Banach ultrapowers and multivalued nonexpansive mappings

The Banach ultrapower construction is applied in fixed point theory for multivalued mappings. We introduce the notion of ultra-asymptotic centers and use it to remove the separability assumption from the results of Domínguez Benavides, Lorenzo Ramírez (2004) and Dhompongsa, Kaewcharoen, Kaewkhao (2006).