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Asymptotic properties of third order delay differential equations. (English) Zbl 0842.34073
The author considers the delay differential equation $\left( {1 \over r_2 (t)} \left( {1 \over r_1 (t)} \left( {u(t) \over r_0 (t)} \right)' \right)' \right)' - p(t)u \bigl( \tau (t) \bigr) = 0, \tag{1}$ where $$r_i$$ $$(i = 0,1,2)$$, $$p, \tau$$ are continuous functions on $$[t_0, \infty)$$, $$r_i (t) > 0$$, $$p(t) > 0$$, $$\tau (t) < t$$ on $$[t_0, \infty)$$, is increasing and $$\lim_{t \to \infty} \tau (t) = \infty$$. In this paper there are proved sufficient conditions for that every solution $$u$$ of (1) is either oscillatory or $$u(t)$$ $$L_i u(t) > 0$$, $$i = 0,1$$, for $$t \geq t_1 \geq t_0$$, where $$L_0 u(t) = u(t)/r_0 (t)$$, $$L_1 u(t) = (L_0 u(t))'/r_1 (t)$$.

MSC:
 34K11 Oscillation theory of functional-differential equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
Keywords:
delay differential equation
Full Text:
References:
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