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 $f$.