Banach ultrapowers and multivalued nonexpansive mappings (Q865303)

Using the notion of the ultrapower of a Banach space [see, e.g., \textit{K. Goebel} and \textit{W. A. Kirk}, ``Topics in metric fixed point theory'' (Cambridge Studies in Advanced Mathematics 28, Cambridge University Press) (1990; Zbl 0708.47031)], the authors prove a fixed point theorem for a (compact convex)-valued nonexpansive \(1-\chi\)-contractive (w.r.t.\ the Hausdorff MNC \(\chi\)) multimap satisfying the inwardness condition on a weakly compact convex subset of a Banach space. As applications, the extensions of some known fixed point results for nonexpansive inward multimaps on bounded closed convex subsets of Banach spaces are presented.