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Convergence of the solutions of the equation $$\dot y(t) = \beta(t)[y(t-\delta)-y(t-\tau)]$$ in the critical case. (English) Zbl 1125.34059
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 $$\tau>\delta$$; $$\beta\in C([t_0-\tau,\infty),\mathbb R^+)$$. 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.

##### MSC:
 34K25 Asymptotic theory of functional-differential equations 34K12 Growth, boundedness, comparison of solutions to functional-differential equations 34K06 Linear functional-differential equations
##### Keywords:
convergent solution; two delayed arguments
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##### References:
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