Asymptotic behavior of nonexpansive mappings in normed linear spaces

LetT be a nonexpansive mapping on a normed linear spaceX. We show that there exists a linear functional.f, ‖f‖=1, such that, for allx∈X, limn→xf(Tnx/n)=limn→x‖Tnx/n‖=α, where α≡infy∈c‖Ty-y‖. This means, ifX is reflexive, that there is a faceF of the ball of radius α to whichTnx/n converges weakly for allx (infz∈fg(Tnx/n-z)→0, for every linear functionalg); ifX is strictly conves as well as reflexive, the convergence is to a point; and ifX satisfies the stronger condition that its dual has Fréchet differentiable norm then the convergence is strong. Furthermore, we show that each of the foregoing conditions on X is satisfied if and only if the associated convergence property holds for all nonexpansiveT.