We show that in all infinite-dimensional normed spaces it is possible to construct a fixed point free continuous map of the unit ball whose measure of noncompactness is bounded by 2. Moreover, for a large class of spaces (containing separable spaces, Hilbert spaces and l-infinity (S)) even the best possible bound 1 is attained for certain measures of noncompactness.
@article{urn:eudml:doc:42727,
title = {Fixed point free maps of a closed ball with small measures of noncompactness.},
journal = {Collectanea Mathematica},
volume = {52},
year = {2001},
pages = {101-116},
zbl = {0996.47054},
mrnumber = {MR1852032},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:42727}
}
Väth, Martin. Fixed point free maps of a closed ball with small measures of noncompactness.. Collectanea Mathematica, Tome 52 (2001) pp. 101-116. http://gdmltest.u-ga.fr/item/urn:eudml:doc:42727/