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Explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case. (English) Zbl 1155.34337
Summary: A scalar linear differential equation with time-dependent delay $\dot x(t)= -a(t)x(t-\tau(t))$ is considered, where $t\in I:[t_{0},\infty ), t_0 \in \Bbb R, a: I \to \Bbb R^+:=(0,\infty)$ is a continuous function and $\tau : I \to \Bbb R^+$ is a continuous function such that $t - \tau (t)>t_{0} - \tau (t_{0})$ if $t>t_{0}$. The goal of our investigation is to give sufficient conditions for the existence of positive solutions as $t\rightarrow \infty$ in the critical case in terms of inequalities on $a$ and $\tau$. A generalization of one known final (in a certain sense) result is given for the case of $\tau$ being not a constant. Analysing this generalization, we show, e.g., that it differs from the original statement with a constant delay since it does not give the best possible result. This is demonstrated on a suitable example.

##### MSC:
 34K05 General theory of functional-differential equations
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##### References:
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