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

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.