Existence of positive solutions of higher order nonlinear neutral differential equations

The neutral differential equation (1.1) dⁿ/dtⁿ [x(t) + h(t)x(t – τ)] + σf (t, x(g(t))) = 0 is considered under the following conditions: n ≥ 2; σ = ±1; τ > 0; h ∈ C[t₀, ∞), lim _{t → ∞} g(t)=∞ ; f ∈ C([t₀, ∞), × (0, ∞)), f(t, u) ≥ 0 for (t, u) ∈ [t₀, ∞)× (0, ∞), and f(t, u) is nondecreasing in u ∈ (0, ∞) for each fixed t ∈ [t₀, ∞). It is shown that, for the case where h(t) > –1 and h(t) = h(t – τ) on [t₀, ∞), equation (1.1) has a positive solution x(t) satisfying x(t) = [c/1+h(t)+o(1)]t^k as t → ≈ for some c > 0 if and only if ∫^∞ t^n-k-1 (t, a[g(t)]^k) dt < ≈ for some a > 0. Here k is an integer with 0 ≤ k ≤ n – 1.