Geodesic
Fixed points and differentiability of the norm
Sbornik. Mathematics, Tome 64 (1989) no. 2, pp. 461-469 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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N. M. Gulevich; S. V. Konyagin; R. V. Rakhmankulov. Fixed points and differentiability of the norm. Sbornik. Mathematics, Tome 64 (1989) no. 2, pp. 461-469. http://geodesic.mathdoc.fr/item/SM_1989_64_2_a11/

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