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-Karlin. In a Banach space which satisfies Opial’s 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 Hilbert space. In many cases, preliminary results on nonexpansive or
quasi-nonexpansive retractions are obtained which play crucial roles in proving the
above theorems.
