Let be a uniformly convex Banach space, be a nonempty closed convex subset of and be a multimap, where is the family of all nonempty compact subsets of . If we denote
then is nonempty and compact for every .
The first main result of the paper (Theorem 3.1) shows that, if is a nonexpansive retract of and if, for each and , the multivalued contraction defined by has a fixed point , then has a fixed point if and only if remains bounded as . Moreover, in this case, converges strongly to a fixed point of as .
A similar result (Theorem 3.2) is then obtained for nonself-multimaps satisfying the inwardness condition in the case of reflexive Banach spaces having a uniformly Gâteaux differentiable norm. Several corollaries of these results are also presented.