Oscillation of first order neutral differential equations. (English) Zbl 0725.34074

In this paper the problem of obtaining necessary and sufficient conditions for oscillation of all solutions of delay equations of the forms \[ (i)\quad \frac{d}{dt}(x(t)-px(t-\tau))+q(t)x(t-\sigma (t))=0 \] and \[ \frac{d}{dt}(x(t)-c(t)x(t-\Delta))+P(t)x(t-\tau)-Q(t)x(t-\sigma)=0 \] is investigated. The results (apart from some technical conditions) state that these equations have oscillatory solutions if and only if the corresponding differential inequalities \[ (i)\quad \frac{d}{dt}(x(t)- px(t-\tau))+q(t)x(t-\sigma (t))\leq 0, \]
\[ (ii)\quad u'(t)+(P(t)-Q(t- \tau +\sigma))u(t-\tau)\leq 0 \] respectively, have no eventually positive solutions.


34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34K40 Neutral functional-differential equations
Full Text: DOI


[1] DOI: 10.1137/0518005 · Zbl 0566.34053
[2] Arino O., J. Differential Equations 18 (1987)
[3] DOI: 10.1017/S0004972700006687 · Zbl 0465.34042
[4] Chuanxi Q., Hirosima. Math. J 22 (1980)
[5] Chuanxi Q., Applicable Analysis 22 (1980)
[6] Erbe L. H., Differential and Integral Equations 1 (1988)
[7] DOI: 10.1080/00036818808839732 · Zbl 0618.34063
[8] DOI: 10.1016/0022-0396(88)90077-0 · Zbl 0669.34069
[9] DOI: 10.1016/0022-247X(87)90045-X · Zbl 0649.34069
[10] Ladde G. S., Oscillation Theory of Differential Equations with Deviating Arguments (1987) · Zbl 0832.34071
[11] Starvoulakis I. D., Hiroshima. Math. J (1987)
[12] Wei Junjie, Acta. Math. SINICA 22 pp 632– (1989)
[13] DOI: 10.1016/0022-247X(89)90110-8 · Zbl 0683.34037
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.