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Oscillations of first-order delay differential equations in a critical state. (English) Zbl 0882.34069
The first-order delay differential equation \[ x'(t)+a(t)x(t-\tau)=0, \qquad \tau>0,\quad t\geq t_0, \] is considered. It has been shown by Myshkis that in the case if \(a(t)\geq0\) and \(\liminf_{t\to\infty} a(t)>{1/(\tau e)}\) all solutions oscillate.
Yu. Domshlak [Funct. Differ. Equ. 2, 59-68 (1994; Zbl 0860.34036)] observed in a previous work that there exist equations of this type with \(\liminf_{t\to\infty} a(t)={1/(\tau e)}\) (critical case) such that all their solutions are oscillatory in spite of the fact that the corresponding limiting equation \(x'(t)+{1\over\tau e}x(t-\tau)=0\) admits the nonoscillatory solution \(x(t)=\exp(-{t\over\tau})\).
In the present paper it is shown that for the arisal of such a situation it suffices that additionally the following equalities are satisfied \[ \liminf_{t\to\infty}\left[\left(a(t)-{1\over\tau e}t^2 \right)\right]= {\tau\over 8e},\qquad \liminf_{t\to\infty}\left[\left(a(t)-{1\over\tau e}t^2 -{\tau\over 8e} \right)\right]> -{\tau\over 8e}. \]
Reviewer: I.Ginchev (Varna)

34K11 Oscillation theory of functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
Full Text: DOI
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