Iterative methods for monotone nonexpansive mappings in uniformly convex spaces

Abstract We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed subset of an ordered space ( X , ∣·∣, ⪯), T : → mapping and x ⪯ ), then sequence averages 1 n ∑ i = 0 − T stretchy="false">( x stretchy="false">) $ \frac{1}{n}\sum\nolimits_{i=0}^{n-1}T^{i}(x) converges weakly to fixed point . As consequence, it shown that Picard’s iteration { n )} also The results are new even Hilbert space. Krasnosel’skiĭ-Mann Halpern schemes studied as well.