The Brouwer fixed point theorem is a key ingredient in the proof that a periodic differential equation has a periodic solution in a set that satisfies a suitable tangency condition on its boundary. The main goal of this note is to show that both results are in fact equivalent.

中文翻译：

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Brouwer不动点定理和微分方程的周期解

Brouwer 不动点定理是证明周期微分方程在其边界上满足适当相切条件的集合中具有周期解的关键要素。本说明的主要目的是表明这两个结果实际上是等价的。