Oscillation of bounded solutions of neutral differential equations. (English) Zbl 0776.34058

This paper deals with the neutral differential equation of the form (1) \(\{a(t)(x(t)-p(t)x(t-\tau))^{(n-1)}\}'+\delta q(t)f(x(\sigma(t)))=0\), \(\delta=\pm 1\), where \(a,p,q\), \(\sigma\in C([t_ 0,\infty),R),a(t)>0,q(t)\geq 0(\not\equiv 0)\) on \([t_ 0,\infty)\), \(\lim_{t\to\infty}\sigma(t)=\infty\). The authors give sufficient conditions under which every bounded solution \(x(t)\) of (1) is: i) oscillatory when \((-1)^ n\delta=1\), ii) either oscillatory or \(\lim_{t\to\infty}x(t)=0\) when \((-1)^ n\delta=-1\).


34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34K40 Neutral functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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