# Oscillation Criteria for Delay and Advanced Differential Equations with Nonmonotone Arguments

Authors: George E. Chatzarakis; Tongxing Li
Journal: Complexity
 (2018)

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## Abstract

We study the oscillatory behavior of differential equations with nonmonotone deviating arguments and nonnegative coefficients. New oscillation criteria, involvinglimsup 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.


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## Body

## 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 aftereffect 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. The time delays add to these models memory effects, taking into account the dependence of the model’s present state on its past history [9]. The 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).The 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)y′t=ryt1-yt-τK,where y(t) denotes the size of the population, in the present time t, y′(t) describes the change of this size, at time t, y(t-τ) is the size, in some past time t-τ, τ>0 is the delay, representing the time for new eggs to hatch, and r is the reproduction rate of the population, while K is the carrying capacity, for the population.Many physiological processes, including the concentration of red blood cells, the concentration ofCO2 in the blood, causing the observed periodic oscillations in the breathing frequency, and the production of new blood cells, in the bone marrow, exhibit oscillations and several DDE models have been proposed to model these processes.Below, we present two applications indicating the relevance of the DDEs we study in this paper to real world problems. The two examples are taken from the areas of physiology and population dynamics.Application 1 (blood cells production [9]).
The production of red and white blood cells, in the bone marrow, is regulated by the level of oxygen, in the blood. A reduction in the number of cells in the blood, as a result of the loss of cells, causes the level of oxygen in the blood to decrease. When the level of oxygen in the blood decreases, a substance is released that in turn leads to the release of blood elements, from the bone marrow. Thus, the concentrationc(t) of cells in the blood stream, at any time t, changes according to the loss of cells and the release of new cells, from the bone marrow. But the bone marrow responds to a reduction in the number of blood cells and the decrease in the level of oxygen, with a delay that is in the order of 6 days. That means the release of new cells, into the blood stream, at time t, depends on the cell concentration, at an earlier time, namely, t-τ, where τ is the delay with which the bone marrow responds to a reduced level of oxygen in the blood. The simplest model of the concentration of the cells in the blood stream can be described by the DDE (2)c′t=λct-τ-γct,where λ represents the flux of cells into the blood stream, γ is the death rate, and τ is the delay. All of them are positive constants. The solutions of the above equation exhibit similar oscillations to the actual oscillatory pattern observed in the concentration of cells in the blood stream.Application 2.
Imagine a biological population composed of adult and juvenile individuals. LetN(t) denote the density of adults at time t. Assume that the length of the juvenile period is exactly h units of time for each individual. Assume that adults produce offspring at a per capita rate α and that their probability per unit of time of dying is μ. Assume that a newborn survives the juvenile period with probability ρ and put t=αρ. Then the dynamics of N can be described by the differential equation (3)N′t=-μNt+rNt-hwhich involves a nonlocal term, rN(t-h) meaning that newborns become adults with some delay. So the time variation of the population density N involves the current as well as the past values of N.The use of DDEs, from the initial application, in population dynamics, has spread to every area of the life sciences: immunology, physiology, epidemiology, and cell growth. The original delay logistic equation has led to several new DDE forms, like Volterra’s integrodifferential equations and neutral DDEs [9], and several new models, from the delayed Hopfield model, in neural networks to the SIR model, in epidemiology [7]. More recently, the idea of state dependent delays has been introduced, involving “a delay that itself is governed by a differential equation that represents adaptation to the system’s state” [9].From the above review of DDEs, in the biological sciences, it is apparent that if DDEs are so extensively used in this area, this is because the dynamics of those equations, namely, the stability and oscillatory properties of the solutions of those equations, replicate the stability and oscillatory patterns, we actually observe in processes, in those areas. Thus, the study of the stability and oscillatory behavior of the solutions of DDEs has become the principal subject of the research on those equations. For more advanced treatises on oscillation theory, the reader is referred to [13–33].In the paper, we consider a differential equation with delay argument of the formEx′t+ptxτt=0,t≥t0,where p is a function of nonnegative real numbers and τ is a function of positive real numbers such that(4)τt<t,t≥t0,limt→∞⁡τt=∞.By asolution of E we understand a continuously differentiable function defined on [τT0,∞) for some T0≥t0 and such that E is satisfied for t≥T0. Such a solution is calledoscillatory if it has arbitrarily large zeros, and otherwise it is callednonoscillatory. An equation isoscillatory if all its solutions oscillate.A parallel problem to that of establishing oscillation criteria for the solutions of equationE is the one concerning the solutions of the advanced differential equation (ADE)E′x′t-qtxσt=0,t≥t0,where q is a function of nonnegative real numbers and σ is a function of positive real numbers such that(5)σt>t,t≥t0.The objective of this paper is to consider the oscillatory dynamics of both delay and advanced differential equations, from the perspective of the qualitative analysis of those equations. In that framework, (i) we formulate new iterative oscillation conditions, for testing whether all solutions of a DDE of the form ofE or an ADE of the form of E′ are oscillatory, (ii) we show that these tests significantly improve on all the previous, iterative, and noniterative oscillation criteria which, briefly, are reviewed in the Historical and Chronological Review, in Section 2, requiring fewer iterations to determine whether an equation of the considered form is oscillatory, and (iii) these criteria apply to a more general class of equations, having nonmonotone arguments τ(t) or σ(t), in contrast to the large majority of the other studies where the criteria apply to equations with nondecreasing arguments.From this point onward, we will use the notation(6)α≔liminft→∞⁡∫τttpsds,β≔liminft→∞⁡∫tσtqsds,Dω≔0,ifω>1e,1-ω-1-2ω-ω22,ifω∈0,1e,LD≔limsupt→∞⁡∫τttpsds,whereτtis  nondecreasing,LA≔limsupt→∞⁡∫tσtqsds,whereσtis  nondecreasing.


