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Oscillatory properties of solutions of second order nonlinear neutral differential inequalities with oscillating coefficients. (English) Zbl 0832.34066

The authors consider the second-order differential inequalities \[ (-1)^\nu x(t)\{z''(t)+ (- 1)^\nu q(t) f(x(h(t)))\}\leq 0,\tag{A\(_\nu\)} \] \(t\geq t_0\geq 0\), where \(\nu= 0\) or \(\nu= 1\), \(z(t)= x(t)+ p(t) x(t- \tau)\), \(0< \tau=\text{const}\), \(p,q,h: [t_0, \infty)\to \mathbb{R}\) are continuous functions, \(\lim_{t\to \infty} h(t)= \infty\), \(p,q\not\equiv 0\) on any subinterval of half line \([t_0, \infty)\), \(f: \mathbb{R}\to \mathbb{R}\) is continuous, \(uf(u)> 0\) for \(u\neq 0\).
Main results of the paper are sufficient conditions under which every bounded solution of \((\text{A}_\nu)\) is either oscillatory or \(\liminf_{t\to \infty} |x(t)|= 0\). Some new aspects in the study of the oscillatory properties of solutions of \((\text{A}_\nu)\) with an oscillatory coefficient \(q\) are presented.

MSC:

34K11 Oscillation theory of functional-differential equations
34K40 Neutral functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A40 Differential inequalities involving functions of a single real variable
34K25 Asymptotic theory of functional-differential equations
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