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This paper presents new criteria for the oscillation of all solutions of the third-order nonlinear delay differential equations with noncanonical operators. Our approach to establishing new criteria essentially simplifies and refines the main results obtained in Džurina and Jadlovská (2018) and Grace et al. (2019). Examples illustrating the importance of our results are presented. • This paper presents new criteria for the oscillation of all solutions of the third-order nonlinear delay differential equations with noncanonical operators. We first transform the studied nonlinear noncanonical third-order delay differential equation into semi-canonical type. This approach simplifies the examination of the considered equation. Moreover, our approach to establishing new criteria substantially improves and extends some known results in the relevant literature. [ABSTRACT FROM AUTHOR]

Abstract We consider existence of positive solutions for a difference equation with continuous time, variable coefficients and delays x (t + 1) − x (t) + ∑ k = 1 m a k (t) x (h k (t)) = 0 , a k (t) ≥ 0 , h k (t) ≤ t , t ≥ 0 , k = 1 , ... , m. We prove that for a fixed h (t) ≢ t , a positive solution may exist for a k exceeding any prescribed M > 0, as well as for constant positive a k with h k (t) ≤ t − n , where n ∈ N is arbitrary and fixed. The point is that for equations with continuous time, non-existence of positive solutions with inf x (t) > 0 on any bounded interval should be considered rather than oscillation. Sufficient conditions when such solutions exist or do not exist are obtained. We also present an analogue of the Grönwall–Bellman inequality for equations with continuous time, and examine the question when the equation has no positive non-increasing solutions. Counterexamples illustrate the role of variable delays. [ABSTRACT FROM AUTHOR]

Abstract This paper mainly deals with the oscillation of nonlinear delay differential equation which is used to describe advertising capital model, analytically and numerically. Firstly, the condition of oscillation of the analytic solution is presented by the technique of the theory of characteristic. Secondly, the asymptotic behavior of non-oscillatory analytic solution is verified. Thirdly, the θ -methods are applied to the mentioned equation, some conditions under which the numerical solution oscillates are obtained. Moreover, it is proved that every non-oscillatory numerical solution tends to the steady state of the model. Finally, some numerical simulations for verifying the theoretical findings are provided. [ABSTRACT FROM AUTHOR]

Abstract The objective of this paper is to offer sufficient conditions for the oscillation and asymptotic behavior of all solutions of second-order half-linear differential equations with advanced argument of the form (r (y ′) α) ′ (t) + q (t) y α (σ (t)) = 0 , when ∫ ∞ r − 1 / α (s) d s < ∞. Our criteria substantially improve, simplify and complement a number of existing ones. The results obtained are illustrated by an example on the Euler type equations. [ABSTRACT FROM AUTHOR]

New oscillation criteria for third-order delay differential equations with noncanonical operators are presented. Contrary to existing results, oscillation of the studied equation is attained via only two conditions. Our criteria not only improve, extend and significantly simplify existing ones, but the newly proposed approach could hopefully serve as a reference in the less-developed theory of noncanonical equations of higher-order. The importance of the results obtained is illustrated via Euler-type equations. [ABSTRACT FROM AUTHOR]

Some results are presented on how oscillation is inherited by chemical reaction networks (CRNs) when they are built in natural ways from smaller oscillatory networks. The main results describe four important ways in which a CRN can be enlarged while preserving its capacity for oscillation. The results are for general CRNs, not necessarily fully open, but lead to an important corollary for fully open networks: if a fully open CRN R with mass action kinetics admits a nondegenerate (resp., linearly stable) periodic orbit, then so do all such CRNs which include R as an induced subnetwork. This claim holds for other classes of kinetics, but fails, in general, for CRNs which are not fully open. Where analogous results for multistationarity can be proved using the implicit function theorem alone, the results here call on regular and singular perturbation theory. Equipped with these results and with the help of some analysis and numerical simulation, lower bounds are put on the proportion of small fully open CRNs capable of stable oscillation under various assumptions on the kinetics. This exploration suggests that small oscillatory motifs are an important source of oscillation in CRNs. [ABSTRACT FROM AUTHOR]

We study asymptotic behavior of nonoscillatory solutions for the third-order differential equations of the form [ r ( t ) ( y ′ ( t ) ) γ ] ′ ′ + p ( t ) y ( t ) = 0 . For property A of studied equations we extend the information about the asymptotic properties of positive decreasing solutions. [ABSTRACT FROM AUTHOR]

