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My problem is regarding Theorem 3 in Chapter 9 of McOwen's PDE text. The Schauder fixed point theorem states that: Let $X$ be a real Banach space. Suppose $A \subset X$ is compact and convex, and assume also $T:A \rightarrow A$ is continuous. Then $T$ has a fixed point in $A$.

Theorem 3 replaces the compactness of $A$ with the compactness of $\overline{T(A)}$ and claims the result still holds. That is, Let $A$ be a closed convex set in a Banach space $X$, and $T: A \rightarrow A$ be continuous such that $\overline{T(A)}$ is compact in $X$. Then $T$ has a fixed point.

The hint is to consider the fact that if $A$ is compact, then its convex hull $co(A)$ is compact. However, I don't see how you can possibly get this new result from the Schauder fixed point theorem or by reusing certain parts of its proof. Could someone explain this to me? It seems the author considers this to be obvious, but I can't figure out what he's getting at.

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up vote 4 down vote accepted

We have that $K := \operatorname{conv}\overline{T(A)}$ is compact and convex, and $T(K) \subseteq K$. So $T|_K$ has a fixed point by Schauder.

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