Fixed points and differentiability of the norm

It is proved that in a (real) uniformly smooth Banach space $X$ a nonexpansive mapping $f\colon X\to X$ has a fixed point if
$$ \inf\{\|x-y\|:x\in f(\partial E),\ y\in X\setminus\operatorname{\overline{co}}E\}>0 $$
for some nonempty closed bounded (not necessarily convex) set $E\subset X$ with boundary $\partial E$ and closed convex hull $\operatorname{\overline{co}}E$.
It is also shown that a nonexpansive mapping $f\colon B\to X$, where $B$ is a closed bounded convex subset of a Hilbert space or a two-dimensional strictly convex Banach space $X$, has a fixed point if
$$ \{x+t(f(x)-x):0<t\leqslant 1\}\cap C\ne\varnothing\quad\text{for all}\quad x\in\partial C $$
for some nonempty closed (not necessarily convex) set $C\subset B$.
Bibliography: 11 titles.