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On existence of nonoscillatory solutions of second order quasilinear neutral differential equations. (English) Zbl 0936.34067

Marušiak, Pavol (ed.) et al., Proceedings of the international scientific conference of mathematics, Žilina, Slovakia, June 30-July 3, 1998. Vol. I. Žilina: EDIS, Žilina University Publisher. 175-182 (1999).
The neutral equation \((*)\) \((L^\alpha_1x(t))'+ f(t,x(g(t)))= 0\) with \[ L_0x(t)= x(t)+ p(t)x(h(t)),\;L^\alpha_1 x(t)= r(t)|L_0 x(t)'|^\alpha\text{sgn}(L_0 x(t))' \] is considered. Two theorems on the existence of at least one solution to \((*)\) with the following properties \[ 0< \liminf_{t\to\infty} |x(t)|,\quad \limsup_{t\to\infty} |x(t)|< \infty, \]
\[ 0< \liminf_{t\to\infty} |x(t)|/\mathbb{R}_0(t)|,\quad \limsup_{t\to\infty} |x(t)|/R_0(t)< \infty, \] with \(R_0(t)= \int^t_a r(s)^{-\alpha-1} ds\), \(t\geq a\) are demonstrated.
For the entire collection see [Zbl 0914.00057].

MSC:

34K40 Neutral functional-differential equations
34K25 Asymptotic theory of functional-differential equations
34K11 Oscillation theory of functional-differential equations
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