119909 be an eventually positive solution of (E) As inthe proof ofTheorem 8 we can show that (76) holds namely

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (84)

Since 120591(119904) le ℎ(119904) inequality (84) gives119909 (120591 (119904)) ge 119909 (ℎ (119904))sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (85)

By Lemma 5 for each 120598 gt 0 there exists a real number 119905120598 suchthat

119909 (ℎ (119905))119909 (119905) gt 1205820 minus 120598 forall119905 ge 119905120598 ge 1199051 (86)

Note that by the nondecreasing nature of the function119909(ℎ(119905))119909(119904) in 119904 it holds1 = 119909 (ℎ (119905))119909 (ℎ (119905)) le 119909 (ℎ (119905))119909 (119904) le 119909 (ℎ (119905))119909 (119905)

119905120598 le ℎ (119905) le 119904 le 119905(87)

In particular for 120598 isin (0 1205820 minus 1) by continuity we concludethat there exists a real number 119905lowast isin (ℎ(119905) 119905] satisfying

1 lt 1205820 minus 120598 = 119909 (ℎ (119905))119909 (119905lowast) (88)

Integrating (E) from 119905lowast to 119905 and using (85) we obtain

119909 (119905) minus 119909 (119905lowast) + 119909 (ℎ (119905)) int119905119905lowast119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(89)

or

int119905119905lowast119901 (119904) exp(intℎ(119904)

120591(119904)119901 (119906)

sdot exp(int119906120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 119909 (119905lowast)119909 (ℎ (119905)) minus 119909 (119905)119909 (ℎ (119905)) (90)

Using (88) andLemma4we deduce that for the 120598 consideredthere exists a real number 1199051015840120598 ge 119905120598 such that

int119905119905lowast119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 11205820 minus 120598 minus 119863 (120572) + 120598(91)

for 119905 ge 1199051015840120598Dividing (E) by119909(119905) integrating from ℎ(119905) to 119905lowast and using

(85) we deduce that

int119905lowastℎ(119905)119901 (119904)sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(92)

Complexity 9

Clearly by means of (36) 119909(ℎ(119904))119909(119904) gt 1205820minus120598 for 119904 ge ℎ(119905) ge1199051015840120598 Hence for all sufficiently large 119905 we conclude that(1205820 minus 120598)int119905

lowast

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(93)

or

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minus 11205820 minus 120598 int119905lowast

ℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904 = 11205820 minus 120598 ln119909 (ℎ (119905))119909 (119905lowast)

= ln (1205820 minus 120598)1205820 minus 120598

(94)

that is

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt ln (1205820 minus 120598)1205820 minus 120598 (95)

Using (91) along with (95) we get

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 + ln (1205820 minus 120598)1205820 minus 120598 minus 119863 (120572) + 120598(96)

which contradicts (83) when 120598 rarr 0This completes the proof of the theorem

Theorem 10 Let ℎ(119905) be defined by (13) If for some 119895 isin Nlim inf119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119890 (97)

where 119875119895 is defined by (38) then all solutions of (E) oscillate

Proof For the sake of contradiction let 119909 be a nonincreasingeventually positive solution and 1199051 gt 1199050 be such that 119909(119905) gt 0and 119909(120591(119905)) gt 0 for all 119905 ge 1199051 We note that wemay obtain (85)as in the proof of Theorem 9

Dividing (E) by 119909(119905) and integrating from ℎ(119905) to 119905 wehave

ln(119909 (ℎ (119905))119909 (119905) ) = int119905

ℎ(119905)119901 (119904) 119909 (120591 (119904))119909 (119904) 119889119904 forall119905 ge 1199052 ge 1199051 (98)

from which in view of 120591(119904) le ℎ(119904) and (85) we get

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(99)

Since 119909 is nonincreasing and ℎ(119904) lt 119904 inequality (99)becomes

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(100)

From (97) it is clear that there exists a constant 119888 gt 0 suchthat

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

ge 119888 gt 1119890 (101)

Choose 1198881015840 such that 119888 gt 1198881015840 gt 1119890 For every 120598 gt 0 such that119888 minus 120598 gt 1198881015840 we haveint119905ℎ(t)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 119888 minus 120598 gt 1198881015840 gt 1119890 (102)

Combining inequalities (100) and (102) we obtain

ln(119909 (ℎ (119905))119909 (119905) ) gt 1198881015840 (103)

or119909 (ℎ (119905))119909 (119905) gt 1198901198881015840 gt 1198901198881015840 gt 1 (104)

which yields

119909 (ℎ (119905)) gt (1198901198881015840) 119909 (119905) (105)

10 Complexity

Following the above steps we can inductively show that forany positive integer 119896

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 for sufficiently large 119905 (106)

Since 1198901198881015840 gt 1 there is a natural number 119896 isin N satisfying119896 gt 2[ln 2 minus ln 1198881015840](1 + ln 1198881015840) such that for 119905 sufficiently large

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 gt ( 21198881015840 )2 (107)

Further (cf [13 24]) for sufficiently large 119905 there exists a realnumber 119905119898 isin (ℎ(119905) 119905) such that

int119905119898ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402 int119905119905119898

119901 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402

(108)

Integrating (E) from ℎ(119905) to 119905119898 using (85) and the fact that119909(119905) gt 0 we obtain119909 (ℎ (119905)) gt 119909 (ℎ (119905119898)) int119905119898

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(109)

which in view of the first inequality in (108) implies that

119909 (ℎ (119905)) gt 11988810158402 119909 (ℎ (119905119898)) (110)

Similarly integrating (E) from 119905119898 to 119905 using (85) and the factthat 119909(119905) gt 0 we have119909 (119905119898) gt 119909 (ℎ (119905)) int119905

119905119898

119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(111)

which in view of the second inequality in (108) yields

119909 (119905119898) gt 11988810158402 119909 (ℎ (119905)) (112)

Combining inequalities (110) and (112) we deduce that

119909 (ℎ (119905119898)) lt 21198881015840 119909 (ℎ (119905)) lt ( 21198881015840 )2 119909 (119905119898) (113)

which contradicts (107)The proof of the theorem is complete

32 ADEs Analogous oscillation conditions to thoseobtained for the delay equation (E) can be derived for the(dual) advanced differential equation (E1015840) by followingsimilar arguments with the ones employed for obtainingTheorems 6minus10Theorem 11 Let 120588(119905) be defined by (27) and for some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1

(114)

where

119876119895 (119905) = 119902 (119905) [1 + int120590(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895minus1 (120585) 119889120585)119889119906)119889119904]

(115)

with 1198760(119905) = 1205820119902(119905) and let 1205820 be the smaller root of theequation 120582 = 119890120573120582 Then all solutions of (E1015840) oscillateTheorem 12 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 minus 119863 (120573)

(116)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillateTheorem 13 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119863 (120573) (117)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate

Complexity 11

Theorem 14 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120573) (118)

where 119876119895 is defined by (115) and 1205820 is the smaller root of theequation 120582 = 119890120573120582 then all solutions of (E1015840) oscillateTheorem 15 Let 120588(119905) be defined by (27) If for some 119895 isin N

lim inf119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119890 (119)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate33 Differential Inequalities A slight modification in theproofs of Theorems 6minus15 leads to the following results aboutdifferential inequalities

Theorem 16 Assume that all the conditions ofTheorem 6 [11] 7 [12] 8 [13] 9 [14] or 10 [15] hold Then(119894) the delay [advanced] differential inequality

1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) le 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) ge 0]

119905 ge 1199050(120)

has no eventually positive solutions(119894119894) the delay [advanced] differential inequality1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) ge 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) le 0]

119905 ge 1199050(121)

has no eventually negative solutions

Remark 17 The oscillation criteria established in this paperall depend on 1205820 (see eg (37) and (71)) in contrast to theconditions obtained in [15 16] and in [17 for m = 1] In factthe left-hand side of conditions (37) and (71) depends on 1205820which is not the casewith the left-hand side of conditions (20)and (21) Since 1205820 gt 1 when 120572 isin (0 1119890] it is obvious that

1198750 (119905) = 1205820119901 (119905) gt 119901 (119905) = 1199010 (119905) (122)

Consequently the left-hand side of conditions (37) and(71) is greater than the corresponding parts of (20) and(21) respectively This is the reason why the conditions inthis paper improve on all known conditions mentioned inSection 2

4 Examples and Comments

The oscillation tests we have proposed and established in themain results involve an iterative procedure We iterativelycompute limsup and liminf on the terms 119875119895(119905) and119876119895(119905) 119895 isinN of a recurrent relation defined on the coefficients and thedeviating argument of an equation of the form (E) or (E1015840)to determine whether that equation is oscillatory But thiscomputation cannot be performed on paper but by meansof a program numerically computing limsup and liminfThe examples below illustrate the significance of our resultsand indicate the high level of improvement in the oscillationcriteria The calculations were performed using MATLABcode

Example 1 Consider the delay differential equation

1199091015840 (119905) + 325119909 (120591 (119905)) = 0 119905 ge 0 (123)

with (see Figure 1(a))

120591 (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]minus4119905 + 40119896 + 9 if 119905 isin [8119896 + 2 8119896 + 3]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 3 8119896 + 4]minus4119905 + 40119896 + 18 if 119905 isin [8119896 + 4 8119896 + 5]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 5 8119896 + 6]minus2119905 + 24119896 + 15 if 119905 isin [8119896 + 6 8119896 + 7]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 7 8119896 + 8]

(124)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (13) we see (Figure 1(b)) that

ℎ (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]8119896 + 1 if 119905 isin [8119896 + 2 8119896 + 195 ]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 195 8119896 + 4]8119896 + 2 if 119905 isin [8119896 + 4 8119896 + 295 ]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 295 8119896 + 6]8119896 + 3 if 119905 isin [8119896 + 6 8119896 + 446 ]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 446 8119896 + 8]

(125)

It is obvious that

120572 = lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 = lim inf

119905rarrinfinint8119896+28119896+1

325119889119904= 012

(126)

12 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

minus1minus2

651 2 3 4 8 9 10 11 12 13 15 16

1415

minus3

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

minus1minus2

t

195 295

1415

7 14651 2 3 4 8 9 10 11 12 13 15 16

446

y = tℎ(t)

(b)

Figure 1 The graphs of 120591(119905) and ℎ(119905)

and therefore the smaller root of 119890012120582 = 120582 is 1205820 = 114765Observe that the function 119865119895 R0 rarr R+ defined as

119865119895 (119905) = int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

(127)

attains its maximum at 119905 = 8119896+446 119896 isin N0 for every 119895 isin NSpecifically

1198651 (119905 = 8119896 + 446 ) = int8119896+446

8119896+3119901 (119904)

sdot exp(int8119896+3120591(119904)

119901 (119906) exp(int119906120591(119906)1198751 (120585) 119889120585) 119889119906)119889119904

(128)

with

1198751 (120585) = 119901 (120585) [1 + int120585120591(120585)119901 (V)

sdot exp(int120585120591(V)119901 (119908) exp(int119908

120591(119908)1205820119901 (119911) 119889119911) 119889119908)119889V]

(129)

Using MATLAB we obtain

1198651 (119905 = 8119896 + 446 ) ≃ 10417 (130)

and therefore

lim sup119905rarrinfin

1198651 (119905) ≃ 10417 gt 1 (131)

Hence condition (37) of Theorem 6 is satisfied for 119895 = 1Consequently all solutions of (123) are oscillatory

Observe however that

LD = lim sup119896rarrinfin

int8119896+4468119896+3

335119889119904 = 052 lt 1120572 = 012 lt 1119890

052 lt 1 + ln 12058201205820 minus 119863 (120572) ≃ 09831(132)

Note that the functionΦ119895 defined by

Φ119895 (119905) = int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

ℎ(119904)119901 (119906) 120595119895 (119906) 119889119906)119889119904

119895 ge 2(133)

attains its maximum at 119905 = 8119896 + 446 119896 isin N0 for every 119895 ge 2Specifically

Φ2 (8119896 + 446 )= int8119896+4468119896+3

119901 (119904) exp(int8119896+3ℎ(119904)

119901 (119904) 1205952 (119906) 119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 exp(int119906

120591(119906)

325 sdot 0 119889119908)119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 sdot 1 119889119906)119889119904= 325 sdot [int

8119896+195

8119896+3exp( 325 int

8119896+3

8119896+1119889119906)119889119904

Complexity 13

+ int8119896+48119896+195

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+2958119896+4

exp( 325 int8119896+3

8119896+2119889119906)119889119904

+ int8119896+68119896+295

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+4468119896+6

exp( 325 int8119896+3

8119896+3119889119906)119889119904] ≃ 057983

(134)

Thus

lim sup119905rarrinfin

Φ2 (119905) ≃ 057983 lt 1 minus 119863 (120572) ≃ 099174 (135)

Also

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int8119896+4468119896+3

325 exp(int8119896+3

120591(119904)

325119889119906)119889119904= 325 sdot lim sup

119905rarrinfin[int8119896+48119896+3

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+58119896+4

exp( 325 int8119896+3

minus4119904+40119896+18119889119906)119889119904

+ int8119896+68119896+5

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+78119896+6

exp( 325 int8119896+3

minus2119904+24119896+15119889119906)119889119904

+ int8119896+4468119896+7

exp( 325 int8119896+3

6119904minus40119896minus41119889119906)119889119904] ≃ 07043

lt 107043 lt 1 minus 119863 (120572) ≃ 099174

(136)

In addition

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)1199011 (119906) 119889119906)119889119904 ≃ 08052

lt 108052 lt 1 minus 119863 (120572) ≃ 099174

(137)

That is none of conditions (8) (9) (12) (14) (for 119895 = 2) (16)(17) (20) (for 119895 = 1) and (21) (for 119895 = 1) is satisfiedComment The improvement of condition (37) over thecorresponding condition (8) is significant approximately10033 We get this measure by comparing the values inthe left-hand side of those conditions Also the improvementover conditions (14) (16) and (20) is very satisfactory

around 7966 479 and 2937 respectively In additioncondition (37) is satisfied from the first iteration whileconditions (14) (20) and (21) need more than one iteration

Example 2 (taken and adapted from [17]) Consider theadvanced differential equation

1199091015840 (119905) minus 3332500119909 (120590 (119905)) = 0 119905 ge 0 (138)

with (see Figure 2(a))

120590 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 2]minus3119905 + 20119896 + 13 if 119905 isin [5119896 + 2 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 4]minus119905 + 10119896 + 13 if 119905 isin [5119896 + 4 5119896 + 5]

(139)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (27) we see (Figure 2(b)) that

120588 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 125]5119896 + 4 if 119905 isin [5119896 + 125 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 38]5119896 + 8 if 119905 isin [5119896 + 38 5119896 + 5]

(140)

It is obvious that

120573 = lim inf119905rarrinfin

int5119896+45119896+3

3332500119889119904 = 01332 (141)

and therefore the smaller root of 11989001332120582 = 120582 is 1205820 = 116839Observe that the function 119866119895 R0 rarr R+ defined as

119866119895 (119905) = int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

(142)

attains its maximum at 119905 = 5119896 + 38 119896 isin N0 for every 119895 isin NSpecifically

1198661 (119905 = 5119896 + 38) = int5119896+85119896+38

119902 (119904)sdot exp(int120590(119904)

5119896+8119902 (119906) exp(int120590(119906)

1199061198761 (120585) 119889120585)119889119906)119889119904

(143)

with

1198761 (120585) = 119902 (120585) [1 + int120590(120585)120585119902 (V)

sdot exp(int120590(V)120585119902 (119908) exp(int120590(119908)

1199081205820119902 (119911) 119889119911)119889119908)119889V]

(144)

14 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

651 2 3 4 8 9 10 11 12 13

1415

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

7

t

14651

2 3 4 8 9 10 11 12 13

38

1415

125

y = t

(t)

(b)

Figure 2 The graphs of 120590(119905) and 120588(119905)

Using MATLAB we obtain

1198661 (119905 = 5119896 + 38) ≃ 09915 (145)

Thereforelim sup119905rarrinfin

1198661 (119905) ≃ 09915 gt 1 minus 119863 (120573) ≃ 09896 (146)

Hence condition (116) of Theorem 12 is satisfied for 119895 = 1Consequently all solutions of (138) oscillate

Observe however that

LA = lim sup119896rarrinfin

int5119896+85119896+38

3332500119889119904 = 055944 lt 1120573 = 01332 lt 1119890 lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int5119896+85119896+38

119902 (119904) exp(int120590(119904)5119896+8

119902 (119906) 119889119906)119889119904= lim sup119896rarrinfin

[int5119896+45119896+38

119902 (119904)sdot exp(int5119904minus20119896minus11

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+5

5119896+4119902 (119904)

sdot exp(intminus119904+10119896+135119896+8

119902 (119906) 119889119906)119889119904 + int5119896+65119896+5

119902 (119904)sdot exp(int5119896+8

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+7

5119896+6119902 (119904)

sdot exp(int4119904minus15119896minus165119896+8

119902 (119906) 119889119906)119889119904 + int5119896+85119896+7

119902 (119904)sdot exp(intminus3119904+20119896+33

5119896+8119902 (119906) 119889119906)119889119904] ≃ 06672 lt 1

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim inf119905rarrinfin

int5119896+45119896+3

119902 (119904) exp(int120590(119904)5119896+4

119902 (119906) 119889119906)119889119904≃ 01893 lt 1119890

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)1199021 (119906) 119889119906)119889119904 ≃ 07196

lt 107196 lt 1 minus 119863 (120573) ≃ 09896

(147)

That is none of conditions (24) (25) (28) (29) (30) (for 119895 =1) and (31) (for 119895 = 1) is satisfiedComment The improvement of condition (116) over thecorresponding condition (24) is significant approximately7723 We get this measure by comparing the values in theleft-hand side of those conditions Also the improvementover conditions (28) and (30) is very satisfactory around4861 and 3778 respectively In addition condition (116)is satisfied from the first iteration while conditions (30) and(31) need more than one iteration

Complexity 15

Remark 3 Similarly one can provide examples illustratingthe other main results

5 Concluding Remarks

In the present paper we have considered the oscillatorydynamics of differential equations having nonmonotonedeviating arguments and nonnegative coefficients New suf-ficient conditions have been established for the oscillation ofall solutions of (E) and (E1015840) These conditions include (37)(71) (75) (83) and (97) and (114) (116) (117) (118) and(119) for (E) and (E1015840) respectively Applying these conditionsinvolves a procedure that checks for oscillations by iterativelycomputing limsup and liminf on terms recursively definedon the equationrsquos coefficients and deviating argument

The main advantage of these conditions is that theyachieve a major improvement over all the related oscillationconditions for (E) [(E1015840)] in the literature For example con-dition (37) [(114)] improves upon the noniterative conditionsthat are reviewed in the introduction namely conditions (8)[(24)] (12) (16)equiv(20) (for 119895 = 1) [(28)equiv(30) (for 119895 = 1)]and (17)equiv(21) (for 119895 = 1) [(31) (for 119895 = 1)] That immediatelybecomes evident by inspecting the left-hand side of (37)[(114)] and the left-hand side of each of the above conditions

The improvement of (37) [(114)] over the other iterativeconditions namely (14) (for j gt 2) (20) (for 119895 gt 1) [(30)(for 119895 gt 1)] and (21) (for 119895 gt 1) [(31) (for 119895 gt 1)] is that itrequires far fewer iterations to establish oscillation than theother conditions

This advantage easily can be verified computationallyby running the MATLAB programs (see Appendix) forcomputing limsup and liminf and comparing the number ofiterations required by each condition to establish oscillationThen we see that we achieve a significant improvement overall known oscillation criteria

Another advantage and a significant departure from thelarge majority of the other studies is that the criteria in thispaper apply to a more general class of equations havingnonmonotone arguments 120591(119905) or 120590(119905) in contrast to mostof the other oscillation criteria that apply to equations withnondecreasing arguments

Appendix

In this appendix for completeness we give the algorithmon MATLAB software used in Example 1 for calculation oflim sup119905rarrinfin1198651(119905) ≃ 10417 For Example 2 the algorithm isomitted since it is similar to the one in Example 1

Algorithm for Example 1

clear clcc = 12119899 = 501205820 = 114765a5 = 19b5 = 703h5 = (b5 minus a5)119899

for i5 = 1 1 119899 + 1x5 = a5 + (i5 minus 1) lowast h5a4 = TFunction(x5)b4 = 19h4 = (b4 minus a4)119899for i4 = 1 1 119899 + 1x4 = a4 + (i4 minus 1) lowast h4a3 = TFunction(x4)b3 = x4h3 = (b3 minus a3)119899for i3 = 1 1 119899 + 1x3 = a3 + (i3 minus 1) lowast h3a2 = TFunction(x3)b2 = x3h2 = (b2 minus a2)119899for i2 = 1 1 119899 + 1x2 = a2 + (i2 minus 1) lowast h2a1 = TFunction(x2)b1 = x3h1 = (b1 minus a1)119899for i1 = 1 1 119899 + 1x1 = a1 + (i1 minus 1) lowast h1f1(i1) = c lowast exp(1205820 lowast c lowast (x1 minus TFunction(x1)))endI1 = f1(1) + f1(119899 + 1)for i1 = 2 2 119899I1 = I1 + f1(i1) lowast 4endfor i1 = 3 2 119899 minus 1I1 = I1 + f1(i1) lowast 2endI1 = I1 lowast h13f2(i2) = c lowast exp(I1)endI2 = f2(1) + f2(119899 + 1)for i2 = 2 2 119899I2 = I2 + f2(i2) lowast 4endfor i2 = 3 2 119899 minus 1I2 = I2 + f2(i2) lowast 2endI2 = I2 lowast h23f3(i3) = c lowast (1 + I2)end

