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

Lalli, B. S.; Zhang, Biggen

Appl. Anal. 39, No. 4, 265-274 (1990).

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.

Reviewer: S.P.Banks (Sheffield)

MSC:

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

Keywords:

neutral delay; differential equations; oscillation; differential inequalities

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