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 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 1n∑i=0n−1Ti(x)$ \frac{1}{n}\sum\nolimits_{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 {Tn(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.

Keywords

• monotone mapping
• nonexpansive mapping
• fixed point
• ergodic theorem
• picard iteration
• ordered banach space
• 47j26
• 46b20
• 47h07
• 54f05