## 2. Historical and Chronological Review

### 2.1. DDEs

The first systematic study for the oscillation of all solutions of equationE was made by Myškis in 1950 [31], when he proved that every solution of E oscillates, if(7)limsupt→∞⁡t-τt<∞,liminft→∞⁡t-τtliminft→∞⁡pt>1e.In 1972, Ladas et al. [27] proved that if(8)LD>1,then all solutions of E are oscillatory.In 1982, Koplatadze and Chanturiya [24] improved (7) to(9)α>1e.Regarding the constant 1/e in (9), it should be remarked that if the inequality (10)∫τttpsds≤1eholds eventually, then, according to [24], E has a nonoscillatory solution.It is apparent that there is a gap between conditions (8) and (9), when (11)limt→∞∫τttpsdsdoes not exist. How to fill this gap is an interesting problem which has been investigated by several authors. For example, in 2000, Jaroš and Stavroulakis [23] proved that if λ0 is the smaller root of the equation λ=eαλ and(12)LD>1+ln⁡λ0λ0-Dα,then all solutions of E oscillate.Now we come to the general case where the argumentτ(t) is nonmonotone. Set(13)ht≔sups≤t⁡τs,t≥t0.Clearly, the function h(t) is nondecreasing and τ(t)≤h(t)<t, for all t≥t0.In 1994, Koplatadze and Kvinikadze [25] proved that if(14)limsupt→∞⁡∫httpsexp⁡∫hshtpuψjududs>1-Dα,where(15)ψ1t=0,ψjt=exp⁡∫τttpuψj-1udu,j≥2,then all solutions of E oscillate.In 2011, Braverman and Karpuz [14] proved that if(16)limsupt→∞⁡∫httpsexp⁡∫τshtpududs>1,then all solutions of E oscillate, while in 2014, Stavroulakis [32] improved (16) to(17)limsupt→∞⁡∫httpsexp⁡∫τshtpududs>1-Dα.In 2016, El-Morshedy and Attia [30] proved that if(18)limsupt→∞⁡∫gttpnsds+Dαexp⁡∫gtt∑j=0n-1pjsds>1,where p0(t)=p(t) and(19)pnt=pn-1t∫gttpn-1sexp⁡∫gstpn-1ududs,n≥1,then all solutions of E are oscillatory. Here, g(t) is a nondecreasing continuous function such that τ(t)≤g(t)≤t,t≥t1, for some t1≥t0. Clearly, g(t) is more general than h(t) defined by (13).Recently, Chatzarakis [15, 16] proved that if, for some j∈N,(20)limsupt→∞⁡∫httpsexp∫τshtpjududs>1or(21)limsupt→∞⁡∫httpsexp⁡∫τshtpjududs>1-Dα,where(22)pjt=pt1+∫τttpsexp⁡∫τshtpj-1ududs,with p0(t)=p(t), then all solutions of E are oscillatory.Lately, Chatzarakis [17] studied a more general form of E; namely,(23)x′t+∑i=1mpitxτit=0,t≥t0,and established sufficient oscillation conditions. Those conditions can lead to (20) and (21) when m=1.


### 2.2. ADEs

By Theorem 2.4.3 [29], if(24)LA>1,then all solutions of E′ are oscillatory.In 1984, Fukagai and Kusano [21] proved that if(25)β>1e,then all solutions of E′ are oscillatory, while if(26)∫tσtqsds≤1efor  all  sufficiently  larget,then E′ has a nonoscillatory solution.Assume that the argumentσ(t) is not necessarily monotone. Set(27)ρt=infs≥t⁡σs,t≥t0.Clearly, the function ρ(t) is nondecreasing and σ(t)≥ρ(t)>t, for all t≥t0.In 2015, Chatzarakis and Öcalan [18] proved that if(28)limsupt→∞⁡∫tρtqsexp⁡∫ρtσsqududs>1,or(29)liminft→∞⁡∫tρtqsexp⁡∫ρtσsqududs>1e,then all solutions of E′ are oscillatory.Recently, Chatzarakis [15, 16] proved that if, for some j∈N,(30)limsupt→∞⁡∫tρtqsexp⁡∫ρtσsqjududs>1,or(31)limsupt→∞⁡∫tρtqsexp⁡∫ρtσsqjududs>1-Dβ,where(32)qjt=qt1+∫tσtqsexp⁡∫ρtσsqj-1ududs,j≥1with q0(t)=q(t), then all solutions of E′ oscillate.Lately, Chatzarakis [17] studied a more general form of E′, namely, (33)x′t-∑i=1mqitxσit=0,t≥t0,and established sufficient oscillation conditions. Those conditions can lead to (30) and (31) when m=1.


## 2.1. DDEs

The first systematic study for the oscillation of all solutions of equationE was made by Myškis in 1950 [31], when he proved that every solution of E oscillates, if(7)limsupt→∞⁡t-τt<∞,liminft→∞⁡t-τtliminft→∞⁡pt>1e.In 1972, Ladas et al. [27] proved that if(8)LD>1,then all solutions of E are oscillatory.In 1982, Koplatadze and Chanturiya [24] improved (7) to(9)α>1e.Regarding the constant 1/e in (9), it should be remarked that if the inequality (10)∫τttpsds≤1eholds eventually, then, according to [24], E has a nonoscillatory solution.It is apparent that there is a gap between conditions (8) and (9), when (11)limt→∞∫τttpsdsdoes not exist. How to fill this gap is an interesting problem which has been investigated by several authors. For example, in 2000, Jaroš and Stavroulakis [23] proved that if λ0 is the smaller root of the equation λ=eαλ and(12)LD>1+ln⁡λ0λ0-Dα,then all solutions of E oscillate.Now we come to the general case where the argumentτ(t) is nonmonotone. Set(13)ht≔sups≤t⁡τs,t≥t0.Clearly, the function h(t) is nondecreasing and τ(t)≤h(t)<t, for all t≥t0.In 1994, Koplatadze and Kvinikadze [25] proved that if(14)limsupt→∞⁡∫httpsexp⁡∫hshtpuψjududs>1-Dα,where(15)ψ1t=0,ψjt=exp⁡∫τttpuψj-1udu,j≥2,then all solutions of E oscillate.In 2011, Braverman and Karpuz [14] proved that if(16)limsupt→∞⁡∫httpsexp⁡∫τshtpududs>1,then all solutions of E oscillate, while in 2014, Stavroulakis [32] improved (16) to(17)limsupt→∞⁡∫httpsexp⁡∫τshtpududs>1-Dα.In 2016, El-Morshedy and Attia [30] proved that if(18)limsupt→∞⁡∫gttpnsds+Dαexp⁡∫gtt∑j=0n-1pjsds>1,where p0(t)=p(t) and(19)pnt=pn-1t∫gttpn-1sexp⁡∫gstpn-1ududs,n≥1,then all solutions of E are oscillatory. Here, g(t) is a nondecreasing continuous function such that τ(t)≤g(t)≤t,t≥t1, for some t1≥t0. Clearly, g(t) is more general than h(t) defined by (13).Recently, Chatzarakis [15, 16] proved that if, for some j∈N,(20)limsupt→∞⁡∫httpsexp∫τshtpjududs>1or(21)limsupt→∞⁡∫httpsexp⁡∫τshtpjududs>1-Dα,where(22)pjt=pt1+∫τttpsexp⁡∫τshtpj-1ududs,with p0(t)=p(t), then all solutions of E are oscillatory.Lately, Chatzarakis [17] studied a more general form of E; namely,(23)x′t+∑i=1mpitxτit=0,t≥t0,and established sufficient oscillation conditions. Those conditions can lead to (20) and (21) when m=1.


## 2.2. ADEs

By Theorem 2.4.3 [29], if(24)LA>1,then all solutions of E′ are oscillatory.In 1984, Fukagai and Kusano [21] proved that if(25)β>1e,then all solutions of E′ are oscillatory, while if(26)∫tσtqsds≤1efor  all  sufficiently  larget,then E′ has a nonoscillatory solution.Assume that the argumentσ(t) is not necessarily monotone. Set(27)ρt=infs≥t⁡σs,t≥t0.Clearly, the function ρ(t) is nondecreasing and σ(t)≥ρ(t)>t, for all t≥t0.In 2015, Chatzarakis and Öcalan [18] proved that if(28)limsupt→∞⁡∫tρtqsexp⁡∫ρtσsqududs>1,or(29)liminft→∞⁡∫tρtqsexp⁡∫ρtσsqududs>1e,then all solutions of E′ are oscillatory.Recently, Chatzarakis [15, 16] proved that if, for some j∈N,(30)limsupt→∞⁡∫tρtqsexp⁡∫ρtσsqjududs>1,or(31)limsupt→∞⁡∫tρtqsexp⁡∫ρtσsqjududs>1-Dβ,where(32)qjt=qt1+∫tσtqsexp⁡∫ρtσsqj-1ududs,j≥1with q0(t)=q(t), then all solutions of E′ oscillate.Lately, Chatzarakis [17] studied a more general form of E′, namely, (33)x′t-∑i=1mqitxσit=0,t≥t0,and established sufficient oscillation conditions. Those conditions can lead to (30) and (31) when m=1.


