On an approach to the analysis of asymptotic properties of solutions of first-order ordinary delay differential equations

Abstract

For the first-order ordinary delay differential equation

$$ u'(t) + p(t)u(r(t)) = 0, $$

where p ∈ L _loc(ℝ₊; ℝ₊), τ ∈ C(ℝ₊; ℝ₊), τ(t) ≤ t for t ∈ ℝ₊, lim_t→+∞ τ(t) = +∞, and ℝ₊:= [0, ∞), we obtain new criteria for the existence of sign-definite and oscillating solutions, thus generalizing some earlier-known results.