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For a function $f$ defined from a domain $K$ to itself, we have a point $x$ in $K$ is said to be a fixed point of $f$ if $f$ maps $x$ to itself. When the domain K is a compact convex set with some additional properties we have theorems like Brouwer Fixed-Point Theorem, Schauder Fixed-Point Theorem etc. which asserts the existence of a fixed point in $K$ for any continuous function $f$ from $K$ to itself. I want to know when the domain $K$ (instead of compact set) is just a closed and bounded convex set, are there any results which gives the existence of a fixed point for a continuous function on $K$. i.e., when does continuous functions defined on a closed and bounded convex set has a fixed point?

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  • $\begingroup$ So to clarify, you're not concerned with finite dimensional vector spaces over R or C where closed and bounded is equivalent to compact? $\endgroup$ – Batman Dec 2 '14 at 13:34
  • $\begingroup$ No, in a general space. $\endgroup$ – supremum Dec 2 '14 at 13:35

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