In this paper, we first prove 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.
"A weak convergence theorem for mean nonexpansive mappings." Rocky Mountain J. Math. 47 (7) 2167 - 2178, 2017. https://doi.org/10.1216/RMJ-2017-47-7-2167