Eventually positive solutions of first order nonlinear differential equations with a deviating argument

The following first order nonlinear differential equation with a deviating argument $ x'(t) + p(t)x(\tau (t))]^\alpha = 0 $ is considered, where α > 0, α ≠ 1, p ∈ Ct ₀; ∞), p(t) > 0 for t ≧ t ₀, τ ∈ Ct ₀; ∞), lim _t→∞ τ(t) = ∞, τ(t) < t for t ≧ t ₀. Every eventually positive solution x(t) satisfying lim _t→∞ x(t) ≧ 0. The structure of solutions x(t) satisfying lim _t→∞ x(t) > 0 is well known. In this paper we study the existence, nonexistence and asymptotic behavior of eventually positive solutions x(t) satisfying lim _t→∞ x(t) = 0.