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In this paper, a new effective technique for the investigation of the higher-order trinomial differential equations y (n) (t) + p (t) y (τ (t)) + q (t) y (σ (t)) = 0 is established. We offer new criteria for so-called property (A) and oscillation of the considered equation. Examples are provided to illustrate the importance of our results. [ABSTRACT FROM AUTHOR]

In this work, we establish the necessary and sufficient conditions for oscillation of a class of functional differential equations of the form ((x(t)+p(t)x(to))' + q(t)(x(tr)) + v(t)(x( - )) = 0 of a neutral coefficient p(t), by using the Knaster-Tarski fixed point theorem and Banach's fixed point theorem. [ABSTRACT FROM AUTHOR]

This article is dedicated to the oscillatory properties of a second order Emden-Fowler delay differential equation with a sublinear neutral term. The presented criteria extend and improve several well-known results reported recently in the literature. Four illustrating examples are included to verify the validity of the results derived. [ABSTRACT FROM AUTHOR]

The aim of this research is to study the oscillatory properties of higher -order delay half linear differential equations with non-canonical operators. Two techniques for establishing new oscillation conditions for all solutions of higher-order differential equations will be presented. The first method to employ the Riccati transformations, which differ from those described in some published works. The second method employs comparison principles with first-order delay differential equations, from which it is straightforward to deduce oscillation for all studied equation solutions. In addition to improving, extending, and significantly simplifying the previously established criteria, the newly proposed criteria have the potential to serve as a benchmark for the theory of delay differential equations of higher order, which is still in its infancy of development. We were able to determine three fundamental theorems regarding the oscillation of this equation. Some examples will be provided to illustrate the findings. [ABSTRACT FROM AUTHOR]

We investigate the oscillation of the fourth-order differential equation for a class of functional differential equations of the neutral type. We obtain a new single-oscillation criterion for the oscillation of all the solutions of our equation. We establish new monotonic properties for some cases of positive solutions of the studied equation. Moreover, we improve these properties by using an iterative method. This development of monotonic properties contributes to obtaining new and more efficient criteria for verifying the oscillation of the equation. The results obtained extend and improve previous findings in the literature by using an Euler-type equation as an example. The importance of the results was clarified by applying them to some special cases of the studied equation. The fourth-order delay differential equations have great practical importance due to their wide applications in civil, mechanical, and aeronautical engineering. Research on this type of equation is still ongoing due to its remarkable importance in many fields. [ABSTRACT FROM AUTHOR]

The distance between consecutive zeros of a first-order differential equation with several variable delays is studied. Here, we show that the distribution of zeros of differential equations with variable delays is not an easy extension of the case of constant delays. We obtain new upper bounds for the distance between zeros of all solutions of a differential equation with several delays, which extend and improve some existing results. Two illustrative examples are given to show the advantages of the proposed results over the known ones. [ABSTRACT FROM AUTHOR]

In this paper, new criteria for a class oscillation of second-order delay differential equations with distributed deviating arguments were established. Our method mainly depends on making sharper estimates for the non-oscillatory solutions of the studied equation. By using the Ricati technique and comparison theorems that compare the studied equations with first-order delay differential equations, we obtained new and less restrictive conditions that ensure the oscillation of all solutions of the studied equation. Further, we give an illustrative example. [ABSTRACT FROM AUTHOR]

In this paper, we aim to study the monotonic properties of the solutions of a class of neutral delay differential equations. The importance of this study lies in the fact that the monotonic properties largely control the study of the oscillation and asymptotic behaviour of the solutions to delay differential equations. Then, by using the new properties, we create improved criteria for testing the oscillation of solutions to the studied equation. We also find new criteria that can be applied more than once. Moreover, we discuss the importance and novelty of the results through the application to a special case of the studied equation. [ABSTRACT FROM AUTHOR]

This study purposes to present some new comparison theorems that guarantee the oscillation of all solutions of even order functional differential equations with a neutral term including both advanced and delayed arguments. The obtained results are based on comparisons with associated first‐order delay differential inequalities and first‐order delay differential equations, and they are applicable to the both cases where the neutral coefficients are unbounded and/or bounded. In the last section, several examples showing the applicability of the results are provided. [ABSTRACT FROM AUTHOR]

