基于Halanay不等式的高等数学微分问题定性分析

Qualitative Analysis of Differential Problems in Advanced Mathematics Based on Halanay Inequality

摘要: Halanay不等式是时滞微分系统中常用的一种微分不等式，常用来分析含有时滞项的函数是如何以指数的方式进行衰减的。本文从Halanay不等式的基本定理和核心性质出发，结合高等数学中函数极限的判定、一阶微分方程的收敛性分析，以及微分不等式的证明问题展开讨论。同时，将Halanay不等式与Gronwall不等式对比，可以清楚直观地发现利用Halanay不等式不需要求出解析解，适用性强，方法简单，在解决高等数学中微分定性分析问题上，能明显简化过程，提高做题效率。

Abstract: The Halanay inequality is a commonly used differential inequality in delayed differential systems, which is mainly applied to analyze the exponential decay characteristics of functions with delay terms. Starting from the fundamental theorem and core properties of the Halanay inequality, this paper discusses the judgment of function limits, the convergence analysis of first-order differential equations and the proof of differential inequalities in advanced mathematics. By comparing the Halanay inequality with the Gronwall inequality, it is clearly concluded that the Halanay inequality features wide applicability and concise operation without solving analytical solutions. It can effectively simplify the solving process and improve efficiency in dealing with qualitative analysis problems of differential equations in advanced mathematics.