Iterative methods for monotone nonexpansive mappings in uniformly convex spaces

We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space (X, parallel to.parallel to, <=), T : C -> C a monotone 1-Lipschitz mapping and x <= T(x), then the sequence of averages 1/n Sigma T-n-1(i-0)i(x) converges weakly to a fixed point of T. As a consequence, it is shown that the sequence of Picard's iteration {T-n(x)} also converges weakly to a fixed point of T. The results are new even in a Hilbert space. The Krasnosel'skii-Mann and the Halpern iteration schemes are studied as well.