Asymptotic behavior of nonexpansive mappings in normed linear spaces

Let T be a nonexpansive mapping on a normed linear space X. We show that there exists a linear functional. f, {norm of matrix}f{norm of matrix}=1, such that, for all x∈X, lim_n→x f(T ⁿ x/n)=lim_n→x{norm of matrix}T ⁿ x/n {norm of matrix}=α, where α≡inf _y∈c {norm of matrix}Ty-y{norm of matrix}. This means, if X is reflexive, that there is a face F of the ball of radius α to which T ⁿ x/n converges weakly for all x (inf_z∈f g(T ⁿ x/n-z)→0, for every linear functional g); if X is strictly conves as well as reflexive, the convergence is to a point; and if X 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 nonexpansive T.