The fixed-point property in Banach spaces containing a copy of $c_0$

Abstract

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.