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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Page 7: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

Complexity 7

Since 120598may be taken arbitrarily small this inequality contra-dicts (37)

This completes the proof of the theorem

Theorem 7 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1 minus 119863 (120572) (71)

where 119875119895 is defined by (38) then all solutions of (E) oscillateProof Assume 119909 is an eventually positive solution of (E)Clearly (67) is satisfied for sufficiently large 119905 Thus

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119905)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus 119909 (119905)119909 (ℎ (119905)) (72)

which implies that

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus lim inf119905rarrinfin

119909 (119905)119909 (ℎ (119905)) (73)

Using Lemmas 3 and 4 it is evident that inequality (35) issatisfied Thus (73) leads to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 minus 119863 (120572) (74)

Since 120598may be taken arbitrarily small this inequality contra-dicts (71)

This completes the proof of the theorem

Theorem 8 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119863 (120572) (75)

where 119875119895 is defined by (38) then all solutions of (E) oscillateProof Assume 119909 is an eventually positive solution of (E)Then as in the proof of Theorem 6 for sufficiently large 119905we conclude that

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (76)

Integrating (E) from ℎ(119905) to 119905 and using (76) we obtain

119909 (119905) minus 119909 (ℎ (119905)) + int119905ℎ(119905)119901 (119904) 119909 (119905)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(77)

or

minus 119909 (ℎ (119905)) + int119905ℎ(119905)119901 (119904) 119909 (119905)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 0(78)

Hence

119909 (ℎ (119905)) [ 119909 (119905)119909 (ℎ (119905)) int119905

ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

minus 1] lt 0(79)

which yields for all sufficiently large 119905int119905ℎ(119905)119901 (119904)sdot exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 119909 (ℎ (119905))119909 (119905)

(80)

8 Complexity

and consequently

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le lim sup119905rarrinfin

119909 (ℎ (119905))119909 (119905) (81)

Taking into account the fact that (35) is satisfied inequality(81) leads to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1119863 (120572) (82)

which contradicts (75) when 120598 rarr 0This completes the proof of the theorem

Theorem 9 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120572) (83)

where 119875119895 is defined by (38) and 1205820 is the smaller root of theequation 120582 = 119890120572120582 then all solutions of (E) oscillateProof Let 119909 be an eventually positive solution of (E) As inthe proof ofTheorem 8 we can show that (76) holds namely

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (84)

Since 120591(119904) le ℎ(119904) inequality (84) gives119909 (120591 (119904)) ge 119909 (ℎ (119904))sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (85)

By Lemma 5 for each 120598 gt 0 there exists a real number 119905120598 suchthat

119909 (ℎ (119905))119909 (119905) gt 1205820 minus 120598 forall119905 ge 119905120598 ge 1199051 (86)

Note that by the nondecreasing nature of the function119909(ℎ(119905))119909(119904) in 119904 it holds1 = 119909 (ℎ (119905))119909 (ℎ (119905)) le 119909 (ℎ (119905))119909 (119904) le 119909 (ℎ (119905))119909 (119905)

119905120598 le ℎ (119905) le 119904 le 119905(87)

In particular for 120598 isin (0 1205820 minus 1) by continuity we concludethat there exists a real number 119905lowast isin (ℎ(119905) 119905] satisfying

1 lt 1205820 minus 120598 = 119909 (ℎ (119905))119909 (119905lowast) (88)

Integrating (E) from 119905lowast to 119905 and using (85) we obtain

119909 (119905) minus 119909 (119905lowast) + 119909 (ℎ (119905)) int119905119905lowast119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(89)

or

int119905119905lowast119901 (119904) exp(intℎ(119904)

120591(119904)119901 (119906)

sdot exp(int119906120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 119909 (119905lowast)119909 (ℎ (119905)) minus 119909 (119905)119909 (ℎ (119905)) (90)

Using (88) andLemma4we deduce that for the 120598 consideredthere exists a real number 1199051015840120598 ge 119905120598 such that

int119905119905lowast119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 11205820 minus 120598 minus 119863 (120572) + 120598(91)

for 119905 ge 1199051015840120598Dividing (E) by119909(119905) integrating from ℎ(119905) to 119905lowast and using

(85) we deduce that

int119905lowastℎ(119905)119901 (119904)sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(92)

Complexity 9

Clearly by means of (36) 119909(ℎ(119904))119909(119904) gt 1205820minus120598 for 119904 ge ℎ(119905) ge1199051015840120598 Hence for all sufficiently large 119905 we conclude that(1205820 minus 120598)int119905

lowast

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(93)

or

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minus 11205820 minus 120598 int119905lowast

ℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904 = 11205820 minus 120598 ln119909 (ℎ (119905))119909 (119905lowast)

= ln (1205820 minus 120598)1205820 minus 120598

(94)

that is

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt ln (1205820 minus 120598)1205820 minus 120598 (95)

Using (91) along with (95) we get

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 + ln (1205820 minus 120598)1205820 minus 120598 minus 119863 (120572) + 120598(96)

which contradicts (83) when 120598 rarr 0This completes the proof of the theorem

Theorem 10 Let ℎ(119905) be defined by (13) If for some 119895 isin Nlim inf119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119890 (97)

where 119875119895 is defined by (38) then all solutions of (E) oscillate

Proof For the sake of contradiction let 119909 be a nonincreasingeventually positive solution and 1199051 gt 1199050 be such that 119909(119905) gt 0and 119909(120591(119905)) gt 0 for all 119905 ge 1199051 We note that wemay obtain (85)as in the proof of Theorem 9

Dividing (E) by 119909(119905) and integrating from ℎ(119905) to 119905 wehave

ln(119909 (ℎ (119905))119909 (119905) ) = int119905

ℎ(119905)119901 (119904) 119909 (120591 (119904))119909 (119904) 119889119904 forall119905 ge 1199052 ge 1199051 (98)

from which in view of 120591(119904) le ℎ(119904) and (85) we get

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(99)

Since 119909 is nonincreasing and ℎ(119904) lt 119904 inequality (99)becomes

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(100)

From (97) it is clear that there exists a constant 119888 gt 0 suchthat

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

ge 119888 gt 1119890 (101)

Choose 1198881015840 such that 119888 gt 1198881015840 gt 1119890 For every 120598 gt 0 such that119888 minus 120598 gt 1198881015840 we haveint119905ℎ(t)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 119888 minus 120598 gt 1198881015840 gt 1119890 (102)

Combining inequalities (100) and (102) we obtain

ln(119909 (ℎ (119905))119909 (119905) ) gt 1198881015840 (103)

or119909 (ℎ (119905))119909 (119905) gt 1198901198881015840 gt 1198901198881015840 gt 1 (104)

which yields

119909 (ℎ (119905)) gt (1198901198881015840) 119909 (119905) (105)

10 Complexity

Following the above steps we can inductively show that forany positive integer 119896

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 for sufficiently large 119905 (106)

Since 1198901198881015840 gt 1 there is a natural number 119896 isin N satisfying119896 gt 2[ln 2 minus ln 1198881015840](1 + ln 1198881015840) such that for 119905 sufficiently large

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 gt ( 21198881015840 )2 (107)

Further (cf [13 24]) for sufficiently large 119905 there exists a realnumber 119905119898 isin (ℎ(119905) 119905) such that

int119905119898ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402 int119905119905119898

119901 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402

(108)

Integrating (E) from ℎ(119905) to 119905119898 using (85) and the fact that119909(119905) gt 0 we obtain119909 (ℎ (119905)) gt 119909 (ℎ (119905119898)) int119905119898

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(109)

which in view of the first inequality in (108) implies that

119909 (ℎ (119905)) gt 11988810158402 119909 (ℎ (119905119898)) (110)

Similarly integrating (E) from 119905119898 to 119905 using (85) and the factthat 119909(119905) gt 0 we have119909 (119905119898) gt 119909 (ℎ (119905)) int119905

119905119898

119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(111)

which in view of the second inequality in (108) yields

119909 (119905119898) gt 11988810158402 119909 (ℎ (119905)) (112)

Combining inequalities (110) and (112) we deduce that

119909 (ℎ (119905119898)) lt 21198881015840 119909 (ℎ (119905)) lt ( 21198881015840 )2 119909 (119905119898) (113)

which contradicts (107)The proof of the theorem is complete

32 ADEs Analogous oscillation conditions to thoseobtained for the delay equation (E) can be derived for the(dual) advanced differential equation (E1015840) by followingsimilar arguments with the ones employed for obtainingTheorems 6minus10Theorem 11 Let 120588(119905) be defined by (27) and for some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1

(114)

where

119876119895 (119905) = 119902 (119905) [1 + int120590(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895minus1 (120585) 119889120585)119889119906)119889119904]

(115)

with 1198760(119905) = 1205820119902(119905) and let 1205820 be the smaller root of theequation 120582 = 119890120573120582 Then all solutions of (E1015840) oscillateTheorem 12 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 minus 119863 (120573)

(116)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillateTheorem 13 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119863 (120573) (117)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate

Complexity 11

Theorem 14 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120573) (118)

where 119876119895 is defined by (115) and 1205820 is the smaller root of theequation 120582 = 119890120573120582 then all solutions of (E1015840) oscillateTheorem 15 Let 120588(119905) be defined by (27) If for some 119895 isin N

lim inf119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119890 (119)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate33 Differential Inequalities A slight modification in theproofs of Theorems 6minus15 leads to the following results aboutdifferential inequalities

Theorem 16 Assume that all the conditions ofTheorem 6 [11] 7 [12] 8 [13] 9 [14] or 10 [15] hold Then(119894) the delay [advanced] differential inequality

1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) le 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) ge 0]

119905 ge 1199050(120)

has no eventually positive solutions(119894119894) the delay [advanced] differential inequality1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) ge 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) le 0]

119905 ge 1199050(121)

has no eventually negative solutions

Remark 17 The oscillation criteria established in this paperall depend on 1205820 (see eg (37) and (71)) in contrast to theconditions obtained in [15 16] and in [17 for m = 1] In factthe left-hand side of conditions (37) and (71) depends on 1205820which is not the casewith the left-hand side of conditions (20)and (21) Since 1205820 gt 1 when 120572 isin (0 1119890] it is obvious that

1198750 (119905) = 1205820119901 (119905) gt 119901 (119905) = 1199010 (119905) (122)

Consequently the left-hand side of conditions (37) and(71) is greater than the corresponding parts of (20) and(21) respectively This is the reason why the conditions inthis paper improve on all known conditions mentioned inSection 2

4 Examples and Comments

The oscillation tests we have proposed and established in themain results involve an iterative procedure We iterativelycompute limsup and liminf on the terms 119875119895(119905) and119876119895(119905) 119895 isinN of a recurrent relation defined on the coefficients and thedeviating argument of an equation of the form (E) or (E1015840)to determine whether that equation is oscillatory But thiscomputation cannot be performed on paper but by meansof a program numerically computing limsup and liminfThe examples below illustrate the significance of our resultsand indicate the high level of improvement in the oscillationcriteria The calculations were performed using MATLABcode

Example 1 Consider the delay differential equation

1199091015840 (119905) + 325119909 (120591 (119905)) = 0 119905 ge 0 (123)

with (see Figure 1(a))

120591 (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]minus4119905 + 40119896 + 9 if 119905 isin [8119896 + 2 8119896 + 3]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 3 8119896 + 4]minus4119905 + 40119896 + 18 if 119905 isin [8119896 + 4 8119896 + 5]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 5 8119896 + 6]minus2119905 + 24119896 + 15 if 119905 isin [8119896 + 6 8119896 + 7]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 7 8119896 + 8]

(124)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (13) we see (Figure 1(b)) that

ℎ (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]8119896 + 1 if 119905 isin [8119896 + 2 8119896 + 195 ]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 195 8119896 + 4]8119896 + 2 if 119905 isin [8119896 + 4 8119896 + 295 ]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 295 8119896 + 6]8119896 + 3 if 119905 isin [8119896 + 6 8119896 + 446 ]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 446 8119896 + 8]

(125)

It is obvious that

120572 = lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 = lim inf

119905rarrinfinint8119896+28119896+1

325119889119904= 012

(126)

12 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

minus1minus2

651 2 3 4 8 9 10 11 12 13 15 16

1415

minus3

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

minus1minus2

t

195 295

1415

7 14651 2 3 4 8 9 10 11 12 13 15 16

446

y = tℎ(t)

(b)

Figure 1 The graphs of 120591(119905) and ℎ(119905)

and therefore the smaller root of 119890012120582 = 120582 is 1205820 = 114765Observe that the function 119865119895 R0 rarr R+ defined as

119865119895 (119905) = int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

(127)

attains its maximum at 119905 = 8119896+446 119896 isin N0 for every 119895 isin NSpecifically

1198651 (119905 = 8119896 + 446 ) = int8119896+446

8119896+3119901 (119904)

sdot exp(int8119896+3120591(119904)

119901 (119906) exp(int119906120591(119906)1198751 (120585) 119889120585) 119889119906)119889119904

(128)

with

1198751 (120585) = 119901 (120585) [1 + int120585120591(120585)119901 (V)

sdot exp(int120585120591(V)119901 (119908) exp(int119908

120591(119908)1205820119901 (119911) 119889119911) 119889119908)119889V]

(129)

Using MATLAB we obtain

1198651 (119905 = 8119896 + 446 ) ≃ 10417 (130)

and therefore

lim sup119905rarrinfin

1198651 (119905) ≃ 10417 gt 1 (131)

Hence condition (37) of Theorem 6 is satisfied for 119895 = 1Consequently all solutions of (123) are oscillatory

Observe however that

LD = lim sup119896rarrinfin

int8119896+4468119896+3

335119889119904 = 052 lt 1120572 = 012 lt 1119890

052 lt 1 + ln 12058201205820 minus 119863 (120572) ≃ 09831(132)

Note that the functionΦ119895 defined by

Φ119895 (119905) = int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

ℎ(119904)119901 (119906) 120595119895 (119906) 119889119906)119889119904

119895 ge 2(133)

attains its maximum at 119905 = 8119896 + 446 119896 isin N0 for every 119895 ge 2Specifically

Φ2 (8119896 + 446 )= int8119896+4468119896+3

119901 (119904) exp(int8119896+3ℎ(119904)

119901 (119904) 1205952 (119906) 119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 exp(int119906

120591(119906)

325 sdot 0 119889119908)119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 sdot 1 119889119906)119889119904= 325 sdot [int

8119896+195

8119896+3exp( 325 int

8119896+3

8119896+1119889119906)119889119904

Complexity 13

+ int8119896+48119896+195

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+2958119896+4

exp( 325 int8119896+3

8119896+2119889119906)119889119904

+ int8119896+68119896+295

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+4468119896+6

exp( 325 int8119896+3

8119896+3119889119906)119889119904] ≃ 057983

(134)

Thus

lim sup119905rarrinfin

Φ2 (119905) ≃ 057983 lt 1 minus 119863 (120572) ≃ 099174 (135)

Also

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int8119896+4468119896+3

325 exp(int8119896+3

120591(119904)

325119889119906)119889119904= 325 sdot lim sup

119905rarrinfin[int8119896+48119896+3

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+58119896+4

exp( 325 int8119896+3

minus4119904+40119896+18119889119906)119889119904

+ int8119896+68119896+5

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+78119896+6

exp( 325 int8119896+3

minus2119904+24119896+15119889119906)119889119904

+ int8119896+4468119896+7

exp( 325 int8119896+3

6119904minus40119896minus41119889119906)119889119904] ≃ 07043

lt 107043 lt 1 minus 119863 (120572) ≃ 099174

(136)

In addition

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)1199011 (119906) 119889119906)119889119904 ≃ 08052

lt 108052 lt 1 minus 119863 (120572) ≃ 099174

(137)

That is none of conditions (8) (9) (12) (14) (for 119895 = 2) (16)(17) (20) (for 119895 = 1) and (21) (for 119895 = 1) is satisfiedComment The improvement of condition (37) over thecorresponding condition (8) is significant approximately10033 We get this measure by comparing the values inthe left-hand side of those conditions Also the improvementover conditions (14) (16) and (20) is very satisfactory

around 7966 479 and 2937 respectively In additioncondition (37) is satisfied from the first iteration whileconditions (14) (20) and (21) need more than one iteration

Example 2 (taken and adapted from [17]) Consider theadvanced differential equation

1199091015840 (119905) minus 3332500119909 (120590 (119905)) = 0 119905 ge 0 (138)

with (see Figure 2(a))

120590 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 2]minus3119905 + 20119896 + 13 if 119905 isin [5119896 + 2 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 4]minus119905 + 10119896 + 13 if 119905 isin [5119896 + 4 5119896 + 5]

(139)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (27) we see (Figure 2(b)) that

120588 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 125]5119896 + 4 if 119905 isin [5119896 + 125 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 38]5119896 + 8 if 119905 isin [5119896 + 38 5119896 + 5]

(140)

It is obvious that

120573 = lim inf119905rarrinfin

int5119896+45119896+3

3332500119889119904 = 01332 (141)

and therefore the smaller root of 11989001332120582 = 120582 is 1205820 = 116839Observe that the function 119866119895 R0 rarr R+ defined as

119866119895 (119905) = int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

(142)

attains its maximum at 119905 = 5119896 + 38 119896 isin N0 for every 119895 isin NSpecifically

1198661 (119905 = 5119896 + 38) = int5119896+85119896+38

119902 (119904)sdot exp(int120590(119904)

5119896+8119902 (119906) exp(int120590(119906)

1199061198761 (120585) 119889120585)119889119906)119889119904

(143)

with

1198761 (120585) = 119902 (120585) [1 + int120590(120585)120585119902 (V)

sdot exp(int120590(V)120585119902 (119908) exp(int120590(119908)

1199081205820119902 (119911) 119889119911)119889119908)119889V]

(144)

14 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

651 2 3 4 8 9 10 11 12 13

1415

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

7

t

14651

2 3 4 8 9 10 11 12 13

38

1415

125

y = t

(t)

(b)

Figure 2 The graphs of 120590(119905) and 120588(119905)

Using MATLAB we obtain

1198661 (119905 = 5119896 + 38) ≃ 09915 (145)

Thereforelim sup119905rarrinfin

1198661 (119905) ≃ 09915 gt 1 minus 119863 (120573) ≃ 09896 (146)

Hence condition (116) of Theorem 12 is satisfied for 119895 = 1Consequently all solutions of (138) oscillate

Observe however that

LA = lim sup119896rarrinfin

int5119896+85119896+38

3332500119889119904 = 055944 lt 1120573 = 01332 lt 1119890 lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int5119896+85119896+38

119902 (119904) exp(int120590(119904)5119896+8

119902 (119906) 119889119906)119889119904= lim sup119896rarrinfin

[int5119896+45119896+38

119902 (119904)sdot exp(int5119904minus20119896minus11

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+5

5119896+4119902 (119904)

sdot exp(intminus119904+10119896+135119896+8

119902 (119906) 119889119906)119889119904 + int5119896+65119896+5

119902 (119904)sdot exp(int5119896+8

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+7

5119896+6119902 (119904)

sdot exp(int4119904minus15119896minus165119896+8

119902 (119906) 119889119906)119889119904 + int5119896+85119896+7

119902 (119904)sdot exp(intminus3119904+20119896+33

5119896+8119902 (119906) 119889119906)119889119904] ≃ 06672 lt 1

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim inf119905rarrinfin

int5119896+45119896+3

119902 (119904) exp(int120590(119904)5119896+4

119902 (119906) 119889119906)119889119904≃ 01893 lt 1119890

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)1199021 (119906) 119889119906)119889119904 ≃ 07196

lt 107196 lt 1 minus 119863 (120573) ≃ 09896

(147)

That is none of conditions (24) (25) (28) (29) (30) (for 119895 =1) and (31) (for 119895 = 1) is satisfiedComment The improvement of condition (116) over thecorresponding condition (24) is significant approximately7723 We get this measure by comparing the values in theleft-hand side of those conditions Also the improvementover conditions (28) and (30) is very satisfactory around4861 and 3778 respectively In addition condition (116)is satisfied from the first iteration while conditions (30) and(31) need more than one iteration

Complexity 15

Remark 3 Similarly one can provide examples illustratingthe other main results

5 Concluding Remarks

In the present paper we have considered the oscillatorydynamics of differential equations having nonmonotonedeviating arguments and nonnegative coefficients New suf-ficient conditions have been established for the oscillation ofall solutions of (E) and (E1015840) These conditions include (37)(71) (75) (83) and (97) and (114) (116) (117) (118) and(119) for (E) and (E1015840) respectively Applying these conditionsinvolves a procedure that checks for oscillations by iterativelycomputing limsup and liminf on terms recursively definedon the equationrsquos coefficients and deviating argument

