Kirk introduced the notion of pointwise eventually asymptotically non- expansive mappings and proved that uniformly convex Banach spaces have the fixed point property for pointwise eventually asymptotically nonexpansive maps. Further, Kirk raised the following question: “Does a Banach space X have the fixed point property for pointwise eventually asymptotically nonexpansive mappings whenever X has the fixed point property for nonexpansive mappings?”. In this paper, we prove that a Banach space X has the fixed point property for pointwise eventually asymptotically nonexpansive maps if X has uniform normal sturcture or X is uniformly convex in every direction with the Maluta constant D(X) < 1. Also, we study the asymptotic behavior of the sequence {T nx} for a pointwise eventually asymptotically nonexpansive map T defined on a nonempty weakly compact convex subset K of a Banach space X whenever X satisfies the uniform Opial condition or X has a weakly continuous duality map.