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

The following first order nonlinear differential equation with a deviating argument, is considered, where α > 0, α ≠ 1, p ∈ C[t₀; ∞), p(t) > 0 for t ≧ t₀, τ ∈ C[t₀; ∞), lim_t→∞τ(t) = ∞, τ(t) < t for t ≧ t₀. 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.