The paper deals with the oscillation and non-oscillation of the Runge–Kutta methods for a differential equation with piecewise continuous arguments of mixed type. The conditions of the oscillation and non-oscillation for the Runge–Kutta method are obtained. It is proved that oscillation of the analytic solution is not preserved by the Runge–Kutta method under any conditions. The conditions under which the non-oscillation of analytic solutions is preserved by the Runge–Kutta method are obtained. Some numerical experiments are given. [ABSTRACT FROM AUTHOR]

In the paper, we establish new technique for investigation of properties of trinomial differential equations with positive and negative term ( b ( t ) ( a ( t ) x ′ ( t ) ) ′ ) ′ + p ( t ) f ( x ( τ ( t ) ) ) − q ( t ) h ( x ( σ ( t ) ) ) = 0 . We offer new criteria for asymptotic properties of nonoscillatory solutions for the studied equations. We support our results with illustrative examples. [ABSTRACT FROM AUTHOR]

This paper is concerned with the oscillation of certain third-order nonlinear delay differential equations with damping. We give new characterizations of oscillation of the third-order equation in terms of oscillation of a related, well-studied, second-order linear differential equation without damping. We also establish new oscillation results for the third-order equation by using the integral averaging technique due to Philos. Numerous examples are given throughout. [ABSTRACT FROM AUTHOR]

This paper is concerned with n th-order nonlinear dynamic equations on time scales of the form ( r ( t ) Φ γ ( x Δ n − 1 ( t ) ) ) Δ + ∑ i = 0 k q i ( t ) Φ α i ( x ( δ i ( t ) ) ) = 0 with n ≥ 2. By discussing the signs of i th-order derivatives of eventually positive solutions for i = 1 , … , n − 1 , and using the generalized Riccati technique and integral averaging technique, we derive new criteria for oscillation and asymptotic behavior of the equation. Our results extend many existing results in the literature. [ABSTRACT FROM AUTHOR]

We study oscillatory behavior of a class of nonlinear second-order neutral differential equations. A new criterion is established that amends related results reported in the literature. [ABSTRACT FROM AUTHOR]

By employing the generalized Riccati transformation, we establish some new oscillation criteria for second order delay dynamic equations on time scales. The obtained results essentially improve the well-know oscillation results for half-linear dynamic equations such as Kamenev-type and Philos-type oscillation criteria. Some interesting examples are given to illustrate the versatility of our results. [ABSTRACT FROM AUTHOR]

We show that adding new chemical species into the reactions of a chemical reaction network (CRN) in such a way that the rank of the network remains unchanged preserves its capacity for multiple nondegenerate equilibria and/or periodic orbits. One consequence is that any bounded nondegenerate behaviours which can occur in a CRN can occur in a CRN with bounded stoichiometric classes. The main result adds to a family of theorems which tell us which enlargements of a CRN preserve its capacity for nontrivial dynamical behaviours. It generalises some earlier claims, and complements similar claims involving the addition of reactions into CRNs. The result gives us information on how ignoring some chemical species, as is common in biochemical modelling, might affect the allowed dynamics in differential equation models of CRNs. We demonstrate the scope and limitations of the main theorem via several examples. These illustrate how we can use the main theorem to predict multistationarity and oscillation in CRNs enlarged with additional species; but also how the enlargements can introduce new behaviours such as additional periodic orbits and new bifurcations. [ABSTRACT FROM AUTHOR]

The study of oscillation theory for fractional differential equations has been initiated by Grace et.al. [14]. In this paper we establish some new criteria for the oscillation of fractional differential equations with the Caputo derivative of the form c D a α x ( t ) = e ( t ) + f ( t , x ( t ) ) , a > 1 , α ∈ ( 1 , 2 ) . We also present the conditions under which all solutions of this equation are asymptotic to at + b as t → ∞ for some real numbers a, b. We shall employ a different technique rather than that in [14]. [ABSTRACT FROM AUTHOR]

This paper is concerned with oscillation and non-oscillation of numerical solutions for nonlinear delay differential equation of hematopoiesis with unimodal production rate. By investigating the roots of characteristic equation, some conditions under which the numerical solution is oscillatory and non-oscillatory are obtained. The properties of non-oscillatory numerical solutions are investigated. To verify our results, we give numerical experiments. [ABSTRACT FROM AUTHOR]

