We consider the delay differential equation
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$
\begin{gathered}
\dot x(t) + p(t)x(t - \tau (t)) = f(t), t \in [0,\infty ) \hfill \\
x(\xi ) = \varphi (\xi ), \xi < 0 \hfill \\
\end{gathered} $
\begin{gathered}
\dot x(t) + p(t)x(t - \tau (t)) = f(t), t \in [0,\infty ) \hfill \\
x(\xi ) = \varphi (\xi ), \xi < 0 \hfill \\
\end{gathered}
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with state dependent impulses. We give sufficient conditions for positivity of solutions of the Cauchy and periodic problems
as well as conditions for positivity of solutions of the initial problem with a condition on the right end of the interval
[0, ω]. We also formulate sufficient conditions for nonoscillation of solutions of the homogeneous equation (
f = 0, ϕ = 0) on the halfline.
The research has been supported by Kamea and Giladi programs of the Ministry of Absorption of the State of Israel