In this work first we transform the semi-noncanonical fourth order neutral delay differential equations into canonical type. This simplifies the investigations of finding the relationships between the solution and its companion function which plays an important role in the oscillation theory of neutral differential equations. Moreover, we improve these relationships based on the monotonic properties of positive solutions. We present new conditions for the oscillation of all solutions of the corresponding equation which improve the oscillation results already reported in the literature. Examples are provided to illustrate the importance of our main results. [ABSTRACT FROM AUTHOR]

The oscillation of a first-order differential equation with several non-monotone delays is proposed. We extend the works of Kwong (1991) and Sficas and Stavroulakis (2003) for equations with several delays. Our results not only essentially improve but also generalize a large number of the existing ones. Using some numerical examples, we illustrate the applicability and effectiveness of our results over many known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in studying the oscillation of differential equations with a damping term and distributed delay. We establish new criteria that guarantee the oscillation of the third-order differential equation in terms of oscillation of the second-order linear differential equation without a damping term. By using the Riccati transformation technique and the principle of comparison, we obtain new results on the oscillation for the studied equation. The results show significant improvement and extend the previous works. Symmetry contributes to determining the correct methods for solving neutral differential equations. Some examples are provided to show the significance of our results. [ABSTRACT FROM AUTHOR]

In this paper, we present a single-condition sharp criterion for the oscillation of the fourth-order linear delay differential equation x (4) (t) + p (t) x (τ (t)) = 0 by employing a novel method of iteratively improved monotonicity properties of nonoscillatory solutions. The result obtained improves a large number of existing ones in the literature. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the asymptotic behavior of solutions to a class of noncanonical third-order Emden–Fowler delay differential equations with a superlinear neutral term. Using a Riccati-type transformation as well as integral criteria, we establish some new sufficient conditions guaranteeing that every solution of the equation considered either oscillates or converges to zero asymptotically. The results are illustrated with two examples. [ABSTRACT FROM AUTHOR]

In this study, new asymptotic properties of positive solutions of the even-order neutral delay differential equation with the noncanonical operator are established. The new properties are of an iterative nature, which allows it to be applied several times. Using these properties, we obtain new criteria to exclude a class from the positive solutions of the studied equation, using the comparison principles. [ABSTRACT FROM AUTHOR]

In this article, we are interested in studying the asymptotic behavior of fourth-order neutral differential equations. Despite the growing interest in studying the oscillatory behavior of delay differential equations of second-order, fourth-order equations have received less attention. We get more than one criterion to check the oscillation by the generalized Riccati method and the integral average technique. Our results are an extension and complement to some results published in the literature. Examples are given to prove the significance of new theorems. [ABSTRACT FROM AUTHOR]

An interesting point in studying the oscillatory behavior of solutions of delay differential equations is the abbreviation of the conditions that ensure the oscillation of all solutions, especially when studying the noncanonical case. Therefore, this study aims to reduce the oscillation conditions of the fourth-order delay differential equations with a noncanonical operator. Moreover, the approach used gives more accurate results when applied to some special cases, as we explained in the examples. [ABSTRACT FROM AUTHOR]

We study the half-linear advanced differential equation (r (t) | y ′ (t) | α - 1 y ′ (t)) ′ + q (t) | y | α - 1 (σ (t)) y (σ (t)) = 0 , t ≥ t 0 > 0 , where α > 0 , r (t) > 0 , q (t) > 0 , σ (t) ≥ t , and R (t) : = ∫ t 0 t r - 1 / α (s) d s → ∞ as t → ∞ . We prove that such an equation is oscillatory if λ ∗ : = lim inf t → ∞ R (σ (t)) R (t) < ∞ and lim inf t → ∞ r 1 / α (t) R α + 1 (t) q (t) > max { α m α (1 - m) λ ∗ - α m : 0 < m < 1 } or lim t → ∞ R (σ (t)) R (t) = ∞ and lim inf t → ∞ r 1 / α (t) R α + 1 (t) q (t) > 0. The obtained criteria can be regarded as a natural extension of the well-known Kneser oscillation criterion for half-linear ordinary differential equations. Our oscillation constant is optimal for the correponding half-linear Euler-type delay differential equation. [ABSTRACT FROM AUTHOR]