This paper is concerned with the distribution of zeros of all solutions of the first-order neutral differential equation x ( t ) + p ( t ) x ( t - τ ) ′ + Q ( t ) x ( t - σ ) = 0 , t ⩾ t 0 , where p ∈ C [ t 0 , ∞ ) , [ 0 , ∞ ) , Q ∈ C [ t 0 , ∞ ) , ( 0 , ∞ ) and τ , σ ∈ R + . New estimations for the distance between adjacent zeros of this neutral equation are obtained via comparison with a corresponding differential inequality. These results extend some known results from the non-neutral to the neutral case and improve other published results as well. [ABSTRACT FROM AUTHOR]

This paper mainly deals with the oscillation of numerical solution for the neutral logistic delay differential equation. Applying the θ -methods to the simplified type of the original equation which is got through an invariant oscillation transformation. It is proven that every non-oscillatory numerical solution tends to the steady state of the model. Moreover, some conditions under which the numerical solution oscillates are obtained. Finally, to demonstrate our theoretical results, some numerical examples are adopted. [ABSTRACT FROM AUTHOR]

Using a matrix linear transformation which preserves oscillatory property, some new oscillation criteria for the following linear matrix Hamiltonian system U ′ = ( A ( t ) - λ ( t ) I ) U + B ( t ) V , V ′ = C ( t ) U + ( μ ( t ) I - A ∗ ( t ) ) V , t ⩾ t 0 are obtained, which improve and generalize some recent results in literature. [ABSTRACT FROM AUTHOR]

A novel family of phase-fitted and P-stable symmetric extended linear multistep (SELM) methods for solving initial value problems of second-order oscillatory differential equations are developed. SELM methods are naturally P-stable. The order conditions are presented. Based on the harmonic fitting condition and order conditions, new explicit SELM methods of order two, order four, order six and order eight, respectively and implicit SELM methods of order four, order six, order eight and order ten, respectively, are derived and the local error for each method is obtained. A new eight-step SELM predictor–corrector pair of order ten is obtained. Numerical experiments are accompanied to show the high accuracy and competence of the new SELM methods compared with some highly efficient symmetric multistep methods in the literature. [ABSTRACT FROM AUTHOR]

A new Philos-type theorem for a class of third-order nonlinear dynamic equations is presented. Among others, the restrictive condition in terms of the commutativity of the jump and delay operators is removed here. [ABSTRACT FROM AUTHOR]

A class of second order nonlinear damped differential equations is studied and several new oscillation conditions are obtained. Our results extend, improve and unify a number of existing results and handle the cases which are not covered by known criteria. [ABSTRACT FROM AUTHOR]

Abstract: Sufficient conditions are established for the oscillation of nth order neutral differential equations of the form where is even integer, . The results obtained extend some of the known results. [Copyright &y& Elsevier]

Abstract: Consider the first-order linear delay differential equation where , for and , and the (discrete analogue) difference equation where denotes the forward difference operator is a sequence of nonnegative real numbers and is a sequence of integers such that for all and . The state-of-the-art on the oscillation of all solutions to these equations are established especially in the case of non-monotone arguments. Examples illustrating the results are given. [Copyright &y& Elsevier]

Abstract: In the paper, we present several functional inequalities and offer their application for higher order advanced differential equations of the form to be oscillatory. The conditions obtained essentially improve, complements and extends many other known results and involves unimprovable constants. [Copyright &y& Elsevier]

Abstract: The inhibitory effects of toxin-producing phytoplankton (TPP) on zooplankton modulate the dynamics of marine plankton. In this article, we employ simple mathematical models to compare theoretically the dynamics of phytoplankton–zooplankton interaction in situations where the TPP are present with those where TPP are absent. We consider two sets of three-component interaction models: one that does not include the effect of TPP and the other that does. The negative effects of TPP on zooplankton is described by a non-linear interaction term. Extensive theoretical analyses of the models have been performed to understand the qualitative behaviour of the model systems around every possible equilibria. The results of local-stability analysis and numerical simulations demonstrate that the two model-systems differ qualitatively with regard to oscillations and stability. The model system that does not include TPP is asymptotically stable around the coexisting equilibria, whereas, the system that includes TPP oscillates for a range of parametric values associated with toxin-inhibition rate and competition coefficients. Our analysis suggests that the qualitative dynamics of the plankton–zooplankton interactions are very likely to alter due to the presence of TPP species, and therefore the effects of TPP should be considered carefully while modelling plankton dynamics. [Copyright &y& Elsevier]

