Abstract
In this paper, we study the asymptotic and oscillatory properties of a certain class of third-order neutral delay differential equations with middle term. We obtain new characterizations of oscillation of the third-order neutral equation in terms of oscillation of a related, well-studied, second-order linear equation without damping. An Example is provided to illustrate the main results.
Highlights
(t) + b (t) y0 (t) + q (t) f ( x (σ (t))) = 0, for t ≥ t0, where y (t) = x (t) + p (t) x (τ (t)), α is a ratio of positive odd integers and f ∈ C (R, R) satisfies f ( x ) ≥ kx α for x 6= 0
In view of [24](Theorem 1), we see that the first-order delay differential Equation (21) has a positive a solution, a contradiction
If (31) hold, it is clear that all conditions of Theorem 5 are satisfied, and every solution of (1), or y0 (t), is oscillatory
Summary
We consider the third-order nonlinear damped neutral differential equation of the form Throughout this paper, we assume the following conditions: (I1 ) r1 , r2 ∈ C ([t0 , ∞) , (0, ∞)) Solutions of (1) existing on some half-line [ Tx , ∞) and satisfying the condition sup{| x (t)| : T ≤ t
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