In this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the oscillation of a class of fourth-order Emden–Fowler delay differential equations with neutral term. Using the Riccati transformation and comparison method, we establish several new oscillation conditions. These new conditions complement a number of results in the literature. We give examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

Differential equations of second order appear in numerous applications such as fluid dynamics, electromagnetism, quantum mechanics, neural networks and the field of time symmetric electrodynamics. The aim of this work is to establish necessary and sufficient conditions for the oscillation of the solutions to a second-order neutral differential equation. First, we have taken a single delay and later the results are generalized for multiple delays. Some examples are given and open problems are presented. [ABSTRACT FROM AUTHOR]

In this paper, necessary and sufficient conditions for asymptotic behavior are established of the solutions to second-order neutral delay differential equations of the form d/dt(r(t)(d/dt[x(t) -- p(t)x(τ(t))])γ) + ... qi(t)fi(x(σi(t))) = 0 for t ≥ to. We consider two cases when fi(u)/uβ is nonincreasing for γ > β, and nondecreasing for β > γ, where β and γ are quotients of two positive odd integers. Our main tool is Lebesgue's dominated convergence theorem. Examples illustrating the applicability of the results are also given, and state an open problem. [ABSTRACT FROM AUTHOR]

In the paper, we study oscillation of the half-linear second order delay differential equations of the form (r(t)(y′(t))α)′+p(t)yα(τ(t))=0. We introduce new monotonic properties of its nonoscillatory solutions and use them for linearization of considered equation which leads to new oscillatory criteria. The presented results essentially improve existing ones. [ABSTRACT FROM AUTHOR]

Consider the first-order linear di¤erential equation with several retarded arguments x' (t)+ Σmi=1 pi (t) x (Τi (t)) = 0, t ≼ t0; where the functions pi, Τi 2 C [t0,1); R+ ; for every i = 1; 2; ... m; Τi (t) t for t t0 and limt!1 Τi (t) = 1. New oscillation criteria which essentially improve known results in the literature are established. An example illustrating the results is given. [ABSTRACT FROM AUTHOR]

We consider a nonautonomous first-order linear differential equation with several delays and nonnegative coefficients. Some new sufficient conditions for the oscillation of all solutions are obtained in the form of an estimate for the lower limit of the sum of integrals of the coefficients. For the equation with one delay, the obtained oscillation conditions sharpen the classical Koplatadze-Chanturiya Theorem. The difference in strength between the new and available oscillation conditions is more significant for the equation with several delays. [ABSTRACT FROM AUTHOR]

We study scalar advanced and delayed differential equations with piecewise constant generalized arguments, in short DEPCAG of mixed type, that is, the arguments are general step functions. It is shown that the argument deviation generates, under certain conditions, oscillations of the solutions, which is an impossible phenomenon for the corresponding equation without the argument deviations. Criteria for existence of periodic solutions of such equations are discussed. New criteria extend and improve related results reported in the literature. The efficiency of our criteria is illustrated via several numerical examples and simulations. [ABSTRACT FROM AUTHOR]

In this work, necessary and sufficient conditions are obtained such that every solution of nonlinear neutral first-order differential equations with several delays of the form (x ⁢ (t) + r ⁢ (t) ⁢ x ⁢ (t - τ) ) ′ + ∑ i = 1 m ϕ i ⁢ (t) ⁢ H ⁢ (x ⁢ (t - σ i)) = f ⁢ (t) \bigl{(}x(t)+r(t)x(t-\tau)\bigr{)}^{\prime}+\sum_{i=1}^{m}\phi_{i}(t)H\bigl{(}% x(t-\sigma_{i})\bigr{)}=f(t) is oscillatory or tends to zero as t → ∞. {t\rightarrow\infty.} This problem is considered in various ranges of the neutral coefficient r. Finally, some illustrating examples are presented to show that feasibility and effectiveness of main results. [ABSTRACT FROM AUTHOR]