• New oscillation criteria have been established. • Dynamic equation over time scale has been considered which is more general. • Examples are given for illustration. • Results improve and extend the ones reported in the literature. A new approach is introduced for the oscillatory solutions of the second order non-linear delay dynamic equations on time scale with a super-linear neutral term. The result are obtained through various integral and differential relations. The Kamenev and Philos-type criteria for oscillation are also discussed. The results improve and extend the ones reported in the literature in the sense that the inequalities are easy to verify and results cover various kinds of equations by choosing different time scales. Examples for particular time scales are given for illustration. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present sufficient conditions for the oscillation and asymptotic behaviour of solutions of delay dynamic equations of the form where is a time scale unbounded above, satisfies either with or with and are unbounded nondecreasing functions such that for all under the condition . [Copyright &y& Elsevier]

Abstract: In this paper, the oscillatory behavior of the second order superlinear dynamic equation is studied under the assumption where in our main theorem is assumed to be a regular time scale, is the quotient of odd positive integers. When the coefficient function is allowed to be negative for arbitrarily large values of t, we establish a sufficient condition for oscillation of all solutions of Eq. (0.1). As special cases, we get that the superlinear differential equation is oscillatory, if and the superlinear difference equation is oscillatory, if [Copyright &y& Elsevier]

Abstract: This paper is concerned with the distribution of zeros of solutions of first order linear delay differential equations with variable coefficients of the form where , . By introducing a class of polynomial functions, we are able to derive new estimates for the lower and upper bounds of the distance between consecutive zeros of solutions of the above equations. We illustrate the obtained results with several examples. [Copyright &y& Elsevier]

Abstract: Sufficient conditions, in terms of coefficient functions, are obtained for non-oscillation of a class of linear third order non-homogeneous difference equations of the form andwhere and . The available methods, usually used for the study of difference equations, are developed appropriately for this purpose. [Copyright &y& Elsevier]

Abstract: For the blood cell production modelwe review known results. Permanence of solutions, oscillation and stability of a positive equilibrium or a positive periodic solution are analyzed for non-autonomous generalizations of this equation, with both concentrated and distributed delays. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to study the asymptotic properties and oscillation of the nth order neutral differential equationsObtained results are based on the new comparison theorems, that permit to reduce the problem of the oscillation of the nth order equation to the oscillation of a set of the first order equation. Obtained comparison principles essentially simplify the examination of studied equations and allow to relax some conditions imposed on the coefficients of (E). [Copyright &y& Elsevier]

Abstract: This paper is concerned with oscillation and asymptotic behavior of a class of third-order nonlinear delay dynamic equations on an arbitrary time scale. A new theorem is presented that improves a number of results reported in the literature. Examples are included to illustrate new results. [Copyright &y& Elsevier]

Abstract: A chemostat model with discrete time delay is proposed to describe oscillatory behavior in microbial continuous culture. Stability and Hopf bifurcation are discussed, including the changing regularity of bifurcation value. Explicit algorithm for determining the direction of Hopf bifurcation and the stability of periodic solution is derived, using the theory of normal form and center manifold. Finally, some numerical simulations are performed to illustrate the analytical results found. [Copyright &y& Elsevier]

Abstract: Some sufficient conditions are obtained for the oscillation of all solutions of a pantograph differential equation with impulsive perturbations of the formOur results reveal the fact that the oscillatory properties of all solutions of impulsive differential equations (∗) and (∗∗) may be caused by the impulsive perturbations (∗∗), though the corresponding differential equations without impulses admit a nonoscillatory solution. Some examples are also given to illustrate the applicability of the results obtained. [Copyright &y& Elsevier]

Abstract: In this work, we analyze the new generalized Emden–Fowler equation with neutral type delays:where . By use of averaging technique and specific analytical skills, some easily-accessible oscillation and asymptotic criteria are established, which have extended the results in the cited literature. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study oscillation and asymptotic behavior of solutions to a third order linear delay differential equationNew comparison theorems deduce oscillation of the given third order delay differential equation via application of known oscillation criteria to associated first order delay differential equations or inequalities. Contrary to most known results, we establish oscillation of all solutions to a given delay differential equation. Conditions in all theorems are not difficult to verify; computer algebra systems can be efficiently used to assist the computations. Main results in the paper are illustrated with a number of examples. [Copyright &y& Elsevier]

