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.


34K25 Asymptotic theory of functional-differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations
34K06 Linear functional-differential equations
Full Text: DOI


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