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Zbl 1155.34337
Dibl{\'\i}k, Josef; Svoboda, Zdeněk; Šmarda, Zdeněk
Explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case. (English)
[J] Comput. Math. Appl. 56, No. 2, 556-564 (2008). ISSN 0898-1221

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 2000:: *34K05 General theory of functional-differential equations

Keywords: positive solution; delayed equation; critical case; infinite delay; $p$-function

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Zentralblatt MATH Berlin [Germany]