The main advantage of these conditions is that theyachieve a major improvement over all the related oscillationconditions for (E) [(E1015840)] in the literature For example con-dition (37) [(114)] improves upon the noniterative conditionsthat are reviewed in the introduction namely conditions (8)[(24)] (12) (16)equiv(20) (for 119895 = 1) [(28)equiv(30) (for 119895 = 1)]and (17)equiv(21) (for 119895 = 1) [(31) (for 119895 = 1)] That immediatelybecomes evident by inspecting the left-hand side of (37)[(114)] and the left-hand side of each of the above conditions

The improvement of (37) [(114)] over the other iterativeconditions namely (14) (for j gt 2) (20) (for 119895 gt 1) [(30)(for 119895 gt 1)] and (21) (for 119895 gt 1) [(31) (for 119895 gt 1)] is that itrequires far fewer iterations to establish oscillation than theother conditions

This advantage easily can be verified computationallyby running the MATLAB programs (see Appendix) forcomputing limsup and liminf and comparing the number ofiterations required by each condition to establish oscillationThen we see that we achieve a significant improvement overall known oscillation criteria

Another advantage and a significant departure from thelarge majority of the other studies is that the criteria in thispaper apply to a more general class of equations havingnonmonotone arguments 120591(119905) or 120590(119905) in contrast to mostof the other oscillation criteria that apply to equations withnondecreasing arguments

Appendix

In this appendix for completeness we give the algorithmon MATLAB software used in Example 1 for calculation oflim sup119905rarrinfin1198651(119905) ≃ 10417 For Example 2 the algorithm isomitted since it is similar to the one in Example 1

Algorithm for Example 1

clear clcc = 12119899 = 501205820 = 114765a5 = 19b5 = 703h5 = (b5 minus a5)119899

for i5 = 1 1 119899 + 1x5 = a5 + (i5 minus 1) lowast h5a4 = TFunction(x5)b4 = 19h4 = (b4 minus a4)119899for i4 = 1 1 119899 + 1x4 = a4 + (i4 minus 1) lowast h4a3 = TFunction(x4)b3 = x4h3 = (b3 minus a3)119899for i3 = 1 1 119899 + 1x3 = a3 + (i3 minus 1) lowast h3a2 = TFunction(x3)b2 = x3h2 = (b2 minus a2)119899for i2 = 1 1 119899 + 1x2 = a2 + (i2 minus 1) lowast h2a1 = TFunction(x2)b1 = x3h1 = (b1 minus a1)119899for i1 = 1 1 119899 + 1x1 = a1 + (i1 minus 1) lowast h1f1(i1) = c lowast exp(1205820 lowast c lowast (x1 minus TFunction(x1)))endI1 = f1(1) + f1(119899 + 1)for i1 = 2 2 119899I1 = I1 + f1(i1) lowast 4endfor i1 = 3 2 119899 minus 1I1 = I1 + f1(i1) lowast 2endI1 = I1 lowast h13f2(i2) = c lowast exp(I1)endI2 = f2(1) + f2(119899 + 1)for i2 = 2 2 119899I2 = I2 + f2(i2) lowast 4endfor i2 = 3 2 119899 minus 1I2 = I2 + f2(i2) lowast 2endI2 = I2 lowast h23f3(i3) = c lowast (1 + I2)end

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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Page 8: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

8 Complexity

and consequently

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le lim sup119905rarrinfin

119909 (ℎ (119905))119909 (119905) (81)

Taking into account the fact that (35) is satisfied inequality(81) leads to

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(int119905120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1119863 (120572) (82)

which contradicts (75) when 120598 rarr 0This completes the proof of the theorem

Theorem 9 Let ℎ(119905) be defined by (13) and 120572 isin (0 1119890] If forsome 119895 isin N

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120572) (83)

where 119875119895 is defined by (38) and 1205820 is the smaller root of theequation 120582 = 119890120572120582 then all solutions of (E) oscillateProof Let 119909 be an eventually positive solution of (E) As inthe proof ofTheorem 8 we can show that (76) holds namely

119909 (120591 (119904))ge 119909 (119905) exp(int119905

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (84)

Since 120591(119904) le ℎ(119904) inequality (84) gives119909 (120591 (119904)) ge 119909 (ℎ (119904))sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906) (85)

By Lemma 5 for each 120598 gt 0 there exists a real number 119905120598 suchthat

119909 (ℎ (119905))119909 (119905) gt 1205820 minus 120598 forall119905 ge 119905120598 ge 1199051 (86)

Note that by the nondecreasing nature of the function119909(ℎ(119905))119909(119904) in 119904 it holds1 = 119909 (ℎ (119905))119909 (ℎ (119905)) le 119909 (ℎ (119905))119909 (119904) le 119909 (ℎ (119905))119909 (119905)

119905120598 le ℎ (119905) le 119904 le 119905(87)

In particular for 120598 isin (0 1205820 minus 1) by continuity we concludethat there exists a real number 119905lowast isin (ℎ(119905) 119905] satisfying

1 lt 1205820 minus 120598 = 119909 (ℎ (119905))119909 (119905lowast) (88)

Integrating (E) from 119905lowast to 119905 and using (85) we obtain

119909 (119905) minus 119909 (119905lowast) + 119909 (ℎ (119905)) int119905119905lowast119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 0(89)

or

int119905119905lowast119901 (119904) exp(intℎ(119904)

120591(119904)119901 (119906)

sdot exp(int119906120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 119909 (119905lowast)119909 (ℎ (119905)) minus 119909 (119905)119909 (ℎ (119905)) (90)

Using (88) andLemma4we deduce that for the 120598 consideredthere exists a real number 1199051015840120598 ge 119905120598 such that

int119905119905lowast119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt 11205820 minus 120598 minus 119863 (120572) + 120598(91)

for 119905 ge 1199051015840120598Dividing (E) by119909(119905) integrating from ℎ(119905) to 119905lowast and using

(85) we deduce that

int119905lowastℎ(119905)119901 (119904)sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(92)

Complexity 9

Clearly by means of (36) 119909(ℎ(119904))119909(119904) gt 1205820minus120598 for 119904 ge ℎ(119905) ge1199051015840120598 Hence for all sufficiently large 119905 we conclude that(1205820 minus 120598)int119905

lowast

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(93)

or

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minus 11205820 minus 120598 int119905lowast

ℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904 = 11205820 minus 120598 ln119909 (ℎ (119905))119909 (119905lowast)

= ln (1205820 minus 120598)1205820 minus 120598

(94)

that is

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt ln (1205820 minus 120598)1205820 minus 120598 (95)

Using (91) along with (95) we get

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 + ln (1205820 minus 120598)1205820 minus 120598 minus 119863 (120572) + 120598(96)

which contradicts (83) when 120598 rarr 0This completes the proof of the theorem

Theorem 10 Let ℎ(119905) be defined by (13) If for some 119895 isin Nlim inf119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119890 (97)

where 119875119895 is defined by (38) then all solutions of (E) oscillate

Proof For the sake of contradiction let 119909 be a nonincreasingeventually positive solution and 1199051 gt 1199050 be such that 119909(119905) gt 0and 119909(120591(119905)) gt 0 for all 119905 ge 1199051 We note that wemay obtain (85)as in the proof of Theorem 9

Dividing (E) by 119909(119905) and integrating from ℎ(119905) to 119905 wehave

ln(119909 (ℎ (119905))119909 (119905) ) = int119905

ℎ(119905)119901 (119904) 119909 (120591 (119904))119909 (119904) 119889119904 forall119905 ge 1199052 ge 1199051 (98)

from which in view of 120591(119904) le ℎ(119904) and (85) we get

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(99)

Since 119909 is nonincreasing and ℎ(119904) lt 119904 inequality (99)becomes

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(100)

From (97) it is clear that there exists a constant 119888 gt 0 suchthat

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

ge 119888 gt 1119890 (101)

Choose 1198881015840 such that 119888 gt 1198881015840 gt 1119890 For every 120598 gt 0 such that119888 minus 120598 gt 1198881015840 we haveint119905ℎ(t)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 119888 minus 120598 gt 1198881015840 gt 1119890 (102)

Combining inequalities (100) and (102) we obtain

ln(119909 (ℎ (119905))119909 (119905) ) gt 1198881015840 (103)

or119909 (ℎ (119905))119909 (119905) gt 1198901198881015840 gt 1198901198881015840 gt 1 (104)

which yields

119909 (ℎ (119905)) gt (1198901198881015840) 119909 (119905) (105)

10 Complexity

Following the above steps we can inductively show that forany positive integer 119896

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 for sufficiently large 119905 (106)

Since 1198901198881015840 gt 1 there is a natural number 119896 isin N satisfying119896 gt 2[ln 2 minus ln 1198881015840](1 + ln 1198881015840) such that for 119905 sufficiently large

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 gt ( 21198881015840 )2 (107)

Further (cf [13 24]) for sufficiently large 119905 there exists a realnumber 119905119898 isin (ℎ(119905) 119905) such that

int119905119898ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402 int119905119905119898

119901 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402

(108)

Integrating (E) from ℎ(119905) to 119905119898 using (85) and the fact that119909(119905) gt 0 we obtain119909 (ℎ (119905)) gt 119909 (ℎ (119905119898)) int119905119898

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(109)

which in view of the first inequality in (108) implies that

119909 (ℎ (119905)) gt 11988810158402 119909 (ℎ (119905119898)) (110)

Similarly integrating (E) from 119905119898 to 119905 using (85) and the factthat 119909(119905) gt 0 we have119909 (119905119898) gt 119909 (ℎ (119905)) int119905

119905119898

119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(111)

which in view of the second inequality in (108) yields

119909 (119905119898) gt 11988810158402 119909 (ℎ (119905)) (112)

Combining inequalities (110) and (112) we deduce that

119909 (ℎ (119905119898)) lt 21198881015840 119909 (ℎ (119905)) lt ( 21198881015840 )2 119909 (119905119898) (113)

which contradicts (107)The proof of the theorem is complete

32 ADEs Analogous oscillation conditions to thoseobtained for the delay equation (E) can be derived for the(dual) advanced differential equation (E1015840) by followingsimilar arguments with the ones employed for obtainingTheorems 6minus10Theorem 11 Let 120588(119905) be defined by (27) and for some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1

(114)

where

119876119895 (119905) = 119902 (119905) [1 + int120590(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895minus1 (120585) 119889120585)119889119906)119889119904]

(115)

with 1198760(119905) = 1205820119902(119905) and let 1205820 be the smaller root of theequation 120582 = 119890120573120582 Then all solutions of (E1015840) oscillateTheorem 12 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 minus 119863 (120573)

(116)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillateTheorem 13 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119863 (120573) (117)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate

Complexity 11

Theorem 14 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120573) (118)

where 119876119895 is defined by (115) and 1205820 is the smaller root of theequation 120582 = 119890120573120582 then all solutions of (E1015840) oscillateTheorem 15 Let 120588(119905) be defined by (27) If for some 119895 isin N

lim inf119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119890 (119)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate33 Differential Inequalities A slight modification in theproofs of Theorems 6minus15 leads to the following results aboutdifferential inequalities

Theorem 16 Assume that all the conditions ofTheorem 6 [11] 7 [12] 8 [13] 9 [14] or 10 [15] hold Then(119894) the delay [advanced] differential inequality

1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) le 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) ge 0]

119905 ge 1199050(120)

has no eventually positive solutions(119894119894) the delay [advanced] differential inequality1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) ge 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) le 0]

119905 ge 1199050(121)

has no eventually negative solutions

Remark 17 The oscillation criteria established in this paperall depend on 1205820 (see eg (37) and (71)) in contrast to theconditions obtained in [15 16] and in [17 for m = 1] In factthe left-hand side of conditions (37) and (71) depends on 1205820which is not the casewith the left-hand side of conditions (20)and (21) Since 1205820 gt 1 when 120572 isin (0 1119890] it is obvious that

1198750 (119905) = 1205820119901 (119905) gt 119901 (119905) = 1199010 (119905) (122)

Consequently the left-hand side of conditions (37) and(71) is greater than the corresponding parts of (20) and(21) respectively This is the reason why the conditions inthis paper improve on all known conditions mentioned inSection 2

4 Examples and Comments

The oscillation tests we have proposed and established in themain results involve an iterative procedure We iterativelycompute limsup and liminf on the terms 119875119895(119905) and119876119895(119905) 119895 isinN of a recurrent relation defined on the coefficients and thedeviating argument of an equation of the form (E) or (E1015840)to determine whether that equation is oscillatory But thiscomputation cannot be performed on paper but by meansof a program numerically computing limsup and liminfThe examples below illustrate the significance of our resultsand indicate the high level of improvement in the oscillationcriteria The calculations were performed using MATLABcode

Example 1 Consider the delay differential equation

1199091015840 (119905) + 325119909 (120591 (119905)) = 0 119905 ge 0 (123)

with (see Figure 1(a))

120591 (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]minus4119905 + 40119896 + 9 if 119905 isin [8119896 + 2 8119896 + 3]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 3 8119896 + 4]minus4119905 + 40119896 + 18 if 119905 isin [8119896 + 4 8119896 + 5]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 5 8119896 + 6]minus2119905 + 24119896 + 15 if 119905 isin [8119896 + 6 8119896 + 7]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 7 8119896 + 8]

(124)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (13) we see (Figure 1(b)) that

ℎ (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]8119896 + 1 if 119905 isin [8119896 + 2 8119896 + 195 ]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 195 8119896 + 4]8119896 + 2 if 119905 isin [8119896 + 4 8119896 + 295 ]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 295 8119896 + 6]8119896 + 3 if 119905 isin [8119896 + 6 8119896 + 446 ]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 446 8119896 + 8]

(125)

It is obvious that

120572 = lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 = lim inf

119905rarrinfinint8119896+28119896+1

325119889119904= 012

(126)

12 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

minus1minus2

651 2 3 4 8 9 10 11 12 13 15 16

1415

minus3

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

minus1minus2

t

195 295

1415

7 14651 2 3 4 8 9 10 11 12 13 15 16

446

y = tℎ(t)

(b)

Figure 1 The graphs of 120591(119905) and ℎ(119905)

and therefore the smaller root of 119890012120582 = 120582 is 1205820 = 114765Observe that the function 119865119895 R0 rarr R+ defined as

119865119895 (119905) = int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

(127)

attains its maximum at 119905 = 8119896+446 119896 isin N0 for every 119895 isin NSpecifically

1198651 (119905 = 8119896 + 446 ) = int8119896+446

8119896+3119901 (119904)

sdot exp(int8119896+3120591(119904)

119901 (119906) exp(int119906120591(119906)1198751 (120585) 119889120585) 119889119906)119889119904

(128)

with

1198751 (120585) = 119901 (120585) [1 + int120585120591(120585)119901 (V)

sdot exp(int120585120591(V)119901 (119908) exp(int119908

120591(119908)1205820119901 (119911) 119889119911) 119889119908)119889V]

(129)

Using MATLAB we obtain

1198651 (119905 = 8119896 + 446 ) ≃ 10417 (130)

and therefore

lim sup119905rarrinfin

1198651 (119905) ≃ 10417 gt 1 (131)

Hence condition (37) of Theorem 6 is satisfied for 119895 = 1Consequently all solutions of (123) are oscillatory

Observe however that

LD = lim sup119896rarrinfin

int8119896+4468119896+3

335119889119904 = 052 lt 1120572 = 012 lt 1119890

052 lt 1 + ln 12058201205820 minus 119863 (120572) ≃ 09831(132)

Note that the functionΦ119895 defined by

Φ119895 (119905) = int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

ℎ(119904)119901 (119906) 120595119895 (119906) 119889119906)119889119904

119895 ge 2(133)

attains its maximum at 119905 = 8119896 + 446 119896 isin N0 for every 119895 ge 2Specifically

Φ2 (8119896 + 446 )= int8119896+4468119896+3

119901 (119904) exp(int8119896+3ℎ(119904)

119901 (119904) 1205952 (119906) 119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 exp(int119906

120591(119906)

325 sdot 0 119889119908)119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 sdot 1 119889119906)119889119904= 325 sdot [int

8119896+195

8119896+3exp( 325 int

8119896+3

8119896+1119889119906)119889119904

Complexity 13

+ int8119896+48119896+195

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+2958119896+4

exp( 325 int8119896+3

8119896+2119889119906)119889119904

+ int8119896+68119896+295

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+4468119896+6

exp( 325 int8119896+3

8119896+3119889119906)119889119904] ≃ 057983

(134)

Thus

lim sup119905rarrinfin

Φ2 (119905) ≃ 057983 lt 1 minus 119863 (120572) ≃ 099174 (135)

Also

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int8119896+4468119896+3

325 exp(int8119896+3

120591(119904)

325119889119906)119889119904= 325 sdot lim sup

119905rarrinfin[int8119896+48119896+3

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+58119896+4

exp( 325 int8119896+3

minus4119904+40119896+18119889119906)119889119904

+ int8119896+68119896+5

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+78119896+6

exp( 325 int8119896+3

minus2119904+24119896+15119889119906)119889119904

+ int8119896+4468119896+7

exp( 325 int8119896+3

6119904minus40119896minus41119889119906)119889119904] ≃ 07043

lt 107043 lt 1 minus 119863 (120572) ≃ 099174

(136)

In addition

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)1199011 (119906) 119889119906)119889119904 ≃ 08052

lt 108052 lt 1 minus 119863 (120572) ≃ 099174

(137)

That is none of conditions (8) (9) (12) (14) (for 119895 = 2) (16)(17) (20) (for 119895 = 1) and (21) (for 119895 = 1) is satisfiedComment The improvement of condition (37) over thecorresponding condition (8) is significant approximately10033 We get this measure by comparing the values inthe left-hand side of those conditions Also the improvementover conditions (14) (16) and (20) is very satisfactory

around 7966 479 and 2937 respectively In additioncondition (37) is satisfied from the first iteration whileconditions (14) (20) and (21) need more than one iteration

Example 2 (taken and adapted from [17]) Consider theadvanced differential equation

1199091015840 (119905) minus 3332500119909 (120590 (119905)) = 0 119905 ge 0 (138)

with (see Figure 2(a))

120590 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 2]minus3119905 + 20119896 + 13 if 119905 isin [5119896 + 2 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 4]minus119905 + 10119896 + 13 if 119905 isin [5119896 + 4 5119896 + 5]

(139)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (27) we see (Figure 2(b)) that

120588 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 125]5119896 + 4 if 119905 isin [5119896 + 125 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 38]5119896 + 8 if 119905 isin [5119896 + 38 5119896 + 5]

(140)

It is obvious that

120573 = lim inf119905rarrinfin

int5119896+45119896+3

3332500119889119904 = 01332 (141)

and therefore the smaller root of 11989001332120582 = 120582 is 1205820 = 116839Observe that the function 119866119895 R0 rarr R+ defined as

119866119895 (119905) = int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

(142)

attains its maximum at 119905 = 5119896 + 38 119896 isin N0 for every 119895 isin NSpecifically

1198661 (119905 = 5119896 + 38) = int5119896+85119896+38

119902 (119904)sdot exp(int120590(119904)

5119896+8119902 (119906) exp(int120590(119906)

1199061198761 (120585) 119889120585)119889119906)119889119904

(143)

with

1198761 (120585) = 119902 (120585) [1 + int120590(120585)120585119902 (V)

sdot exp(int120590(V)120585119902 (119908) exp(int120590(119908)

1199081205820119902 (119911) 119889119911)119889119908)119889V]

(144)

14 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

651 2 3 4 8 9 10 11 12 13

1415

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

7

t

14651

2 3 4 8 9 10 11 12 13

38

1415

125

y = t

(t)

(b)

Figure 2 The graphs of 120590(119905) and 120588(119905)

Using MATLAB we obtain

1198661 (119905 = 5119896 + 38) ≃ 09915 (145)

Thereforelim sup119905rarrinfin

1198661 (119905) ≃ 09915 gt 1 minus 119863 (120573) ≃ 09896 (146)

Hence condition (116) of Theorem 12 is satisfied for 119895 = 1Consequently all solutions of (138) oscillate

Observe however that

LA = lim sup119896rarrinfin

int5119896+85119896+38

3332500119889119904 = 055944 lt 1120573 = 01332 lt 1119890 lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int5119896+85119896+38

119902 (119904) exp(int120590(119904)5119896+8

119902 (119906) 119889119906)119889119904= lim sup119896rarrinfin

[int5119896+45119896+38

119902 (119904)sdot exp(int5119904minus20119896minus11

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+5

5119896+4119902 (119904)

sdot exp(intminus119904+10119896+135119896+8

119902 (119906) 119889119906)119889119904 + int5119896+65119896+5

119902 (119904)sdot exp(int5119896+8

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+7

5119896+6119902 (119904)

sdot exp(int4119904minus15119896minus165119896+8

119902 (119906) 119889119906)119889119904 + int5119896+85119896+7

119902 (119904)sdot exp(intminus3119904+20119896+33

5119896+8119902 (119906) 119889119906)119889119904] ≃ 06672 lt 1

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim inf119905rarrinfin

int5119896+45119896+3

119902 (119904) exp(int120590(119904)5119896+4

119902 (119906) 119889119906)119889119904≃ 01893 lt 1119890

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)1199021 (119906) 119889119906)119889119904 ≃ 07196

lt 107196 lt 1 minus 119863 (120573) ≃ 09896

(147)

