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Convergence of the solutions of the equation y ˙(t)=β(t)[y(t-δ)-y(t-τ)] 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)=β(t)[y(t-δ)-y(t-τ)],

where δ and τ are positive with τ>δ; βC([t 0 -τ,), + ). The authors especially deal with the so called critical case with respect to the function β 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:
34K25Asymptotic theory of functional-differential equations
34K12Growth, boundedness, comparison of solutions of functional-differential equations
34K06Linear functional-differential equations