I'm wondering if there exists some extension of the Kakutani-Ky Fan Theorem
Theorem. Let $K$ be a nonempty, compact and convex subset of a locally convex space $X$. Let $f:K\longrightarrow 2^K$ be upper semicontinuous, such that $f(x)$ is nonempty, convex and closed for every $x\in K$. Then f has a fixed point $\bar x\in K$.
maybe without the convexity assumption of $f(x)$, or with some assumption such that $f(x)$ is homeomorphic to a convex set. I did some research in literature but, except some old articles (e.g. I. L. Glicksberg, A Further Generalization of the Kakutani Fixed Point Theorem, with Application to Nash Equilibrium Points and J-P. Penot, Fixed point theorems without convexity) which do not suit me anyway, i didn't find anything.
May you suggest me, if there are, some books or articles where I can find a similar result?
Thank you
Here is a counter-example to show that convexity of $f$ is necessary. Let $C$ be the unit circle in $\mathbb R^2$, let $K$ be the closed ball of radius $2$ in $\mathbb R^2$. Define $f(x) = (x + C)\cap K$. Then $x\not\in f(x)$ for all $x$.