## 3. Main Results

### 3.1. DDEs

In our main results, we state theorems, establishing new sufficient oscillation conditions. For the proofs of those theorems, we use the following lemmas.Lemma 3 (see [19, Lemma 2.1.1]).
Assume thath(t) is defined by (13). Then(34)α≔liminft→∞⁡∫τttpsds=liminft→∞⁡∫httpsds.Lemma 4 (see [19, Lemma 2.1.3]).
Assume thath(t) is defined by (13), α∈(0,1/e], and x(t) is an eventually positive solution of E. Then(35)liminft→∞⁡xtxht≥Dα.Lemma 5 (see [26]).
Assume thath(t) is defined by (13), α∈(0,1/e], and x(t) is an eventually positive solution of E. Then(36)liminft→∞⁡xhtxt≥λ0,where λ0 is the smaller root of the equation λ=eαλ.Theorem 6.
Leth(t) be defined by (13) and for some j∈N(37)limsupt→∞⁡∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξdξduds>1,where(38)Pjt=pt1+∫τttpsexp⁡∫τstpuexp⁡∫τuuPj-1ξdξdudswith P0(t)=λ0p(t), and let λ0 be the smaller root of the equation λ=eαλ. Then all solutions of E oscillate.Proof.
Assume, for the sake of contradiction, that there exists a nonoscillatory solutionx(t) of E. Since -x(t) is also a solution of E, we can confine our discussion only to the case where the solution x(t) is eventually positive. Then there exists a real number t1>t0 such that x(t),xτ(t)>0 for all t≥t1. Thus, from E we have (39)x′t=-ptxτt≤0∀t≥t1,which means that x(t) is an eventually nonincreasing function of positive numbers. Taking into account the fact that τ(t)≤h(t), E implies that(40)x′t+ptxht≤0,t≥t1.Observe that (36) implies that, for each ϵ>0, there exists a real number tϵ such that(41)xhtxt>λ0-ϵ∀t≥tϵ≥t1.Combining inequalities (40) and (41), we obtain (42)x′t+ptλ0-ϵxt≤0,t≥tϵ,or(43)x′t+P0t,ϵxt≤0,t≥tϵ,where(44)P0t,ϵ=ptλ0-ϵ.Applying the Grönwall inequality in (43), we conclude that(45)xs≥xtexp⁡∫stP0ξ,ϵdξ,t≥s≥tϵ.Now we divide E by xt>0 and integrate on s,t, so(46)-∫stx′uxudu=∫stpuxτuxudu,or(47)ln⁡xsxt=∫stpuxτuxudu,t≥s≥tϵ.Since τ(u)<u, equality (47) gives (48)ln⁡xsxt=∫stpuxτuxudu≥∫stpuxuxuexp⁡∫τuuP0ξ,ϵdξdu=∫stpuexp⁡∫τuuP0ξ,ϵdξdu,or(49)xs≥xtexp⁡∫stpuexp⁡∫τuuP0ξ,ϵdξdu.Substituting τs for s in (49), we get(50)xτs≥xtexp⁡∫τstpuexp∫τuuP0ξ,ϵdξdu.Integrating E from τ(t) to t, we have(51)xt-xτt+∫τttpsxτsds=0.Combining (50) and (51), we obtain (52)xt-xτt+xt∫τttpsexp⁡∫τstpuexp⁡∫τuuP0ξ,ϵdξduds≤0.Multiplying inequality (52) by p(t), we find(53)ptxt-ptxτt+ptxt∫τttpsexp⁡∫τstpuexp⁡∫τuuP0ξ,ϵdξduds≤0,which, in view of E, becomes (54)x′t+ptxt+ptxt∫τttpsexp⁡∫τstpuexp⁡∫τuuP0ξ,ϵdξduds≤0.Hence, for sufficiently large t, (55)x′t+pt1+∫τttpsexp⁡∫τstpuexp⁡∫τuuP0ξ,ϵdξdudsxt≤0,or(56)x′t+P1t,ϵxt≤0,where (57)P1t,ϵ=pt1+∫τttpsexp⁡∫τstpuexp∫τuuP0ξ,ϵdξduds.Clearly (56) resembles (43), if we replace P0 by P1. Thus, integrating (56) on s,t yields(58)xs≥xtexp⁡∫stP1ξ,ϵdξ.Repeating steps (45) through (50), we can see that x satisfies the inequality(59)xτs≥xtexp∫τstpuexp⁡∫τuuP1ξ,ϵdξdu.Combining now (51) and (59), we obtain (60)xt-xτt+xt∫τttpsexp⁡∫τstpuexp⁡∫τuuP1ξ,ϵdξduds≤0.Multiplying inequality (60) by p(t), as before, we find (61)x′t+pt1+∫τttpsexp∫τstpuexp⁡∫τuuP1ξ,ϵdξdudsxt≤0.Therefore, for sufficiently large t, we have (62)x′t+P2t,ϵxt≤0,where (63)P2t,ϵ=pt1+∫τttpsexp⁡∫τstpuexp⁡∫τuuP1ξ,ϵdξduds.It becomes apparent, now, that, by repeating the above steps, we can build inequalities on x′(t) with progressively higher indices Pj(t,ϵ),j∈N. In general, for sufficiently large t, the positive solution x(t) satisfies the inequality(64)x′t+Pjt,ϵxt≤0,j∈N,where (65)Pjt,ϵ=pt1+∫τttpsexp⁡∫τstpuexp⁡∫τuuPj-1ξ,ϵdξduds.Proceeding to final step, we recall that h(t), defined by (13), is a nondecreasing function. Since τs≤hs≤ht, we have (66)xτs≥xhtexp⁡∫τshtpuexp⁡∫τuuPjξ,ϵdξdu.Hence(67)xt-xht+xht∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξ,ϵdξduds≤0,or (68)xht∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξ,ϵdξduds-1<0.Thus (69)∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξ,ϵdξduds-1<0.Taking the limit as t→∞, we have (70)limsupt→∞⁡∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξ,ϵdξduds≤1.Since ϵ may be taken arbitrarily small, this inequality contradicts (37).
This completes the proof of the theorem.Theorem 7.
Leth(t) be defined by (13) and α∈(0,1/e]. If for some j∈N(71)limsupt→∞⁡∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξdξduds>1-Dα,where Pj is defined by (38), then all solutions of E oscillate.Proof.
Assumex is an eventually positive solution of E. Clearly, (67) is satisfied for sufficiently large t. Thus, (72)∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξ,ϵdξduds≤1-xtxht,which implies that(73)limsupt→∞⁡∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξ,ϵdξduds≤1-liminft→∞⁡xtxht.Using Lemmas 3 and 4, it is evident that inequality (35) is satisfied. Thus, (73) leads to (74)limsupt→∞⁡∫httpsexp∫τshtpuexp∫τuuPjξ,ϵdξduds≤1-Dα.Since ϵ may be taken arbitrarily small, this inequality contradicts (71).
This completes the proof of the theorem.Theorem 8.
Leth(t) be defined by (13) and α∈(0,1/e]. If for some j∈N(75)limsupt→∞⁡∫httpsexp⁡∫τstpuexp⁡∫τuuPjξdξduds>1Dα,where Pj is defined by (38), then all solutions of E oscillate.Proof.
Assumex is an eventually positive solution of E. Then, as in the proof of Theorem 6, for sufficiently large t, we conclude that(76)xτs≥xtexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξdu.Integrating E from h(t) to t and using (76), we obtain (77)xt-xht+∫httpsxtexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξduds≤0,or (78)-xht+∫httpsxtexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξduds<0.Hence (79)xhtxtxht∫httpsexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξduds-1<0,which yields, for all sufficiently large t, (80)∫httpsexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξduds<xhtxtand consequently(81)limsupt→∞⁡∫httpsexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξduds≤limsupt→∞⁡xhtxt.Taking into account the fact that (35) is satisfied, inequality (81) leads to (82)limsupt→∞⁡∫httpsexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξduds≤1Dα,which contradicts (75), when ϵ→0.
This completes the proof of the theorem.Theorem 9.
Leth(t) be defined by (13) and α∈(0,1/e]. If for some j∈N(83)limsupt→∞⁡∫httpsexp⁡∫τshspuexp⁡∫τuuPjξdξduds>1+ln⁡λ0λ0-Dα,where Pj is defined by (38) and λ0 is the smaller root of the equation λ=eαλ, then all solutions of E oscillate.Proof.
Letx be an eventually positive solution of E. As in the proof of Theorem 8, we can show that (76) holds; namely, (84)xτs≥xtexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξdu.Since τ(s)≤h(s), inequality (84) gives(85)xτs≥xhsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξdu.By Lemma 5, for each ϵ>0, there exists a real number tϵ such that(86)xhtxt>λ0-ϵ∀t≥tϵ≥t1.Note that, by the nondecreasing nature of the function x(h(t))/x(s) in s, it holds (87)1=xhtxht≤xhtxs≤xhtxt,tϵ≤ht≤s≤t.In particular, for ϵ∈0,λ0-1, by continuity, we conclude that there exists a real number t∗∈(h(t),t] satisfying(88)1<λ0-ϵ=xhtxt∗.Integrating E from t∗ to t and using (85), we obtain (89)xt-xt∗+xht∫t∗tpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds≤0,or(90)∫t∗tpsexp∫τshspuexp∫τuuPjξ,ϵdξduds≤xt∗xht-xtxht.Using (88) and Lemma 4, we deduce that, for the ϵ considered, there exists a real number tϵ′≥tϵ such that(91)∫t∗tpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds<1λ0-ϵ-Dα+ϵfor t≥tϵ′.
DividingE by x(t), integrating from h(t) to t∗, and using (85), we deduce that(92)∫htt∗psxhsxsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds≤-∫htt∗x′sxsds.Clearly, by means of (36), x(h(s))/x(s)>λ0-ϵ, for s≥h(t)≥tϵ′. Hence, for all sufficiently large t, we conclude that (93)λ0-ϵ∫htt∗psexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds<-∫htt∗x′sxsdsor(94)∫htt∗psexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds<-1λ0-ϵ∫htt∗x′sxsds=1λ0-ϵln⁡xhtxt∗=ln⁡λ0-ϵλ0-ϵ;that is,(95)∫htt∗psexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds<ln⁡λ0-ϵλ0-ϵ.Using (91) along with (95), we get(96)limsupt→∞⁡∫httpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds≤1+ln⁡λ0-ϵλ0-ϵ-Dα+ϵ,which contradicts (83), when ϵ→0.
This completes the proof of the theorem.Theorem 10.
Leth(t) be defined by (13). If for some j∈N(97)liminft→∞⁡∫httpsexp⁡∫τshspuexp⁡∫τuuPjξdξduds>1e,where Pj is defined by (38), then all solutions of E oscillate.Proof.
For the sake of contradiction, letx be a nonincreasing eventually positive solution and t1>t0 be such that x(t)>0 and xτ(t)>0 for all t≥t1. We note that we may obtain (85) as in the proof of Theorem 9.
DividingE by x(t) and integrating from h(t) to t, we have (98)ln⁡xhtxt=∫httpsxτsxsds∀t≥t2≥t1,from which, in view of τ(s)≤h(s) and (85), we get (99)ln⁡xhtxt≥∫httpsxhsxsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds.Since x is nonincreasing and h(s)<s, inequality (99) becomes(100)ln⁡xhtxt≥∫httpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds.From (97), it is clear that there exists a constant c>0 such that (101)∫httpsexp⁡∫τshspuexp⁡∫τuuPjξdξduds≥c>1e.Choose c′ such that c>c′>1/e. For every ϵ>0, such that c-ϵ>c′, we have(102)∫httpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds>c-ϵ>c′>1e.Combining inequalities (100) and (102), we obtain (103)ln⁡xhtxt>c′,or (104)xhtxt>ec′>ec′>1,which yields (105)xht>ec′xt.Following the above steps, we can inductively show that, for any positive integer k, (106)xhtxt>ec′kfor  sufficiently  larget.Since ec′>1, there is a natural number k∈N, satisfying k>2ln⁡2-ln⁡c′/1+ln⁡c′ such that for t sufficiently large(107)xhtxt>ec′k>2c′2.Further (cf. [13, 24]), for sufficiently large t, there exists a real number tm∈(h(t),t), such that(108)∫httmpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds>c′2,∫tmtpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds>c′2.Integrating E from h(t) to tm, using (85) and the fact that x(t)>0, we obtain (109)xht>xhtm∫httmpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds,which, in view of the first inequality in (108), implies that(110)xht>c′2xhtm.Similarly, integrating E from tm to t, using (85) and the fact that x(t)>0, we have (111)xtm>xht∫tmtpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds,which, in view of the second inequality in (108), yields(112)xtm>c′2xht.Combining inequalities (110) and (112), we deduce that (113)xhtm<2c′xht<2c′2xtm,which contradicts (107).
The proof of the theorem is complete.


