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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]

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]

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]