That is none of conditions (24) (25) (28) (29) (30) (for 119895 =1) and (31) (for 119895 = 1) is satisfiedComment The improvement of condition (116) over thecorresponding condition (24) is significant approximately7723 We get this measure by comparing the values in theleft-hand side of those conditions Also the improvementover conditions (28) and (30) is very satisfactory around4861 and 3778 respectively In addition condition (116)is satisfied from the first iteration while conditions (30) and(31) need more than one iteration

Complexity 15

Remark 3 Similarly one can provide examples illustratingthe other main results

5 Concluding Remarks

In the present paper we have considered the oscillatorydynamics of differential equations having nonmonotonedeviating arguments and nonnegative coefficients New suf-ficient conditions have been established for the oscillation ofall solutions of (E) and (E1015840) These conditions include (37)(71) (75) (83) and (97) and (114) (116) (117) (118) and(119) for (E) and (E1015840) respectively Applying these conditionsinvolves a procedure that checks for oscillations by iterativelycomputing limsup and liminf on terms recursively definedon the equationrsquos coefficients and deviating argument

The main advantage of these conditions is that theyachieve a major improvement over all the related oscillationconditions for (E) [(E1015840)] in the literature For example con-dition (37) [(114)] improves upon the noniterative conditionsthat are reviewed in the introduction namely conditions (8)[(24)] (12) (16)equiv(20) (for 119895 = 1) [(28)equiv(30) (for 119895 = 1)]and (17)equiv(21) (for 119895 = 1) [(31) (for 119895 = 1)] That immediatelybecomes evident by inspecting the left-hand side of (37)[(114)] and the left-hand side of each of the above conditions

The improvement of (37) [(114)] over the other iterativeconditions namely (14) (for j gt 2) (20) (for 119895 gt 1) [(30)(for 119895 gt 1)] and (21) (for 119895 gt 1) [(31) (for 119895 gt 1)] is that itrequires far fewer iterations to establish oscillation than theother conditions

This advantage easily can be verified computationallyby running the MATLAB programs (see Appendix) forcomputing limsup and liminf and comparing the number ofiterations required by each condition to establish oscillationThen we see that we achieve a significant improvement overall known oscillation criteria

Another advantage and a significant departure from thelarge majority of the other studies is that the criteria in thispaper apply to a more general class of equations havingnonmonotone arguments 120591(119905) or 120590(119905) in contrast to mostof the other oscillation criteria that apply to equations withnondecreasing arguments

Appendix

In this appendix for completeness we give the algorithmon MATLAB software used in Example 1 for calculation oflim sup119905rarrinfin1198651(119905) ≃ 10417 For Example 2 the algorithm isomitted since it is similar to the one in Example 1

Algorithm for Example 1

clear clcc = 12119899 = 501205820 = 114765a5 = 19b5 = 703h5 = (b5 minus a5)119899

for i5 = 1 1 119899 + 1x5 = a5 + (i5 minus 1) lowast h5a4 = TFunction(x5)b4 = 19h4 = (b4 minus a4)119899for i4 = 1 1 119899 + 1x4 = a4 + (i4 minus 1) lowast h4a3 = TFunction(x4)b3 = x4h3 = (b3 minus a3)119899for i3 = 1 1 119899 + 1x3 = a3 + (i3 minus 1) lowast h3a2 = TFunction(x3)b2 = x3h2 = (b2 minus a2)119899for i2 = 1 1 119899 + 1x2 = a2 + (i2 minus 1) lowast h2a1 = TFunction(x2)b1 = x3h1 = (b1 minus a1)119899for i1 = 1 1 119899 + 1x1 = a1 + (i1 minus 1) lowast h1f1(i1) = c lowast exp(1205820 lowast c lowast (x1 minus TFunction(x1)))endI1 = f1(1) + f1(119899 + 1)for i1 = 2 2 119899I1 = I1 + f1(i1) lowast 4endfor i1 = 3 2 119899 minus 1I1 = I1 + f1(i1) lowast 2endI1 = I1 lowast h13f2(i2) = c lowast exp(I1)endI2 = f2(1) + f2(119899 + 1)for i2 = 2 2 119899I2 = I2 + f2(i2) lowast 4endfor i2 = 3 2 119899 minus 1I2 = I2 + f2(i2) lowast 2endI2 = I2 lowast h23f3(i3) = c lowast (1 + I2)end

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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Page 9: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

Complexity 9

Clearly by means of (36) 119909(ℎ(119904))119909(119904) gt 1205820minus120598 for 119904 ge ℎ(119905) ge1199051015840120598 Hence for all sufficiently large 119905 we conclude that(1205820 minus 120598)int119905

lowast

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minusint119905lowastℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904(93)

or

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt minus 11205820 minus 120598 int119905lowast

ℎ(119905)

1199091015840 (119904)119909 (119904) 119889119904 = 11205820 minus 120598 ln119909 (ℎ (119905))119909 (119905lowast)

= ln (1205820 minus 120598)1205820 minus 120598

(94)

that is

int119905lowastℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

lt ln (1205820 minus 120598)1205820 minus 120598 (95)

Using (91) along with (95) we get

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

le 1 + ln (1205820 minus 120598)1205820 minus 120598 minus 119863 (120572) + 120598(96)

which contradicts (83) when 120598 rarr 0This completes the proof of the theorem

Theorem 10 Let ℎ(119905) be defined by (13) If for some 119895 isin Nlim inf119905rarrinfin

int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

gt 1119890 (97)

where 119875119895 is defined by (38) then all solutions of (E) oscillate

Proof For the sake of contradiction let 119909 be a nonincreasingeventually positive solution and 1199051 gt 1199050 be such that 119909(119905) gt 0and 119909(120591(119905)) gt 0 for all 119905 ge 1199051 We note that wemay obtain (85)as in the proof of Theorem 9

Dividing (E) by 119909(119905) and integrating from ℎ(119905) to 119905 wehave

ln(119909 (ℎ (119905))119909 (119905) ) = int119905

ℎ(119905)119901 (119904) 119909 (120591 (119904))119909 (119904) 119889119904 forall119905 ge 1199052 ge 1199051 (98)

from which in view of 120591(119904) le ℎ(119904) and (85) we get

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot 119909 (ℎ (119904))119909 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(99)

Since 119909 is nonincreasing and ℎ(119904) lt 119904 inequality (99)becomes

ln(119909 (ℎ (119905))119909 (119905) ) ge int119905

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(100)

From (97) it is clear that there exists a constant 119888 gt 0 suchthat

int119905ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

ge 119888 gt 1119890 (101)

Choose 1198881015840 such that 119888 gt 1198881015840 gt 1119890 For every 120598 gt 0 such that119888 minus 120598 gt 1198881015840 we haveint119905ℎ(t)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 119888 minus 120598 gt 1198881015840 gt 1119890 (102)

Combining inequalities (100) and (102) we obtain

ln(119909 (ℎ (119905))119909 (119905) ) gt 1198881015840 (103)

or119909 (ℎ (119905))119909 (119905) gt 1198901198881015840 gt 1198901198881015840 gt 1 (104)

which yields

119909 (ℎ (119905)) gt (1198901198881015840) 119909 (119905) (105)

10 Complexity

Following the above steps we can inductively show that forany positive integer 119896

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 for sufficiently large 119905 (106)

Since 1198901198881015840 gt 1 there is a natural number 119896 isin N satisfying119896 gt 2[ln 2 minus ln 1198881015840](1 + ln 1198881015840) such that for 119905 sufficiently large

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 gt ( 21198881015840 )2 (107)

Further (cf [13 24]) for sufficiently large 119905 there exists a realnumber 119905119898 isin (ℎ(119905) 119905) such that

int119905119898ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402 int119905119905119898

119901 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402

(108)

Integrating (E) from ℎ(119905) to 119905119898 using (85) and the fact that119909(119905) gt 0 we obtain119909 (ℎ (119905)) gt 119909 (ℎ (119905119898)) int119905119898

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(109)

which in view of the first inequality in (108) implies that

119909 (ℎ (119905)) gt 11988810158402 119909 (ℎ (119905119898)) (110)

Similarly integrating (E) from 119905119898 to 119905 using (85) and the factthat 119909(119905) gt 0 we have119909 (119905119898) gt 119909 (ℎ (119905)) int119905

119905119898

119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(111)

which in view of the second inequality in (108) yields

119909 (119905119898) gt 11988810158402 119909 (ℎ (119905)) (112)

Combining inequalities (110) and (112) we deduce that

119909 (ℎ (119905119898)) lt 21198881015840 119909 (ℎ (119905)) lt ( 21198881015840 )2 119909 (119905119898) (113)

which contradicts (107)The proof of the theorem is complete

32 ADEs Analogous oscillation conditions to thoseobtained for the delay equation (E) can be derived for the(dual) advanced differential equation (E1015840) by followingsimilar arguments with the ones employed for obtainingTheorems 6minus10Theorem 11 Let 120588(119905) be defined by (27) and for some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1

(114)

where

119876119895 (119905) = 119902 (119905) [1 + int120590(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895minus1 (120585) 119889120585)119889119906)119889119904]

(115)

with 1198760(119905) = 1205820119902(119905) and let 1205820 be the smaller root of theequation 120582 = 119890120573120582 Then all solutions of (E1015840) oscillateTheorem 12 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 minus 119863 (120573)

(116)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillateTheorem 13 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119863 (120573) (117)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate

Complexity 11

Theorem 14 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120573) (118)

where 119876119895 is defined by (115) and 1205820 is the smaller root of theequation 120582 = 119890120573120582 then all solutions of (E1015840) oscillateTheorem 15 Let 120588(119905) be defined by (27) If for some 119895 isin N

lim inf119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119890 (119)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate33 Differential Inequalities A slight modification in theproofs of Theorems 6minus15 leads to the following results aboutdifferential inequalities

Theorem 16 Assume that all the conditions ofTheorem 6 [11] 7 [12] 8 [13] 9 [14] or 10 [15] hold Then(119894) the delay [advanced] differential inequality

1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) le 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) ge 0]

119905 ge 1199050(120)

has no eventually positive solutions(119894119894) the delay [advanced] differential inequality1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) ge 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) le 0]

119905 ge 1199050(121)

has no eventually negative solutions

Remark 17 The oscillation criteria established in this paperall depend on 1205820 (see eg (37) and (71)) in contrast to theconditions obtained in [15 16] and in [17 for m = 1] In factthe left-hand side of conditions (37) and (71) depends on 1205820which is not the casewith the left-hand side of conditions (20)and (21) Since 1205820 gt 1 when 120572 isin (0 1119890] it is obvious that

1198750 (119905) = 1205820119901 (119905) gt 119901 (119905) = 1199010 (119905) (122)

Consequently the left-hand side of conditions (37) and(71) is greater than the corresponding parts of (20) and(21) respectively This is the reason why the conditions inthis paper improve on all known conditions mentioned inSection 2

4 Examples and Comments

The oscillation tests we have proposed and established in themain results involve an iterative procedure We iterativelycompute limsup and liminf on the terms 119875119895(119905) and119876119895(119905) 119895 isinN of a recurrent relation defined on the coefficients and thedeviating argument of an equation of the form (E) or (E1015840)to determine whether that equation is oscillatory But thiscomputation cannot be performed on paper but by meansof a program numerically computing limsup and liminfThe examples below illustrate the significance of our resultsand indicate the high level of improvement in the oscillationcriteria The calculations were performed using MATLABcode

Example 1 Consider the delay differential equation

1199091015840 (119905) + 325119909 (120591 (119905)) = 0 119905 ge 0 (123)

with (see Figure 1(a))

120591 (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]minus4119905 + 40119896 + 9 if 119905 isin [8119896 + 2 8119896 + 3]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 3 8119896 + 4]minus4119905 + 40119896 + 18 if 119905 isin [8119896 + 4 8119896 + 5]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 5 8119896 + 6]minus2119905 + 24119896 + 15 if 119905 isin [8119896 + 6 8119896 + 7]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 7 8119896 + 8]

(124)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (13) we see (Figure 1(b)) that

ℎ (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]8119896 + 1 if 119905 isin [8119896 + 2 8119896 + 195 ]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 195 8119896 + 4]8119896 + 2 if 119905 isin [8119896 + 4 8119896 + 295 ]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 295 8119896 + 6]8119896 + 3 if 119905 isin [8119896 + 6 8119896 + 446 ]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 446 8119896 + 8]

(125)

It is obvious that

120572 = lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 = lim inf

119905rarrinfinint8119896+28119896+1

325119889119904= 012

(126)

12 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

minus1minus2

651 2 3 4 8 9 10 11 12 13 15 16

1415

minus3

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

minus1minus2

t

195 295

1415

7 14651 2 3 4 8 9 10 11 12 13 15 16

446

y = tℎ(t)

(b)

Figure 1 The graphs of 120591(119905) and ℎ(119905)

and therefore the smaller root of 119890012120582 = 120582 is 1205820 = 114765Observe that the function 119865119895 R0 rarr R+ defined as

119865119895 (119905) = int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

(127)

attains its maximum at 119905 = 8119896+446 119896 isin N0 for every 119895 isin NSpecifically

1198651 (119905 = 8119896 + 446 ) = int8119896+446

8119896+3119901 (119904)

sdot exp(int8119896+3120591(119904)

119901 (119906) exp(int119906120591(119906)1198751 (120585) 119889120585) 119889119906)119889119904

(128)

with

1198751 (120585) = 119901 (120585) [1 + int120585120591(120585)119901 (V)

sdot exp(int120585120591(V)119901 (119908) exp(int119908

120591(119908)1205820119901 (119911) 119889119911) 119889119908)119889V]

(129)

Using MATLAB we obtain

1198651 (119905 = 8119896 + 446 ) ≃ 10417 (130)

and therefore

lim sup119905rarrinfin

1198651 (119905) ≃ 10417 gt 1 (131)

Hence condition (37) of Theorem 6 is satisfied for 119895 = 1Consequently all solutions of (123) are oscillatory

Observe however that

LD = lim sup119896rarrinfin

int8119896+4468119896+3

335119889119904 = 052 lt 1120572 = 012 lt 1119890

052 lt 1 + ln 12058201205820 minus 119863 (120572) ≃ 09831(132)

Note that the functionΦ119895 defined by

Φ119895 (119905) = int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

ℎ(119904)119901 (119906) 120595119895 (119906) 119889119906)119889119904

119895 ge 2(133)

attains its maximum at 119905 = 8119896 + 446 119896 isin N0 for every 119895 ge 2Specifically

Φ2 (8119896 + 446 )= int8119896+4468119896+3

119901 (119904) exp(int8119896+3ℎ(119904)

119901 (119904) 1205952 (119906) 119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 exp(int119906

120591(119906)

325 sdot 0 119889119908)119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 sdot 1 119889119906)119889119904= 325 sdot [int

8119896+195

8119896+3exp( 325 int

8119896+3

8119896+1119889119906)119889119904

Complexity 13

+ int8119896+48119896+195

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+2958119896+4

exp( 325 int8119896+3

8119896+2119889119906)119889119904

+ int8119896+68119896+295

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+4468119896+6

exp( 325 int8119896+3

8119896+3119889119906)119889119904] ≃ 057983

(134)

Thus

lim sup119905rarrinfin

Φ2 (119905) ≃ 057983 lt 1 minus 119863 (120572) ≃ 099174 (135)

Also

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int8119896+4468119896+3

325 exp(int8119896+3

120591(119904)

325119889119906)119889119904= 325 sdot lim sup

119905rarrinfin[int8119896+48119896+3

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+58119896+4

exp( 325 int8119896+3

minus4119904+40119896+18119889119906)119889119904

+ int8119896+68119896+5

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+78119896+6

exp( 325 int8119896+3

minus2119904+24119896+15119889119906)119889119904

+ int8119896+4468119896+7

exp( 325 int8119896+3

6119904minus40119896minus41119889119906)119889119904] ≃ 07043

lt 107043 lt 1 minus 119863 (120572) ≃ 099174

(136)

In addition

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)1199011 (119906) 119889119906)119889119904 ≃ 08052

lt 108052 lt 1 minus 119863 (120572) ≃ 099174

(137)

That is none of conditions (8) (9) (12) (14) (for 119895 = 2) (16)(17) (20) (for 119895 = 1) and (21) (for 119895 = 1) is satisfiedComment The improvement of condition (37) over thecorresponding condition (8) is significant approximately10033 We get this measure by comparing the values inthe left-hand side of those conditions Also the improvementover conditions (14) (16) and (20) is very satisfactory

around 7966 479 and 2937 respectively In additioncondition (37) is satisfied from the first iteration whileconditions (14) (20) and (21) need more than one iteration

Example 2 (taken and adapted from [17]) Consider theadvanced differential equation

1199091015840 (119905) minus 3332500119909 (120590 (119905)) = 0 119905 ge 0 (138)

with (see Figure 2(a))

120590 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 2]minus3119905 + 20119896 + 13 if 119905 isin [5119896 + 2 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 4]minus119905 + 10119896 + 13 if 119905 isin [5119896 + 4 5119896 + 5]

(139)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (27) we see (Figure 2(b)) that

120588 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 125]5119896 + 4 if 119905 isin [5119896 + 125 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 38]5119896 + 8 if 119905 isin [5119896 + 38 5119896 + 5]

(140)

It is obvious that

120573 = lim inf119905rarrinfin

int5119896+45119896+3

3332500119889119904 = 01332 (141)

and therefore the smaller root of 11989001332120582 = 120582 is 1205820 = 116839Observe that the function 119866119895 R0 rarr R+ defined as

119866119895 (119905) = int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

(142)

attains its maximum at 119905 = 5119896 + 38 119896 isin N0 for every 119895 isin NSpecifically

1198661 (119905 = 5119896 + 38) = int5119896+85119896+38

119902 (119904)sdot exp(int120590(119904)

5119896+8119902 (119906) exp(int120590(119906)

1199061198761 (120585) 119889120585)119889119906)119889119904

(143)

with

1198761 (120585) = 119902 (120585) [1 + int120590(120585)120585119902 (V)

sdot exp(int120590(V)120585119902 (119908) exp(int120590(119908)

1199081205820119902 (119911) 119889119911)119889119908)119889V]

(144)

14 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

651 2 3 4 8 9 10 11 12 13

1415

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

7

t

14651

2 3 4 8 9 10 11 12 13

38

1415

125

y = t

(t)

(b)

Figure 2 The graphs of 120590(119905) and 120588(119905)

Using MATLAB we obtain

1198661 (119905 = 5119896 + 38) ≃ 09915 (145)

Thereforelim sup119905rarrinfin

1198661 (119905) ≃ 09915 gt 1 minus 119863 (120573) ≃ 09896 (146)

Hence condition (116) of Theorem 12 is satisfied for 119895 = 1Consequently all solutions of (138) oscillate

Observe however that

LA = lim sup119896rarrinfin

int5119896+85119896+38

3332500119889119904 = 055944 lt 1120573 = 01332 lt 1119890 lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int5119896+85119896+38

119902 (119904) exp(int120590(119904)5119896+8

119902 (119906) 119889119906)119889119904= lim sup119896rarrinfin

[int5119896+45119896+38

119902 (119904)sdot exp(int5119904minus20119896minus11

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+5

5119896+4119902 (119904)

sdot exp(intminus119904+10119896+135119896+8

119902 (119906) 119889119906)119889119904 + int5119896+65119896+5

119902 (119904)sdot exp(int5119896+8

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+7

5119896+6119902 (119904)

sdot exp(int4119904minus15119896minus165119896+8

119902 (119906) 119889119906)119889119904 + int5119896+85119896+7

119902 (119904)sdot exp(intminus3119904+20119896+33

5119896+8119902 (119906) 119889119906)119889119904] ≃ 06672 lt 1

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim inf119905rarrinfin

int5119896+45119896+3

119902 (119904) exp(int120590(119904)5119896+4

119902 (119906) 119889119906)119889119904≃ 01893 lt 1119890

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)1199021 (119906) 119889119906)119889119904 ≃ 07196

lt 107196 lt 1 minus 119863 (120573) ≃ 09896

(147)

That is none of conditions (24) (25) (28) (29) (30) (for 119895 =1) and (31) (for 119895 = 1) is satisfiedComment The improvement of condition (116) over thecorresponding condition (24) is significant approximately7723 We get this measure by comparing the values in theleft-hand side of those conditions Also the improvementover conditions (28) and (30) is very satisfactory around4861 and 3778 respectively In addition condition (116)is satisfied from the first iteration while conditions (30) and(31) need more than one iteration

Complexity 15

Remark 3 Similarly one can provide examples illustratingthe other main results

5 Concluding Remarks

In the present paper we have considered the oscillatorydynamics of differential equations having nonmonotonedeviating arguments and nonnegative coefficients New suf-ficient conditions have been established for the oscillation ofall solutions of (E) and (E1015840) These conditions include (37)(71) (75) (83) and (97) and (114) (116) (117) (118) and(119) for (E) and (E1015840) respectively Applying these conditionsinvolves a procedure that checks for oscillations by iterativelycomputing limsup and liminf on terms recursively definedon the equationrsquos coefficients and deviating argument

The main advantage of these conditions is that theyachieve a major improvement over all the related oscillationconditions for (E) [(E1015840)] in the literature For example con-dition (37) [(114)] improves upon the noniterative conditionsthat are reviewed in the introduction namely conditions (8)[(24)] (12) (16)equiv(20) (for 119895 = 1) [(28)equiv(30) (for 119895 = 1)]and (17)equiv(21) (for 119895 = 1) [(31) (for 119895 = 1)] That immediatelybecomes evident by inspecting the left-hand side of (37)[(114)] and the left-hand side of each of the above conditions

The improvement of (37) [(114)] over the other iterativeconditions namely (14) (for j gt 2) (20) (for 119895 gt 1) [(30)(for 119895 gt 1)] and (21) (for 119895 gt 1) [(31) (for 119895 gt 1)] is that itrequires far fewer iterations to establish oscillation than theother conditions

This advantage easily can be verified computationallyby running the MATLAB programs (see Appendix) forcomputing limsup and liminf and comparing the number ofiterations required by each condition to establish oscillationThen we see that we achieve a significant improvement overall known oscillation criteria

Another advantage and a significant departure from thelarge majority of the other studies is that the criteria in thispaper apply to a more general class of equations havingnonmonotone arguments 120591(119905) or 120590(119905) in contrast to mostof the other oscillation criteria that apply to equations withnondecreasing arguments

Appendix

In this appendix for completeness we give the algorithmon MATLAB software used in Example 1 for calculation oflim sup119905rarrinfin1198651(119905) ≃ 10417 For Example 2 the algorithm isomitted since it is similar to the one in Example 1

Algorithm for Example 1

clear clcc = 12119899 = 501205820 = 114765a5 = 19b5 = 703h5 = (b5 minus a5)119899

for i5 = 1 1 119899 + 1x5 = a5 + (i5 minus 1) lowast h5a4 = TFunction(x5)b4 = 19h4 = (b4 minus a4)119899for i4 = 1 1 119899 + 1x4 = a4 + (i4 minus 1) lowast h4a3 = TFunction(x4)b3 = x4h3 = (b3 minus a3)119899for i3 = 1 1 119899 + 1x3 = a3 + (i3 minus 1) lowast h3a2 = TFunction(x3)b2 = x3h2 = (b2 minus a2)119899for i2 = 1 1 119899 + 1x2 = a2 + (i2 minus 1) lowast h2a1 = TFunction(x2)b1 = x3h1 = (b1 minus a1)119899for i1 = 1 1 119899 + 1x1 = a1 + (i1 minus 1) lowast h1f1(i1) = c lowast exp(1205820 lowast c lowast (x1 minus TFunction(x1)))endI1 = f1(1) + f1(119899 + 1)for i1 = 2 2 119899I1 = I1 + f1(i1) lowast 4endfor i1 = 3 2 119899 minus 1I1 = I1 + f1(i1) lowast 2endI1 = I1 lowast h13f2(i2) = c lowast exp(I1)endI2 = f2(1) + f2(119899 + 1)for i2 = 2 2 119899I2 = I2 + f2(i2) lowast 4endfor i2 = 3 2 119899 minus 1I2 = I2 + f2(i2) lowast 2endI2 = I2 lowast h23f3(i3) = c lowast (1 + I2)end

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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Page 10: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

10 Complexity

Following the above steps we can inductively show that forany positive integer 119896

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 for sufficiently large 119905 (106)

Since 1198901198881015840 gt 1 there is a natural number 119896 isin N satisfying119896 gt 2[ln 2 minus ln 1198881015840](1 + ln 1198881015840) such that for 119905 sufficiently large

119909 (ℎ (119905))119909 (119905) gt (1198901198881015840)119896 gt ( 21198881015840 )2 (107)

Further (cf [13 24]) for sufficiently large 119905 there exists a realnumber 119905119898 isin (ℎ(119905) 119905) such that

int119905119898ℎ(119905)119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402 int119905119905119898

119901 (119904) exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

gt 11988810158402

(108)

Integrating (E) from ℎ(119905) to 119905119898 using (85) and the fact that119909(119905) gt 0 we obtain119909 (ℎ (119905)) gt 119909 (ℎ (119905119898)) int119905119898

ℎ(119905)119901 (119904)

sdot exp(intℎ(119904)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(109)

which in view of the first inequality in (108) implies that

119909 (ℎ (119905)) gt 11988810158402 119909 (ℎ (119905119898)) (110)

Similarly integrating (E) from 119905119898 to 119905 using (85) and the factthat 119909(119905) gt 0 we have119909 (119905119898) gt 119909 (ℎ (119905)) int119905

119905119898

119901 (119904)sdot exp(intℎ(119904)

120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585 120598) 119889120585) 119889119906)119889119904

(111)

which in view of the second inequality in (108) yields

119909 (119905119898) gt 11988810158402 119909 (ℎ (119905)) (112)

Combining inequalities (110) and (112) we deduce that

119909 (ℎ (119905119898)) lt 21198881015840 119909 (ℎ (119905)) lt ( 21198881015840 )2 119909 (119905119898) (113)

which contradicts (107)The proof of the theorem is complete

32 ADEs Analogous oscillation conditions to thoseobtained for the delay equation (E) can be derived for the(dual) advanced differential equation (E1015840) by followingsimilar arguments with the ones employed for obtainingTheorems 6minus10Theorem 11 Let 120588(119905) be defined by (27) and for some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1

(114)

where

119876119895 (119905) = 119902 (119905) [1 + int120590(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895minus1 (120585) 119889120585)119889119906)119889119904]

(115)

with 1198760(119905) = 1205820119902(119905) and let 1205820 be the smaller root of theequation 120582 = 119890120573120582 Then all solutions of (E1015840) oscillateTheorem 12 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 minus 119863 (120573)

(116)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillateTheorem 13 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)119905119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119863 (120573) (117)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate

Complexity 11

Theorem 14 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120573) (118)

where 119876119895 is defined by (115) and 1205820 is the smaller root of theequation 120582 = 119890120573120582 then all solutions of (E1015840) oscillateTheorem 15 Let 120588(119905) be defined by (27) If for some 119895 isin N

lim inf119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119890 (119)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate33 Differential Inequalities A slight modification in theproofs of Theorems 6minus15 leads to the following results aboutdifferential inequalities

Theorem 16 Assume that all the conditions ofTheorem 6 [11] 7 [12] 8 [13] 9 [14] or 10 [15] hold Then(119894) the delay [advanced] differential inequality

1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) le 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) ge 0]

119905 ge 1199050(120)

has no eventually positive solutions(119894119894) the delay [advanced] differential inequality1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) ge 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) le 0]

119905 ge 1199050(121)

has no eventually negative solutions

Remark 17 The oscillation criteria established in this paperall depend on 1205820 (see eg (37) and (71)) in contrast to theconditions obtained in [15 16] and in [17 for m = 1] In factthe left-hand side of conditions (37) and (71) depends on 1205820which is not the casewith the left-hand side of conditions (20)and (21) Since 1205820 gt 1 when 120572 isin (0 1119890] it is obvious that

1198750 (119905) = 1205820119901 (119905) gt 119901 (119905) = 1199010 (119905) (122)

Consequently the left-hand side of conditions (37) and(71) is greater than the corresponding parts of (20) and(21) respectively This is the reason why the conditions inthis paper improve on all known conditions mentioned inSection 2

4 Examples and Comments

The oscillation tests we have proposed and established in themain results involve an iterative procedure We iterativelycompute limsup and liminf on the terms 119875119895(119905) and119876119895(119905) 119895 isinN of a recurrent relation defined on the coefficients and thedeviating argument of an equation of the form (E) or (E1015840)to determine whether that equation is oscillatory But thiscomputation cannot be performed on paper but by meansof a program numerically computing limsup and liminfThe examples below illustrate the significance of our resultsand indicate the high level of improvement in the oscillationcriteria The calculations were performed using MATLABcode

Example 1 Consider the delay differential equation

1199091015840 (119905) + 325119909 (120591 (119905)) = 0 119905 ge 0 (123)

with (see Figure 1(a))

120591 (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]minus4119905 + 40119896 + 9 if 119905 isin [8119896 + 2 8119896 + 3]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 3 8119896 + 4]minus4119905 + 40119896 + 18 if 119905 isin [8119896 + 4 8119896 + 5]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 5 8119896 + 6]minus2119905 + 24119896 + 15 if 119905 isin [8119896 + 6 8119896 + 7]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 7 8119896 + 8]

(124)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (13) we see (Figure 1(b)) that

ℎ (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]8119896 + 1 if 119905 isin [8119896 + 2 8119896 + 195 ]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 195 8119896 + 4]8119896 + 2 if 119905 isin [8119896 + 4 8119896 + 295 ]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 295 8119896 + 6]8119896 + 3 if 119905 isin [8119896 + 6 8119896 + 446 ]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 446 8119896 + 8]

(125)

It is obvious that

120572 = lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 = lim inf

119905rarrinfinint8119896+28119896+1

325119889119904= 012

(126)

12 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

minus1minus2

651 2 3 4 8 9 10 11 12 13 15 16

1415

minus3

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

minus1minus2

t

195 295

1415

7 14651 2 3 4 8 9 10 11 12 13 15 16

446

y = tℎ(t)

(b)

Figure 1 The graphs of 120591(119905) and ℎ(119905)

and therefore the smaller root of 119890012120582 = 120582 is 1205820 = 114765Observe that the function 119865119895 R0 rarr R+ defined as

119865119895 (119905) = int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

(127)

attains its maximum at 119905 = 8119896+446 119896 isin N0 for every 119895 isin NSpecifically

1198651 (119905 = 8119896 + 446 ) = int8119896+446

8119896+3119901 (119904)

sdot exp(int8119896+3120591(119904)

119901 (119906) exp(int119906120591(119906)1198751 (120585) 119889120585) 119889119906)119889119904

(128)

with

1198751 (120585) = 119901 (120585) [1 + int120585120591(120585)119901 (V)

sdot exp(int120585120591(V)119901 (119908) exp(int119908

120591(119908)1205820119901 (119911) 119889119911) 119889119908)119889V]

(129)

Using MATLAB we obtain

1198651 (119905 = 8119896 + 446 ) ≃ 10417 (130)

and therefore

lim sup119905rarrinfin

1198651 (119905) ≃ 10417 gt 1 (131)

Hence condition (37) of Theorem 6 is satisfied for 119895 = 1Consequently all solutions of (123) are oscillatory

Observe however that

LD = lim sup119896rarrinfin

int8119896+4468119896+3

335119889119904 = 052 lt 1120572 = 012 lt 1119890

052 lt 1 + ln 12058201205820 minus 119863 (120572) ≃ 09831(132)

Note that the functionΦ119895 defined by

Φ119895 (119905) = int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

ℎ(119904)119901 (119906) 120595119895 (119906) 119889119906)119889119904

119895 ge 2(133)

attains its maximum at 119905 = 8119896 + 446 119896 isin N0 for every 119895 ge 2Specifically

Φ2 (8119896 + 446 )= int8119896+4468119896+3

119901 (119904) exp(int8119896+3ℎ(119904)

119901 (119904) 1205952 (119906) 119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 exp(int119906

120591(119906)

325 sdot 0 119889119908)119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 sdot 1 119889119906)119889119904= 325 sdot [int

8119896+195

8119896+3exp( 325 int

8119896+3

8119896+1119889119906)119889119904

Complexity 13

+ int8119896+48119896+195

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+2958119896+4

exp( 325 int8119896+3

8119896+2119889119906)119889119904

+ int8119896+68119896+295

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+4468119896+6

exp( 325 int8119896+3

8119896+3119889119906)119889119904] ≃ 057983

(134)

Thus

lim sup119905rarrinfin

Φ2 (119905) ≃ 057983 lt 1 minus 119863 (120572) ≃ 099174 (135)

Also

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int8119896+4468119896+3

325 exp(int8119896+3

120591(119904)

325119889119906)119889119904= 325 sdot lim sup

119905rarrinfin[int8119896+48119896+3

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+58119896+4

exp( 325 int8119896+3

minus4119904+40119896+18119889119906)119889119904

+ int8119896+68119896+5

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+78119896+6

exp( 325 int8119896+3

minus2119904+24119896+15119889119906)119889119904

+ int8119896+4468119896+7

exp( 325 int8119896+3

6119904minus40119896minus41119889119906)119889119904] ≃ 07043

lt 107043 lt 1 minus 119863 (120572) ≃ 099174

(136)

In addition

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)1199011 (119906) 119889119906)119889119904 ≃ 08052

lt 108052 lt 1 minus 119863 (120572) ≃ 099174

(137)

That is none of conditions (8) (9) (12) (14) (for 119895 = 2) (16)(17) (20) (for 119895 = 1) and (21) (for 119895 = 1) is satisfiedComment The improvement of condition (37) over thecorresponding condition (8) is significant approximately10033 We get this measure by comparing the values inthe left-hand side of those conditions Also the improvementover conditions (14) (16) and (20) is very satisfactory

around 7966 479 and 2937 respectively In additioncondition (37) is satisfied from the first iteration whileconditions (14) (20) and (21) need more than one iteration

Example 2 (taken and adapted from [17]) Consider theadvanced differential equation

1199091015840 (119905) minus 3332500119909 (120590 (119905)) = 0 119905 ge 0 (138)

with (see Figure 2(a))

120590 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 2]minus3119905 + 20119896 + 13 if 119905 isin [5119896 + 2 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 4]minus119905 + 10119896 + 13 if 119905 isin [5119896 + 4 5119896 + 5]

(139)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (27) we see (Figure 2(b)) that

120588 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 125]5119896 + 4 if 119905 isin [5119896 + 125 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 38]5119896 + 8 if 119905 isin [5119896 + 38 5119896 + 5]

(140)

It is obvious that

120573 = lim inf119905rarrinfin

int5119896+45119896+3

3332500119889119904 = 01332 (141)

and therefore the smaller root of 11989001332120582 = 120582 is 1205820 = 116839Observe that the function 119866119895 R0 rarr R+ defined as

119866119895 (119905) = int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

(142)

attains its maximum at 119905 = 5119896 + 38 119896 isin N0 for every 119895 isin NSpecifically

1198661 (119905 = 5119896 + 38) = int5119896+85119896+38

119902 (119904)sdot exp(int120590(119904)

5119896+8119902 (119906) exp(int120590(119906)

1199061198761 (120585) 119889120585)119889119906)119889119904

(143)

with

1198761 (120585) = 119902 (120585) [1 + int120590(120585)120585119902 (V)

sdot exp(int120590(V)120585119902 (119908) exp(int120590(119908)

1199081205820119902 (119911) 119889119911)119889119908)119889V]

(144)

14 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

651 2 3 4 8 9 10 11 12 13

1415

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

7

t

14651

2 3 4 8 9 10 11 12 13

38

1415

125

y = t

(t)

(b)

Figure 2 The graphs of 120590(119905) and 120588(119905)

Using MATLAB we obtain

1198661 (119905 = 5119896 + 38) ≃ 09915 (145)

Thereforelim sup119905rarrinfin

1198661 (119905) ≃ 09915 gt 1 minus 119863 (120573) ≃ 09896 (146)

Hence condition (116) of Theorem 12 is satisfied for 119895 = 1Consequently all solutions of (138) oscillate

Observe however that

LA = lim sup119896rarrinfin

int5119896+85119896+38

3332500119889119904 = 055944 lt 1120573 = 01332 lt 1119890 lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int5119896+85119896+38

119902 (119904) exp(int120590(119904)5119896+8

119902 (119906) 119889119906)119889119904= lim sup119896rarrinfin

[int5119896+45119896+38

119902 (119904)sdot exp(int5119904minus20119896minus11

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+5

5119896+4119902 (119904)

sdot exp(intminus119904+10119896+135119896+8

119902 (119906) 119889119906)119889119904 + int5119896+65119896+5

119902 (119904)sdot exp(int5119896+8

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+7

5119896+6119902 (119904)

sdot exp(int4119904minus15119896minus165119896+8

119902 (119906) 119889119906)119889119904 + int5119896+85119896+7

119902 (119904)sdot exp(intminus3119904+20119896+33

5119896+8119902 (119906) 119889119906)119889119904] ≃ 06672 lt 1

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim inf119905rarrinfin

int5119896+45119896+3

119902 (119904) exp(int120590(119904)5119896+4

119902 (119906) 119889119906)119889119904≃ 01893 lt 1119890

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)1199021 (119906) 119889119906)119889119904 ≃ 07196

lt 107196 lt 1 minus 119863 (120573) ≃ 09896

(147)

That is none of conditions (24) (25) (28) (29) (30) (for 119895 =1) and (31) (for 119895 = 1) is satisfiedComment The improvement of condition (116) over thecorresponding condition (24) is significant approximately7723 We get this measure by comparing the values in theleft-hand side of those conditions Also the improvementover conditions (28) and (30) is very satisfactory around4861 and 3778 respectively In addition condition (116)is satisfied from the first iteration while conditions (30) and(31) need more than one iteration

Complexity 15

Remark 3 Similarly one can provide examples illustratingthe other main results

5 Concluding Remarks

In the present paper we have considered the oscillatorydynamics of differential equations having nonmonotonedeviating arguments and nonnegative coefficients New suf-ficient conditions have been established for the oscillation ofall solutions of (E) and (E1015840) These conditions include (37)(71) (75) (83) and (97) and (114) (116) (117) (118) and(119) for (E) and (E1015840) respectively Applying these conditionsinvolves a procedure that checks for oscillations by iterativelycomputing limsup and liminf on terms recursively definedon the equationrsquos coefficients and deviating argument

The main advantage of these conditions is that theyachieve a major improvement over all the related oscillationconditions for (E) [(E1015840)] in the literature For example con-dition (37) [(114)] improves upon the noniterative conditionsthat are reviewed in the introduction namely conditions (8)[(24)] (12) (16)equiv(20) (for 119895 = 1) [(28)equiv(30) (for 119895 = 1)]and (17)equiv(21) (for 119895 = 1) [(31) (for 119895 = 1)] That immediatelybecomes evident by inspecting the left-hand side of (37)[(114)] and the left-hand side of each of the above conditions

The improvement of (37) [(114)] over the other iterativeconditions namely (14) (for j gt 2) (20) (for 119895 gt 1) [(30)(for 119895 gt 1)] and (21) (for 119895 gt 1) [(31) (for 119895 gt 1)] is that itrequires far fewer iterations to establish oscillation than theother conditions

This advantage easily can be verified computationallyby running the MATLAB programs (see Appendix) forcomputing limsup and liminf and comparing the number ofiterations required by each condition to establish oscillationThen we see that we achieve a significant improvement overall known oscillation criteria

Another advantage and a significant departure from thelarge majority of the other studies is that the criteria in thispaper apply to a more general class of equations havingnonmonotone arguments 120591(119905) or 120590(119905) in contrast to mostof the other oscillation criteria that apply to equations withnondecreasing arguments

Appendix

In this appendix for completeness we give the algorithmon MATLAB software used in Example 1 for calculation oflim sup119905rarrinfin1198651(119905) ≃ 10417 For Example 2 the algorithm isomitted since it is similar to the one in Example 1

Algorithm for Example 1

clear clcc = 12119899 = 501205820 = 114765a5 = 19b5 = 703h5 = (b5 minus a5)119899

for i5 = 1 1 119899 + 1x5 = a5 + (i5 minus 1) lowast h5a4 = TFunction(x5)b4 = 19h4 = (b4 minus a4)119899for i4 = 1 1 119899 + 1x4 = a4 + (i4 minus 1) lowast h4a3 = TFunction(x4)b3 = x4h3 = (b3 minus a3)119899for i3 = 1 1 119899 + 1x3 = a3 + (i3 minus 1) lowast h3a2 = TFunction(x3)b2 = x3h2 = (b2 minus a2)119899for i2 = 1 1 119899 + 1x2 = a2 + (i2 minus 1) lowast h2a1 = TFunction(x2)b1 = x3h1 = (b1 minus a1)119899for i1 = 1 1 119899 + 1x1 = a1 + (i1 minus 1) lowast h1f1(i1) = c lowast exp(1205820 lowast c lowast (x1 minus TFunction(x1)))endI1 = f1(1) + f1(119899 + 1)for i1 = 2 2 119899I1 = I1 + f1(i1) lowast 4endfor i1 = 3 2 119899 minus 1I1 = I1 + f1(i1) lowast 2endI1 = I1 lowast h13f2(i2) = c lowast exp(I1)endI2 = f2(1) + f2(119899 + 1)for i2 = 2 2 119899I2 = I2 + f2(i2) lowast 4endfor i2 = 3 2 119899 minus 1I2 = I2 + f2(i2) lowast 2endI2 = I2 lowast h23f3(i3) = c lowast (1 + I2)end

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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Page 11: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

Complexity 11

Theorem 14 Let 120588(119905) be defined by (27) and 120573 isin (0 1119890] Iffor some 119895 isin N

lim sup119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1 + ln 12058201205820 minus 119863 (120573) (118)

where 119876119895 is defined by (115) and 1205820 is the smaller root of theequation 120582 = 119890120573120582 then all solutions of (E1015840) oscillateTheorem 15 Let 120588(119905) be defined by (27) If for some 119895 isin N

lim inf119905rarrinfin

int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119904)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

gt 1119890 (119)

where 119876119895 is defined by (115) then all solutions of (E1015840) oscillate33 Differential Inequalities A slight modification in theproofs of Theorems 6minus15 leads to the following results aboutdifferential inequalities

