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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
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References:

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