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Oscillation of solutions to second-order half-linear differential equations with neutral terms. (English) Zbl 1294.34066

Summary: We are concerned with the oscillation of the second-order neutral differential equation \[ (r(t)|z'(t)|^{\alpha-1} z'(t))'+ q(t)|x(\sigma(t))|^{\alpha-1} x(\sigma(t))= 0, \] where \(z(t):= x(t)+ \sum^m_{i=1} p_i(t) x(\tau_i(t))\), and
(H1) \(m> 1\) is an integer, \(q\in C[t_0,\infty)\), \(r,p_i,\tau_i,\sigma\in C^1[t_0, \infty)\);
(H2) \(\alpha\geq 1\), \(r(t)> 0\), \(q(t)> 0\), \(0\leq p_i(t)\leq a_i<\infty\) for \(i= 1,2,\dots, m\);
(H3) \(\lim_{t\to\infty} \sigma(t)= \infty\), \(\tau_i\circ\sigma= \sigma\circ\tau_i\), \(\tau_i'(t)\geq \lambda_i> 0\) for \(i= 1,2,\dots, m\).
Also we assume that \[ \lim_{t\to\infty}< \infty,\quad R(t):= \int^t_{t_0} {1\over r^{1/\alpha}(s)}\,ds. \]

MSC:

34K11 Oscillation theory of functional-differential equations
34K40 Neutral functional-differential equations
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