Theorem 16 Assume that all the conditions ofTheorem 6 [11] 7 [12] 8 [13] 9 [14] or 10 [15] hold Then(119894) the delay [advanced] differential inequality

1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) le 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) ge 0]

119905 ge 1199050(120)

has no eventually positive solutions(119894119894) the delay [advanced] differential inequality1199091015840 (119905) + 119901 (119905) 119909 (120591 (119905)) ge 0[1199091015840 (119905) minus 119902 (119905) 119909 (120590 (119905)) le 0]

119905 ge 1199050(121)

has no eventually negative solutions

Remark 17 The oscillation criteria established in this paperall depend on 1205820 (see eg (37) and (71)) in contrast to theconditions obtained in [15 16] and in [17 for m = 1] In factthe left-hand side of conditions (37) and (71) depends on 1205820which is not the casewith the left-hand side of conditions (20)and (21) Since 1205820 gt 1 when 120572 isin (0 1119890] it is obvious that

1198750 (119905) = 1205820119901 (119905) gt 119901 (119905) = 1199010 (119905) (122)

Consequently the left-hand side of conditions (37) and(71) is greater than the corresponding parts of (20) and(21) respectively This is the reason why the conditions inthis paper improve on all known conditions mentioned inSection 2

4 Examples and Comments

The oscillation tests we have proposed and established in themain results involve an iterative procedure We iterativelycompute limsup and liminf on the terms 119875119895(119905) and119876119895(119905) 119895 isinN of a recurrent relation defined on the coefficients and thedeviating argument of an equation of the form (E) or (E1015840)to determine whether that equation is oscillatory But thiscomputation cannot be performed on paper but by meansof a program numerically computing limsup and liminfThe examples below illustrate the significance of our resultsand indicate the high level of improvement in the oscillationcriteria The calculations were performed using MATLABcode

Example 1 Consider the delay differential equation

1199091015840 (119905) + 325119909 (120591 (119905)) = 0 119905 ge 0 (123)

with (see Figure 1(a))

120591 (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]minus4119905 + 40119896 + 9 if 119905 isin [8119896 + 2 8119896 + 3]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 3 8119896 + 4]minus4119905 + 40119896 + 18 if 119905 isin [8119896 + 4 8119896 + 5]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 5 8119896 + 6]minus2119905 + 24119896 + 15 if 119905 isin [8119896 + 6 8119896 + 7]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 7 8119896 + 8]

(124)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (13) we see (Figure 1(b)) that

ℎ (119905) =

119905 minus 1 if 119905 isin [8119896 8119896 + 2]8119896 + 1 if 119905 isin [8119896 + 2 8119896 + 195 ]5119905 minus 32119896 minus 18 if 119905 isin [8119896 + 195 8119896 + 4]8119896 + 2 if 119905 isin [8119896 + 4 8119896 + 295 ]5119905 minus 32119896 minus 27 if 119905 isin [8119896 + 295 8119896 + 6]8119896 + 3 if 119905 isin [8119896 + 6 8119896 + 446 ]6119905 minus 40119896 minus 41 if 119905 isin [8119896 + 446 8119896 + 8]

(125)

It is obvious that

120572 = lim inf119905rarrinfin

int119905120591(119905)119901 (119904) 119889119904 = lim inf

119905rarrinfinint8119896+28119896+1

325119889119904= 012

(126)

12 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

minus1minus2

651 2 3 4 8 9 10 11 12 13 15 16

1415

minus3

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

minus1minus2

t

195 295

1415

7 14651 2 3 4 8 9 10 11 12 13 15 16

446

y = tℎ(t)

(b)

Figure 1 The graphs of 120591(119905) and ℎ(119905)

and therefore the smaller root of 119890012120582 = 120582 is 1205820 = 114765Observe that the function 119865119895 R0 rarr R+ defined as

119865119895 (119905) = int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

(127)

attains its maximum at 119905 = 8119896+446 119896 isin N0 for every 119895 isin NSpecifically

1198651 (119905 = 8119896 + 446 ) = int8119896+446

8119896+3119901 (119904)

sdot exp(int8119896+3120591(119904)

119901 (119906) exp(int119906120591(119906)1198751 (120585) 119889120585) 119889119906)119889119904

(128)

with

1198751 (120585) = 119901 (120585) [1 + int120585120591(120585)119901 (V)

sdot exp(int120585120591(V)119901 (119908) exp(int119908

120591(119908)1205820119901 (119911) 119889119911) 119889119908)119889V]

(129)

Using MATLAB we obtain

1198651 (119905 = 8119896 + 446 ) ≃ 10417 (130)

and therefore

lim sup119905rarrinfin

1198651 (119905) ≃ 10417 gt 1 (131)

Hence condition (37) of Theorem 6 is satisfied for 119895 = 1Consequently all solutions of (123) are oscillatory

Observe however that

LD = lim sup119896rarrinfin

int8119896+4468119896+3

335119889119904 = 052 lt 1120572 = 012 lt 1119890

052 lt 1 + ln 12058201205820 minus 119863 (120572) ≃ 09831(132)

Note that the functionΦ119895 defined by

Φ119895 (119905) = int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

ℎ(119904)119901 (119906) 120595119895 (119906) 119889119906)119889119904

119895 ge 2(133)

attains its maximum at 119905 = 8119896 + 446 119896 isin N0 for every 119895 ge 2Specifically

Φ2 (8119896 + 446 )= int8119896+4468119896+3

119901 (119904) exp(int8119896+3ℎ(119904)

119901 (119904) 1205952 (119906) 119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 exp(int119906

120591(119906)

325 sdot 0 119889119908)119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 sdot 1 119889119906)119889119904= 325 sdot [int

8119896+195

8119896+3exp( 325 int

8119896+3

8119896+1119889119906)119889119904

Complexity 13

+ int8119896+48119896+195

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+2958119896+4

exp( 325 int8119896+3

8119896+2119889119906)119889119904

+ int8119896+68119896+295

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+4468119896+6

exp( 325 int8119896+3

8119896+3119889119906)119889119904] ≃ 057983

(134)

Thus

lim sup119905rarrinfin

Φ2 (119905) ≃ 057983 lt 1 minus 119863 (120572) ≃ 099174 (135)

Also

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int8119896+4468119896+3

325 exp(int8119896+3

120591(119904)

325119889119906)119889119904= 325 sdot lim sup

119905rarrinfin[int8119896+48119896+3

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+58119896+4

exp( 325 int8119896+3

minus4119904+40119896+18119889119906)119889119904

+ int8119896+68119896+5

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+78119896+6

exp( 325 int8119896+3

minus2119904+24119896+15119889119906)119889119904

+ int8119896+4468119896+7

exp( 325 int8119896+3

6119904minus40119896minus41119889119906)119889119904] ≃ 07043

lt 107043 lt 1 minus 119863 (120572) ≃ 099174

(136)

In addition

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)1199011 (119906) 119889119906)119889119904 ≃ 08052

lt 108052 lt 1 minus 119863 (120572) ≃ 099174

(137)

That is none of conditions (8) (9) (12) (14) (for 119895 = 2) (16)(17) (20) (for 119895 = 1) and (21) (for 119895 = 1) is satisfiedComment The improvement of condition (37) over thecorresponding condition (8) is significant approximately10033 We get this measure by comparing the values inthe left-hand side of those conditions Also the improvementover conditions (14) (16) and (20) is very satisfactory

around 7966 479 and 2937 respectively In additioncondition (37) is satisfied from the first iteration whileconditions (14) (20) and (21) need more than one iteration

Example 2 (taken and adapted from [17]) Consider theadvanced differential equation

1199091015840 (119905) minus 3332500119909 (120590 (119905)) = 0 119905 ge 0 (138)

with (see Figure 2(a))

120590 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 2]minus3119905 + 20119896 + 13 if 119905 isin [5119896 + 2 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 4]minus119905 + 10119896 + 13 if 119905 isin [5119896 + 4 5119896 + 5]

(139)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (27) we see (Figure 2(b)) that

120588 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 125]5119896 + 4 if 119905 isin [5119896 + 125 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 38]5119896 + 8 if 119905 isin [5119896 + 38 5119896 + 5]

(140)

It is obvious that

120573 = lim inf119905rarrinfin

int5119896+45119896+3

3332500119889119904 = 01332 (141)

and therefore the smaller root of 11989001332120582 = 120582 is 1205820 = 116839Observe that the function 119866119895 R0 rarr R+ defined as

119866119895 (119905) = int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

(142)

attains its maximum at 119905 = 5119896 + 38 119896 isin N0 for every 119895 isin NSpecifically

1198661 (119905 = 5119896 + 38) = int5119896+85119896+38

119902 (119904)sdot exp(int120590(119904)

5119896+8119902 (119906) exp(int120590(119906)

1199061198761 (120585) 119889120585)119889119906)119889119904

(143)

with

1198761 (120585) = 119902 (120585) [1 + int120590(120585)120585119902 (V)

sdot exp(int120590(V)120585119902 (119908) exp(int120590(119908)

1199081205820119902 (119911) 119889119911)119889119908)119889V]

(144)

14 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

651 2 3 4 8 9 10 11 12 13

1415

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

7

t

14651

2 3 4 8 9 10 11 12 13

38

1415

125

y = t

(t)

(b)

Figure 2 The graphs of 120590(119905) and 120588(119905)

Using MATLAB we obtain

1198661 (119905 = 5119896 + 38) ≃ 09915 (145)

Thereforelim sup119905rarrinfin

1198661 (119905) ≃ 09915 gt 1 minus 119863 (120573) ≃ 09896 (146)

Hence condition (116) of Theorem 12 is satisfied for 119895 = 1Consequently all solutions of (138) oscillate

Observe however that

LA = lim sup119896rarrinfin

int5119896+85119896+38

3332500119889119904 = 055944 lt 1120573 = 01332 lt 1119890 lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int5119896+85119896+38

119902 (119904) exp(int120590(119904)5119896+8

119902 (119906) 119889119906)119889119904= lim sup119896rarrinfin

[int5119896+45119896+38

119902 (119904)sdot exp(int5119904minus20119896minus11

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+5

5119896+4119902 (119904)

sdot exp(intminus119904+10119896+135119896+8

119902 (119906) 119889119906)119889119904 + int5119896+65119896+5

119902 (119904)sdot exp(int5119896+8

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+7

5119896+6119902 (119904)

sdot exp(int4119904minus15119896minus165119896+8

119902 (119906) 119889119906)119889119904 + int5119896+85119896+7

119902 (119904)sdot exp(intminus3119904+20119896+33

5119896+8119902 (119906) 119889119906)119889119904] ≃ 06672 lt 1

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim inf119905rarrinfin

int5119896+45119896+3

119902 (119904) exp(int120590(119904)5119896+4

119902 (119906) 119889119906)119889119904≃ 01893 lt 1119890

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)1199021 (119906) 119889119906)119889119904 ≃ 07196

lt 107196 lt 1 minus 119863 (120573) ≃ 09896

(147)

That is none of conditions (24) (25) (28) (29) (30) (for 119895 =1) and (31) (for 119895 = 1) is satisfiedComment The improvement of condition (116) over thecorresponding condition (24) is significant approximately7723 We get this measure by comparing the values in theleft-hand side of those conditions Also the improvementover conditions (28) and (30) is very satisfactory around4861 and 3778 respectively In addition condition (116)is satisfied from the first iteration while conditions (30) and(31) need more than one iteration

Complexity 15

Remark 3 Similarly one can provide examples illustratingthe other main results

5 Concluding Remarks

In the present paper we have considered the oscillatorydynamics of differential equations having nonmonotonedeviating arguments and nonnegative coefficients New suf-ficient conditions have been established for the oscillation ofall solutions of (E) and (E1015840) These conditions include (37)(71) (75) (83) and (97) and (114) (116) (117) (118) and(119) for (E) and (E1015840) respectively Applying these conditionsinvolves a procedure that checks for oscillations by iterativelycomputing limsup and liminf on terms recursively definedon the equationrsquos coefficients and deviating argument

The main advantage of these conditions is that theyachieve a major improvement over all the related oscillationconditions for (E) [(E1015840)] in the literature For example con-dition (37) [(114)] improves upon the noniterative conditionsthat are reviewed in the introduction namely conditions (8)[(24)] (12) (16)equiv(20) (for 119895 = 1) [(28)equiv(30) (for 119895 = 1)]and (17)equiv(21) (for 119895 = 1) [(31) (for 119895 = 1)] That immediatelybecomes evident by inspecting the left-hand side of (37)[(114)] and the left-hand side of each of the above conditions

The improvement of (37) [(114)] over the other iterativeconditions namely (14) (for j gt 2) (20) (for 119895 gt 1) [(30)(for 119895 gt 1)] and (21) (for 119895 gt 1) [(31) (for 119895 gt 1)] is that itrequires far fewer iterations to establish oscillation than theother conditions

This advantage easily can be verified computationallyby running the MATLAB programs (see Appendix) forcomputing limsup and liminf and comparing the number ofiterations required by each condition to establish oscillationThen we see that we achieve a significant improvement overall known oscillation criteria

Another advantage and a significant departure from thelarge majority of the other studies is that the criteria in thispaper apply to a more general class of equations havingnonmonotone arguments 120591(119905) or 120590(119905) in contrast to mostof the other oscillation criteria that apply to equations withnondecreasing arguments

Appendix

In this appendix for completeness we give the algorithmon MATLAB software used in Example 1 for calculation oflim sup119905rarrinfin1198651(119905) ≃ 10417 For Example 2 the algorithm isomitted since it is similar to the one in Example 1

Algorithm for Example 1

clear clcc = 12119899 = 501205820 = 114765a5 = 19b5 = 703h5 = (b5 minus a5)119899

for i5 = 1 1 119899 + 1x5 = a5 + (i5 minus 1) lowast h5a4 = TFunction(x5)b4 = 19h4 = (b4 minus a4)119899for i4 = 1 1 119899 + 1x4 = a4 + (i4 minus 1) lowast h4a3 = TFunction(x4)b3 = x4h3 = (b3 minus a3)119899for i3 = 1 1 119899 + 1x3 = a3 + (i3 minus 1) lowast h3a2 = TFunction(x3)b2 = x3h2 = (b2 minus a2)119899for i2 = 1 1 119899 + 1x2 = a2 + (i2 minus 1) lowast h2a1 = TFunction(x2)b1 = x3h1 = (b1 minus a1)119899for i1 = 1 1 119899 + 1x1 = a1 + (i1 minus 1) lowast h1f1(i1) = c lowast exp(1205820 lowast c lowast (x1 minus TFunction(x1)))endI1 = f1(1) + f1(119899 + 1)for i1 = 2 2 119899I1 = I1 + f1(i1) lowast 4endfor i1 = 3 2 119899 minus 1I1 = I1 + f1(i1) lowast 2endI1 = I1 lowast h13f2(i2) = c lowast exp(I1)endI2 = f2(1) + f2(119899 + 1)for i2 = 2 2 119899I2 = I2 + f2(i2) lowast 4endfor i2 = 3 2 119899 minus 1I2 = I2 + f2(i2) lowast 2endI2 = I2 lowast h23f3(i3) = c lowast (1 + I2)end

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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Page 12: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

12 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

minus1minus2

651 2 3 4 8 9 10 11 12 13 15 16

1415

minus3

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

minus1minus2

t

195 295

1415

7 14651 2 3 4 8 9 10 11 12 13 15 16

446

y = tℎ(t)

(b)

Figure 1 The graphs of 120591(119905) and ℎ(119905)

and therefore the smaller root of 119890012120582 = 120582 is 1205820 = 114765Observe that the function 119865119895 R0 rarr R+ defined as

119865119895 (119905) = int119905ℎ(119905)119901 (119904)

sdot exp(intℎ(119905)120591(119904)119901 (119906) exp(int119906

120591(119906)119875119895 (120585) 119889120585) 119889119906)119889119904

(127)

attains its maximum at 119905 = 8119896+446 119896 isin N0 for every 119895 isin NSpecifically

1198651 (119905 = 8119896 + 446 ) = int8119896+446

8119896+3119901 (119904)

sdot exp(int8119896+3120591(119904)

119901 (119906) exp(int119906120591(119906)1198751 (120585) 119889120585) 119889119906)119889119904

(128)

with

1198751 (120585) = 119901 (120585) [1 + int120585120591(120585)119901 (V)

sdot exp(int120585120591(V)119901 (119908) exp(int119908

120591(119908)1205820119901 (119911) 119889119911) 119889119908)119889V]

(129)

Using MATLAB we obtain

1198651 (119905 = 8119896 + 446 ) ≃ 10417 (130)

and therefore

lim sup119905rarrinfin

1198651 (119905) ≃ 10417 gt 1 (131)

Hence condition (37) of Theorem 6 is satisfied for 119895 = 1Consequently all solutions of (123) are oscillatory

Observe however that

LD = lim sup119896rarrinfin

int8119896+4468119896+3

335119889119904 = 052 lt 1120572 = 012 lt 1119890

052 lt 1 + ln 12058201205820 minus 119863 (120572) ≃ 09831(132)

Note that the functionΦ119895 defined by

Φ119895 (119905) = int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

ℎ(119904)119901 (119906) 120595119895 (119906) 119889119906)119889119904

119895 ge 2(133)

attains its maximum at 119905 = 8119896 + 446 119896 isin N0 for every 119895 ge 2Specifically

Φ2 (8119896 + 446 )= int8119896+4468119896+3

119901 (119904) exp(int8119896+3ℎ(119904)

119901 (119904) 1205952 (119906) 119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 exp(int119906

120591(119906)

325 sdot 0 119889119908)119889119906)119889119904= int8119896+4468119896+3

325 exp(int8119896+3

ℎ(119904)

325 sdot 1 119889119906)119889119904= 325 sdot [int

8119896+195

8119896+3exp( 325 int

8119896+3

8119896+1119889119906)119889119904

Complexity 13

+ int8119896+48119896+195

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+2958119896+4

exp( 325 int8119896+3

8119896+2119889119906)119889119904

+ int8119896+68119896+295

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+4468119896+6

exp( 325 int8119896+3

8119896+3119889119906)119889119904] ≃ 057983

(134)

Thus

lim sup119905rarrinfin

Φ2 (119905) ≃ 057983 lt 1 minus 119863 (120572) ≃ 099174 (135)

Also

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int8119896+4468119896+3

325 exp(int8119896+3

120591(119904)

325119889119906)119889119904= 325 sdot lim sup

119905rarrinfin[int8119896+48119896+3

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+58119896+4

exp( 325 int8119896+3

minus4119904+40119896+18119889119906)119889119904

+ int8119896+68119896+5

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+78119896+6

exp( 325 int8119896+3

minus2119904+24119896+15119889119906)119889119904

+ int8119896+4468119896+7

exp( 325 int8119896+3

6119904minus40119896minus41119889119906)119889119904] ≃ 07043

lt 107043 lt 1 minus 119863 (120572) ≃ 099174

(136)

In addition

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)1199011 (119906) 119889119906)119889119904 ≃ 08052

lt 108052 lt 1 minus 119863 (120572) ≃ 099174

(137)

That is none of conditions (8) (9) (12) (14) (for 119895 = 2) (16)(17) (20) (for 119895 = 1) and (21) (for 119895 = 1) is satisfiedComment The improvement of condition (37) over thecorresponding condition (8) is significant approximately10033 We get this measure by comparing the values inthe left-hand side of those conditions Also the improvementover conditions (14) (16) and (20) is very satisfactory

around 7966 479 and 2937 respectively In additioncondition (37) is satisfied from the first iteration whileconditions (14) (20) and (21) need more than one iteration

Example 2 (taken and adapted from [17]) Consider theadvanced differential equation

1199091015840 (119905) minus 3332500119909 (120590 (119905)) = 0 119905 ge 0 (138)

with (see Figure 2(a))

120590 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 2]minus3119905 + 20119896 + 13 if 119905 isin [5119896 + 2 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 4]minus119905 + 10119896 + 13 if 119905 isin [5119896 + 4 5119896 + 5]

(139)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (27) we see (Figure 2(b)) that

120588 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 125]5119896 + 4 if 119905 isin [5119896 + 125 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 38]5119896 + 8 if 119905 isin [5119896 + 38 5119896 + 5]

(140)

It is obvious that

120573 = lim inf119905rarrinfin

int5119896+45119896+3

3332500119889119904 = 01332 (141)

and therefore the smaller root of 11989001332120582 = 120582 is 1205820 = 116839Observe that the function 119866119895 R0 rarr R+ defined as

119866119895 (119905) = int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

(142)

attains its maximum at 119905 = 5119896 + 38 119896 isin N0 for every 119895 isin NSpecifically

1198661 (119905 = 5119896 + 38) = int5119896+85119896+38

119902 (119904)sdot exp(int120590(119904)

5119896+8119902 (119906) exp(int120590(119906)

1199061198761 (120585) 119889120585)119889119906)119889119904

(143)

with

1198761 (120585) = 119902 (120585) [1 + int120590(120585)120585119902 (V)

sdot exp(int120590(V)120585119902 (119908) exp(int120590(119908)

