FIXED POINTS AND DIFFERENTIABILITY OF THE NORM
N. M. Gulevich, Sergei Vladimirovich Konyagin, R V Rakhmankulov · Mathematics of the USSR-Sbornik · 1989
It is proved that in a (real) uniformly smooth Banach space a nonexpansive mapping has a fixed point if for some nonempty closed bounded (not necessarily convex) set with boundary and closed convex hull . It is also shown that a nonexpansive mapping , where is a closed bounded convex subset of a Hilbert space or a two-dimensional strictly convex Banach space , has a fixed point if for all for some nonempty closed (not necessarily convex) set .Bibliography: 11 titles.