<p style='text-indent:20px;'>A nonlinear version of Halanay's inequality is studied in this paper as a sufficient condition for the convergence of functions to the origin, uniformly with respect to bounded sets of initial values. The same result is provided in the case of forcing terms, for the uniform convergence to suitable neighborhoods of the origin. Related Lyapunov methods for the global uniform asymptotic stability and the input-to-state stability of systems described by retarded functional differential equations, with possibly nonconstant time delays, are provided. The relationship with the Razumikhin methodology is shown.</p>

中文翻译：

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Halanay 不等式的非线性版本，用于均匀收敛到原点

<p style='text-indent:20px;'>本文研究了Halanay不等式的非线性版本，作为函数收敛到原点的充分条件，一致地关于有界的初始值集。在强制项的情况下提供了相同的结果，以均匀收敛到原点的合适邻域。提供了用于全局均匀渐近稳定性和由延迟泛函微分方程描述的系统的输入到状态稳定性的相关 Lyapunov 方法，可能具有非恒定时间延迟。显示了与 Razumikhin 方法的关系。</p>