Abstract In this work, necessary and sufficient conditions are obtained such that every solution of nonlinear neutral first-order differential equations with several delays of the form ( x ⁢ ( t ) + r ⁢ ( t ) ⁢ x ⁢ ( t - τ ) ) ′ + ∑ i = 1 m ϕ i ⁢ ( t ) ⁢ H ⁢ ( x ⁢ ( t - σ i ) ) = f ⁢ ( t ) \bigl{(}x(t)+r(t)x(t-\tau)\bigr{)}^{\prime}+\sum_{i=1}^{m}\phi_{i}(t)H\bigl{(}% x(t-\sigma_{i})\bigr{)}=f(t) is oscillatory or tends to zero as t → ∞ . {t\rightarrow\infty.} This problem is considered in various ranges of the neutral coefficient r. Finally, some illustrating examples are presented to show that feasibility and effectiveness of main results.

中文翻译：

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具有若干中性时滞的非线性一阶强迫微分方程振荡的充要条件

摘要 在这项工作中，获得了充分必要条件，使得非线性中性一阶微分方程的每个解都具有以下形式的几个延迟 ( x ⁢ ( t ) + r ⁢ ( t ) ⁢ x ⁢ ( t - τ ) ) ′ + ∑ i = 1 m ϕ i ⁢ ( t ) ⁢ H ⁢ ( x ⁢ ( t - σ i ) ) = f ⁢ ( t ) \bigl{(}x(t)+r(t)x(t- \tau)\bigr{)}^{\prime}+\sum_{i=1}^{m}\phi_{i}(t)H\bigl{(}% x(t-\sigma_{i}) \bigr{)}=f(t) 是振荡的或随着 t → ∞ 趋于零。{t\rightarrow\infty.} 这个问题在中性系数 r 的各种范围内被考虑。最后，给出了一些说明性的例子来说明主要结果的可行性和有效性。