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

The following first order nonlinear differential equation with a deviating argument \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ x'(t) + p(t)[x(\tau (t))]^\alpha = 0 $$\end{document} is considered, where α > 0, α ≠ 1, p ∈ C[t0; ∞), p(t) > 0 for t ≧ t0, τ ∈ C[t0; ∞), limt→∞τ(t) = ∞, τ(t) < t for t ≧ t0. Every eventually positive solution x(t) satisfying limt→∞x(t) ≧ 0. The structure of solutions x(t) satisfying limt→∞x(t) > 0 is well known. In this paper we study the existence, nonexistence and asymptotic behavior of eventually positive solutions x(t) satisfying limt→∞x(t) = 0.