1199081205820119902 (119911) 119889119911)119889119908)119889V]

(144)

14 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

651 2 3 4 8 9 10 11 12 13

1415

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

7

t

14651

2 3 4 8 9 10 11 12 13

38

1415

125

y = t

(t)

(b)

Figure 2 The graphs of 120590(119905) and 120588(119905)

Using MATLAB we obtain

1198661 (119905 = 5119896 + 38) ≃ 09915 (145)

Thereforelim sup119905rarrinfin

1198661 (119905) ≃ 09915 gt 1 minus 119863 (120573) ≃ 09896 (146)

Hence condition (116) of Theorem 12 is satisfied for 119895 = 1Consequently all solutions of (138) oscillate

Observe however that

LA = lim sup119896rarrinfin

int5119896+85119896+38

3332500119889119904 = 055944 lt 1120573 = 01332 lt 1119890 lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int5119896+85119896+38

119902 (119904) exp(int120590(119904)5119896+8

119902 (119906) 119889119906)119889119904= lim sup119896rarrinfin

[int5119896+45119896+38

119902 (119904)sdot exp(int5119904minus20119896minus11

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+5

5119896+4119902 (119904)

sdot exp(intminus119904+10119896+135119896+8

119902 (119906) 119889119906)119889119904 + int5119896+65119896+5

119902 (119904)sdot exp(int5119896+8

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+7

5119896+6119902 (119904)

sdot exp(int4119904minus15119896minus165119896+8

119902 (119906) 119889119906)119889119904 + int5119896+85119896+7

119902 (119904)sdot exp(intminus3119904+20119896+33

5119896+8119902 (119906) 119889119906)119889119904] ≃ 06672 lt 1

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim inf119905rarrinfin

int5119896+45119896+3

119902 (119904) exp(int120590(119904)5119896+4

119902 (119906) 119889119906)119889119904≃ 01893 lt 1119890

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)1199021 (119906) 119889119906)119889119904 ≃ 07196

lt 107196 lt 1 minus 119863 (120573) ≃ 09896

(147)

That is none of conditions (24) (25) (28) (29) (30) (for 119895 =1) and (31) (for 119895 = 1) is satisfiedComment The improvement of condition (116) over thecorresponding condition (24) is significant approximately7723 We get this measure by comparing the values in theleft-hand side of those conditions Also the improvementover conditions (28) and (30) is very satisfactory around4861 and 3778 respectively In addition condition (116)is satisfied from the first iteration while conditions (30) and(31) need more than one iteration

Complexity 15

Remark 3 Similarly one can provide examples illustratingthe other main results

5 Concluding Remarks

In the present paper we have considered the oscillatorydynamics of differential equations having nonmonotonedeviating arguments and nonnegative coefficients New suf-ficient conditions have been established for the oscillation ofall solutions of (E) and (E1015840) These conditions include (37)(71) (75) (83) and (97) and (114) (116) (117) (118) and(119) for (E) and (E1015840) respectively Applying these conditionsinvolves a procedure that checks for oscillations by iterativelycomputing limsup and liminf on terms recursively definedon the equationrsquos coefficients and deviating argument

The main advantage of these conditions is that theyachieve a major improvement over all the related oscillationconditions for (E) [(E1015840)] in the literature For example con-dition (37) [(114)] improves upon the noniterative conditionsthat are reviewed in the introduction namely conditions (8)[(24)] (12) (16)equiv(20) (for 119895 = 1) [(28)equiv(30) (for 119895 = 1)]and (17)equiv(21) (for 119895 = 1) [(31) (for 119895 = 1)] That immediatelybecomes evident by inspecting the left-hand side of (37)[(114)] and the left-hand side of each of the above conditions

The improvement of (37) [(114)] over the other iterativeconditions namely (14) (for j gt 2) (20) (for 119895 gt 1) [(30)(for 119895 gt 1)] and (21) (for 119895 gt 1) [(31) (for 119895 gt 1)] is that itrequires far fewer iterations to establish oscillation than theother conditions

This advantage easily can be verified computationallyby running the MATLAB programs (see Appendix) forcomputing limsup and liminf and comparing the number ofiterations required by each condition to establish oscillationThen we see that we achieve a significant improvement overall known oscillation criteria

Another advantage and a significant departure from thelarge majority of the other studies is that the criteria in thispaper apply to a more general class of equations havingnonmonotone arguments 120591(119905) or 120590(119905) in contrast to mostof the other oscillation criteria that apply to equations withnondecreasing arguments

Appendix

In this appendix for completeness we give the algorithmon MATLAB software used in Example 1 for calculation oflim sup119905rarrinfin1198651(119905) ≃ 10417 For Example 2 the algorithm isomitted since it is similar to the one in Example 1

Algorithm for Example 1

clear clcc = 12119899 = 501205820 = 114765a5 = 19b5 = 703h5 = (b5 minus a5)119899

for i5 = 1 1 119899 + 1x5 = a5 + (i5 minus 1) lowast h5a4 = TFunction(x5)b4 = 19h4 = (b4 minus a4)119899for i4 = 1 1 119899 + 1x4 = a4 + (i4 minus 1) lowast h4a3 = TFunction(x4)b3 = x4h3 = (b3 minus a3)119899for i3 = 1 1 119899 + 1x3 = a3 + (i3 minus 1) lowast h3a2 = TFunction(x3)b2 = x3h2 = (b2 minus a2)119899for i2 = 1 1 119899 + 1x2 = a2 + (i2 minus 1) lowast h2a1 = TFunction(x2)b1 = x3h1 = (b1 minus a1)119899for i1 = 1 1 119899 + 1x1 = a1 + (i1 minus 1) lowast h1f1(i1) = c lowast exp(1205820 lowast c lowast (x1 minus TFunction(x1)))endI1 = f1(1) + f1(119899 + 1)for i1 = 2 2 119899I1 = I1 + f1(i1) lowast 4endfor i1 = 3 2 119899 minus 1I1 = I1 + f1(i1) lowast 2endI1 = I1 lowast h13f2(i2) = c lowast exp(I1)endI2 = f2(1) + f2(119899 + 1)for i2 = 2 2 119899I2 = I2 + f2(i2) lowast 4endfor i2 = 3 2 119899 minus 1I2 = I2 + f2(i2) lowast 2endI2 = I2 lowast h23f3(i3) = c lowast (1 + I2)end

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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Page 13: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

Complexity 13

+ int8119896+48119896+195

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+2958119896+4

exp( 325 int8119896+3

8119896+2119889119906)119889119904

+ int8119896+68119896+295

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+4468119896+6

exp( 325 int8119896+3

8119896+3119889119906)119889119904] ≃ 057983

(134)

Thus

lim sup119905rarrinfin

Φ2 (119905) ≃ 057983 lt 1 minus 119863 (120572) ≃ 099174 (135)

Also

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)119901 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int8119896+4468119896+3

325 exp(int8119896+3

120591(119904)

325119889119906)119889119904= 325 sdot lim sup

119905rarrinfin[int8119896+48119896+3

exp( 325 int8119896+3

5119904minus32119896minus18119889119906)119889119904

+ int8119896+58119896+4

exp( 325 int8119896+3

minus4119904+40119896+18119889119906)119889119904

+ int8119896+68119896+5

exp( 325 int8119896+3

5119904minus32119896minus27119889119906)119889119904

+ int8119896+78119896+6

exp( 325 int8119896+3

minus2119904+24119896+15119889119906)119889119904

+ int8119896+4468119896+7

exp( 325 int8119896+3

6119904minus40119896minus41119889119906)119889119904] ≃ 07043

lt 107043 lt 1 minus 119863 (120572) ≃ 099174

(136)

In addition

lim sup119905rarrinfin

int119905ℎ(119905)119901 (119904) exp(intℎ(119905)

120591(119904)1199011 (119906) 119889119906)119889119904 ≃ 08052

lt 108052 lt 1 minus 119863 (120572) ≃ 099174

(137)

That is none of conditions (8) (9) (12) (14) (for 119895 = 2) (16)(17) (20) (for 119895 = 1) and (21) (for 119895 = 1) is satisfiedComment The improvement of condition (37) over thecorresponding condition (8) is significant approximately10033 We get this measure by comparing the values inthe left-hand side of those conditions Also the improvementover conditions (14) (16) and (20) is very satisfactory

around 7966 479 and 2937 respectively In additioncondition (37) is satisfied from the first iteration whileconditions (14) (20) and (21) need more than one iteration

Example 2 (taken and adapted from [17]) Consider theadvanced differential equation

1199091015840 (119905) minus 3332500119909 (120590 (119905)) = 0 119905 ge 0 (138)

with (see Figure 2(a))

120590 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 2]minus3119905 + 20119896 + 13 if 119905 isin [5119896 + 2 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 4]minus119905 + 10119896 + 13 if 119905 isin [5119896 + 4 5119896 + 5]

(139)

where 119896 isin N0 and N0 is the set of nonnegative integersBy (27) we see (Figure 2(b)) that

120588 (119905) =

5119896 + 3 if 119905 isin [5119896 5119896 + 1]4119905 minus 15119896 minus 1 if 119905 isin [5119896 + 1 5119896 + 125]5119896 + 4 if 119905 isin [5119896 + 125 5119896 + 3]5119905 minus 20119896 minus 11 if 119905 isin [5119896 + 3 5119896 + 38]5119896 + 8 if 119905 isin [5119896 + 38 5119896 + 5]

(140)

It is obvious that

120573 = lim inf119905rarrinfin

int5119896+45119896+3

3332500119889119904 = 01332 (141)

and therefore the smaller root of 11989001332120582 = 120582 is 1205820 = 116839Observe that the function 119866119895 R0 rarr R+ defined as

119866119895 (119905) = int120588(119905)119905119902 (119904)

sdot exp(int120590(119904)120588(119905)119902 (119906) exp(int120590(119906)

119906119876119895 (120585) 119889120585)119889119906)119889119904

(142)

attains its maximum at 119905 = 5119896 + 38 119896 isin N0 for every 119895 isin NSpecifically

1198661 (119905 = 5119896 + 38) = int5119896+85119896+38

119902 (119904)sdot exp(int120590(119904)

5119896+8119902 (119906) exp(int120590(119906)

1199061198761 (120585) 119889120585)119889119906)119889119904

(143)

with

1198761 (120585) = 119902 (120585) [1 + int120590(120585)120585119902 (V)

sdot exp(int120590(V)120585119902 (119908) exp(int120590(119908)

1199081205820119902 (119911) 119889119911)119889119908)119889V]

(144)

14 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

651 2 3 4 8 9 10 11 12 13

1415

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

7

t

14651

2 3 4 8 9 10 11 12 13

38

1415

125

y = t

(t)

(b)

Figure 2 The graphs of 120590(119905) and 120588(119905)

Using MATLAB we obtain

1198661 (119905 = 5119896 + 38) ≃ 09915 (145)

Thereforelim sup119905rarrinfin

1198661 (119905) ≃ 09915 gt 1 minus 119863 (120573) ≃ 09896 (146)

Hence condition (116) of Theorem 12 is satisfied for 119895 = 1Consequently all solutions of (138) oscillate

Observe however that

LA = lim sup119896rarrinfin

int5119896+85119896+38

3332500119889119904 = 055944 lt 1120573 = 01332 lt 1119890 lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int5119896+85119896+38

119902 (119904) exp(int120590(119904)5119896+8

119902 (119906) 119889119906)119889119904= lim sup119896rarrinfin

[int5119896+45119896+38

119902 (119904)sdot exp(int5119904minus20119896minus11

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+5

5119896+4119902 (119904)

sdot exp(intminus119904+10119896+135119896+8

119902 (119906) 119889119906)119889119904 + int5119896+65119896+5

119902 (119904)sdot exp(int5119896+8

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+7

5119896+6119902 (119904)

sdot exp(int4119904minus15119896minus165119896+8

119902 (119906) 119889119906)119889119904 + int5119896+85119896+7

119902 (119904)sdot exp(intminus3119904+20119896+33

5119896+8119902 (119906) 119889119906)119889119904] ≃ 06672 lt 1

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim inf119905rarrinfin

int5119896+45119896+3

119902 (119904) exp(int120590(119904)5119896+4

119902 (119906) 119889119906)119889119904≃ 01893 lt 1119890

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)1199021 (119906) 119889119906)119889119904 ≃ 07196

lt 107196 lt 1 minus 119863 (120573) ≃ 09896

(147)

That is none of conditions (24) (25) (28) (29) (30) (for 119895 =1) and (31) (for 119895 = 1) is satisfiedComment The improvement of condition (116) over thecorresponding condition (24) is significant approximately7723 We get this measure by comparing the values in theleft-hand side of those conditions Also the improvementover conditions (28) and (30) is very satisfactory around4861 and 3778 respectively In addition condition (116)is satisfied from the first iteration while conditions (30) and(31) need more than one iteration

Complexity 15

Remark 3 Similarly one can provide examples illustratingthe other main results

5 Concluding Remarks

In the present paper we have considered the oscillatorydynamics of differential equations having nonmonotonedeviating arguments and nonnegative coefficients New suf-ficient conditions have been established for the oscillation ofall solutions of (E) and (E1015840) These conditions include (37)(71) (75) (83) and (97) and (114) (116) (117) (118) and(119) for (E) and (E1015840) respectively Applying these conditionsinvolves a procedure that checks for oscillations by iterativelycomputing limsup and liminf on terms recursively definedon the equationrsquos coefficients and deviating argument

The main advantage of these conditions is that theyachieve a major improvement over all the related oscillationconditions for (E) [(E1015840)] in the literature For example con-dition (37) [(114)] improves upon the noniterative conditionsthat are reviewed in the introduction namely conditions (8)[(24)] (12) (16)equiv(20) (for 119895 = 1) [(28)equiv(30) (for 119895 = 1)]and (17)equiv(21) (for 119895 = 1) [(31) (for 119895 = 1)] That immediatelybecomes evident by inspecting the left-hand side of (37)[(114)] and the left-hand side of each of the above conditions

The improvement of (37) [(114)] over the other iterativeconditions namely (14) (for j gt 2) (20) (for 119895 gt 1) [(30)(for 119895 gt 1)] and (21) (for 119895 gt 1) [(31) (for 119895 gt 1)] is that itrequires far fewer iterations to establish oscillation than theother conditions

This advantage easily can be verified computationallyby running the MATLAB programs (see Appendix) forcomputing limsup and liminf and comparing the number ofiterations required by each condition to establish oscillationThen we see that we achieve a significant improvement overall known oscillation criteria

Another advantage and a significant departure from thelarge majority of the other studies is that the criteria in thispaper apply to a more general class of equations havingnonmonotone arguments 120591(119905) or 120590(119905) in contrast to mostof the other oscillation criteria that apply to equations withnondecreasing arguments

Appendix

In this appendix for completeness we give the algorithmon MATLAB software used in Example 1 for calculation oflim sup119905rarrinfin1198651(119905) ≃ 10417 For Example 2 the algorithm isomitted since it is similar to the one in Example 1

Algorithm for Example 1

clear clcc = 12119899 = 501205820 = 114765a5 = 19b5 = 703h5 = (b5 minus a5)119899

for i5 = 1 1 119899 + 1x5 = a5 + (i5 minus 1) lowast h5a4 = TFunction(x5)b4 = 19h4 = (b4 minus a4)119899for i4 = 1 1 119899 + 1x4 = a4 + (i4 minus 1) lowast h4a3 = TFunction(x4)b3 = x4h3 = (b3 minus a3)119899for i3 = 1 1 119899 + 1x3 = a3 + (i3 minus 1) lowast h3a2 = TFunction(x3)b2 = x3h2 = (b2 minus a2)119899for i2 = 1 1 119899 + 1x2 = a2 + (i2 minus 1) lowast h2a1 = TFunction(x2)b1 = x3h1 = (b1 minus a1)119899for i1 = 1 1 119899 + 1x1 = a1 + (i1 minus 1) lowast h1f1(i1) = c lowast exp(1205820 lowast c lowast (x1 minus TFunction(x1)))endI1 = f1(1) + f1(119899 + 1)for i1 = 2 2 119899I1 = I1 + f1(i1) lowast 4endfor i1 = 3 2 119899 minus 1I1 = I1 + f1(i1) lowast 2endI1 = I1 lowast h13f2(i2) = c lowast exp(I1)endI2 = f2(1) + f2(119899 + 1)for i2 = 2 2 119899I2 = I2 + f2(i2) lowast 4endfor i2 = 3 2 119899 minus 1I2 = I2 + f2(i2) lowast 2endI2 = I2 lowast h23f3(i3) = c lowast (1 + I2)end

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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Page 14: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

14 Complexity

1

5

7

2

6

8

1213

t

4

10

14

3

7

9

11

651 2 3 4 8 9 10 11 12 13

1415

y = t

(t)

(a)

1

5

2

6

8

1213

4

10

3

7

9

11

7

t

14651

2 3 4 8 9 10 11 12 13

38

1415

125

y = t

(t)

(b)

Figure 2 The graphs of 120590(119905) and 120588(119905)

Using MATLAB we obtain

1198661 (119905 = 5119896 + 38) ≃ 09915 (145)

Thereforelim sup119905rarrinfin

1198661 (119905) ≃ 09915 gt 1 minus 119863 (120573) ≃ 09896 (146)

Hence condition (116) of Theorem 12 is satisfied for 119895 = 1Consequently all solutions of (138) oscillate

Observe however that

LA = lim sup119896rarrinfin

int5119896+85119896+38

3332500119889119904 = 055944 lt 1120573 = 01332 lt 1119890 lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim sup119896rarrinfin

int5119896+85119896+38

119902 (119904) exp(int120590(119904)5119896+8

119902 (119906) 119889119906)119889119904= lim sup119896rarrinfin

[int5119896+45119896+38

119902 (119904)sdot exp(int5119904minus20119896minus11

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+5

5119896+4119902 (119904)

sdot exp(intminus119904+10119896+135119896+8

119902 (119906) 119889119906)119889119904 + int5119896+65119896+5

119902 (119904)sdot exp(int5119896+8

5119896+8119902 (119906) 119889119906)119889119904 + int5119896+7

5119896+6119902 (119904)

sdot exp(int4119904minus15119896minus165119896+8

119902 (119906) 119889119906)119889119904 + int5119896+85119896+7

119902 (119904)sdot exp(intminus3119904+20119896+33

5119896+8119902 (119906) 119889119906)119889119904] ≃ 06672 lt 1

lim inf119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)119902 (119906) 119889119906)119889119904

= lim inf119905rarrinfin

int5119896+45119896+3

119902 (119904) exp(int120590(119904)5119896+4

119902 (119906) 119889119906)119889119904≃ 01893 lt 1119890

lim sup119905rarrinfin

int120588(119905)119905119902 (119904) exp(int120590(119904)

120588(119905)1199021 (119906) 119889119906)119889119904 ≃ 07196

lt 107196 lt 1 minus 119863 (120573) ≃ 09896

(147)

That is none of conditions (24) (25) (28) (29) (30) (for 119895 =1) and (31) (for 119895 = 1) is satisfiedComment The improvement of condition (116) over thecorresponding condition (24) is significant approximately7723 We get this measure by comparing the values in theleft-hand side of those conditions Also the improvementover conditions (28) and (30) is very satisfactory around4861 and 3778 respectively In addition condition (116)is satisfied from the first iteration while conditions (30) and(31) need more than one iteration

Complexity 15

Remark 3 Similarly one can provide examples illustratingthe other main results

5 Concluding Remarks

In the present paper we have considered the oscillatorydynamics of differential equations having nonmonotonedeviating arguments and nonnegative coefficients New suf-ficient conditions have been established for the oscillation ofall solutions of (E) and (E1015840) These conditions include (37)(71) (75) (83) and (97) and (114) (116) (117) (118) and(119) for (E) and (E1015840) respectively Applying these conditionsinvolves a procedure that checks for oscillations by iterativelycomputing limsup and liminf on terms recursively definedon the equationrsquos coefficients and deviating argument

The main advantage of these conditions is that theyachieve a major improvement over all the related oscillationconditions for (E) [(E1015840)] in the literature For example con-dition (37) [(114)] improves upon the noniterative conditionsthat are reviewed in the introduction namely conditions (8)[(24)] (12) (16)equiv(20) (for 119895 = 1) [(28)equiv(30) (for 119895 = 1)]and (17)equiv(21) (for 119895 = 1) [(31) (for 119895 = 1)] That immediatelybecomes evident by inspecting the left-hand side of (37)[(114)] and the left-hand side of each of the above conditions

The improvement of (37) [(114)] over the other iterativeconditions namely (14) (for j gt 2) (20) (for 119895 gt 1) [(30)(for 119895 gt 1)] and (21) (for 119895 gt 1) [(31) (for 119895 gt 1)] is that itrequires far fewer iterations to establish oscillation than theother conditions

This advantage easily can be verified computationallyby running the MATLAB programs (see Appendix) forcomputing limsup and liminf and comparing the number ofiterations required by each condition to establish oscillationThen we see that we achieve a significant improvement overall known oscillation criteria

Another advantage and a significant departure from thelarge majority of the other studies is that the criteria in thispaper apply to a more general class of equations havingnonmonotone arguments 120591(119905) or 120590(119905) in contrast to mostof the other oscillation criteria that apply to equations withnondecreasing arguments

