Abstract
We consider the nonlinear neutral functional differential equation with continuous arguments. We will develop oscillatory and asymptotic properties of the solutions.
Highlights
Several authors [2, 3, 4, 5, 6, 7, 12, 13, 14] have studied the oscillation theory of second-order and higher-order neutral functional differential equations, in which the highest-order derivative of the unknown function is evaluated both at the present state and at one or more past or future states
We extend these results to nth-order nonlinear neutral equations with continuous arguments b (n−1)
All bounded solutions of this problem are oscillatory
Summary
Several authors [2, 3, 4, 5, 6, 7, 12, 13, 14] have studied the oscillation theory of second-order and higher-order neutral functional differential equations, in which the highest-order derivative of the unknown function is evaluated both at the present state and at one or more past or future states. Suppose that the following conditions hold: (a) r (t) ∈ C([t0, ∞), R), r (t) ∈ C1, r (t) > 0, and ∞(dt/r (t)) = ∞; (b) p(t, μ) ∈ C([t0, ∞) × [a, b], R), 0 ≤ p(t, μ); (c) τ(t, μ) ∈ C([t0, ∞) × [a, b], R), τ(t, μ) ≤ t and limt→∞ minμ∈[a,b] τ(t, μ) = ∞; (d) q(t, ξ) ∈ C([t0, ∞) × [c, d], R) and q(t, ξ) > 0; (e) f (x) ∈ C(R, R) and xf (x) > 0 for x = 0; (f) σ (t, ξ) ∈ C([t0, ∞) × [c, d], R), and lim min σ (t, ξ) = ∞. The following results hold: (i) there exists a T > 0 such that for δ = 1, z(t)z(n−1)(t) > 0, t ≥ T , (2.1)
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