Oscillation Criteria for Delay and Advanced Differential Equations with Nonmonotone Arguments

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, ormechanical 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 andmaturation periods into the model and account for oscillations, in the population of Daphnia,


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][2][3][4][5][6][7][8][9][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, 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.
Many physiological processes, including the concentration of red blood cells, the concentration of CO 2 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 concentration () of cells in the blood stream, at any time , 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 , depends on the cell concentration, at an earlier time, namely,  − , where  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 where  represents the flux of cells into the blood stream,  is the death rate, and  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.Let () denote the density of adults at time .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  and that their probability per unit of time of dying is .Assume that a newborn survives the juvenile period with probability  and put  = .Then the dynamics of  can be described by the differential equation which involves a nonlocal term, ( − ℎ) meaning that newborns become adults with some delay.So the time variation of the population density  involves the current as well as the past values of .
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 .
In the paper, we consider a differential equation with delay argument of the form where  is a function of nonnegative real numbers and  is a function of positive real numbers such that By a solution of (E) we understand a continuously differentiable function defined on [( 0 ), ∞) for some  0 ≥  0 and such that (E) is satisfied for  ≥  0 .Such a solution is called oscillatory if it has arbitrarily large zeros, and otherwise it is called nonoscillatory.An equation is oscillatory if all its solutions oscillate.
A parallel problem to that of establishing oscillation criteria for the solutions of equation (E) is the one concerning the solutions of the advanced differential equation (ADE) where  is a function of nonnegative real numbers and  is a function of positive real numbers such that 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 of (E) 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 () or (), 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 LD fl lim sup In 1972, Ladas et al. [27] proved that if then all solutions of (E) are oscillatory.
It is apparent that there is a gap between conditions (8) and ( 9), when lim →∞ ∫  ()  ()  (11) does 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  0 is the smaller root of the equation  =   and then all solutions of (E) oscillate.Now we come to the general case where the argument () is nonmonotone.Set Clearly, the function ℎ() is nondecreasing and () ≤ ℎ() < , for all  ≥  0 .
In 1994, Koplatadze and Kvinikadze [25] proved that if lim sup where then all solutions of (E) oscillate.

ADEs
. By Theorem 2.4.3 [29], if then all solutions of (E  ) are oscillatory.
In 1984, Fukagai and Kusano [21] proved that if then all solutions of (E  ) are oscillatory, while if then (E  ) has a nonoscillatory solution.

DDEs.
In our main results, we state theorems, establishing new sufficient oscillation conditions.For the proofs of those theorems, we use the following lemmas.
Proof.Assume, for the sake of contradiction, that there exists a nonoscillatory solution () of (E).Since −() is also a solution of (E), we can confine our discussion only to the case where the solution () is eventually positive.Then there exists a real number  1 >  0 such that (), (()) > 0 for all  ≥  1 .Thus, from (E) we have which means that () is an eventually nonincreasing function of positive numbers.Taking into account the fact that () ≤ ℎ(), (E) implies that Observe that (36) implies that, for each  > 0, there exists a real number   such that Combining inequalities (40) and (41), we obtain or where Applying the Grönwall inequality in (43), we conclude that or Substituting () for  in (49), we get Integrating (E) from () to , we have or where Multiplying inequality (60) by (), as before, we find Therefore, for sufficiently large , we have where It becomes apparent, now, that, by repeating the above steps, we can build inequalities on   () with progressively higher indices   (, ),  ∈ N. In general, for sufficiently large , the positive solution () satisfies the inequality where Proceeding to final step, we recall that ℎ(), defined by ( 13), is a nondecreasing function.Since () ≤ ℎ() ≤ ℎ(), we have (, ) ) ) . (66) Hence ( Since  may be taken arbitrarily small, this inequality contradicts (37).This completes the proof of the theorem.
Proof.Assume  is an eventually positive solution of (E).
Clearly, (67) is satisfied for sufficiently large .Thus, ( Since  may be taken arbitrarily small, this inequality contradicts (71).This completes the proof of the theorem.
Proof.Assume  is an eventually positive solution of (E).
Then, as in the proof of Theorem 6, for sufficiently large , we conclude that (, ) ) ) . (76) Integrating (E) from ℎ() to  and using (76), we obtain where   is defined by (38) and  0 is the smaller root of the equation  =   , then all solutions of (E) oscillate.
Proof.Let  be an eventually positive solution of (E).As in the proof of Theorem In particular, for  ∈ (0,  0 − 1), by continuity, we conclude that there exists a real number  * ∈ (ℎ(), ] satisfying Integrating (E) from  * to  and using (85), we obtain (90) Using (88) and Lemma 4, we deduce that, for the  considered, there exists a real number    ≥   such that for  ≥    .Dividing (E) by (), integrating from ℎ() to  * , and using (85), we deduce that which contradicts (83), when  → 0. This completes the proof of the theorem.
Theorem 10.Let ℎ() be defined by (13).If for some  ∈ N where   is defined by ( 38), then all solutions of (E) oscillate.
Proof.For the sake of contradiction, let  be a nonincreasing eventually positive solution and  1 >  0 be such that () > 0 and (()) > 0 for all  ≥  1 .We note that we may obtain (85) as in the proof of Theorem which, in view of the first inequality in (108), implies that Similarly, integrating (E) from   to , using (85) and the fact that () > 0, we have which, in view of the second inequality in (108), yields Combining inequalities ( 110) and ( 112), we deduce that which contradicts (107).The proof of the theorem is complete.

ADEs.
Analogous oscillation conditions to those obtained for the delay equation (E) 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.Let () be defined by (27) and for some  ∈ N lim sup where with  0 () =  0 (), and let  0 be the smaller root of the equation  =   .Then all solutions of (E  ) oscillate.
Theorem 13.Let () be defined by (27) and where   is defined by ( 115), then all solutions of (E  ) oscillate.
Theorem 14.Let () be defined by (27) where   is defined by ( 115) and  0 is the smaller root of the equation  =   , then all solutions of (E  ) oscillate.
Theorem 15.Let () be defined by (27).If for some  ∈ N where   is defined by ( 115), then all solutions of (E  ) oscillate.

Differential Inequalities.
A slight modification in the proofs of Theorems 6−15 leads to the following results about differential inequalities.
Remark 17.The oscillation criteria established in this paper all depend on  0 (see, e.g., (37) and ( 71)) in contrast to the conditions obtained in [15,16] and in [17, for m = 1].In fact, the left-hand side of conditions (37) and (71) depends on  0 , which is not the case with the left-hand side of conditions (20) and (21).Since  0 > 1 when  ∈ (0, 1/], it is obvious that 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.

Examples and Comments
The oscillation tests we have proposed and established, in the main results, involve an iterative procedure.We iteratively compute limsup and liminf on the terms   () and   (),  ∈ 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.
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.
Example 2 (taken and adapted from [17]).Consider the advanced differential equation with (see Figure 2(a)) where  ∈ N 0 and N 0 is the set of nonnegative integers.By (27), we see (Figure 2 It is obvious that and therefore, the smaller root of  0.1332 =  is  0 = 1.16839.
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.

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 of (E) 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 improvement of (37) [( 114)] over the other iterative conditions, namely, (14) (for j > 2), (20) (for  > 1) [( 30) (for  > 1)], and (21) (for  > 1) [(31) (for  > 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 computing limsup 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 arguments () or (), in contrast to most of the other oscillation criteria that apply to equations with nondecreasing arguments.

Complexity 15 Remark 3 .
Similarly, one can provide examples, illustrating the other main results.