### 3.2. ADEs

Analogous oscillation conditions to those obtained for the delay equationE can be derived for the (dual) advanced differential equation E′ by following similar arguments with the ones employed for obtaining Theorems 6−10.Theorem 11.
Letρ(t) be defined by (27) and for some j∈N(114)limsupt→∞⁡∫tρtqsexp⁡∫ρtσsquexp⁡∫uσuQjξdξduds>1,where(115)Qjt=qt1+∫tσtqsexp⁡∫tσsquexp⁡∫uσuQj-1ξdξdudswith Q0(t)=λ0q(t), and let λ0 be the smaller root of the equation λ=eβλ. Then all solutions of E′ oscillate.Theorem 12.
Letρ(t) be defined by (27) and β∈(0,1/e]. If for some j∈N(116)limsupt→∞⁡∫tρtqsexp⁡∫ρtσsquexp⁡∫uσuQjξdξduds>1-Dβ,where Qj is defined by (115), then all solutions of E′ oscillate.Theorem 13.
Letρ(t) be defined by (27) and β∈(0,1/e]. If for some j∈N(117)limsupt→∞⁡∫tρtqsexp⁡∫tσsquexp⁡∫uσuQjξdξduds>1Dβ,where Qj is defined by (115), then all solutions of E′ oscillate.Theorem 14.
Letρ(t) be defined by (27) and β∈(0,1/e]. If for some j∈N(118)limsupt→∞⁡∫tρtqsexp⁡∫ρsσsquexp⁡∫uσuQjξdξduds>1+ln⁡λ0λ0-Dβ,where Qj is defined by (115) and λ0 is the smaller root of the equation λ=eβλ, then all solutions of E′ oscillate.Theorem 15.
Letρ(t) be defined by (27). If for some j∈N(119)liminft→∞⁡∫tρtqsexp⁡∫ρsσsquexp⁡∫uσuQjξdξduds>1e,where Qj is defined by (115), then all solutions of E′ oscillate.


### 3.3. Differential Inequalities

A slight modification in the proofs of Theorems6−15 leads to the following results about differential inequalities.Theorem 16.
Assume that all the conditions of Theorem6 [11], 7 [12], 8 [13], 9 [14], or 10 [15] hold. Then
( i ) the delay [advanced] differential inequality (120)x′t+ptxτt≤0x′t-qtxσt≥0,t≥t0has no eventually positive solutions;
( i i ) the delay [advanced] differential inequality (121)x′t+ptxτt≥0x′t-qtxσt≤0,t≥t0has no eventually negative solutions.Remark 17.
The oscillation criteria established in this paper all depend onλ0 (see, e.g., (37) and (71)) in contrast to the conditions obtained in [15, 16] and in [17, form = 1]. In fact, the left-hand side of conditions (37) and (71) depends on λ0, which is not the case with the left-hand side of conditions (20) and (21). Since λ0>1 when α∈(0,1/e], it is obvious that (122)P0t=λ0pt>pt=p0t.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 in this paper improve on all known conditions mentioned in Section 2.


