Table of Contents Author Guidelines Submit a Manuscript
Complexity
Volume 2018 (2018), Article ID 8237634, 18 pages
https://doi.org/10.1155/2018/8237634
Research Article

Oscillation Criteria for Delay and Advanced Differential Equations with Nonmonotone Arguments

1Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education (ASPETE), N. Heraklio, 14121 Athens, Greece
2School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
3School of Information Science and Engineering, Linyi University, Linyi, Shandong 276005, China

Correspondence should be addressed to Tongxing Li; moc.361@7002xgnotil

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. This 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.

Abstract

We study the oscillatory behavior of differential equations with nonmonotone deviating arguments and nonnegative coefficients. New oscillation criteria, involving and , 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 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, [110] 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 , 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 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 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 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, 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 [1333].

In the paper, we consider a differential equation with delay argument of the formwhere is a function of nonnegative real numbers and is a function of positive real numbers such that

By a solution of we understand a continuously differentiable function defined on for some and such that is satisfied for . 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 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 or an ADE of the form of 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

2. Historical and Chronological Review

2.1. DDEs

The first systematic study for the oscillation of all solutions of equation was made by Myškis in 1950 [31], when he proved that every solution of oscillates, ifIn 1972, Ladas et al. [27] proved that ifthen all solutions of are oscillatory.

In 1982, Koplatadze and Chanturiya [24] improved (7) toRegarding the constant in (9), it should be remarked that if the inequality holds eventually, then, according to [24], has a nonoscillatory solution.

It is apparent that there is a gap between conditions (8) and (9), when 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 is the smaller root of the equation andthen all solutions of oscillate.

Now we come to the general case where the argument is nonmonotone. SetClearly, the function is nondecreasing and , for all .

In 1994, Koplatadze and Kvinikadze [25] proved that ifwherethen all solutions of oscillate.

In 2011, Braverman and Karpuz [14] proved that ifthen all solutions of oscillate, while in 2014, Stavroulakis [32] improved (16) toIn 2016, El-Morshedy and Attia [30] proved that ifwhere andthen all solutions of are oscillatory. Here, is a nondecreasing continuous function such that , for some . Clearly, is more general than defined by (13).

Recently, Chatzarakis [15, 16] proved that if, for some ,orwherewith , then all solutions of are oscillatory.

Lately, Chatzarakis [17] studied a more general form of ; namely,and established sufficient oscillation conditions. Those conditions can lead to (20) and (21) when .

2.2. ADEs

By Theorem 2.4.3 [29], ifthen all solutions of are oscillatory.

In 1984, Fukagai and Kusano [21] proved that ifthen all solutions of are oscillatory, while ifthen has a nonoscillatory solution.

Assume that the argument is not necessarily monotone. SetClearly, the function is nondecreasing and , for all .

In 2015, Chatzarakis and Öcalan [18] proved that iforthen all solutions of are oscillatory.

Recently, Chatzarakis [15, 16] proved that if, for some ,orwherewith , then all solutions of oscillate.

Lately, Chatzarakis [17] studied a more general form of , namely, and established sufficient oscillation conditions. Those conditions can lead to (30) and (31) when .

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 that is defined by (13). Then

Lemma 4 (see [19, Lemma 2.1.3]). Assume that is defined by (13), , and is an eventually positive solution of . Then

Lemma 5 (see [26]). Assume that is defined by (13), , and is an eventually positive solution of . Thenwhere is the smaller root of the equation .

Theorem 6. Let be defined by (13) and for some wherewith , and let be the smaller root of the equation . Then all solutions of oscillate.

Proof. Assume, for the sake of contradiction, that there exists a nonoscillatory solution of . Since is also a solution of , we can confine our discussion only to the case where the solution is eventually positive. Then there exists a real number such that for all . Thus, from we have which means that is an eventually nonincreasing function of positive numbers. Taking into account the fact that , implies thatObserve that (36) implies that, for each , there exists a real number such thatCombining inequalities (40) and (41), we obtain orwhereApplying the Grönwall inequality in (43), we conclude thatNow we divide by and integrate on , soorSince , equality (47) gives orSubstituting for in (49), we getIntegrating from to , we haveCombining (50) and (51), we obtain Multiplying inequality (52) by , we findwhich, in view of , becomes Hence, for sufficiently large , orwhere Clearly (56) resembles (43), if we replace by . Thus, integrating (56) on yieldsRepeating steps (45) through (50), we can see that satisfies the inequalityCombining now (51) and (59), we obtain 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 . In general, for sufficiently large , the positive solution satisfies the inequalitywhere Proceeding to final step, we recall that , defined by (13), is a nondecreasing function. Since , we have Henceor Thus Taking the limit as , we have Since may be taken arbitrarily small, this inequality contradicts (37).
This completes the proof of the theorem.

