Fixed point theorem for a nonconvex set in a Banach space

Question

Generally speaking, I am looking for a generalization of the Schauder fixed point theorem, which applies to the situation described briefly below.

All references I read (e.g. E. Zeider 'Nonlinear Functional Analysis and its Applications I Fixed Point theorems') provide only standard version of the theorem.

My situation is as follows , $X$ is a Banach space (a subspace of continuous functions on the interval $[0,1]$), $f\colon X\to X$ is a continuous map. There exist a set $A$ (I describe it below) such that $$ f(A)\subset A, $$ where $A$ is NON convex. STILL I know a lot about $A$, it is a set of continuous functions on the interval $[0,1]$, close, bounded. And e.g. $A$ contains a convex set $C$, and there is a homotopy between any function in $A$ and a function in $C$. My intuition is that there exists a fixed point of $f$ within $A$.

I cannot apply the more general Lefshetz theorem, as I do not have direct access to $f$.

Answer 1

Your guess is false. One way to see this is to note that each infinite-dimensional Banach space $B$ contains a closed bounded subset $A$ which is homeomorphic to $[0,\infty)$. Now, take a map $f: A\to A$ which corresponds to the shift map $t\mapsto t+1$ in $[0,\infty)$ and use the Tietze extension theorem to extend $f$ to a continuous map $B\to A$. Clearly, $f$ has no fixed points. Take $C$ to be a singleton in $A$. Clearly, $C$ a strong deformation retract of $A$, providing homotopies that you want to have.