## 3.1. DDEs

In our main results, we state theorems, establishing new sufficient oscillation conditions. For the proofs of those theorems, we use the following lemmas.Lemma 3 (see [19, Lemma 2.1.1]).
Assume thath(t) is defined by (13). Then(34)α≔liminft→∞⁡∫τttpsds=liminft→∞⁡∫httpsds.Lemma 4 (see [19, Lemma 2.1.3]).
Assume thath(t) is defined by (13), α∈(0,1/e], and x(t) is an eventually positive solution of E. Then(35)liminft→∞⁡xtxht≥Dα.Lemma 5 (see [26]).
Assume thath(t) is defined by (13), α∈(0,1/e], and x(t) is an eventually positive solution of E. Then(36)liminft→∞⁡xhtxt≥λ0,where λ0 is the smaller root of the equation λ=eαλ.Theorem 6.
Leth(t) be defined by (13) and for some j∈N(37)limsupt→∞⁡∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξdξduds>1,where(38)Pjt=pt1+∫τttpsexp⁡∫τstpuexp⁡∫τuuPj-1ξdξdudswith P0(t)=λ0p(t), and let λ0 be the smaller root of the equation λ=eαλ. Then all solutions of E oscillate.Proof.
Assume, for the sake of contradiction, that there exists a nonoscillatory solutionx(t) of E. Since -x(t) is also a solution of E, we can confine our discussion only to the case where the solution x(t) is eventually positive. Then there exists a real number t1>t0 such that x(t),xτ(t)>0 for all t≥t1. Thus, from E we have (39)x′t=-ptxτt≤0∀t≥t1,which means that x(t) is an eventually nonincreasing function of positive numbers. Taking into account the fact that τ(t)≤h(t), E implies that(40)x′t+ptxht≤0,t≥t1.Observe that (36) implies that, for each ϵ>0, there exists a real number tϵ such that(41)xhtxt>λ0-ϵ∀t≥tϵ≥t1.Combining inequalities (40) and (41), we obtain (42)x′t+ptλ0-ϵxt≤0,t≥tϵ,or(43)x′t+P0t,ϵxt≤0,t≥tϵ,where(44)P0t,ϵ=ptλ0-ϵ.Applying the Grönwall inequality in (43), we conclude that(45)xs≥xtexp⁡∫stP0ξ,ϵdξ,t≥s≥tϵ.Now we divide E by xt>0 and integrate on s,t, so(46)-∫stx′uxudu=∫stpuxτuxudu,or(47)ln⁡xsxt=∫stpuxτuxudu,t≥s≥tϵ.Since τ(u)<u, equality (47) gives (48)ln⁡xsxt=∫stpuxτuxudu≥∫stpuxuxuexp⁡∫τuuP0ξ,ϵdξdu=∫stpuexp⁡∫τuuP0ξ,ϵdξdu,or(49)xs≥xtexp⁡∫stpuexp⁡∫τuuP0ξ,ϵdξdu.Substituting τs for s in (49), we get(50)xτs≥xtexp⁡∫τstpuexp∫τuuP0ξ,ϵdξdu.Integrating E from τ(t) to t, we have(51)xt-xτt+∫τttpsxτsds=0.Combining (50) and (51), we obtain (52)xt-xτt+xt∫τttpsexp⁡∫τstpuexp⁡∫τuuP0ξ,ϵdξduds≤0.Multiplying inequality (52) by p(t), we find(53)ptxt-ptxτt+ptxt∫τttpsexp⁡∫τstpuexp⁡∫τuuP0ξ,ϵdξduds≤0,which, in view of E, becomes (54)x′t+ptxt+ptxt∫τttpsexp⁡∫τstpuexp⁡∫τuuP0ξ,ϵdξduds≤0.Hence, for sufficiently large t, (55)x′t+pt1+∫τttpsexp⁡∫τstpuexp⁡∫τuuP0ξ,ϵdξdudsxt≤0,or(56)x′t+P1t,ϵxt≤0,where (57)P1t,ϵ=pt1+∫τttpsexp⁡∫τstpuexp∫τuuP0ξ,ϵdξduds.Clearly (56) resembles (43), if we replace P0 by P1. Thus, integrating (56) on s,t yields(58)xs≥xtexp⁡∫stP1ξ,ϵdξ.Repeating steps (45) through (50), we can see that x satisfies the inequality(59)xτs≥xtexp∫τstpuexp⁡∫τuuP1ξ,ϵdξdu.Combining now (51) and (59), we obtain (60)xt-xτt+xt∫τttpsexp⁡∫τstpuexp⁡∫τuuP1ξ,ϵdξduds≤0.Multiplying inequality (60) by p(t), as before, we find (61)x′t+pt1+∫τttpsexp∫τstpuexp⁡∫τuuP1ξ,ϵdξdudsxt≤0.Therefore, for sufficiently large t, we have (62)x′t+P2t,ϵxt≤0,where (63)P2t,ϵ=pt1+∫τttpsexp⁡∫τstpuexp⁡∫τuuP1ξ,ϵdξduds.It becomes apparent, now, that, by repeating the above steps, we can build inequalities on x′(t) with progressively higher indices Pj(t,ϵ),j∈N. In general, for sufficiently large t, the positive solution x(t) satisfies the inequality(64)x′t+Pjt,ϵxt≤0,j∈N,where (65)Pjt,ϵ=pt1+∫τttpsexp⁡∫τstpuexp⁡∫τuuPj-1ξ,ϵdξduds.Proceeding to final step, we recall that h(t), defined by (13), is a nondecreasing function. Since τs≤hs≤ht, we have (66)xτs≥xhtexp⁡∫τshtpuexp⁡∫τuuPjξ,ϵdξdu.Hence(67)xt-xht+xht∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξ,ϵdξduds≤0,or (68)xht∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξ,ϵdξduds-1<0.Thus (69)∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξ,ϵdξduds-1<0.Taking the limit as t→∞, we have (70)limsupt→∞⁡∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξ,ϵdξduds≤1.Since ϵ may be taken arbitrarily small, this inequality contradicts (37).
This completes the proof of the theorem.Theorem 7.
Leth(t) be defined by (13) and α∈(0,1/e]. If for some j∈N(71)limsupt→∞⁡∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξdξduds>1-Dα,where Pj is defined by (38), then all solutions of E oscillate.Proof.
Assumex is an eventually positive solution of E. Clearly, (67) is satisfied for sufficiently large t. Thus, (72)∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξ,ϵdξduds≤1-xtxht,which implies that(73)limsupt→∞⁡∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξ,ϵdξduds≤1-liminft→∞⁡xtxht.Using Lemmas 3 and 4, it is evident that inequality (35) is satisfied. Thus, (73) leads to (74)limsupt→∞⁡∫httpsexp∫τshtpuexp∫τuuPjξ,ϵdξduds≤1-Dα.Since ϵ may be taken arbitrarily small, this inequality contradicts (71).
This completes the proof of the theorem.Theorem 8.
Leth(t) be defined by (13) and α∈(0,1/e]. If for some j∈N(75)limsupt→∞⁡∫httpsexp⁡∫τstpuexp⁡∫τuuPjξdξduds>1Dα,where Pj is defined by (38), then all solutions of E oscillate.Proof.
