Abstract
The main purpose of this paper is to obtain criteria for the oscillation of all solutions of a third-order half-linear neutral differential equation. The main result in this paper is an oscillation theorem obtained by comparing the equation under investigation to two first order linear delay differential equations. An additional result is obtained by using a Riccati transformation technique. Examples are provided to show the importance of the main results.
Highlights
We study the oscillation of all solutions of the third-order neutral differential equation ( g(t)((h(t)z0 (t))0 )α )0 + f (t)yα (t) = 0, t ≥ t0, (1)
Σ ∈ C1 ([t0, ∞), R) with σ (t) ≤ t, and limt→∞ σ (t) = ∞; p, f ∈ C ([t0, ∞), [0, ∞)), 0 < p(t) ≤ p < 1, and f does not vanish identically; α is a ratio of odd positive integers; g, h ∈ C ([t0, ∞), (0, ∞)) and satisfy t0 α g (t) t0 dt = ∞
The main result in this paper is an oscillation theorem obtained by comparing the equation under investigation to two first order linear delay differential equations
Summary
We study the oscillation of all solutions of the third-order neutral differential equation ( g(t)((h(t)z0 (t))0 )α )0 + f (t)yα (t) = 0, t ≥ t0 , (1). Where z(t) = y(t) + p(t)y(σ (t)), subject to the following assumptions: ( H1 ) ( H2 ) ( H3 ) ( H4 ). Σ ∈ C1 ([t0 , ∞), R) with σ (t) ≤ t, and limt→∞ σ (t) = ∞; p, f ∈ C ([t0 , ∞), [0, ∞)), 0 < p(t) ≤ p < 1, and f does not vanish identically;. Α is a ratio of odd positive integers; g, h ∈ C ([t0 , ∞), (0, ∞)) and satisfy Z ∞ t0 α g (t) dt =
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