Explicit exponential stability tests are obtained for the scalar neutral differential equation $\dot{x}\left(t\right)-a\left(t\right)\dot{x}\left(g\left(t\right)\right)=-\sum _{k=1}^m{b}_k\left(t\right)x\left(h_k\left(t\right)\right),$ together with exponential estimates for its solutions.

Estimates for solutions of a non-homogeneous neutral equation are also obtained, they are valid on every finite segment, thus describing both asymptotic and transient behavior. For neutral differential equations, exponential estimates are obtained here for the first time. Both the coefficients and the delays are assumed to be measurable, not necessarily continuous functions.

中文翻译：

---

线性中立型微分方程的解估计和稳定性测试

标量中立型微分方程的显式指数稳定性测试 $\dot{X}（Ť）-\mathrm{一种}（Ť）\dot{X}（G（Ť））=-\sum _{ķ=\mathrm{1个}}^{米}{b}_{ķ}（Ť）X（H_{ķ}（Ť）），$ 及其解决方案的指数估算。

还获得了非齐次中立方程解的估计，它们在每个有限段上都是有效的，因此描述了渐近行为和瞬态行为。对于中性微分方程式，此处是首次获得指数估计。假设系数和延迟都是可测量的，不一定是连续函数。