Assumex is an eventually positive solution of E. Then, as in the proof of Theorem 6, for sufficiently large t, we conclude that(76)xτs≥xtexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξdu.Integrating E from h(t) to t and using (76), we obtain (77)xt-xht+∫httpsxtexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξduds≤0,or (78)-xht+∫httpsxtexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξduds<0.Hence (79)xhtxtxht∫httpsexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξduds-1<0,which yields, for all sufficiently large t, (80)∫httpsexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξduds<xhtxtand consequently(81)limsupt→∞⁡∫httpsexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξduds≤limsupt→∞⁡xhtxt.Taking into account the fact that (35) is satisfied, inequality (81) leads to (82)limsupt→∞⁡∫httpsexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξduds≤1Dα,which contradicts (75), when ϵ→0.
This completes the proof of the theorem.Theorem 9.
Leth(t) be defined by (13) and α∈(0,1/e]. If for some j∈N(83)limsupt→∞⁡∫httpsexp⁡∫τshspuexp⁡∫τuuPjξdξduds>1+ln⁡λ0λ0-Dα,where Pj is defined by (38) and λ0 is the smaller root of the equation λ=eαλ, then all solutions of E oscillate.Proof.
Letx be an eventually positive solution of E. As in the proof of Theorem 8, we can show that (76) holds; namely, (84)xτs≥xtexp⁡∫τstpuexp⁡∫τuuPjξ,ϵdξdu.Since τ(s)≤h(s), inequality (84) gives(85)xτs≥xhsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξdu.By Lemma 5, for each ϵ>0, there exists a real number tϵ such that(86)xhtxt>λ0-ϵ∀t≥tϵ≥t1.Note that, by the nondecreasing nature of the function x(h(t))/x(s) in s, it holds (87)1=xhtxht≤xhtxs≤xhtxt,tϵ≤ht≤s≤t.In particular, for ϵ∈0,λ0-1, by continuity, we conclude that there exists a real number t∗∈(h(t),t] satisfying(88)1<λ0-ϵ=xhtxt∗.Integrating E from t∗ to t and using (85), we obtain (89)xt-xt∗+xht∫t∗tpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds≤0,or(90)∫t∗tpsexp∫τshspuexp∫τuuPjξ,ϵdξduds≤xt∗xht-xtxht.Using (88) and Lemma 4, we deduce that, for the ϵ considered, there exists a real number tϵ′≥tϵ such that(91)∫t∗tpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds<1λ0-ϵ-Dα+ϵfor t≥tϵ′.
DividingE by x(t), integrating from h(t) to t∗, and using (85), we deduce that(92)∫htt∗psxhsxsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds≤-∫htt∗x′sxsds.Clearly, by means of (36), x(h(s))/x(s)>λ0-ϵ, for s≥h(t)≥tϵ′. Hence, for all sufficiently large t, we conclude that (93)λ0-ϵ∫htt∗psexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds<-∫htt∗x′sxsdsor(94)∫htt∗psexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds<-1λ0-ϵ∫htt∗x′sxsds=1λ0-ϵln⁡xhtxt∗=ln⁡λ0-ϵλ0-ϵ;that is,(95)∫htt∗psexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds<ln⁡λ0-ϵλ0-ϵ.Using (91) along with (95), we get(96)limsupt→∞⁡∫httpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds≤1+ln⁡λ0-ϵλ0-ϵ-Dα+ϵ,which contradicts (83), when ϵ→0.
This completes the proof of the theorem.Theorem 10.
Leth(t) be defined by (13). If for some j∈N(97)liminft→∞⁡∫httpsexp⁡∫τshspuexp⁡∫τuuPjξdξduds>1e,where Pj is defined by (38), then all solutions of E oscillate.Proof.
For the sake of contradiction, letx be a nonincreasing eventually positive solution and t1>t0 be such that x(t)>0 and xτ(t)>0 for all t≥t1. We note that we may obtain (85) as in the proof of Theorem 9.
DividingE by x(t) and integrating from h(t) to t, we have (98)ln⁡xhtxt=∫httpsxτsxsds∀t≥t2≥t1,from which, in view of τ(s)≤h(s) and (85), we get (99)ln⁡xhtxt≥∫httpsxhsxsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds.Since x is nonincreasing and h(s)<s, inequality (99) becomes(100)ln⁡xhtxt≥∫httpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds.From (97), it is clear that there exists a constant c>0 such that (101)∫httpsexp⁡∫τshspuexp⁡∫τuuPjξdξduds≥c>1e.Choose c′ such that c>c′>1/e. For every ϵ>0, such that c-ϵ>c′, we have(102)∫httpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds>c-ϵ>c′>1e.Combining inequalities (100) and (102), we obtain (103)ln⁡xhtxt>c′,or (104)xhtxt>ec′>ec′>1,which yields (105)xht>ec′xt.Following the above steps, we can inductively show that, for any positive integer k, (106)xhtxt>ec′kfor  sufficiently  larget.Since ec′>1, there is a natural number k∈N, satisfying k>2ln⁡2-ln⁡c′/1+ln⁡c′ such that for t sufficiently large(107)xhtxt>ec′k>2c′2.Further (cf. [13, 24]), for sufficiently large t, there exists a real number tm∈(h(t),t), such that(108)∫httmpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds>c′2,∫tmtpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds>c′2.Integrating E from h(t) to tm, using (85) and the fact that x(t)>0, we obtain (109)xht>xhtm∫httmpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds,which, in view of the first inequality in (108), implies that(110)xht>c′2xhtm.Similarly, integrating E from tm to t, using (85) and the fact that x(t)>0, we have (111)xtm>xht∫tmtpsexp⁡∫τshspuexp⁡∫τuuPjξ,ϵdξduds,which, in view of the second inequality in (108), yields(112)xtm>c′2xht.Combining inequalities (110) and (112), we deduce that (113)xhtm<2c′xht<2c′2xtm,which contradicts (107).
The proof of the theorem is complete.


