达布定理和比较定理思想在中值定理类问题中的运用

微分达布定理表明区间内的导函数具有介值性,这使得我们在考虑一些数学分析问题时,往往可以不需要最高阶导函数的连续性.而在微分方程理论中,比较定理的思想对于解的估计非常重要.本文利用比较定理的思想将中值定理类问题转化为微分方程解的估计问题,对于在数学分析的学习中提高学生的认识和兴趣很有意义.

Treating Generalizations of Mean Value Theorems by Using Darboux's Theorem and Comparison Theorems

Darboux's Theorem shows that any derived function poses intermediate-value property.This enable us to solve many problems of mathematical analysis without assuming the continuous of the highest order derivatives.On the other hand, comparison theorems are very important in theories of differential equations.In this paper, it is shown that many generalizations of mean value theorems can be transformed to the estimates of solutions of differential equations by approaches related to comparison theorems.This will be helpful and interesting for students in studying mathematical analysis.