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Zbl 1125.34059
Dibl{\'\i}k, Josef; Ružičková, Miroslava
Convergence of the solutions of the equation $\dot y(t) = \beta(t)[y(t-\delta)-y(t-\tau)]$ in the critical case.
(English)
[J] J. Math. Anal. Appl. 331, No. 2, 1361-1370 (2007). ISSN 0022-247X

This paper deals with the asymptotic behavior of a first order linear homogeneous differential equation with double delay of the form $$y'(t)=\beta(t)[y(t-\delta)-y(t-\tau)],$$ where $\delta$ and $\tau$ are positive with $\tau>\delta$; $\beta\in C([t_0-\tau,\infty),\Bbb R^+)$. The authors especially deal with the so called critical case with respect to the function $\beta$ which separates the case when all solutions are convergent and the case when there exist divergent solutions. For coefficients below the critical function, a strictly increasing and bounded solution is constructed, which characterizes the asymptotic convergence of all solutions.
[Meng Fan (Changchun)]
MSC 2000:
*34K25 Asymptotic theory of functional-differential equations
34K12 Properties of solutions of functional-differential equations
34K06 Linear functional-differential equations

Keywords: convergent solution; two delayed arguments

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