ABSTRACT
In this chapter we describe some of the recent developments in the oscillation and nonoscillation theory of higher order delay differential equations.
In Section 6.2, we consider nonlinear neutral delay differential equations with variable coefficients of the form
dn
dtn
( x(t)− P (t)x(t− τ)
) +Q(t)f
( x(t− δ(t))
) = 0
under the assumption P (t∗ + kτ) ≤ 1 for k ∈ N
and establish a comparison theorem for oscillation. Some necessary and/or sufficient conditions for all solutions to be oscillatory are presented. In Section 6.3, we consider linear neutral delay differential equations of the form
dn
dtn
( x(t)− px(t− τ)
) +
qi(t)x(t− σi) = 0
under the assumption p ≥ 1 or 0 ≤ p < 1, respectively. Some oscillation criteria are presented. In Section 6.4, we are concerned with the asymptotic behavior of nonoscillatory solutions of nonlinear neutral differential equations of the form
dn
dtn
( x(t)−
Pi(t)x(t− ri) )
+ δ k∑ j=1
Qj(t)fj ( x (hj(t))
) = 0
under the condition ∑m i=1 |Pi(t)| ≤ λ < 1, where pi(t) is allowed to oscillate about
zero. Section 6.5 deals with the nonlinear neutral delay differential equation
dn
dtn
( x(t)− p(t)x(t− τ)
) + q(t)
∣∣∣x(t− σi)∣∣∣αi sgnx(t− σi) = 0 under the conditions p(t) ≥ 1 and 0 ≤ p(t) ≤ 1. We present some sufficient conditions for all solutions to be oscillatory. In Section 6.6, we consider the existence of positive solutions of the neutral delay differential equation
dn
dtn
( x(t)− px(t− τ)
) + q(t)x(g(t)) = 0,
where p 6= 1 and p = 1, respectively. In Sections 6.7 and 6.8, we give a classification scheme of eventually positive solutions of our two type equations in terms of their asymptotic magnitude and provide necessary and/or sufficient conditions for the existence of these solutions.
