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Oscillation of differential equation of neutral type. (English) Zbl 0831.34076
The author considers the neutral differential equation \[ \biggl[ x(t) - p(t)x \bigl( \sigma (t) \bigr) \biggr]' + q(t)x \bigl( \tau (t) \bigr) = 0,\;t \geq t_0, \tag{1} \] where \(p,q, \sigma, \tau\) are continuous functions, \(\sigma\) and \(\tau\) tend to \(\infty\) as \(t \to \infty\) and \(\sigma (t)\) is strictly increasing. Sufficient conditions are obtained for the oscillation of all solutions of (1) and also the asymptotic behaviour of the nonoscillatory solutions of (1) is studied. It should be noted that some of the results in the paper are already known (Theorem 1 for example) but most of them, especially those in which the author makes use of the inverse of \(\sigma (t)\), are new. A typical result of this type is the following theorem: “Suppose that \(1 \leq p(t)\), \(0 \leq q(t)\), \(\sigma (t) < t\), \(\sigma^{-1} (\tau (t)) > t\) for \(t \geq t_0\) and \[ \liminf \int_t^{ \sigma^{-1} (\tau (t))} {q(s) \over p \biggl( \sigma^{-1} \bigl( \tau (s) \bigr) \biggr)} ds > {1 \over e}. \] Then every solution of (1) is oscillatory.” In the last part of the paper the author studies the asymptotic behaviour of the nonoscillatory solutions. He proves that under certain conditions any nonoscillatory solution of (1) tends to zero as \(t \to \infty\).
Reviewer: V.Petrov (Plovdiv)

34K11 Oscillation theory of functional-differential equations
34K40 Neutral functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations