A remark on the caristi’s fixed point theorem and the brouwer fixed point theorem

Abstract

It is well-known that a partial order induced from a lower semi-continuous map gives us a clear picture of a proof of the Caristi’s fixed point theorem. The proof utilized Zorn’s lemma to guarantee the existence of a minimal element which turns out to be a desired fixed point. The proof cannot be carried over to prove the Brouwer fixed point theorem. We show that making an idea of ordering, we get a proof of the later one.