# zbMATH — the first resource for mathematics

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)

##### MSC:
 34K11 Oscillation theory of functional-differential equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
##### Keywords:
delay differential equations; oscillations
Full Text:
##### References:
  Domshlak Y., Sturmian Comparison Method in investigation of the be-havior. of solutions for Differential-Operator Equations. (1986)  Domshalk Y., Functional Differential Equational. Israe-Seminar 2 pp 59–  Elbert A., Recent trends in differential equations. World Sci. Ser. Appl. Anal. World Sci. Publishing co. 1 pp 163– (1992)  Elbert A., T. R. N{$$\deg$$}0. 228 Proc. Amer. Math. Soc. 128 pp 1503– (1995)  Hille E., Trans. Amer. Math. Sco. 64 pp 234– (1948)  Kneser A., Math. Ann. 42 pp 409– (1893) · JFM 25.0522.01  Koplatadze R. G., Differential Uravnenija ’nye 18 pp 1463– (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.