The main objective of this article is to improve and complement some of the oscillation criteria published recently in the literature for third order differential equation of the form

$$ \bigl( r(t) \bigl( z^{\prime \prime }(t) \bigr) ^{\alpha } \bigr) ^{\prime }+q(t)f \bigl(x \bigl(\sigma (t) \bigr) \bigr)=0,\quad t\geq t_{0}>0, $$

where \(z(t)=x(t)+p(t)x(\tau (t))\) and α is a ratio of odd positive integers in the two cases \(\int _{t_{0}}^{\infty }r^{\frac{-1}{\alpha } }(s)\,\mathrm {d}s<\infty \) and \(\int _{t_{0}}^{\infty }r^{\frac{-1}{\alpha } }(s)\,\mathrm {d}s=\infty \). Some illustrative examples are presented.

中文翻译：

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三阶非线性中立型微分方程解的振动性

本文的主要目的是改进和补充最近发表在文献中的三阶微分方程形式的振动准则

$$ \ bigl（r（t）\ bigl（z ^ {\ prime \ prime}（t）\ bigr）^ {\ alpha} \ bigr）^ {\ prime} + q（t）f \ bigl（x \ bigl（\ sigma（t）\ bigr）\ bigr）= 0，\ quad t \ geq t_ {0}> 0，$$

其中\（z（t）= x（t）+ p（t）x（\ tau（t））\）和α是两种情况下\（\ int _ {t_ {0} } ^ {\ infty} r ^ {\ frac {-1} {\ alpha}}（s）\，\ mathrm {d} s <\ infty \）和\（\ int _ {t_ {0}} ^ { \ infty} r ^ {\ frac {-1} {\ alpha}}（s）\，\ mathrm {d} s = \ infty \）。给出了一些说明性的例子。