Theorem 7. Let be defined by (13) and . If for some where is defined by (38), then all solutions of oscillate.

Proof. Assume is an eventually positive solution of . Clearly, (67) is satisfied for sufficiently large . Thus, which implies thatUsing Lemmas 3 and 4, it is evident that inequality (35) is satisfied. Thus, (73) leads to Since may be taken arbitrarily small, this inequality contradicts (71).
This completes the proof of the theorem.

Theorem 8. Let be defined by (13) and . If for some where is defined by (38), then all solutions of oscillate.

Proof. Assume is an eventually positive solution of . Then, as in the proof of Theorem 6, for sufficiently large , we conclude thatIntegrating from to and using (76), we obtain or Hence which yields, for all sufficiently large , and consequentlyTaking into account the fact that (35) is satisfied, inequality (81) leads to which contradicts (75), when .
This completes the proof of the theorem.

Theorem 9. Let be defined by (13) and . If for some where is defined by (38) and is the smaller root of the equation , then all solutions of oscillate.

Proof. Let be an eventually positive solution of . As in the proof of Theorem 8, we can show that (76) holds; namely, Since , inequality (84) givesBy Lemma 5, for each , there exists a real number such thatNote that, by the nondecreasing nature of the function in , it holds In particular, for , by continuity, we conclude that there exists a real number satisfyingIntegrating from to and using (85), we obtain orUsing (88) and Lemma 4, we deduce that, for the considered, there exists a real number such thatfor .
Dividing by , integrating from to , and using (85), we deduce thatClearly, by means of (36), , for . Hence, for all sufficiently large , we conclude that orthat is,Using (91) along with (95), we getwhich contradicts (83), when .
This completes the proof of the theorem.

Theorem 10. Let be defined by (13). If for some where is defined by (38), then all solutions of oscillate.

Proof. For the sake of contradiction, let be a nonincreasing eventually positive solution and be such that and for all . We note that we may obtain (85) as in the proof of Theorem 9.
Dividing by and integrating from to , we have from which, in view of and (85), we get Since is nonincreasing and , inequality (99) becomesFrom (97), it is clear that there exists a constant such that Choose such that . For every , such that , we haveCombining inequalities (100) and (102), we obtain or which yields Following the above steps, we can inductively show that, for any positive integer , Since , there is a natural number , satisfying such that for sufficiently largeFurther (cf. [13, 24]), for sufficiently large , there exists a real number , such thatIntegrating from to , using (85) and the fact that , we obtain which, in view of the first inequality in (108), implies thatSimilarly, integrating from to , using (85) and the fact that , we have which, in view of the second inequality in (108), yieldsCombining inequalities (110) and (112), we deduce that which contradicts (107).
The proof of the theorem is complete.

3.2. ADEs

Analogous oscillation conditions to those obtained for the delay equation can be derived for the (dual) advanced differential equation by following similar arguments with the ones employed for obtaining Theorems 610.

Theorem 11. Let be defined by (27) and for some wherewith , and let be the smaller root of the equation . Then all solutions of oscillate.

Theorem 12. Let be defined by (27) and . If for some where is defined by (115), then all solutions of oscillate.

Theorem 13. Let be defined by (27) and . If for some where is defined by (115), then all solutions of oscillate.

Theorem 14. Let be defined by (27) and . If for some where is defined by (115) and is the smaller root of the equation , then all solutions of oscillate.

Theorem 15. Let be defined by (27). If for some where is defined by (115), then all solutions of oscillate.

3.3. Differential Inequalities

A slight modification in the proofs of Theorems 615 leads to the following results about differential inequalities.

Theorem 16. Assume that all the conditions of Theorem 6 [11], 7 [12], 8 [13], 9 [14], or 10 [15] hold. Then
the delay [advanced] differential inequality has no eventually positive solutions;
the delay [advanced] differential inequality