整篇論文大約分成三個部分，首先，先介紹Ekeland變分原理的定理內容及其提出的動機，接著再介紹兩種證明的方法證明Ekeland變分原理，第一種是利用定歸納數列的方式，第二種是利用耗損性動態系統證明，並且藉Ekeland變分原理提出了兩個corollary，這兩個corollary對於下一節的證明有很大的功用。最後，是整篇的最重要部分，我將介紹兩個與Ekeland變分原理等價的定理，分別是Takahashi定理和Caristi 固定點定理，同時再介紹證明這三者等價的方法，我的方法是先證明Ekeland變分原理和Takahashi定理等價，再證明Takahashi定理和Caristi 固定點定理等價，如此，三者即等價。

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