The common fixed point theory of singlevalued mappings and multivalued mappings

First, in a locally convex topological vector space, a theorem is proved which extends fixed point theorems by Lau and Fan-Glicksberg. In a strictly convex Banach space, a similar result is obtained, which is a generalization of the fixed point theorem by Bohnenblust-Earlin. In a Banach space which satisfies Opias condition, a fixed point theorem is given that generalizes both results by Holmes-Lau-Lim and Lami Dozo. In a uniformly convex Banach space, a similar theorem is considered which extends Lim's fixed point theorem. Finally, the existence of common fixed points of a quasi-nonexpansive mapping and a multivalued nonexpansive mapping is established by an elementary constructive method in a Hubert space. In many cases, preliminary results on nonexpansive or quasi-nonexpansive retractions are obtained which play crucial roles in proving the above theorems.