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Oscillation criteria for certain even order quasi-linear neutral differential equations with deviating arguments. (English) Zbl 1131.34319
The authors study the oscillatory behaviour of $n$-th order neutral differential equations of the form $$(r(t)\vert (x(t)+p(t)x(t-\tau))^{(n-1)}\vert^{\alpha-1}(x(t)+p(t)x(t-\tau))^{(n-1)})' +q(t)f(x(\sigma(t)))=0,\tag1$$ where $t\geq t_0$, $n$ is an even integer, $\alpha>0$ and $\tau\geq 0$ are constants. Throughout the paper they assume that: ($H_1$) $p,q\in C([t_0,\infty),\Bbb R))$, $0\leq p(t)\leq 1$, $q(t)\geq 0$; ($H_2$) $r\in C^1[t_0,\infty)$, $r(t)>0$, $r'(t)\geq 0$, $R(t)=\int_{t_0}^tr^{-1/\alpha}(s)ds\to\infty$ as $t\to\infty$; ($H_3$) $f\in C(\Bbb R,\Bbb R)$, $\frac{f(x)}{\vert x\vert^{\alpha-1}x}\geq \beta>0$ for $x\neq 0$ ($\beta$ is a constant); ($H_4$) $\sigma\in C^1[t_0,\infty)$, $\sigma(t)\leq t$, $\sigma'(t)>0$, $\lim_{t\to\infty}\sigma(t)=\infty$. Main result of the paper (Theorem 2.1) gives sufficient conditions for equation (1) to be oscillatory and it is applied to the second-order neutral differential equation (2) $(\vert(x(t)+p(t)x(t-\tau))'\vert^{\alpha-1}(x(t)+p(t)x(t-\tau))')'+q(t)\vert x(\sigma(t))\vert^{\alpha-1}x(\sigma(t))=0$. The purpose of the paper is to improve and extend several known results.

MSC:
34K11Oscillation theory of functional-differential equations
34K40Neutral functional-differential equations
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References:
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