## 3.2. ADEs

Analogous oscillation conditions to those obtained for the delay equationE can be derived for the (dual) advanced differential equation E′ by following similar arguments with the ones employed for obtaining Theorems 6−10.Theorem 11.
Letρ(t) be defined by (27) and for some j∈N(114)limsupt→∞⁡∫tρtqsexp⁡∫ρtσsquexp⁡∫uσuQjξdξduds>1,where(115)Qjt=qt1+∫tσtqsexp⁡∫tσsquexp⁡∫uσuQj-1ξdξdudswith Q0(t)=λ0q(t), and let λ0 be the smaller root of the equation λ=eβλ. Then all solutions of E′ oscillate.Theorem 12.
Letρ(t) be defined by (27) and β∈(0,1/e]. If for some j∈N(116)limsupt→∞⁡∫tρtqsexp⁡∫ρtσsquexp⁡∫uσuQjξdξduds>1-Dβ,where Qj is defined by (115), then all solutions of E′ oscillate.Theorem 13.
Letρ(t) be defined by (27) and β∈(0,1/e]. If for some j∈N(117)limsupt→∞⁡∫tρtqsexp⁡∫tσsquexp⁡∫uσuQjξdξduds>1Dβ,where Qj is defined by (115), then all solutions of E′ oscillate.Theorem 14.
Letρ(t) be defined by (27) and β∈(0,1/e]. If for some j∈N(118)limsupt→∞⁡∫tρtqsexp⁡∫ρsσsquexp⁡∫uσuQjξdξduds>1+ln⁡λ0λ0-Dβ,where Qj is defined by (115) and λ0 is the smaller root of the equation λ=eβλ, then all solutions of E′ oscillate.Theorem 15.
Letρ(t) be defined by (27). If for some j∈N(119)liminft→∞⁡∫tρtqsexp⁡∫ρsσsquexp⁡∫uσuQjξdξduds>1e,where Qj is defined by (115), then all solutions of E′ oscillate.


## 3.3. Differential Inequalities

A slight modification in the proofs of Theorems6−15 leads to the following results about differential inequalities.Theorem 16.
Assume that all the conditions of Theorem6 [11], 7 [12], 8 [13], 9 [14], or 10 [15] hold. Then
( i ) the delay [advanced] differential inequality (120)x′t+ptxτt≤0x′t-qtxσt≥0,t≥t0has no eventually positive solutions;
( i i ) the delay [advanced] differential inequality (121)x′t+ptxτt≥0x′t-qtxσt≤0,t≥t0has no eventually negative solutions.Remark 17.
The oscillation criteria established in this paper all depend onλ0 (see, e.g., (37) and (71)) in contrast to the conditions obtained in [15, 16] and in [17, form = 1]. In fact, the left-hand side of conditions (37) and (71) depends on λ0, which is not the case with the left-hand side of conditions (20) and (21). Since λ0>1 when α∈(0,1/e], it is obvious that (122)P0t=λ0pt>pt=p0t.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 in this paper improve on all known conditions mentioned in Section 2.


