Iterative methods for monotone nonexpansive mappings in uniformly convex spaces
Rahul Shukla, Andrzej Wiśnicki · Advances in Nonlinear Analysis · 2021
Abstract 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 , ∣·∣, ⪯), T : C → C a monotone 1-Lipschitz mapping and x ⪯ T ( x ), then the sequence of averages 1 n ∑ i = 0 n − 1 T i ( x ) $ \frac{1}{n}\sum olimits_{i=0}^{n-1}T^{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’skiĭ-Mann and the Halpern iteration schemes are studied as well.