On a theorem of Helly

We consider a group of problems related to the well-known Helly theorem on the intersections of convex bodies. We introduce convex subsets $K(f)$ of a compact convex set $K$ defined by the relation
$$ K(f)=\operatorname{co}\biggl\{\frac N{N+1}x+\frac 1{N+1}f(x)\biggr\} \quad(x\in K\subset\mathbb R^N), $$
where $f\colon K\to K$ are continuous mappings, and prove that the intersection $\bigcap_{f\in F}K(f)$ is not empty; here $F$ is the set of all continuous mappings $f\colon K\to K$.