It is shown that a closed convex bounded subset of a Banach space is weakly compact if and only if it has the generic fixed point property for continuous affine mappings. The class of continuous affine mappings can be replaced by the class of affine mappings which are uniformly Lipschitzian with some constant M > 1 in the case of c0, the class of affine mappings which are uniformly Lipschitzian with some constant M > √6 in the case of quasi-reflexive James’ space J and the class of nonexpansive affine mappings in the case of L-embedded spaces.Dirección General de Enseñanza SuperiorJunta de AndalucíaState Committee for Scientific Research (Poland
We prove that every Banach space containing an isomorphic copy of c 0 fails to have the fixed-point property for asymptotically nonexpansive mappings with respect to some locally convex topology which is coarser than the weak topology. If the copy of c 0 is asymptotically isometric, this result can be improved, because we can prove the failure of the fixed-point property for nonexpansive mappings.
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