  We prove the existence of a fixed point for mappings which satisfy some
asymptotic nonexpansive conditions in Banach spaces which are either nearly
uniformly convex or they satisfy that asymptotic centers of bounded sequences
are compact. Nominally, we consider pointwise eventually nonexpansive mappings,
pointwise asymptotically nonexpansive mappings and asymptotically type
nonexpansive. We do not assume the existence of a continuous iterated, solving
some long-standing open questions about existence of a fixed point for these
mappings in absence of continuity.