In this paper, we study the second-order linear delay differential equation of the form (E) y ′ ′ (t) + p (t) y (τ (t)) = 0. We establish new oscillation criteria for (1.1), which improve a number of related ones in the literature. Our approach essentially involves establishing stronger monotonicity properties for the positive solutions of (1.1) than those presented in known works. Two main approaches for the investigation of (1.1) will be used, namely a comparison principle with first-order delay differential inequalities and generalization of very effective Koplatadze's technique. We illustrate the improvement over the known results by applying and comparing our method with the other known results for (1.1). [ABSTRACT FROM AUTHOR]

We obtain new nonimprovable conditions of oscillation of all solutions to a linear differential equation with several variable delays and positive coefficients. These conditions have the form of the upper and lower limits of the sum of integrals of the coefficients over the sets that are determined only by the delay corresponding to this coefficient. These results differ from the well-known results in which a coarsening of the integration set is assumed. [ABSTRACT FROM AUTHOR]

We consider a class of autonomous functional-differential equations for which the properties of oscillation of all solutions and positiveness of the fundamental solution are mutually complementary. [ABSTRACT FROM AUTHOR]

We consider explicit conditions for all solutions to linear scalar differential equations with several variable delays to be oscillatory. The considered conditions have the form of inequalities bounding the upper limit of the sum of integrals of coefficients over a subset of the real semiaxis, by the constant 1 from below. The main result is a new oscillation condition, which sharpens several known conditions of the kind. Some results are presented in the form of counterexamples. [ABSTRACT FROM AUTHOR]

We investigate the oscillation of a first order delay differential equation with non-negative coefficient and non-monotone arguments. New oscillation criterion of lim sup type is obtained. An example is given to show the applicability and strength of the obtained condition over known ones. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to study asymptotic properties of the third-order delay differential equation (r2(t)(r1(t)(y'(t))γ)')' + p(t) (y1(t))γ + q(t)f (y(τ(t))) = 0. (E) Employing comparison principles with a suitable first order delay differential equation we shall establish criteria for all nonoscillatory solutions of (E) to converge to zero, while oscillation of a couple of first order delay differential equations yields oscillation of (E). An example is provided to illustrate the main results. [ABSTRACT FROM AUTHOR]

Consider the first-order linear differential equation with several retarded arguments x′(t) + Σi=1mpi(t)x(τi(t)) = 0, t≥t0, where the functions pi, τi ∈ C([t0,∞),ℝ+) for every i = 1,2, ... ,m, τi(t) ≤ t for t ≥ t0 and limt 8 τi(t) =∞. A survey on the oscillation of all solutions to this equation is presented in the case of several non-monotone arguments and especially when well-known oscillation conditions are not satisfied. Examples illustrating the results are given. [ABSTRACT FROM AUTHOR]

Abstract: The -Laplace equations have some applications in continuum mechanics. On the basis of this background detail, we study oscillation and asymptotic behavior of solutions to two classes of higher-order delay damped differential equations with -Laplacian like operators. Some new criteria are presented that improve the related contributions to the subject. Several examples are provided to illustrate the relevance of new theorems. [Copyright &y& Elsevier]

The objective of this study is to establish new sufficient criteria for oscillation of solutions of even-order delay Emden-Fowler differential equations with neutral term r ı y ı + m ı y g ı n − 1 γ ′ + ∑ i = 1 j q i ı y γ μ i ı = 0 . We use Riccati transformation and the comparison with first-order differential inequalities to obtain theses criteria. Moreover, the presented oscillation conditions essentially simplify and extend known criteria in the literature. To show the importance of our results, we provide some examples. Symmetry plays an essential role in determining the correct methods for solutions to differential equations. [ABSTRACT FROM AUTHOR]

In this work, by using the comparison method and Riccati transformation, we obtain some oscillation criteria of solutions of delay differential equations of fourth-order in canonical form. These criteria complement those results in the literature. We give two examples to illustrate the main results. Symmetry plays an essential role in determining the correct methods for solutions to differential equations. [ABSTRACT FROM AUTHOR]

