Abstract The main result of this paper is a straightforward oscillation test for linear differential equations whose coefficients consist of products of periodic and slowly varying continuous functions. The attention is paid to the case when the slowly varying parts of the coefficients are unbounded. In particular, using the presented oscillation test, we identify a very general type of conditionally oscillatory equations together with the threshold value of the coefficients (the critical setting on the border of oscillation and non-oscillation). Hence, our results supply a class of linear equations with solved oscillation behaviour which can be used as testing equations and as a starting point to the oscillation theory of the corresponding half-linear and more general non-linear and partial differential equations with unbounded coefficients.

中文翻译：

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新的条件振荡类方程，其系数包含缓慢变化和周期函数

摘要 本文的主要结果是线性微分方程的直接振荡测试，其系数由周期函数和缓慢变化的连续函数的乘积组成。注意当系数的缓慢变化部分是无界时的情况。特别是，使用所提出的振荡测试，我们确定了一种非常通用的条件振荡方程以及系数的阈值（振荡和非振荡边界上的关键设置）。因此，我们的结果提供了一类具有求解振荡行为的线性方程，可用作测试方程，并作为相应的半线性和更一般的非线性和具有无界系数的偏微分方程的振荡理论的起点。