A note on the Mann iteration for $k$-strict pseudocontractions in Banach spaces

TL;DR

This paper explores the properties of the Mann iteration for $k$-strict pseudocontractions in Banach spaces, providing simplified convergence proofs, rates of asymptotic regularity, and insights into space characteristics.

Contribution

It introduces a new characterization of 2-uniform smoothness and simplifies convergence proofs for iterative methods in Banach spaces.

Findings

Characterization of 2-uniform smoothness via a variant function

Simplified proofs of convergence theorems

Derived rates of asymptotic regularity

Abstract

We show that a variant of a function defined by Cholamjiak and Suantai can be used to characterize 2-uniform smoothness. We then obtain greatly simplified proofs of a convergence theorem of Marino and Xu and of one of Zhou using a generalization of a lemma of Browder and Petryshyn and the aforementioned characterization. We also obtain more easily rates of asymptotic regularity corresponding to the studied iterations. Finally, we derive a way to relate two constants which are characteristic to 2-uniformly smooth spaces.