In this work, new criteria for the oscillatory behavior of even-order delay differential equations with neutral term are established by comparison technique, Riccati transformation and integral averaging method. The presented results essentially extend and simplify known conditions in the literature. To prove the validity of our results, we give some examples. [ABSTRACT FROM AUTHOR]

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]

We establish some oscillation criteria for a forced second-order differential equation with impulses. These results extend some well-known results for forced second-order impulsive differential equations with delay. [ABSTRACT FROM AUTHOR]

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]

The purpose of this paper is to give oscillation criteria for the third order delay nonlinear differential equation [a2(t){(a1(t)(x' (t))α1)'}α2]' + q(t)f(x(g(t))) = 0, via comparison with some first differential equations whose oscillatory characters are known. Our results generalize and improve some known results for oscillation of third order nonlinear differential equations. Some examples are given to illustrate the main results. [ABSTRACT FROM AUTHOR]

Abstract: Han et al. [Z. Han, T. Li, S. Sun, C. Zhang, Oscillation behavior of third order neutral Emden–Fowler delay dynamic equations on time-scales, Adv. Differential Equations 2010 (2010). Article ID 586312, 23 pages] gave some criteria for the oscillation and asymptotic behavior of solutions of the third order neutral delay dynamic equation on a time scale , where is the quotient of odd positive integers. In this paper, the authors establish some new criteria for the oscillation of this equation that improve and unify those of Han et al. [Copyright &y& Elsevier]

This paper is concerned with oscillations of numerical solutions for the nonlinear delay differential equation of population dynamics. The equation proposed by Mackey and Glass for a ”dynamic disease” involves respiratory disorders and its solution resembles the envelope of lung ventilation for pathological breathing, called Cheyne-Stokes respiration. Some conditions under which the numerical solution is oscillatory are obtained. The properties of non-oscillatory numerical solutions are investigated. To verify our results, we give numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, new oscillatory properties for fourth-order delay differential equations with p-Laplacian-like operators are established, using the Riccati transformation and comparison method. Moreover, our results are an extension and complement to previous results in the literature. We provide some examples to examine the applicability of our results. [ABSTRACT FROM AUTHOR]

In this study, a new oscillation criterion for the fourth-order neutral delay differential equation r u x u + p u x δ u ‴ α ′ + q u x β ϕ u = 0 , u ≥ u 0 is established. By introducing a Riccati substitution, we obtain a new criterion for oscillation without requiring the existence of the unknown function. Furthermore, the new criterion improves and complements the previous results in the literature. The results obtained are illustrated by an example. [ABSTRACT FROM AUTHOR]

The main purpose of this study is aimed at developing new criteria of the iterative nature to test the asymptotic and oscillation of nonlinear neutral delay differential equations of third order with noncanonical operator (a (ι) [ (b (ι) x (ι) + p (ι) x (ι − τ) ′) ′ ] β ) ′ + ∫ c d q (ι , μ) x β (σ (ι , μ)) d μ = 0 , where ι ≥ ι 0 and w (ι) : = x (ι) + p (ι) x (ι − τ) . New oscillation results are established by using the generalized Riccati technique under the assumption of ∫ ι 0 ι a − 1 / β (s) d s < ∫ ι 0 ι 1 b (s) d s = ∞ as ι → ∞ . Our new results complement the related contributions to the subject. An example is given to prove the significance of new theorem. [ABSTRACT FROM AUTHOR]

In this paper, new oscillation conditions for the 2nd-order noncanonical neutral differential equation (a 0 t ((u t + a 1 t u g 0 t ) ′ ) β) ′ + a 2 t u β g 1 t = 0 , where t ≥ t 0 , are established. Using Riccati substitution and comparison with an equation of the first-order, we obtain criteria that ensure the oscillation of the studied equation. Furthermore, we complement and improve the previous results in the literature. [ABSTRACT FROM AUTHOR]

In this paper we study the oscillatory and global asymptotic stability of a single neuron model with two delays and a general activation function. New sufficient conditions for the oscillation and nonoscillation of the model are given. We obtain both delay-dependent and delay-independent global asymptotic stability criteria. Some of our results are new even for models with one delay. [ABSTRACT FROM AUTHOR]