  In this paper, we prove first that the iterates of a mean nonexpansive map
defined on a weakly compact, convex set converge weakly to a fixed point in the
presence of Opial's property and asymptotic regularity at a point. Next, we
prove the analogous result for closed, convex (not necessarily bounded) subsets
of uniformly convex Opial spaces. These results generalize the classical
theorems for nonexpansive maps of Browder and Petryshyn in Hilbert space and
Opial in reflexive spaces satisfying Opial's condition.