## 4. Examples and Comments

The oscillation tests we have proposed and established, in the main results, involve an iterative procedure. We iteratively computelimsup and liminf on the terms Pj(t) and Qj(t),j∈N of a recurrent relation defined on the coefficients and the deviating argument of an equation of the form E or E′ to determine whether that equation is oscillatory. But this computation cannot be performed on paper, but by means of a program, numerically computing limsup and liminf. The examples below illustrate the significance of our results and indicate the high level of improvement in the oscillation criteria. The calculations were performed using MATLAB code.Example 1.
Consider the delay differential equation(123)x′t+325xτt=0,t≥0,with (see Figure 1(a)) (124)τt=t-1,ift∈8k,8k+2-4t+40k+9,ift∈8k+2,8k+35t-32k-18,ift∈8k+3,8k+4-4t+40k+18,ift∈8k+4,8k+55t-32k-27,ift∈8k+5,8k+6-2t+24k+15,ift∈8k+6,8k+76t-40k-41,ift∈8k+7,8k+8,where k∈N0 and N0 is the set of nonnegative integers.
By (13), we see (Figure 1(b)) that (125)ht=t-1,ift∈8k,8k+28k+1,ift∈8k+2,8k+1955t-32k-18,ift∈8k+195,8k+48k+2,ift∈8k+4,8k+2955t-32k-27,ift∈8k+295,8k+68k+3,ift∈8k+6,8k+4466t-40k-41,ift∈8k+446,8k+8.It is obvious that (126)α=liminft→∞⁡∫τttpsds=liminft→∞⁡∫8k+18k+2325ds=0.12and therefore, the smaller root of e0.12λ=λ is λ0=1.14765.
Observe that the functionFj:R0→R+ defined as (127)Fjt=∫httpsexp⁡∫τshtpuexp⁡∫τuuPjξdξdudsattains its maximum at t=8k+44/6, k∈N0, for every j∈N. Specifically, (128)F1t=8k+446=∫8k+38k+44/6psexp⁡∫τs8k+3puexp⁡∫τuuP1ξdξdudswith (129)P1ξ=pξ1+∫τξξpvexp⁡∫τvξpwexp⁡∫τwwλ0pzdzdwdv.Using MATLAB, we obtain (130)F1t=8k+446≃1.0417and therefore (131)limsupt→∞⁡F1t≃1.0417>1.Hence, condition (37) of Theorem 6 is satisfied, for j=1. Consequently, all solutions of (123) are oscillatory.
Observe, however, that(132)LD=limsupk→∞⁡∫8k+38k+44/6335ds=0.52<1,α=0.12<1e,0.52<1+ln⁡λ0λ0-Dα≃0.9831.Note that the function Φj defined by (133)Φjt=∫httpsexp⁡∫hshtpuψjududs,j≥2,attains its maximum at t=8k+44/6, k∈N0, for every j≥2. Specifically, (134)Φ28k+446=∫8k+38k+44/6psexp⁡∫hs8k+3psψ2ududs=∫8k+38k+44/6325exp⁡∫hs8k+3325exp⁡∫τuu325·0dwduds=∫8k+38k+44/6325exp⁡∫hs8k+3325·1duds=325·∫8k+38k+19/5exp⁡325∫8k+18k+3duds+∫8k+19/58k+4exp⁡325∫5s-32k-188k+3duds+∫8k+48k+29/5exp⁡325∫8k+28k+3duds+∫8k+29/58k+6exp⁡325∫5s-32k-278k+3duds+∫8k+68k+44/6exp⁡325∫8k+38k+3duds≃0.57983.Thus (135)limsupt→∞⁡Φ2t≃0.57983<1-Dα≃0.99174.Also (136)limsupt→∞⁡∫httpsexp⁡∫τshtpududs=limsupk→∞⁡∫8k+38k+44/6325exp⁡∫τs8k+3325duds=325·limsupt→∞⁡∫8k+38k+4exp⁡325∫5s-32k-188k+3duds+∫8k+48k+5exp⁡325∫-4s+40k+188k+3duds+∫8k+58k+6exp⁡325∫5s-32k-278k+3duds+∫8k+68k+7exp⁡325∫-2s+24k+158k+3duds+∫8k+78k+44/6exp⁡325∫6s-40k-418k+3duds≃0.7043<1,0.7043<1-Dα≃0.99174.In addition, (137)limsupt→∞⁡∫httpsexp⁡∫τshtp1ududs≃0.8052<1,0.8052<1-Dα≃0.99174.That is, none of conditions (8), (9), (12), (14) (for j=2), (16), (17), (20) (for j=1), and (21) (for j=1) is satisfied.
Comment. The improvement of condition (37) over the corresponding condition (8) is significant, approximately 100.33%. We get this measure by comparing the values, in the left-hand side of those conditions. Also, the improvement over conditions (14), (16), and (20) is very satisfactory, around 79.66%, 47.9%, and 29.37%, respectively. In addition, condition (37) is satisfied from the first iteration, while conditions (14), (20), and (21) need more than one iteration.Figure 1
The graphs ofτ(t) and h(t).
(a) (b)Example 2 (taken and adapted from [17]).
Consider the advanced differential equation(138)x′t-3332500xσt=0,t≥0,with (see Figure 2(a)) (139)σt=5k+3,ift∈5k,5k+14t-15k-1,ift∈5k+1,5k+2-3t+20k+13,ift∈5k+2,5k+35t-20k-11,ift∈5k+3,5k+4-t+10k+13,ift∈5k+4,5k+5,where k∈N0 and N0 is the set of nonnegative integers.
By (27), we see (Figure 2(b)) that (140)ρt=5k+3,ift∈5k,5k+14t-15k-1,ift∈5k+1,5k+1.255k+4,ift∈5k+1.25,5k+35t-20k-11,ift∈5k+3,5k+3.85k+8,ift∈5k+3.8,5k+5.It is obvious that (141)β=liminft→∞⁡∫5k+35k+43332500ds=0.1332and therefore, the smaller root of e0.1332λ=λ is λ0=1.16839.
Observe, that the functionGj:R0→R+ defined as (142)Gjt=∫tρtqsexp⁡∫ρtσsquexp⁡∫uσuQjξdξdudsattains its maximum at t=5k+3.8, k∈N0, for every j∈N. Specifically, (143)G1t=5k+3.8=∫5k+3.85k+8qsexp⁡∫5k+8σsquexp⁡∫uσuQ1ξdξdudswith (144)Q1ξ=qξ1+∫ξσξqvexp⁡∫ξσvqwexp⁡∫wσwλ0qzdzdwdv.Using MATLAB, we obtain (145)G1t=5k+3.8≃0.9915.Therefore (146)limsupt→∞⁡G1t≃0.9915>1-Dβ≃0.9896.Hence, condition (116) of Theorem 12 is satisfied, for j=1. Consequently, all solutions of (138) oscillate.
Observe, however, that(147)LA=limsupk→∞⁡∫5k+3.85k+83332500ds=0.55944<1,β=0.1332<1e,limsupt→∞⁡∫tρtqsexp⁡∫ρtσsqududs=limsupk→∞⁡∫5k+3.85k+8qsexp⁡∫5k+8σsqududs=limsupk→∞⁡∫5k+3.85k+4qsexp⁡∫5k+85s-20k-11qududs+∫5k+45k+5qsexp⁡∫5k+8-s+10k+13qududs+∫5k+55k+6qsexp⁡∫5k+85k+8qududs+∫5k+65k+7qsexp⁡∫5k+84s-15k-16qududs+∫5k+75k+8qsexp⁡∫5k+8-3s+20k+33qududs≃0.6672<1,liminft→∞⁡∫tρtqsexp⁡∫ρtσsqududs=liminft→∞⁡∫5k+35k+4qsexp⁡∫5k+4σsqududs≃0.1893<1e,limsupt→∞⁡∫tρtqsexp⁡∫ρtσsq1ududs≃0.7196<1,0.7196<1-Dβ≃0.9896.That is, none of conditions (24), (25), (28), (29), (30) (for j=1), and (31) (for j=1) is satisfied.
Comment. The improvement of condition (116) over the corresponding condition (24) is significant, approximately 77.23%. We get this measure by comparing the values, in the left-hand side of those conditions. Also, the improvement over conditions (28) and (30) is very satisfactory, around 48.61% and 37.78%, respectively. In addition, condition (116) is satisfied from the first iteration, while conditions (30) and (31) need more than one iteration.Figure 2
The graphs ofσ(t) and ρ(t).
(a) (b)Remark 3.
Similarly, one can provide examples, illustrating the other main results.


## 5. Concluding Remarks

In the present paper, we have considered the oscillatory dynamics of differential equations, having nonmonotone deviating arguments and nonnegative coefficients. New sufficient conditions have been established, for the oscillation of all solutions ofE and E′. These conditions include (37), (71), (75), (83), and (97) and (114), (116), (117), (118), and (119), for E and E′, respectively. Applying these conditions involves a procedure that checks for oscillations by iteratively computing limsup and liminf, on terms recursively defined on the equation’s coefficients and deviating argument.The main advantage of these conditions is that they achieve a major improvement over all the related oscillation conditions forE [E′], in the literature. For example, condition (37) [(114)] improves upon the noniterative conditions that are reviewed in the introduction, namely, conditions (8) [(24)], (12), (16)≡(20) (for j=1) [(28)≡(30) (for j=1)], and (17)≡(21) (for j=1) [(31) (for j=1)]. That immediately becomes 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 iterative conditions, namely, (14) (for j>2), (20) (for j>1) [(30) (for j>1)], and (21) (for j>1) [(31) (for j>1)], is that it requires far fewer iterations to establish oscillation than the other conditions.This advantage, easily, can be verified computationally, by running the MATLAB programs (see Appendix), for computinglimsup and liminf and comparing the number of iterations required by each condition to establish oscillation. Then we see that we achieve a significant improvement over all known oscillation criteria.Another advantage and a significant departure from the large majority of the other studies is that the criteria in this paper apply to a more general class of equations, having nonmonotone argumentsτ(t) or σ(t), in contrast to most of the other oscillation criteria that apply to equations with nondecreasing arguments.



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