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In this paper, we offer new technique for investigation of the third-order linear differential equation with mixed argument y ‴ (t) + p (t) y (τ (t)) = 0. (E) We establish new criteria for property A of (E) which fulfill the gap in the oscillation theory. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the oscillatory behavior of half-linear functional differential equations with deviating argument of the form (r(t)(y′(t))α)′=p(t)yα(τ(t)).(E) We introduce new technique based on monotonic properties of nonoscillatory solutions to offer new oscillatory criteria for (E). We will show that presented results essentially improve existing ones even for linear differential equations. [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]

New oscillation criteria for the second‐order Emden–Fowler delay differential equation with a sublinear neutral term are presented. An essential feature of our results is that oscillation of the studied equation is ensured via only one condition. Furthermore, as opposed to the results by Agarwal et al. (Ann. Mat. Pura Appl. (4) 193 (2014), no. 6, 1861–1875), Li and Rogovchenko (Math. Nachr. 288 (2015), no. 10, 1150–1162; Monatsh. Math. 184 (2017), no. 3, 489–500), and Xu (Monatsh. Math. 150 (2007), no. 2, 157–171), new criteria can be applied to Emden–Fowler delay differential equations with noncanonical operators and a sublinear neutral term. Our results essentially improve, extend, and simplify some known ones reported in the literature. The results are illustrated with examples. [ABSTRACT FROM AUTHOR]

In the paper, new single-condition criteria for the oscillation of all solutions to a second-order half-linear delay differential equation in noncanonical form are obtained, relaxing a traditionally posed assumption that the delay function is nondecreasing. The oscillation constant is best possible in the sense that the strict inequality cannot be replaced by the nonstrict one without affecting the validity of the theorem. This sharp result is new even in the linear case and, to the best of our knowledge, improves all the existing results reporting in the literature so far. The advantage of our approach is the simplicity of the proof, only based on sequentially improved monotonicities of a positive solution. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to improve recent oscillation results for the second-order half-linear delay differential equation [r(t) (y'(t)γ)' + q(t)yγ (τ(t)) =0, t ≥ t0, under the condition ... dt/r1/γ(t) < ∞. Our approach is essentially based on establishing sharper estimates for positive solutions of the studied equation than those used in known works. Two examples illustrating the results are given. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to show that the canonical operator L3 given by L3(⋅)=(r2(r1(⋅)')')' where the functions ri(t)∈C([t0,∞),[0,∞)) satisfy ... can be written in a certain \textit{strongly noncanonical} form L3(⋅)≡b3(b2(b1(b0(⋅))')')', such that the functions bi(t)∈C([t0,∞),[0,∞)) satisfy ... . We study some relations between canonical and strongly noncanonical operators, showing the advantage of this reverse approach based on the use of a noncanonical representation of L3 in the study of oscillatory and asymptotic properties of third-order delay differential equations. [ABSTRACT FROM AUTHOR]

We establish a new technique for deducing oscillation of the second order advanced differential equation (r(t)u'(t))' + p(t)u(σ(t)) = 0 with help of a suitable equation of the form r(t)u'(t))' + q(t)u(t) = 0 The comparison principle obtained fills the gap in the theory of oscillation and essentially improves existing criteria. Our technique is based on new monotonic properties of nonoscillatory solutions, and iterated exponentiation is employed. The results are supported with several illustrative examples. [ABSTRACT FROM AUTHOR]

This paper deals with the oscillation of the fourth-order linear delay differential equation with a negative middle term under the assumption that all solutions of the auxiliary third-order differential equation are nonoscillatory. Examples are included to illustrate the importance of results obtained. [ABSTRACT FROM AUTHOR]

In the paper, we elaborate new technique for the investigation of the asymptotic properties for third order 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 easily verifiable criteria for property A. We support our results with illustrative examples. [ABSTRACT FROM AUTHOR]

We establish sufficient conditions for the third-order nonlinear advanced differential equation (a(t)[(b(t)y'(t))']γ') -p(t)ƒ(y(σ(t))) = 0 to have property B or to be oscillatory. These conditions are based on monotonic properties and estimates of non-oscillatory solutions, and essentially improve known results for differential equations with deviating arguments and for ordinary differential equations. [ABSTRACT FROM AUTHOR]

The objective of this article is to study the oscillation properties of the solutions to the fourth-order linear trinomial delay differential equation y(4) (t) + p(t)y' (t) + q(t)y(τ(t)) = 0. Applying suitable comparison principles, we present new criteria for oscillation. In contrast with the existing results, we establish oscillation of all solutions, and essentially simplify the examination process for oscillation. An example is included to illustrate the importance of results obtained. [ABSTRACT FROM AUTHOR]

In the paper, we offer a new technique for investigation of properties of trinomial differential equations with positive and negative terms (b(t) (a{t)x'{t))')' + p{t)f{x{τ{t))) - q{t)h{x{σ(t))) = 0. We offer criteria for every solution to be oscillatory. We support our results with illustrative examples. [ABSTRACT FROM AUTHOR]

In this article, we study the oscillation of solutions to the third-order neutral differential equations (a(t)([x(t) ± p(t)x(δ(t))]''α)' + q(t)xα(τ(t)) = 0. Sufficient conditions are established so that every solution is either oscillatory or converges to zero. In particular, we extend the results obtain in [1] for a(t) non-decreasing, to the non-increasing case. [ABSTRACT FROM AUTHOR]

In this article, we use the Riccati transformation technique and some inequalities, to establish oscillation theorems for all solutions to evenorder quasilinear neutral differential equation ([(x(t)+p(t)x(Τ(t)))(n-1)]γ)' +q(t)xγ (σ(t)) = 0, t≥ t ₀ Our main results are illustrated with examples. [ABSTRACT FROM AUTHOR]

The objective of this paper is to study asymptotic properties of the third-order neutral differential equation . We will establish two kinds of sufficient conditions which ensure that either all nonoscillatory solutions of (E) converge to zero or all solutions of (E) are oscillatory. Some examples are considered to illustrate the main results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study properties of the third order trinomial delay differential equation by transforming this equation to binomial second/third order differential equation. Employing suitable comparison theorems we establish new results on asymptotic behavior of solutions of the studied equation. [Copyright &y& Elsevier]

The aim of this paper is to study asymptotic properties of the solutions of the third order delay differential equation (*) (1/r(t)y'(t))" - p(t)y'(t) + g(t) y(r(t)) = 0. Using suitable comparison theorem we study properties of Eq. (*) with help of the oscillation of the second order differential equation. [ABSTRACT FROM AUTHOR]

In this paper, the authors present some new results for the oscillation of the second order nonlinear neutral differential equations of the form. Easily verifiable criteria are obtained that are also new for differential equations without neutral term i.e. for p( t)≡0. [ABSTRACT FROM AUTHOR]