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Existence of strictly decreasing positive solutions of linear differential equations of neutral type

  • * Corresponding author: Josef Diblík

    * Corresponding author: Josef Diblík 
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  • The paper is concerned with a linear neutral differential equation

    $ \dot y(t) = -c(t)y(t-\tau(t))+d(t)\dot y(t-\delta(t)) $

    where $ c\colon [t_0,\infty)\to (0,\infty) $, $ d\colon [t_0,\infty)\to [0,\infty) $, $ t_0\in {\Bbb{R}} $ and $ \tau, \delta \colon [t_0,\infty)\to (0,r] $, $ r\in{\mathbb{R}} $, $ r>0 $ are continuous functions. A new criterion is given for the existence of positive strictly decreasing solutions. The proof is based on the Rybakowski variant of a topological Ważewski principle suitable for differential equations of the delayed type. Unlike in the previous investigations known, this time the progress is achieved by using a special system of initial functions satisfying a so-called sewing condition. The result obtained is extended to more general equations. Comparisons with known results are given as well.

    Mathematics Subject Classification: Primary: 34K40, 34K25; Secondary: 34K12.


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