Weak convergence of orbits of nonlinear operators in reflexive Banach spaces

Let K be a closed convex subset of a reflexive Banach space X. We consider self-mappings of K which are bounded on bounded subsets of K and satisfy a relaxed form of nonexpansivity with respect to a given convex function f. The family of these operators is endowed with the topology of uniform convergence on bounded subsets of K. We show that "almost all" such operators T share the property that they have a fixed point z_T such that, for any x ∈ K, the orbit {T^kx}_k=1^∞ converges weakly to Z_T. Here the meaning of "almost all" is in the sense of Baire's categories: the collection of all those operators which do not have this property is contained in a countable union of nowhere dense sets.