Abstract: For the delay differential equationwe demonstrate that the inequality is not sufficient for oscillation. Moreover, for any A >0 the relation , generally, does not imply oscillation. A similar result is obtained for difference equations. In terms of the maximal argument, new sufficient oscillation conditions are presented for differential and difference equations. [Copyright &y& Elsevier]

Abstract: This paper is concerned with upper bounds for the distances between adjacent zeros of solutions of first-order linear delay differential equations with variable coefficients of the form where τ >0 and p(t)∈ C([t ∗,∞), R). By introducing a sequence of testing functions, we are able to derive upper bounds that are better than those in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the oscillations in a class of forced second-order differential equations with nonlinear damping terms. By using classical variational principle and averaging technique, several new interval oscillation criteria for the equations are established, which improve and extend some known results. Example is also given to illustrate the results. [Copyright &y& Elsevier]

Abstract: In this paper, sufficient conditions are obtained for oscillation/non-oscillation of solutions of second order difference equations, involving generalized difference, of the form.andwhere Δ a is defined by Δ a y(n)= y(n +1)− ay(n), a ≠0. Necessary and sufficient conditions are obtained for oscillation of the first equation with a >0 and q(n)= q, a non-zero constant. The signs of q(n) and a play an important role for oscillation/non-oscillation of these two equations. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the oscillation of second order superlinear differential equations of the formA new oscillation criterion is established under quite general assumptions. Our results are based on the information on a sequence of subintervals of [t 0,∞) only, rather than on the whole half-line. Our methodology is somewhat different from that of previous authors . Our results generalize and extend some earlier results of Yu and Lu and Meng . An example is also given to illustrate the results. [Copyright &y& Elsevier]

Abstract: The purpose of this paper to establish oscillation criteria for second order nonlinear dynamic equation on an arbitrary time scale , where γ is a quotient of odd positive integers and r is a positive rd-continuous function on . The function satisfies g(t)⩾ t and lim t→∞ g(t)=∞ and . We establish some new sufficient conditions such that the above equation is oscillatory by using generalized Riccati transformation. Our results in the special cases when and involve and improve some oscillation results for second-order differential and difference equations; and when and our oscillation results are essentially new. Some examples illustrating the importance of our results are included. [Copyright &y& Elsevier]

Abstract: In this paper, we study the difference equation: where . We give some sufficient conditions which guarantee that all solutions of the equation are oscillatory, or that the positive equilibrium of the equation is globally asymptotically stable. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to the study of the oscillatory behavior of solutions of nonlinear hyperbolic equations with functional arguments by using integral averaging method and a generalized Riccati technique. First, we establish oscillation results for nonlinear hyperbolic equations by applying oscillation criteria for Riccati inequality. Secondly, we present interval oscillation results for nonlinear hyperbolic equations. These results improve and extend oscillation results of Cui and Xu . [ABSTRACT FROM AUTHOR]

Abstract: In an epidemiological model, time spent in one compartment is often modeled by a delay in the model. In general the presence of delay in differential equations can change the stability of an equilibrium to instability and causes the appearance of oscillatory solutions. In this paper we consider a SIS epidemiological model with demographic effects: birth, mortality and mortality caused by infection. The delay is the period of infection. We define the concept of oscillation in the sense that solutions of the model studied fluctuate around a steady state. Our goal is to show that in this model, there are oscillating solutions for certain parameters values. We determine a large set of initial data for which solutions of this model are slowly oscillating. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the oscillation of second-order nonlinear neutral dynamic equations of the formon a time scale . By using a generalized Riccati technique and integral averaging techniques, we establish new oscillation results which handle some cases not covered by known criteria. [Copyright &y& Elsevier]

Abstract: In this paper, some sufficient conditions are established for the oscillation of second-order neutral differential equations where and . On the other hand, some new oscillation criteria are established for the second-order nonlinear neutral delay differential equations where . The results obtained here complement and correct some known results in . Some examples are given to illustrate the main results. [Copyright &y& Elsevier]

Abstract: In this paper, necessary and sufficient conditions for the asymptotic behaviour of all bounded solutions of a class of nonlinear higher-order neutral dynamic equations are stated related to various ranges of the coefficient associated with the neutral part, in one of which the coefficient is allowed to oscillate and not tend to zero asymptotically. The new results improve and generalize the known results stated for difference equations and differential equations, and answer some problems left open for the case for arbitrary time scales. Some certain examples, which the results in the literature fail to deliver an answer to, are given to illustrate the applicability and significance of the new results. [Copyright &y& Elsevier]