In this note, weakly p-summable (resp. weakly p-summable and Dunford-Pettis) sequences in a Banach space are used to obtain a characterization of weak normal structure of order p (resp. Right normal structure of order p). It is proved that a Banach space has weak normal structure of order p (resp. Right normal structure of order p) if and only if it has the weak fixed point property of order p (resp. Right fixed point property of order p) for non-expansive mappings with respect to orbits.
