The asymptotics of nonexpansive iterations

Let D be a closed subset of a Banach space X and T: D → D a nonexpansive mapping. Conditions are given (on the space X) for T to satisfy the following property of ergodic type: $\text{{T}{}^{\mathrm{n}}\text{x}\text{n}\text{}}$ converges (either weakly or strongly) to a vector v. Rather unexpectedly, D is not assumed to be convex, nor is I – T assumed to satisfy any range condition. In addition, it is shown that ?v is the unique point of least norm in the closure of R(I – T) if and only if I – T satisfies a certain range condition at infinity. Several interesting applications to accretive operator and nonlinear semigroup theory are also included.