Appendix

In this appendix for completeness we give the algorithmon MATLAB software used in Example 1 for calculation oflim sup119905rarrinfin1198651(119905) ≃ 10417 For Example 2 the algorithm isomitted since it is similar to the one in Example 1

Algorithm for Example 1

clear clcc = 12119899 = 501205820 = 114765a5 = 19b5 = 703h5 = (b5 minus a5)119899

for i5 = 1 1 119899 + 1x5 = a5 + (i5 minus 1) lowast h5a4 = TFunction(x5)b4 = 19h4 = (b4 minus a4)119899for i4 = 1 1 119899 + 1x4 = a4 + (i4 minus 1) lowast h4a3 = TFunction(x4)b3 = x4h3 = (b3 minus a3)119899for i3 = 1 1 119899 + 1x3 = a3 + (i3 minus 1) lowast h3a2 = TFunction(x3)b2 = x3h2 = (b2 minus a2)119899for i2 = 1 1 119899 + 1x2 = a2 + (i2 minus 1) lowast h2a1 = TFunction(x2)b1 = x3h1 = (b1 minus a1)119899for i1 = 1 1 119899 + 1x1 = a1 + (i1 minus 1) lowast h1f1(i1) = c lowast exp(1205820 lowast c lowast (x1 minus TFunction(x1)))endI1 = f1(1) + f1(119899 + 1)for i1 = 2 2 119899I1 = I1 + f1(i1) lowast 4endfor i1 = 3 2 119899 minus 1I1 = I1 + f1(i1) lowast 2endI1 = I1 lowast h13f2(i2) = c lowast exp(I1)endI2 = f2(1) + f2(119899 + 1)for i2 = 2 2 119899I2 = I2 + f2(i2) lowast 4endfor i2 = 3 2 119899 minus 1I2 = I2 + f2(i2) lowast 2endI2 = I2 lowast h23f3(i3) = c lowast (1 + I2)end

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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

Page 15: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

Complexity 15

Remark 3 Similarly one can provide examples illustratingthe other main results

5 Concluding Remarks

In the present paper we have considered the oscillatorydynamics of differential equations having nonmonotonedeviating arguments and nonnegative coefficients New suf-ficient conditions have been established for the oscillation ofall solutions of (E) and (E1015840) These conditions include (37)(71) (75) (83) and (97) and (114) (116) (117) (118) and(119) for (E) and (E1015840) respectively Applying these conditionsinvolves a procedure that checks for oscillations by iterativelycomputing limsup and liminf on terms recursively definedon the equationrsquos coefficients and deviating argument

The main advantage of these conditions is that theyachieve a major improvement over all the related oscillationconditions for (E) [(E1015840)] in the literature For example con-dition (37) [(114)] improves upon the noniterative conditionsthat are reviewed in the introduction namely conditions (8)[(24)] (12) (16)equiv(20) (for 119895 = 1) [(28)equiv(30) (for 119895 = 1)]and (17)equiv(21) (for 119895 = 1) [(31) (for 119895 = 1)] That immediatelybecomes evident by inspecting the left-hand side of (37)[(114)] and the left-hand side of each of the above conditions

The improvement of (37) [(114)] over the other iterativeconditions namely (14) (for j gt 2) (20) (for 119895 gt 1) [(30)(for 119895 gt 1)] and (21) (for 119895 gt 1) [(31) (for 119895 gt 1)] is that itrequires far fewer iterations to establish oscillation than theother conditions

This advantage easily can be verified computationallyby running the MATLAB programs (see Appendix) forcomputing limsup and liminf and comparing the number ofiterations required by each condition to establish oscillationThen we see that we achieve a significant improvement overall known oscillation criteria

Another advantage and a significant departure from thelarge majority of the other studies is that the criteria in thispaper apply to a more general class of equations havingnonmonotone arguments 120591(119905) or 120590(119905) in contrast to mostof the other oscillation criteria that apply to equations withnondecreasing arguments

Appendix

In this appendix for completeness we give the algorithmon MATLAB software used in Example 1 for calculation oflim sup119905rarrinfin1198651(119905) ≃ 10417 For Example 2 the algorithm isomitted since it is similar to the one in Example 1

Algorithm for Example 1

clear clcc = 12119899 = 501205820 = 114765a5 = 19b5 = 703h5 = (b5 minus a5)119899

for i5 = 1 1 119899 + 1x5 = a5 + (i5 minus 1) lowast h5a4 = TFunction(x5)b4 = 19h4 = (b4 minus a4)119899for i4 = 1 1 119899 + 1x4 = a4 + (i4 minus 1) lowast h4a3 = TFunction(x4)b3 = x4h3 = (b3 minus a3)119899for i3 = 1 1 119899 + 1x3 = a3 + (i3 minus 1) lowast h3a2 = TFunction(x3)b2 = x3h2 = (b2 minus a2)119899for i2 = 1 1 119899 + 1x2 = a2 + (i2 minus 1) lowast h2a1 = TFunction(x2)b1 = x3h1 = (b1 minus a1)119899for i1 = 1 1 119899 + 1x1 = a1 + (i1 minus 1) lowast h1f1(i1) = c lowast exp(1205820 lowast c lowast (x1 minus TFunction(x1)))endI1 = f1(1) + f1(119899 + 1)for i1 = 2 2 119899I1 = I1 + f1(i1) lowast 4endfor i1 = 3 2 119899 minus 1I1 = I1 + f1(i1) lowast 2endI1 = I1 lowast h13f2(i2) = c lowast exp(I1)endI2 = f2(1) + f2(119899 + 1)for i2 = 2 2 119899I2 = I2 + f2(i2) lowast 4endfor i2 = 3 2 119899 minus 1I2 = I2 + f2(i2) lowast 2endI2 = I2 lowast h23f3(i3) = c lowast (1 + I2)end

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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

Page 16: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

16 Complexity

I3 = f3(1) + f3(119899 + 1)for i3 = 2 2 119899I3 = I3 + f3(i3) lowast 4endfor i3 = 3 2 119899 minus 1I3 = I3 + f3(i3) lowast 2endI3 = I3 lowast h33f4(i4) = c lowast exp(I3)endI4 = f4(1) + f4(119899 + 1)for i4 = 2 2 119899I4 = I4 + f4(i4) lowast 4endfor i4 = 3 2 119899 minus 1I4 = I4 + f4(i4) lowast 2endI4 = I4 lowast h43f5(i5) = c lowast exp(I4)endI5 = f5(1) + f5(119899 + 1)for i5 = 2 2 119899I5 = I5 + f5(i5) lowast 4

endfor i5 = 3 2 119899 minus 1I5 = I5 + f5(i5) lowast 2endI5 = I5 lowast h53

Algorithms for functions 120591(119905) and ℎ(119905)function[a] = TFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1endif (119903 gt= 2) ampamp (119903 lt 3)a = minus4 lowast x + 40 lowast k + 9endif (119903 gt= 3) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18endif (119903 gt= 4) ampamp (119903 lt 5)a = minus4 lowast x + 40 lowast k + 18end

if (119903 gt= 5) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 7)a = minus2 lowast x + 24 lowast k + 15end

if (119903 gt= 7) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

function[a] = HFunction(x)119903 = mod(119909 8)k = floor(1199098)if (119903 gt= 0) ampamp (119903 lt 2)a = x minus 1end

if (119903 gt= 2) ampamp (119903 lt 195)a = 8 lowast k + 1end

if (119903 gt= 195) ampamp (119903 lt 4)a = 5 lowast x minus 32 lowast k minus 18end

if (119903 gt= 4) ampamp (119903 lt 295)a = 8 lowast k + 2end

if (119903 gt= 295) ampamp (119903 lt 6)a = 5 lowast x minus 32 lowast k minus 27end

if (119903 gt= 6) ampamp (119903 lt 446)a = 8 lowast k + 3end

if (119903 gt= 446) ampamp (119903 lt 8)a = 6 lowast x minus 40 lowast k minus 41end

end

Conflicts of Interest

The authors declare that they have no conflicts of interest

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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

Page 17: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

Complexity 17

Acknowledgments

The research of the second author is supported by theNational Natural Science Foundation of China (Grant no61503171) China Postdoctoral Science Foundation (Grantno 2015M582091) Natural Science Foundation of Shan-dong Province (Grant no ZR2016JL021) Doctoral Scien-tific Research Foundation of Linyi University (Grant noLYDX2015BS001) and the Applied Mathematics Enhance-ment Program of Linyi University

References

[1] M Bani-Yaghoub ldquoAnalysis and applications of delay differen-tial equations in biology and medicinerdquo httpsarxivorgabs170104173

[2] G A Bocharov and F A Rihan ldquoNumerical modelling inbiosciences using delay differential equationsrdquo Journal of Com-putational and Applied Mathematics vol 125 no 1-2 pp 183ndash199 2000

[3] F Brauer and C Castillo-Chavez Mathematical Models inPopulation Biology and Epidemiology vol 40 of Texts in AppliedMathematics Springer New York NY USA 2nd edition 2012

[4] U Forys ldquoMarchukrsquos model of immune system dynamics withapplication to tumour growthrdquo Journal of Theoretical Medicinevol 4 no 1 pp 85ndash93 2002

[5] K Gopalsamy Stability and Oscillations in Delay DifferentialEquations of Population Dynamics vol 74 of Mathematics andIts Applications Kluwer Academic Publishers Dordrecht TheNetherlands 1992

[6] G E Hutchinson ldquoCircular causal systems in ecologyrdquo Annalsof the New York Academy of Sciences vol 50 no 4 pp 221ndash2461948

[7] F A Rihan and M N Anwar ldquoQualitative analysis of delayedSIR epidemic model with a saturated incidence raterdquo Interna-tional Journal of Differential Equations vol 2012 Article ID408637 13 pages 2012

[8] F A Rihan D H Abdel Rahman S Lakshmanan and AS Alkhajeh ldquoA time delay model of tumourndashimmune systeminteractions global dynamics parameter estimation sensitivityanalysisrdquo Applied Mathematics and Computation vol 232 pp606ndash623 2014

[9] F A Rihan and B F Rihan ldquoNumerical modelling of biolog-ical systems with memory using delay differential equationsrdquoApplied Mathematics amp Information Sciences vol 9 no 3 pp1645ndash1658 2015

[10] H Smith An Introduction to Delay Differential Equations withApplications to the Life Sciences Springer New York NY USA2011

[11] R D Driver ldquoCan the future influence the presentrdquo PhysicalReview D Particles Fields Gravitation and Cosmology vol 19no 4 pp 1098ndash1107 1979

[12] L E Elrsquosgolrsquots and S B Norkin Introduction to the Theory andApplication of Differential Equations with Deviating ArgumentsAcademic Press New York NY USA 1973

[13] E BravermanG E Chatzarakis and I P Stavroulakis ldquoIterativeoscillation tests for differential equations with several non-monotone argumentsrdquo Advances in Difference Equations vol2016 Article ID 87 18 pages 2016

[14] E Braverman and B Karpuz ldquoOn oscillation of differentialand difference equations with non-monotone delaysrdquo Applied

Mathematics and Computation vol 218 no 7 pp 3880ndash38872011

[15] G E Chatzarakis ldquoDifferential equations with non-monotonearguments iterative oscillation resultsrdquo Journal ofMathematicaland Computational Science vol 6 no 5 pp 953ndash964 2016

[16] G E Chatzarakis ldquoOn oscillation of differential equations withnon-monotone deviating argumentsrdquoMediterranean Journal ofMathematics vol 14 no 2 Article ID 82 17 pages 2017

[17] G E Chatzarakis ldquoOscillations caused by several non-monotone deviating argumentsrdquo Differential Equations ampApplications vol 9 no 3 pp 285ndash310 2017

[18] G E Chatzarakis and O Ocalan ldquoOscillations of differentialequations with several non-monotone advanced argumentsrdquoDynamical Systems vol 30 no 3 pp 310ndash323 2015

[19] L H Erbe Q Kong and B G Zhang Oscillation Theory forFunctional Differential Equations Marcel Dekker New YorkNY USA 1995

[20] L H Erbe and B G Zhang ldquoOscillation for first order lineardifferential equations with deviating argumentsrdquo Differentialand Integral Equations International Journal for Theory andApplications vol 1 no 3 pp 305ndash314 1988

[21] N Fukagai and T Kusano ldquoOscillation theory of first orderfunctional differential equations with deviating argumentsrdquoAnnali di Matematica Pura ed Applicata Serie Quarta vol 136no 1 pp 95ndash117 1984

[22] I Gyori and G Ladas Oscillation Theory of Delay Differen-tial Equations Oxford Mathematical Monographs ClarendonPress New York NY USA 1991

[23] J Jaros and I P Stavroulakis ldquoOscillation tests for delayequationsrdquo Rocky Mountain Journal of Mathematics vol 29 no1 pp 197ndash207 1999

[24] R G Koplatadze and T A Chanturiya ldquoOscillating andmonotone solutions of first-order differential equations withdeviating argumentrdquoDifferentsialrsquonye Uravneniya vol 18 no 8pp 1463ndash1465 1472 1982

[25] R Koplatadze and G Kvinikadze ldquoOn the oscillation of solu-tions of first order delay differential inequalities and equationsrdquoGeorgian Mathematical Journal vol 1 no 6 pp 675ndash685 1994

[26] M K Kwong ldquoOscillation of first-order delay equationsrdquoJournal of Mathematical Analysis and Applications vol 156 no1 pp 274ndash286 1991

[27] G Ladas V Lakshmikantham and J S Papadakis ldquoOscillationsof higher-order retarded differential equations generated by theretarded argumentrdquo Delay and functional differential equationsand their applications (Proc Conf Park City Utah 1972) pp219ndash231 1972

[28] G S Ladde ldquoOscillations caused by retarded perturbationsof first order linear ordinary differential equationsrdquo Atti dellaAccademia Nazionale dei Lincei Rendiconti della Classe diScienze Fisiche Matematiche e Naturali vol 63 no 5 pp 351ndash359 (1978) 1977

[29] G S Ladde V Lakshmikantham and B G Zhang OscillationTheory of Differential Equations with Deviating Argumentsvol 110 of Monographs and Textbooks in Pure and AppliedMathematics Marcel Dekker New York NY USA 1987

[30] H A El-Morshedy and E R Attia ldquoNew oscillation criterionfor delay differential equationswith non-monotone argumentsrdquoApplied Mathematics Letters vol 54 pp 54ndash59 2016

[31] A D Myskis ldquoLinear homogeneous differential equations ofthe first order with retarded argumentrdquoAkademiya Nauk SSSR iMoskovskoeMatematicheskoeObshchestvoUspekhiMatematich-eskikh Nauk vol 5 no 2(36) pp 160ndash162 1950

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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

Page 18: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

18 Complexity

[32] I P Stavroulakis ldquoOscillation criteria for delay and differenceequations with non-monotone argumentsrdquo Applied Mathemat-ics and Computation vol 226 pp 661ndash672 2014

[33] D Zhou ldquoOn some problems on oscillation of functional differ-ential equations of first orderrdquo Journal of Shandong University(Natural Science) vol 25 no 4 pp 434ndash442 1990

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

Page 19: Oscillation Criteria for Delay and Advanced Differential ...downloads.hindawi.com/journals/complexity/2018/8237634.pdf · Oscillation Criteria for Delay and Advanced Differential

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


On the Oscillation of Neutral Differential Equations · PDF fileOn the Oscillation of Neutral Differential Equations CH. ... function is evaluated both at the present state t and at

On the Oscillation of Neutral Differential Equations · PDF fileOn the Oscillation of Neutral Differential Equations CH. ... function is evaluated both at the present state t and at

DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION · DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION JEAN-PHILIPPE LESSARD Abstract. In these lecture notes, we demonstrate how rigorous

DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION · DELAY DIFFERENTIAL EQUATIONS AND CONTINUATION JEAN-PHILIPPE LESSARD Abstract. In these lecture notes, we demonstrate how rigorous

Delay Differential Equations

Delay Differential Equations

Delay-Differential Equations. Tools for Epidemics Modelling

Delay-Differential Equations. Tools for Epidemics Modelling

Oscillation Criteria for Second-Order Delay, Difference ...

Oscillation Criteria for Second-Order Delay, Difference ...

A Computer Virus Propagation Model Using Delay Differential …airccse.org/journal/cnc/6514cnc08.pdf · A COMPUTER VIRUS PROPAGATION MODEL USING DELAY DIFFERENTIAL EQUATIONS WITH

A Computer Virus Propagation Model Using Delay Differential …airccse.org/journal/cnc/6514cnc08.pdf · A COMPUTER VIRUS PROPAGATION MODEL USING DELAY DIFFERENTIAL EQUATIONS WITH

Numerical Solution of Delay Differential Equations - ResearchGate

Numerical Solution of Delay Differential Equations - ResearchGate

IEEE Standard 1500 Based Interconnect Diagnosis for Delay ... · Oscillation Ring Test Scheme for Interconnect Detection and Diagnosis Oscillation Ring Test Scheme Test Architecture

IEEE Standard 1500 Based Interconnect Diagnosis for Delay ... · Oscillation Ring Test Scheme for Interconnect Detection and Diagnosis Oscillation Ring Test Scheme Test Architecture

Applications of Delay Differential Equations in Biological ...downloads.hindawi.com/journals/complexity/2018/4584389.pdf · Editorial Applications of Delay Differential Equations

Applications of Delay Differential Equations in Biological ...downloads.hindawi.com/journals/complexity/2018/4584389.pdf · Editorial Applications of Delay Differential Equations

Chapter 3 Differential-Delay Equationspi.math.cornell.edu/~rand/randpdf/DDE-chapter3.pdfChapter 3 1 Differential-Delay Equations 2 Richard Rand 3 Abstract Periodic motions in DDE (Differential-Delay

Chapter 3 Differential-Delay Equationspi.math.cornell.edu/~rand/randpdf/DDE-chapter3.pdfChapter 3 1 Differential-Delay Equations 2 Richard Rand 3 Abstract Periodic motions in DDE (Differential-Delay

OSCILLATION FOR ADVANCED DIFFERENTIAL EQUATIONS WITH OSCILLATING COEFFICIENTS · 2019. 8. 1. · OSCILLATION FOR ADVANCED DIFFERENTIAL EQUATIONS WITH OSCILLATING COEFFICIENTS XIANYI

OSCILLATION FOR ADVANCED DIFFERENTIAL EQUATIONS WITH OSCILLATING COEFFICIENTS · 2019. 8. 1. · OSCILLATION FOR ADVANCED DIFFERENTIAL EQUATIONS WITH OSCILLATING COEFFICIENTS XIANYI

Oscillation of Solutions to Fractional Partial Differential Equations … · Oscillation of Solutions to Fractional Partial Differential Equations with Several Delays Yongfu Xiong,

Oscillation of Solutions to Fractional Partial Differential Equations … · Oscillation of Solutions to Fractional Partial Differential Equations with Several Delays Yongfu Xiong,

INTEGRAL OPERATORS AND DELAY DIFFERENTIAL EQUATIONS 1. Introduction. Delay

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NON-NORMALITY IN SCALAR DELAY DIFFERENTIAL EQUATIONS …bueler.github.io/papers/jnstrohMS.pdf · 1 Delay Differential Equations 1.1 Background Delaydifferential equations(DDE) aredifferential

NON-NORMALITY IN SCALAR DELAY DIFFERENTIAL EQUATIONS …bueler.github.io/papers/jnstrohMS.pdf · 1 Delay Differential Equations 1.1 Background Delaydifferential equations(DDE) aredifferential

Stability analysis of impulsive fractional differential systems with delay

Stability analysis of impulsive fractional differential systems with delay

A DELAY DIFFERENTIAL EQUATIONS MODEL FOR DISEASE ...

A DELAY DIFFERENTIAL EQUATIONS MODEL FOR DISEASE ...

SOLVING SOME DELAY DIFFERENTIAL EQUATIONS WITH COMPUTER ...rcorless/frames/9561/CorlessHeffernanTMS.pdf · SOLVING SOME DELAY DIFFERENTIAL EQUATIONS WITH COMPUTER ALGEBRA JANE M.

SOLVING SOME DELAY DIFFERENTIAL EQUATIONS WITH COMPUTER ...rcorless/frames/9561/CorlessHeffernanTMS.pdf · SOLVING SOME DELAY DIFFERENTIAL EQUATIONS WITH COMPUTER ALGEBRA JANE M.

OSCILLATION AND NONOSCILLATION OF SOLUTIONS OF … · OSCILLATION AND NONOSCILLATION OF SOLUTIONS OF SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH INTEGRABLE COEFFICIENTS^) BY JAMES

OSCILLATION AND NONOSCILLATION OF SOLUTIONS OF … · OSCILLATION AND NONOSCILLATION OF SOLUTIONS OF SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH INTEGRABLE COEFFICIENTS^) BY JAMES

Nonoscillation & Oscillation. Theory for Functional Differential Equations. - Agarwal R. (Marcel Dekker). 2004

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A Two Delay Differential Equations Model of Blood Partial ... · A Two Delay Differential Equations Model of Blood Partial Pressures in Human Cardiovascular-Respiratory System ...

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