Nonoscillation and positivity of solutions to first order state-dependent differential equations with impulses in variable moments

The state-dependent delay differential equationover(x, ̇) (t) + underover(∑, i = 1, m) p_i (t) x (t - (H_i x) (t)) = f (t), t ∈ [0, ∞), x (ξ) = φ (ξ), ξ < 0, with state-dependent impulses is under consideration. Sufficient conditions for positivity of solutions to the Cauchy and periodic problems as well as conditions for positivity of solutions to the problem with a condition on the right end of the interval [0, ω] are given. Sufficient conditions for nonoscillation of solutions to the homogeneous equation (f = 0, φ = 0) on the half-line are formulated.